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RFC 8017 - PKCS #1: RSA Cryptography Specifications Version 2.2


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Internet Engineering Task Force (IETF)                  K. Moriarty, Ed.
Request for Comments: 8017                               EMC Corporation
Obsoletes: 3447                                               B. Kaliski
Category: Informational                                         Verisign
ISSN: 2070-1721                                               J. Jonsson
                                                               Subset AB
                                                                A. Rusch
                                                                     RSA
                                                           November 2016

          PKCS #1: RSA Cryptography Specifications Version 2.2

Abstract

   This document provides recommendations for the implementation of
   public-key cryptography based on the RSA algorithm, covering
   cryptographic primitives, encryption schemes, signature schemes with
   appendix, and ASN.1 syntax for representing keys and for identifying
   the schemes.

   This document represents a republication of PKCS #1 v2.2 from RSA
   Laboratories' Public-Key Cryptography Standards (PKCS) series.  By
   publishing this RFC, change control is transferred to the IETF.

   This document also obsoletes RFC 3447.

Status of This Memo

   This document is not an Internet Standards Track specification; it is
   published for informational purposes.

   This document is a product of the Internet Engineering Task Force
   (IETF).  It represents the consensus of the IETF community.  It has
   received public review and has been approved for publication by the
   Internet Engineering Steering Group (IESG).  Not all documents
   approved by the IESG are a candidate for any level of Internet
   Standard; see Section 2 of RFC 7841.

   Information about the current status of this document, any errata,
   and how to provide feedback on it may be obtained at
   http://www.rfc-editor.org/info/rfc8017.

Copyright Notice

   Copyright (c) 2016 IETF Trust and the persons identified as the
   document authors.  All rights reserved.

   This document is subject to BCP 78 and the IETF Trust's Legal
   Provisions Relating to IETF Documents
   (http://trustee.ietf.org/license-info) in effect on the date of
   publication of this document.  Please review these documents
   carefully, as they describe your rights and restrictions with respect
   to this document.  Code Components extracted from this document must
   include Simplified BSD License text as described in Section 4.e of
   the Trust Legal Provisions and are provided without warranty as
   described in the Simplified BSD License.

Table of Contents

   1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   4
     1.1.  Requirements Language . . . . . . . . . . . . . . . . . .   5
   2.  Notation  . . . . . . . . . . . . . . . . . . . . . . . . . .   6
   3.  Key Types . . . . . . . . . . . . . . . . . . . . . . . . . .   8
     3.1.  RSA Public Key  . . . . . . . . . . . . . . . . . . . . .   8
     3.2.  RSA Private Key . . . . . . . . . . . . . . . . . . . . .   9
   4.  Data Conversion Primitives  . . . . . . . . . . . . . . . . .  11
     4.1.  I2OSP . . . . . . . . . . . . . . . . . . . . . . . . . .  11
     4.2.  OS2IP . . . . . . . . . . . . . . . . . . . . . . . . . .  12
   5.  Cryptographic Primitives  . . . . . . . . . . . . . . . . . .  12
     5.1.  Encryption and Decryption Primitives  . . . . . . . . . .  12
       5.1.1.  RSAEP . . . . . . . . . . . . . . . . . . . . . . . .  13
       5.1.2.  RSADP . . . . . . . . . . . . . . . . . . . . . . . .  13
     5.2.  Signature and Verification Primitives . . . . . . . . . .  15
       5.2.1.  RSASP1  . . . . . . . . . . . . . . . . . . . . . . .  15
       5.2.2.  RSAVP1  . . . . . . . . . . . . . . . . . . . . . . .  16
   6.  Overview of Schemes . . . . . . . . . . . . . . . . . . . . .  17
   7.  Encryption Schemes  . . . . . . . . . . . . . . . . . . . . .  18
     7.1.  RSAES-OAEP  . . . . . . . . . . . . . . . . . . . . . . .  19
       7.1.1.  Encryption Operation  . . . . . . . . . . . . . . . .  22
       7.1.2.  Decryption Operation  . . . . . . . . . . . . . . . .  25
     7.2.  RSAES-PKCS1-v1_5  . . . . . . . . . . . . . . . . . . . .  27
       7.2.1.  Encryption Operation  . . . . . . . . . . . . . . . .  28
       7.2.2.  Decryption Operation  . . . . . . . . . . . . . . . .  29
   8.  Signature Scheme with Appendix  . . . . . . . . . . . . . . .  31
     8.1.  RSASSA-PSS  . . . . . . . . . . . . . . . . . . . . . . .  32
       8.1.1.  Signature Generation Operation  . . . . . . . . . . .  33
       8.1.2.  Signature Verification Operation  . . . . . . . . . .  34
     8.2.  RSASSA-PKCS1-v1_5 . . . . . . . . . . . . . . . . . . . .  35
       8.2.1.  Signature Generation Operation  . . . . . . . . . . .  36
       8.2.2.  Signature Verification Operation  . . . . . . . . . .  37
   9.  Encoding Methods for Signatures with Appendix . . . . . . . .  39
     9.1.  EMSA-PSS  . . . . . . . . . . . . . . . . . . . . . . . .  40
       9.1.1.  Encoding Operation  . . . . . . . . . . . . . . . . .  42
       9.1.2.  Verification Operation  . . . . . . . . . . . . . . .  44
     9.2.  EMSA-PKCS1-v1_5 . . . . . . . . . . . . . . . . . . . . .  45
   10. Security Considerations . . . . . . . . . . . . . . . . . . .  47
   11. References  . . . . . . . . . . . . . . . . . . . . . . . . .  48
     11.1.  Normative References . . . . . . . . . . . . . . . . . .  48
     11.2.  Informative References . . . . . . . . . . . . . . . . .  48

   Appendix A.  ASN.1 Syntax . . . . . . . . . . . . . . . . . . . .  54
     A.1.  RSA Key Representation  . . . . . . . . . . . . . . . . .  54
       A.1.1.  RSA Public Key Syntax . . . . . . . . . . . . . . . .  54
       A.1.2.  RSA Private Key Syntax  . . . . . . . . . . . . . . .  55
     A.2.  Scheme Identification . . . . . . . . . . . . . . . . . .  57
       A.2.1.  RSAES-OAEP  . . . . . . . . . . . . . . . . . . . . .  57
       A.2.2.  RSAES-PKCS-v1_5 . . . . . . . . . . . . . . . . . . .  60
       A.2.3.  RSASSA-PSS  . . . . . . . . . . . . . . . . . . . . .  60
       A.2.4.  RSASSA-PKCS-v1_5  . . . . . . . . . . . . . . . . . .  62
   Appendix B.  Supporting Techniques  . . . . . . . . . . . . . . .  63
     B.1.  Hash Functions  . . . . . . . . . . . . . . . . . . . . .  63
     B.2.  Mask Generation Functions . . . . . . . . . . . . . . . .  66
       B.2.1.  MGF1  . . . . . . . . . . . . . . . . . . . . . . . .  67
   Appendix C.  ASN.1 Module . . . . . . . . . . . . . . . . . . . .  68
   Appendix D.  Revision History of PKCS #1  . . . . . . . . . . . .  76
   Appendix E.  About PKCS . . . . . . . . . . . . . . . . . . . . .  77
   Acknowledgements  . . . . . . . . . . . . . . . . . . . . . . . .  78
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . .  78

1.  Introduction

   This document provides recommendations for the implementation of
   public-key cryptography based on the RSA algorithm [RSA], covering
   the following aspects:

   o  Cryptographic primitives

   o  Encryption schemes

   o  Signature schemes with appendix

   o  ASN.1 syntax for representing keys and for identifying the schemes

   The recommendations are intended for general application within
   computer and communications systems and as such include a fair amount
   of flexibility.  It is expected that application standards based on
   these specifications may include additional constraints.  The
   recommendations are intended to be compatible with the standards IEEE
   1363 [IEEE1363], IEEE 1363a [IEEE1363A], and ANSI X9.44 [ANSIX944].

   This document supersedes PKCS #1 version 2.1 [RFC3447] but includes
   compatible techniques.

   The organization of this document is as follows:

   o  Section 1 is an introduction.

   o  Section 2 defines some notation used in this document.

   o  Section 3 defines the RSA public and private key types.

   o  Sections 4 and 5 define several primitives, or basic mathematical
      operations.  Data conversion primitives are in Section 4, and
      cryptographic primitives (encryption-decryption and signature-
      verification) are in Section 5.

   o  Sections 6, 7, and 8 deal with the encryption and signature
      schemes in this document.  Section 6 gives an overview.  Along
      with the methods found in PKCS #1 v1.5, Section 7 defines an
      encryption scheme based on Optimal Asymmetric Encryption Padding
      (OAEP) [OAEP], and Section 8 defines a signature scheme with
      appendix based on the Probabilistic Signature Scheme (PSS)
      [RSARABIN] [PSS].

   o  Section 9 defines the encoding methods for the signature schemes
      in Section 8.

   o  Appendix A defines the ASN.1 syntax for the keys defined in
      Section 3 and the schemes in Sections 7 and 8.

   o  Appendix B defines the hash functions and the mask generation
      function (MGF) used in this document, including ASN.1 syntax for
      the techniques.

   o  Appendix C gives an ASN.1 module.

   o  Appendices D and E outline the revision history of PKCS #1 and
      provide general information about the Public-Key Cryptography
      Standards.

   This document represents a republication of PKCS #1 v2.2 [PKCS1_22]
   from RSA Laboratories' Public-Key Cryptography Standards (PKCS)
   series.

1.1.  Requirements Language

   The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
   "SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this
   document are to be interpreted as described in [RFC2119].

2.  Notation

   The notation in this document includes:

      c              ciphertext representative, an integer between 0 and
                     n-1

      C              ciphertext, an octet string

      d              RSA private exponent

      d_i            additional factor r_i's CRT exponent,
                     a positive integer such that

                       e * d_i == 1 (mod (r_i-1)), i = 3, ..., u

      dP             p's CRT exponent, a positive integer such that

                       e * dP == 1 (mod (p-1))

      dQ             q's CRT exponent, a positive integer such that

                       e * dQ == 1 (mod (q-1))

      e              RSA public exponent

      EM             encoded message, an octet string

      emBits         (intended) length in bits of an encoded message EM

      emLen          (intended) length in octets of an encoded message
                     EM

      GCD(. , .)     greatest common divisor of two nonnegative integers

      Hash           hash function

      hLen           output length in octets of hash function Hash

      k              length in octets of the RSA modulus n

      K              RSA private key

      L              optional RSAES-OAEP label, an octet string

      LCM(., ..., .) least common multiple of a list of nonnegative
                     integers

      m              message representative, an integer between 0 and
                     n-1

      M              message, an octet string

      mask           MGF output, an octet string

      maskLen        (intended) length of the octet string mask

      MGF            mask generation function

      mgfSeed        seed from which mask is generated, an octet string

      mLen           length in octets of a message M

      n              RSA modulus, n = r_1 * r_2 * ... * r_u , u >= 2

      (n, e)         RSA public key

      p, q           first two prime factors of the RSA modulus n

      qInv           CRT coefficient, a positive integer less than
                     p such that q * qInv == 1 (mod p)

      r_i            prime factors of the RSA modulus n, including
                     r_1 = p, r_2 = q, and additional factors if any

      s              signature representative, an integer between 0 and
                     n-1

      S              signature, an octet string

      sLen           length in octets of the EMSA-PSS salt

      t_i            additional prime factor r_i's CRT coefficient, a
                     positive integer less than r_i such that

                       r_1 * r_2 * ... * r_(i-1) * t_i == 1 (mod r_i) ,

                     i = 3, ... , u

      u              number of prime factors of the RSA modulus, u >= 2

      x              a nonnegative integer

      X              an octet string corresponding to x

      xLen           (intended) length of the octet string X

      0x             indicator of hexadecimal representation of an octet
                     or an octet string: "0x48" denotes the octet with
                     hexadecimal value 48; "(0x)48 09 0e" denotes the
                     string of three consecutive octets with hexadecimal
                     values 48, 09, and 0e, respectively

      \lambda(n)     LCM(r_1-1, r_2-1, ... , r_u-1)

      \xor           bit-wise exclusive-or of two octet strings

      \ceil(.)       ceiling function; \ceil(x) is the smallest integer
                     larger than or equal to the real number x

      ||             concatenation operator

      ==             congruence symbol; a == b (mod n) means that the
                     integer n divides the integer a - b

   Note: The Chinese Remainder Theorem (CRT) can be applied in a non-
   recursive as well as a recursive way.  In this document, a recursive
   approach following Garner's algorithm [GARNER] is used.  See also
   Note 1 in Section 3.2.

3.  Key Types

   Two key types are employed in the primitives and schemes defined in
   this document: RSA public key and RSA private key.  Together, an RSA
   public key and an RSA private key form an RSA key pair.

   This specification supports so-called "multi-prime" RSA where the
   modulus may have more than two prime factors.  The benefit of multi-
   prime RSA is lower computational cost for the decryption and
   signature primitives, provided that the CRT is used.  Better
   performance can be achieved on single processor platforms, but to a
   greater extent on multiprocessor platforms, where the modular
   exponentiations involved can be done in parallel.

   For a discussion on how multi-prime affects the security of the RSA
   cryptosystem, the reader is referred to [SILVERMAN].

3.1.  RSA Public Key

   For the purposes of this document, an RSA public key consists of two
   components:

         n        the RSA modulus, a positive integer
         e        the RSA public exponent, a positive integer

   In a valid RSA public key, the RSA modulus n is a product of u
   distinct odd primes r_i, i = 1, 2, ..., u, where u >= 2, and the RSA
   public exponent e is an integer between 3 and n - 1 satisfying
   GCD(e,\lambda(n)) = 1, where \lambda(n) = LCM(r_1 - 1, ..., r_u - 1).
   By convention, the first two primes r_1 and r_2 may also be denoted p
   and q, respectively.

   A recommended syntax for interchanging RSA public keys between
   implementations is given in Appendix A.1.1; an implementation's
   internal representation may differ.

3.2.  RSA Private Key

   For the purposes of this document, an RSA private key may have either
   of two representations.

   1.  The first representation consists of the pair (n, d), where the
       components have the following meanings:

            n       the RSA modulus, a positive integer
            d       the RSA private exponent, a positive integer

   2.  The second representation consists of a quintuple (p, q, dP, dQ,
       qInv) and a (possibly empty) sequence of triplets (r_i, d_i,
       t_i), i = 3, ..., u, one for each prime not in the quintuple,
       where the components have the following meanings:

            p      the first factor, a positive integer
            q      the second factor, a positive integer
            dP     the first factor's CRT exponent, a positive integer
            dQ     the second factor's CRT exponent, a positive integer
            qInv   the (first) CRT coefficient, a positive integer
            r_i    the i-th factor, a positive integer
            d_i    the i-th factor's CRT exponent, a positive integer
            t_i    the i-th factor's CRT coefficient, a positive integer

   In a valid RSA private key with the first representation, the RSA
   modulus n is the same as in the corresponding RSA public key and is
   the product of u distinct odd primes r_i, i = 1, 2, ..., u, where u
   >= 2.  The RSA private exponent d is a positive integer less than n
   satisfying

      e * d == 1 (mod \lambda(n)),

   where e is the corresponding RSA public exponent and \lambda(n) is
   defined as in Section 3.1.

   In a valid RSA private key with the second representation, the two
   factors p and q are the first two prime factors of the RSA modulus n
   (i.e., r_1 and r_2); the CRT exponents dP and dQ are positive
   integers less than p and q, respectively, satisfying

      e * dP == 1 (mod (p-1))

      e * dQ == 1 (mod (q-1)) ,

   and the CRT coefficient qInv is a positive integer less than p
   satisfying

      q * qInv == 1 (mod p).

   If u > 2, the representation will include one or more triplets (r_i,
   d_i, t_i), i = 3, ..., u.  The factors r_i are the additional prime
   factors of the RSA modulus n.  Each CRT exponent d_i (i = 3, ..., u)
   satisfies

      e * d_i == 1 (mod (r_i - 1)).

   Each CRT coefficient t_i (i = 3, ..., u) is a positive integer less
   than r_i satisfying

      R_i * t_i == 1 (mod r_i) ,

   where R_i = r_1 * r_2 * ... * r_(i-1).

   A recommended syntax for interchanging RSA private keys between
   implementations, which includes components from both representations,
   is given in Appendix A.1.2; an implementation's internal
   representation may differ.

   Notes:

   1.  The definition of the CRT coefficients here and the formulas that
       use them in the primitives in Section 5 generally follow Garner's
       algorithm [GARNER] (see also Algorithm 14.71 in [HANDBOOK]).
       However, for compatibility with the representations of RSA
       private keys in PKCS #1 v2.0 and previous versions, the roles of
       p and q are reversed compared to the rest of the primes.  Thus,
       the first CRT coefficient, qInv, is defined as the inverse of q
       mod p, rather than as the inverse of R_1 mod r_2, i.e., of
       p mod q.

   2.  Quisquater and Couvreur [FASTDEC] observed the benefit of
       applying the CRT to RSA operations.

4.  Data Conversion Primitives

   Two data conversion primitives are employed in the schemes defined in
   this document:

   o  I2OSP - Integer-to-Octet-String primitive

   o  OS2IP - Octet-String-to-Integer primitive

   For the purposes of this document, and consistent with ASN.1 syntax,
   an octet string is an ordered sequence of octets (eight-bit bytes).
   The sequence is indexed from first (conventionally, leftmost) to last
   (rightmost).  For purposes of conversion to and from integers, the
   first octet is considered the most significant in the following
   conversion primitives.

4.1.  I2OSP

   I2OSP converts a nonnegative integer to an octet string of a
   specified length.

   I2OSP (x, xLen)

   Input:

      x        nonnegative integer to be converted

      xLen     intended length of the resulting octet string

   Output:

         X corresponding octet string of length xLen

   Error:  "integer too large"

   Steps:

      1.  If x >= 256^xLen, output "integer too large" and stop.

      2.  Write the integer x in its unique xLen-digit representation in
          base 256:

             x = x_(xLen-1) 256^(xLen-1) + x_(xLen-2) 256^(xLen-2) + ...
             + x_1 256 + x_0,

          where 0 <= x_i < 256 (note that one or more leading digits
          will be zero if x is less than 256^(xLen-1)).

      3.  Let the octet X_i have the integer value x_(xLen-i) for 1 <= i
          <= xLen.  Output the octet string

             X = X_1 X_2 ... X_xLen.

4.2.  OS2IP

   OS2IP converts an octet string to a nonnegative integer.

   OS2IP (X)

   Input:  X octet string to be converted

   Output:  x corresponding nonnegative integer

   Steps:

      1.  Let X_1 X_2 ... X_xLen be the octets of X from first to last,
          and let x_(xLen-i) be the integer value of the octet X_i for 1
          <= i <= xLen.

      2.  Let x = x_(xLen-1) 256^(xLen-1) + x_(xLen-2) 256^(xLen-2) +
          ...  + x_1 256 + x_0.

      3.  Output x.

5.  Cryptographic Primitives

   Cryptographic primitives are basic mathematical operations on which
   cryptographic schemes can be built.  They are intended for
   implementation in hardware or as software modules and are not
   intended to provide security apart from a scheme.

   Four types of primitive are specified in this document, organized in
   pairs: encryption and decryption; and signature and verification.

   The specifications of the primitives assume that certain conditions
   are met by the inputs, in particular that RSA public and private keys
   are valid.

5.1.  Encryption and Decryption Primitives

   An encryption primitive produces a ciphertext representative from a
   message representative under the control of a public key, and a
   decryption primitive recovers the message representative from the
   ciphertext representative under the control of the corresponding
   private key.

   One pair of encryption and decryption primitives is employed in the
   encryption schemes defined in this document and is specified here:
   RSA Encryption Primitive (RSAEP) / RSA Decryption Primitive (RSADP).
   RSAEP and RSADP involve the same mathematical operation, with
   different keys as input.  The primitives defined here are the same as
   Integer Factorization Encryption Primitive using RSA (IFEP-RSA) /
   Integer Factorization Decryption Primitive using RSA (IFDP-RSA) in
   IEEE 1363 [IEEE1363] (except that support for multi-prime RSA has
   been added) and are compatible with PKCS #1 v1.5.

   The main mathematical operation in each primitive is exponentiation.

5.1.1.  RSAEP

   RSAEP ((n, e), m)

   Input:

         (n, e) RSA public key

         m message representative, an integer between 0 and n - 1

   Output:  c ciphertext representative, an integer between 0 and n - 1

   Error:  "message representative out of range"

   Assumption:  RSA public key (n, e) is valid

   Steps:

      1.  If the message representative m is not between 0 and n - 1,
          output "message representative out of range" and stop.

      2.  Let c = m^e mod n.

      3.  Output c.

5.1.2.  RSADP

   RSADP (K, c)

   Input:

         K RSA private key, where K has one of the following forms:

         +  a pair (n, d)

         +  a quintuple (p, q, dP, dQ, qInv) and a possibly empty
            sequence of triplets (r_i, d_i, t_i), i = 3, ..., u

         c ciphertext representative, an integer between 0 and n - 1

   Output:  m message representative, an integer between 0 and n - 1

   Error:  "ciphertext representative out of range"

   Assumption:  RSA private key K is valid

   Steps:

      1.  If the ciphertext representative c is not between 0 and n - 1,
          output "ciphertext representative out of range" and stop.

      2.  The message representative m is computed as follows.

          a.  If the first form (n, d) of K is used, let m = c^d mod n.

          b.  If the second form (p, q, dP, dQ, qInv) and (r_i, d_i,
              t_i) of K is used, proceed as follows:

              i.   Let m_1 = c^dP mod p and m_2 = c^dQ mod q.

              ii.  If u > 2, let m_i = c^(d_i) mod r_i, i = 3, ..., u.

              iii. Let h = (m_1 - m_2) * qInv mod p.

              iv.  Let m = m_2 + q * h.

              v.   If u > 2, let R = r_1 and for i = 3 to u do

                   1.  Let R = R * r_(i-1).

                   2.  Let h = (m_i - m) * t_i mod r_i.

                   3.  Let m = m + R * h.

      3.  Output m.

   Note: Step 2.b can be rewritten as a single loop, provided that one
   reverses the order of p and q.  For consistency with PKCS #1 v2.0,
   however, the first two primes p and q are treated separately from the
   additional primes.

5.2.  Signature and Verification Primitives

   A signature primitive produces a signature representative from a
   message representative under the control of a private key, and a
   verification primitive recovers the message representative from the
   signature representative under the control of the corresponding
   public key.  One pair of signature and verification primitives is
   employed in the signature schemes defined in this document and is
   specified here: RSA Signature Primitive, version 1 (RSASP1) / RSA
   Verification Primitive, version 1 (RSAVP1).

   The primitives defined here are the same as Integer Factorization
   Signature Primitive using RSA, version 1 (IFSP-RSA1) / Integer
   Factorization Verification Primitive using RSA, version 1 (IFVP-RSA1)
   in IEEE 1363 [IEEE1363] (except that support for multi-prime RSA has
   been added) and are compatible with PKCS #1 v1.5.

   The main mathematical operation in each primitive is exponentiation,
   as in the encryption and decryption primitives of Section 5.1.
   RSASP1 and RSAVP1 are the same as RSADP and RSAEP except for the
   names of their input and output arguments; they are distinguished as
   they are intended for different purposes.

5.2.1.  RSASP1

   RSASP1 (K, m)

   Input:

      K        RSA private key, where K has one of the following forms:
               - a pair (n, d)
               - a quintuple (p, q, dP, dQ, qInv) and a (possibly empty)
                 sequence of triplets (r_i, d_i, t_i), i = 3, ..., u
      m        message representative, an integer between 0 and n - 1

   Output:

      s        signature representative, an integer between 0 and n - 1

   Error:  "message representative out of range"

   Assumption:  RSA private key K is valid

   Steps:

      1.  If the message representative m is not between 0 and n - 1,
          output "message representative out of range" and stop.

      2.  The signature representative s is computed as follows.

          a.  If the first form (n, d) of K is used, let s = m^d mod n.

          b.  If the second form (p, q, dP, dQ, qInv) and (r_i, d_i,
              t_i) of K is used, proceed as follows:

              1.  Let s_1 = m^dP mod p and s_2 = m^dQ mod q.

              2.  If u > 2, let s_i = m^(d_i) mod r_i, i = 3, ..., u.

              3.  Let h = (s_1 - s_2) * qInv mod p.

              4.  Let s = s_2 + q * h.

              5.  If u > 2, let R = r_1 and for i = 3 to u do

                  a.  Let R = R * r_(i-1).

                  b.  Let h = (s_i - s) * t_i mod r_i.

                  c.  Let s = s + R * h.

      3.  Output s.

   Note: Step 2.b can be rewritten as a single loop, provided that one
   reverses the order of p and q.  For consistency with PKCS #1 v2.0,
   however, the first two primes p and q are treated separately from the
   additional primes.

5.2.2.  RSAVP1

   RSAVP1 ((n, e), s)

   Input:

         (n, e) RSA public key

         s signature representative, an integer between 0 and n - 1

   Output:

         m message representative, an integer between 0 and n - 1

   Error:  "signature representative out of range"

   Assumption:  RSA public key (n, e) is valid

   Steps:

      1.  If the signature representative s is not between 0 and n - 1,
          output "signature representative out of range" and stop.

      2.  Let m = s^e mod n.

      3.  Output m.

6.  Overview of Schemes

   A scheme combines cryptographic primitives and other techniques to
   achieve a particular security goal.  Two types of scheme are
   specified in this document: encryption schemes and signature schemes
   with appendix.

   The schemes specified in this document are limited in scope in that
   their operations consist only of steps to process data with an RSA
   public or private key, and they do not include steps for obtaining or
   validating the key.  Thus, in addition to the scheme operations, an
   application will typically include key management operations by which
   parties may select RSA public and private keys for a scheme
   operation.  The specific additional operations and other details are
   outside the scope of this document.

   As was the case for the cryptographic primitives (Section 5), the
   specifications of scheme operations assume that certain conditions
   are met by the inputs, in particular that RSA public and private keys
   are valid.  The behavior of an implementation is thus unspecified
   when a key is invalid.  The impact of such unspecified behavior
   depends on the application.  Possible means of addressing key
   validation include explicit key validation by the application; key
   validation within the public-key infrastructure; and assignment of
   liability for operations performed with an invalid key to the party
   who generated the key.

   A generally good cryptographic practice is to employ a given RSA key
   pair in only one scheme.  This avoids the risk that vulnerability in
   one scheme may compromise the security of the other and may be
   essential to maintain provable security.  While RSAES-PKCS1-v1_5

   (Section 7.2) and RSASSA-PKCS1-v1_5 (Section 8.2) have traditionally
   been employed together without any known bad interactions (indeed,
   this is the model introduced by PKCS #1 v1.5), such a combined use of
   an RSA key pair is NOT RECOMMENDED for new applications.

   To illustrate the risks related to the employment of an RSA key pair
   in more than one scheme, suppose an RSA key pair is employed in both
   RSAES-OAEP (Section 7.1) and RSAES-PKCS1-v1_5.  Although RSAES-OAEP
   by itself would resist attack, an opponent might be able to exploit a
   weakness in the implementation of RSAES-PKCS1-v1_5 to recover
   messages encrypted with either scheme.  As another example, suppose
   an RSA key pair is employed in both RSASSA-PSS (Section 8.1) and
   RSASSA-PKCS1-v1_5.  Then the security proof for RSASSA-PSS would no
   longer be sufficient since the proof does not account for the
   possibility that signatures might be generated with a second scheme.
   Similar considerations may apply if an RSA key pair is employed in
   one of the schemes defined here and in a variant defined elsewhere.

7.  Encryption Schemes

   For the purposes of this document, an encryption scheme consists of
   an encryption operation and a decryption operation, where the
   encryption operation produces a ciphertext from a message with a
   recipient's RSA public key, and the decryption operation recovers the
   message from the ciphertext with the recipient's corresponding RSA
   private key.

   An encryption scheme can be employed in a variety of applications.  A
   typical application is a key establishment protocol, where the
   message contains key material to be delivered confidentially from one
   party to another.  For instance, PKCS #7 [RFC2315] employs such a
   protocol to deliver a content-encryption key from a sender to a
   recipient; the encryption schemes defined here would be suitable key-
   encryption algorithms in that context.

   Two encryption schemes are specified in this document: RSAES-OAEP and
   RSAES-PKCS1-v1_5.  RSAES-OAEP is REQUIRED to be supported for new
   applications; RSAES-PKCS1-v1_5 is included only for compatibility
   with existing applications.

   The encryption schemes given here follow a general model similar to
   that employed in IEEE 1363 [IEEE1363], combining encryption and
   decryption primitives with an encoding method for encryption.  The
   encryption operations apply a message encoding operation to a message
   to produce an encoded message, which is then converted to an integer
   message representative.  An encryption primitive is applied to the
   message representative to produce the ciphertext.  Reversing this,
   the decryption operations apply a decryption primitive to the

   ciphertext to recover a message representative, which is then
   converted to an octet-string-encoded message.  A message decoding
   operation is applied to the encoded message to recover the message
   and verify the correctness of the decryption.

   To avoid implementation weaknesses related to the way errors are
   handled within the decoding operation (see [BLEICHENBACHER] and
   [MANGER]), the encoding and decoding operations for RSAES-OAEP and
   RSAES-PKCS1-v1_5 are embedded in the specifications of the respective
   encryption schemes rather than defined in separate specifications.
   Both encryption schemes are compatible with the corresponding schemes
   in PKCS #1 v2.1.

7.1.  RSAES-OAEP

   RSAES-OAEP combines the RSAEP and RSADP primitives (Sections 5.1.1
   and 5.1.2) with the EME-OAEP encoding method (Step 2 in
   Section 7.1.1, and Step 3 in Section 7.1.2).  EME-OAEP is based on
   Bellare and Rogaway's Optimal Asymmetric Encryption scheme [OAEP].
   It is compatible with the Integer Factorization Encryption Scheme
   (IFES) defined in IEEE 1363 [IEEE1363], where the encryption and
   decryption primitives are IFEP-RSA and IFDP-RSA and the message
   encoding method is EME-OAEP.  RSAES-OAEP can operate on messages of
   length up to k - 2hLen -2 octets, where hLen is the length of the
   output from the underlying hash function and k is the length in
   octets of the recipient's RSA modulus.

   Assuming that computing e-th roots modulo n is infeasible and the
   mask generation function in RSAES-OAEP has appropriate properties,
   RSAES-OAEP is semantically secure against adaptive chosen-ciphertext
   attacks.  This assurance is provable in the sense that the difficulty
   of breaking RSAES-OAEP can be directly related to the difficulty of
   inverting the RSA function, provided that the mask generation
   function is viewed as a black box or random oracle; see [FOPS] and
   the note below for further discussion.

   Both the encryption and the decryption operations of RSAES-OAEP take
   the value of a label L as input.  In this version of PKCS #1, L is
   the empty string; other uses of the label are outside the scope of
   this document.  See Appendix A.2.1 for the relevant ASN.1 syntax.

   RSAES-OAEP is parameterized by the choice of hash function and mask
   generation function.  This choice should be fixed for a given RSA
   key.  Suggested hash and mask generation functions are given in
   Appendix B.

   Note: Past results have helpfully clarified the security properties
   of the OAEP encoding method [OAEP]  (roughly the procedure described
   in Step 2 in Section 7.1.1).  The background is as follows.  In 1994,
   Bellare and Rogaway [OAEP] introduced a security concept that they
   denoted plaintext awareness (PA94).  They proved that if a
   deterministic public-key encryption primitive (e.g., RSAEP) is hard
   to invert without the private key, then the corresponding OAEP-based
   encryption scheme is plaintext aware (in the random oracle model),
   meaning roughly that an adversary cannot produce a valid ciphertext
   without actually "knowing" the underlying plaintext.  Plaintext
   awareness of an encryption scheme is closely related to the
   resistance of the scheme against chosen-ciphertext attacks.  In such
   attacks, an adversary is given the opportunity to send queries to an
   oracle simulating the decryption primitive.  Using the results of
   these queries, the adversary attempts to decrypt a challenge
   ciphertext.

   However, there are two flavors of chosen-ciphertext attacks, and PA94
   implies security against only one of them.  The difference relies on
   what the adversary is allowed to do after she is given the challenge
   ciphertext.  The indifferent attack scenario (denoted CCA1) does not
   admit any queries to the decryption oracle after the adversary is
   given the challenge ciphertext, whereas the adaptive scenario
   (denoted CCA2) does (except that the decryption oracle refuses to
   decrypt the challenge ciphertext once it is published).  In 1998,
   Bellare and Rogaway, together with Desai and Pointcheval [PA98], came
   up with a new, stronger notion of plaintext awareness (PA98) that
   does imply security against CCA2.

   To summarize, there have been two potential sources for
   misconception: that PA94 and PA98 are equivalent concepts, or that
   CCA1 and CCA2 are equivalent concepts.  Either assumption leads to
   the conclusion that the Bellare-Rogaway paper implies security of
   OAEP against CCA2, which it does not.

   (Footnote: It might be fair to mention that PKCS #1 v2.0 cites [OAEP]
   and claims that "a chosen ciphertext attack is ineffective against a
   plaintext-aware encryption scheme such as RSAES-OAEP" without
   specifying the kind of plaintext awareness or chosen ciphertext
   attack considered.)

   OAEP has never been proven secure against CCA2; in fact, Victor Shoup
   [SHOUP] has demonstrated that such a proof does not exist in the
   general case.  Put briefly, Shoup showed that an adversary in the
   CCA2 scenario who knows how to partially invert the encryption
   primitive but does not know how to invert it completely may well be
   able to break the scheme.  For example, one may imagine an attacker
   who is able to break RSAES-OAEP if she knows how to recover all but

   the first 20 bytes of a random integer encrypted with RSAEP.  Such an
   attacker does not need to be able to fully invert RSAEP, because she
   does not use the first 20 octets in her attack.

   Still, RSAES-OAEP is secure against CCA2, which was proved by
   Fujisaki, Okamoto, Pointcheval, and Stern [FOPS] shortly after the
   announcement of Shoup's result.  Using clever lattice reduction
   techniques, they managed to show how to invert RSAEP completely given
   a sufficiently large part of the pre-image.  This observation,
   combined with a proof that OAEP is secure against CCA2 if the
   underlying encryption primitive is hard to partially invert, fills
   the gap between what Bellare and Rogaway proved about RSAES-OAEP and
   what some may have believed that they proved.  Somewhat
   paradoxically, we are hence saved by an ostensible weakness in RSAEP
   (i.e., the whole inverse can be deduced from parts of it).

   Unfortunately, however, the security reduction is not efficient for
   concrete parameters.  While the proof successfully relates an
   adversary A against the CCA2 security of RSAES-OAEP to an algorithm I
   inverting RSA, the probability of success for I is only approximately
   \epsilon^2 / 2^18, where \epsilon is the probability of success for
   A.

   (Footnote: In [FOPS], the probability of success for the inverter was
   \epsilon^2 / 4.  The additional factor 1 / 2^16 is due to the eight
   fixed zero bits at the beginning of the encoded message EM, which are
   not present in the variant of OAEP considered in [FOPS].  (A must be
   applied twice to invert RSA, and each application corresponds to a
   factor 1 / 2^8.))

   In addition, the running time for I is approximately t^2, where t is
   the running time of the adversary.  The consequence is that we cannot
   exclude the possibility that attacking RSAES-OAEP is considerably
   easier than inverting RSA for concrete parameters.  Still, the
   existence of a security proof provides some assurance that the
   RSAES-OAEP construction is sounder than ad hoc constructions such as
   RSAES-PKCS1-v1_5.

   Hybrid encryption schemes based on the RSA Key Encapsulation
   Mechanism (RSA-KEM) paradigm offer tight proofs of security directly
   applicable to concrete parameters; see [ISO18033] for discussion.
   Future versions of PKCS #1 may specify schemes based on this
   paradigm.

7.1.1.  Encryption Operation

   RSAES-OAEP-ENCRYPT ((n, e), M, L)

   Options:

      Hash     hash function (hLen denotes the length in octets of
               the hash function output)
      MGF      mask generation function

   Input:

      (n, e)   recipient's RSA public key (k denotes the length in
               octets of the RSA modulus n)
      M        message to be encrypted, an octet string of length mLen,
               where mLen <= k - 2hLen - 2
      L        optional label to be associated with the message; the
               default value for L, if L is not provided, is the empty
               string

   Output:

      C        ciphertext, an octet string of length k

   Errors:  "message too long"; "label too long"

   Assumption:  RSA public key (n, e) is valid

   Steps:

      1.  Length checking:

          a.  If the length of L is greater than the input limitation
              for the hash function (2^61 - 1 octets for SHA-1), output
              "label too long" and stop.

          b.  If mLen > k - 2hLen - 2, output "message too long" and
              stop.

      2.  EME-OAEP encoding (see Figure 1 below):

          a.  If the label L is not provided, let L be the empty string.
              Let lHash = Hash(L), an octet string of length hLen (see
              the note below).

          b.  Generate a padding string PS consisting of k - mLen -
              2hLen - 2 zero octets.  The length of PS may be zero.

          c.  Concatenate lHash, PS, a single octet with hexadecimal
              value 0x01, and the message M to form a data block DB of
              length k - hLen - 1 octets as

                 DB = lHash || PS || 0x01 || M.

          d.  Generate a random octet string seed of length hLen.

          e.  Let dbMask = MGF(seed, k - hLen - 1).

          f.  Let maskedDB = DB \xor dbMask.

          g.  Let seedMask = MGF(maskedDB, hLen).

          h.  Let maskedSeed = seed \xor seedMask.

          i.  Concatenate a single octet with hexadecimal value 0x00,
              maskedSeed, and maskedDB to form an encoded message EM of
              length k octets as

                 EM = 0x00 || maskedSeed || maskedDB.

      3.  RSA encryption:

          a.  Convert the encoded message EM to an integer message
              representative m (see Section 4.2):

                 m = OS2IP (EM).

          b.  Apply the RSAEP encryption primitive (Section 5.1.1) to
              the RSA public key (n, e) and the message representative m
              to produce an integer ciphertext representative c:

                 c = RSAEP ((n, e), m).

          c.  Convert the ciphertext representative c to a ciphertext C
              of length k octets (see Section 4.1):

                 C = I2OSP (c, k).

      4.  Output the ciphertext C.

      _________________________________________________________________

                                +----------+------+--+-------+
                           DB = |  lHash   |  PS  |01|   M   |
                                +----------+------+--+-------+
                                               |
                     +----------+              |
                     |   seed   |              |
                     +----------+              |
                           |                   |
                           |-------> MGF ---> xor
                           |                   |
                  +--+     V                   |
                  |00|    xor <----- MGF <-----|
                  +--+     |                   |
                    |      |                   |
                    V      V                   V
                  +--+----------+----------------------------+
            EM =  |00|maskedSeed|          maskedDB          |
                  +--+----------+----------------------------+
      _________________________________________________________________

                   Figure 1: EME-OAEP Encoding Operation

   Notes:

   -  lHash is the hash of the optional label L.

   -  The decoding operation follows reverse steps to recover M and
      verify lHash and PS.

   -  If L is the empty string, the corresponding hash value lHash has
      the following hexadecimal representation for different choices of
      Hash:

      SHA-1:   (0x)da39a3ee 5e6b4b0d 3255bfef 95601890 afd80709
      SHA-256: (0x)e3b0c442 98fc1c14 9afbf4c8 996fb924 27ae41e4 649b934c
                   a495991b 7852b855
      SHA-384: (0x)38b060a7 51ac9638 4cd9327e b1b1e36a 21fdb711 14be0743
                   4c0cc7bf 63f6e1da 274edebf e76f65fb d51ad2f1 4898b95b
      SHA-512: (0x)cf83e135 7eefb8bd f1542850 d66d8007 d620e405 0b5715dc
                   83f4a921 d36ce9ce 47d0d13c 5d85f2b0 ff8318d2 877eec2f
                   63b931bd 47417a81 a538327a f927da3e

7.1.2.  Decryption Operation

   RSAES-OAEP-DECRYPT (K, C, L)

   Options:

      Hash     hash function (hLen denotes the length in octets of
               the hash function output)
      MGF      mask generation function

   Input:

      K        recipient's RSA private key (k denotes the length in
               octets of the RSA modulus n), where k >= 2hLen + 2
      C        ciphertext to be decrypted, an octet string of length k
      L        optional label whose association with the message is to
               be verified; the default value for L, if L is not
               provided, is the empty string

   Output:

      M        message, an octet string of length mLen, where
               mLen <= k - 2hLen - 2

   Error:  "decryption error"

   Steps:

      1.  Length checking:

          a.  If the length of L is greater than the input limitation
              for the hash function (2^61 - 1 octets for SHA-1), output
              "decryption error" and stop.

          b.  If the length of the ciphertext C is not k octets, output
              "decryption error" and stop.

          c.  If k < 2hLen + 2, output "decryption error" and stop.

      2.  RSA decryption:

          a.  Convert the ciphertext C to an integer ciphertext
              representative c (see Section 4.2):

                 c = OS2IP (C).

          b.  Apply the RSADP decryption primitive (Section 5.1.2) to
              the RSA private key K and the ciphertext representative c
              to produce an integer message representative m:

                 m = RSADP (K, c).

              If RSADP outputs "ciphertext representative out of range"
              (meaning that c >= n), output "decryption error" and stop.

          c.  Convert the message representative m to an encoded message
              EM of length k octets (see Section 4.1):

                 EM = I2OSP (m, k).

      3.  EME-OAEP decoding:

          a.  If the label L is not provided, let L be the empty string.
              Let lHash = Hash(L), an octet string of length hLen (see
              the note in Section 7.1.1).

          b.  Separate the encoded message EM into a single octet Y, an
              octet string maskedSeed of length hLen, and an octet
              string maskedDB of length k - hLen - 1 as

                 EM = Y || maskedSeed || maskedDB.

          c.  Let seedMask = MGF(maskedDB, hLen).

          d.  Let seed = maskedSeed \xor seedMask.

          e.  Let dbMask = MGF(seed, k - hLen - 1).

          f.  Let DB = maskedDB \xor dbMask.

          g.  Separate DB into an octet string lHash' of length hLen, a
              (possibly empty) padding string PS consisting of octets
              with hexadecimal value 0x00, and a message M as

                 DB = lHash' || PS || 0x01 || M.

              If there is no octet with hexadecimal value 0x01 to
              separate PS from M, if lHash does not equal lHash', or if
              Y is nonzero, output "decryption error" and stop.  (See
              the note below.)

      4.  Output the message M.

      Note: Care must be taken to ensure that an opponent cannot
      distinguish the different error conditions in Step 3.g, whether by
      error message or timing, and, more generally, that an opponent
      cannot learn partial information about the encoded message EM.
      Otherwise, an opponent may be able to obtain useful information
      about the decryption of the ciphertext C, leading to a chosen-
      ciphertext attack such as the one observed by Manger [MANGER].

7.2.  RSAES-PKCS1-v1_5

   RSAES-PKCS1-v1_5 combines the RSAEP and RSADP primitives (Sections
   5.1.1 and 5.1.2) with the EME-PKCS1-v1_5 encoding method (Step 2 in
   Section 7.2.1, and Step 3 in Section 7.2.2).  It is mathematically
   equivalent to the encryption scheme in PKCS #1 v1.5.
   RSAES-PKCS1-v1_5 can operate on messages of length up to k - 11
   octets (k is the octet length of the RSA modulus), although care
   should be taken to avoid certain attacks on low-exponent RSA due to
   Coppersmith, Franklin, Patarin, and Reiter when long messages are
   encrypted (see the third bullet in the notes below and [LOWEXP];
   [NEWATTACK] contains an improved attack).  As a general rule, the use
   of this scheme for encrypting an arbitrary message, as opposed to a
   randomly generated key, is NOT RECOMMENDED.

   It is possible to generate valid RSAES-PKCS1-v1_5 ciphertexts without
   knowing the corresponding plaintexts, with a reasonable probability
   of success.  This ability can be exploited in a chosen-ciphertext
   attack as shown in [BLEICHENBACHER].  Therefore, if RSAES-PKCS1-v1_5
   is to be used, certain easily implemented countermeasures should be
   taken to thwart the attack found in [BLEICHENBACHER].  Typical
   examples include the addition of structure to the data to be encoded,
   rigorous checking of PKCS #1 v1.5 conformance (and other redundancy)
   in decrypted messages, and the consolidation of error messages in a
   client-server protocol based on PKCS #1 v1.5.  These can all be
   effective countermeasures and do not involve changes to a protocol
   based on PKCS #1 v1.5.  See [BKS] for a further discussion of these
   and other countermeasures.  It has recently been shown that the
   security of the SSL/TLS handshake protocol [RFC5246], which uses
   RSAES-PKCS1-v1_5 and certain countermeasures, can be related to a
   variant of the RSA problem; see [RSATLS] for discussion.

   Note: The following passages describe some security recommendations
   pertaining to the use of RSAES-PKCS1-v1_5.  Recommendations from PKCS
   #1 v1.5 are included as well as new recommendations motivated by
   cryptanalytic advances made in the intervening years.

   o  It is RECOMMENDED that the pseudorandom octets in Step 2 in
      Section 7.2.1 be generated independently for each encryption
      process, especially if the same data is input to more than one
      encryption process.  Haastad's results [HAASTAD] are one
      motivation for this recommendation.

   o  The padding string PS in Step 2 in Section 7.2.1 is at least eight
      octets long, which is a security condition for public-key
      operations that makes it difficult for an attacker to recover data
      by trying all possible encryption blocks.

   o  The pseudorandom octets can also help thwart an attack due to
      Coppersmith et al.  [LOWEXP] (see [NEWATTACK] for an improvement
      of the attack) when the size of the message to be encrypted is
      kept small.  The attack works on low-exponent RSA when similar
      messages are encrypted with the same RSA public key.  More
      specifically, in one flavor of the attack, when two inputs to
      RSAEP agree on a large fraction of bits (8/9) and low-exponent RSA
      (e = 3) is used to encrypt both of them, it may be possible to
      recover both inputs with the attack.  Another flavor of the attack
      is successful in decrypting a single ciphertext when a large
      fraction (2/3) of the input to RSAEP is already known.  For
      typical applications, the message to be encrypted is short (e.g.,
      a 128-bit symmetric key), so not enough information will be known
      or common between two messages to enable the attack.  However, if
      a long message is encrypted, or if part of a message is known,
      then the attack may be a concern.  In any case, the RSAES-OAEP
      scheme overcomes the attack.

7.2.1.  Encryption Operation

   RSAES-PKCS1-V1_5-ENCRYPT ((n, e), M)

   Input:

      (n, e)   recipient's RSA public key (k denotes the length in
               octets of the modulus n)
      M        message to be encrypted, an octet string of length
               mLen, where mLen <= k - 11

   Output:

      C        ciphertext, an octet string of length k

   Error:  "message too long"

   Steps:

      1.  Length checking: If mLen > k - 11, output "message too long"
          and stop.

      2.  EME-PKCS1-v1_5 encoding:

          a.  Generate an octet string PS of length k - mLen - 3
              consisting of pseudo-randomly generated nonzero octets.
              The length of PS will be at least eight octets.

          b.  Concatenate PS, the message M, and other padding to form
              an encoded message EM of length k octets as

                 EM = 0x00 || 0x02 || PS || 0x00 || M.

      3.  RSA encryption:

          a.  Convert the encoded message EM to an integer message
              representative m (see Section 4.2):

                 m = OS2IP (EM).

          b.  Apply the RSAEP encryption primitive (Section 5.1.1) to
              the RSA public key (n, e) and the message representative m
              to produce an integer ciphertext representative c:

                 c = RSAEP ((n, e), m).

          c.  Convert the ciphertext representative c to a ciphertext C
              of length k octets (see Section 4.1):

                 C = I2OSP (c, k).

      4.  Output the ciphertext C.

7.2.2.  Decryption Operation

   RSAES-PKCS1-V1_5-DECRYPT (K, C)

   Input:

      K        recipient's RSA private key
      C        ciphertext to be decrypted, an octet string of length k,
               where k is the length in octets of the RSA modulus n

   Output:

      M        message, an octet string of length at most k - 11

   Error:  "decryption error"

   Steps:

      1.  Length checking: If the length of the ciphertext C is not k
          octets (or if k < 11), output "decryption error" and stop.

      2.  RSA decryption:

          a.  Convert the ciphertext C to an integer ciphertext
              representative c (see Section 4.2):

                 c = OS2IP (C).

          b.  Apply the RSADP decryption primitive (Section 5.1.2) to
              the RSA private key (n, d) and the ciphertext
              representative c to produce an integer message
              representative m:

                 m = RSADP ((n, d), c).

              If RSADP outputs "ciphertext representative out of range"
              (meaning that c >= n), output "decryption error" and stop.

          c.  Convert the message representative m to an encoded message
              EM of length k octets (see Section 4.1):

                 EM = I2OSP (m, k).

      3.  EME-PKCS1-v1_5 decoding: Separate the encoded message EM into
          an octet string PS consisting of nonzero octets and a message
          M as

             EM = 0x00 || 0x02 || PS || 0x00 || M.

          If the first octet of EM does not have hexadecimal value 0x00,
          if the second octet of EM does not have hexadecimal value
          0x02, if there is no octet with hexadecimal value 0x00 to
          separate PS from M, or if the length of PS is less than 8
          octets, output "decryption error" and stop.  (See the note
          below.)

      4.  Output M.

      Note: Care shall be taken to ensure that an opponent cannot
      distinguish the different error conditions in Step 3, whether by
      error message or timing.  Otherwise, an opponent may be able to
      obtain useful information about the decryption of the ciphertext
      C, leading to a strengthened version of Bleichenbacher's attack
      [BLEICHENBACHER]; compare to Manger's attack [MANGER].

8.  Signature Scheme with Appendix

   For the purposes of this document, a signature scheme with appendix
   consists of a signature generation operation and a signature
   verification operation, where the signature generation operation
   produces a signature from a message with a signer's RSA private key,
   and the signature verification operation verifies the signature on
   the message with the signer's corresponding RSA public key.  To
   verify a signature constructed with this type of scheme, it is
   necessary to have the message itself.  In this way, signature schemes
   with appendix are distinguished from signature schemes with message
   recovery, which are not supported in this document.

   A signature scheme with appendix can be employed in a variety of
   applications.  For instance, the signature schemes with appendix
   defined here would be suitable signature algorithms for X.509
   certificates [ISO9594].  Related signature schemes could be employed
   in PKCS #7 [RFC2315], although for technical reasons the current
   version of PKCS #7 separates a hash function from a signature scheme,
   which is different than what is done here; see the note in
   Appendix A.2.3 for more discussion.

   Two signature schemes with appendix are specified in this document:
   RSASSA-PSS and RSASSA-PKCS1-v1_5.  Although no attacks are known
   against RSASSA-PKCS1-v1_5, in the interest of increased robustness,
   RSASSA-PSS is REQUIRED in new applications.  RSASSA-PKCS1-v1_5 is
   included only for compatibility with existing applications.

   The signature schemes with appendix given here follow a general model
   similar to that employed in IEEE 1363 [IEEE1363], combining signature
   and verification primitives with an encoding method for signatures.
   The signature generation operations apply a message encoding
   operation to a message to produce an encoded message, which is then
   converted to an integer message representative.  A signature
   primitive is applied to the message representative to produce the
   signature.  Reversing this, the signature verification operations
   apply a signature verification primitive to the signature to recover
   a message representative, which is then converted to an octet-string-
   encoded message.  A verification operation is applied to the message
   and the encoded message to determine whether they are consistent.

   If the encoding method is deterministic (e.g., EMSA-PKCS1-v1_5), the
   verification operation may apply the message encoding operation to
   the message and compare the resulting encoded message to the
   previously derived encoded message.  If there is a match, the
   signature is considered valid.  If the method is randomized (e.g.,
   EMSA-PSS), the verification operation is typically more complicated.
   For example, the verification operation in EMSA-PSS extracts the
   random salt and a hash output from the encoded message and checks
   whether the hash output, the salt, and the message are consistent;
   the hash output is a deterministic function in terms of the message
   and the salt.  For both signature schemes with appendix defined in
   this document, the signature generation and signature verification
   operations are readily implemented as "single-pass" operations if the
   signature is placed after the message.  See PKCS #7 [RFC2315] for an
   example format in the case of RSASSA-PKCS1-v1_5.

8.1.  RSASSA-PSS

   RSASSA-PSS combines the RSASP1 and RSAVP1 primitives with the
   EMSA-PSS encoding method.  It is compatible with the Integer
   Factorization Signature Scheme with Appendix (IFSSA) as amended in
   IEEE 1363a [IEEE1363A], where the signature and verification
   primitives are IFSP-RSA1 and IFVP-RSA1 as defined in IEEE 1363
   [IEEE1363], and the message encoding method is EMSA4.  EMSA4 is
   slightly more general than EMSA-PSS as it acts on bit strings rather
   than on octet strings.  EMSA-PSS is equivalent to EMSA4 restricted to
   the case that the operands as well as the hash and salt values are
   octet strings.

   The length of messages on which RSASSA-PSS can operate is either
   unrestricted or constrained by a very large number, depending on the
   hash function underlying the EMSA-PSS encoding method.

   Assuming that computing e-th roots modulo n is infeasible and the
   hash and mask generation functions in EMSA-PSS have appropriate
   properties, RSASSA-PSS provides secure signatures.  This assurance is
   provable in the sense that the difficulty of forging signatures can
   be directly related to the difficulty of inverting the RSA function,
   provided that the hash and mask generation functions are viewed as
   black boxes or random oracles.  The bounds in the security proof are
   essentially "tight", meaning that the success probability and running
   time for the best forger against RSASSA-PSS are very close to the
   corresponding parameters for the best RSA inversion algorithm; see
   [RSARABIN] [PSSPROOF] [JONSSON] for further discussion.

   In contrast to the RSASSA-PKCS1-v1_5 signature scheme, a hash
   function identifier is not embedded in the EMSA-PSS encoded message,
   so in theory it is possible for an adversary to substitute a

   different (and potentially weaker) hash function than the one
   selected by the signer.  Therefore, it is RECOMMENDED that the
   EMSA-PSS mask generation function be based on the same hash function.
   In this manner, the entire encoded message will be dependent on the
   hash function, and it will be difficult for an opponent to substitute
   a different hash function than the one intended by the signer.  This
   matching of hash functions is only for the purpose of preventing hash
   function substitution and is not necessary if hash function
   substitution is addressed by other means (e.g., the verifier accepts
   only a designated hash function).  See [HASHID] for further
   discussion of these points.  The provable security of RSASSA-PSS does
   not rely on the hash function in the mask generation function being
   the same as the hash function applied to the message.

   RSASSA-PSS is different from other RSA-based signature schemes in
   that it is probabilistic rather than deterministic, incorporating a
   randomly generated salt value.  The salt value enhances the security
   of the scheme by affording a "tighter" security proof than
   deterministic alternatives such as Full Domain Hashing (FDH); see
   [RSARABIN] for discussion.  However, the randomness is not critical
   to security.  In situations where random generation is not possible,
   a fixed value or a sequence number could be employed instead, with
   the resulting provable security similar to that of FDH [FDH].

8.1.1.  Signature Generation Operation

   RSASSA-PSS-SIGN (K, M)

   Input:

      K        signer's RSA private key
      M        message to be signed, an octet string

   Output:

      S        signature, an octet string of length k, where k is the
               length in octets of the RSA modulus n

   Errors:  "message too long;" "encoding error"

   Steps:

      1.  EMSA-PSS encoding: Apply the EMSA-PSS encoding operation
          (Section 9.1.1) to the message M to produce an encoded message
          EM of length \ceil ((modBits - 1)/8) octets such that the bit
          length of the integer OS2IP (EM) (see Section 4.2) is at most
          modBits - 1, where modBits is the length in bits of the RSA
          modulus n:

             EM = EMSA-PSS-ENCODE (M, modBits - 1).

          Note that the octet length of EM will be one less than k if
          modBits - 1 is divisible by 8 and equal to k otherwise.  If
          the encoding operation outputs "message too long", output
          "message too long" and stop.  If the encoding operation
          outputs "encoding error", output "encoding error" and stop.

      2.  RSA signature:

          a.  Convert the encoded message EM to an integer message
              representative m (see Section 4.2):

                 m = OS2IP (EM).

          b.  Apply the RSASP1 signature primitive (Section 5.2.1) to
              the RSA private key K and the message representative m to
              produce an integer signature representative s:

                 s = RSASP1 (K, m).

          c.  Convert the signature representative s to a signature S of
              length k octets (see Section 4.1):

                 S = I2OSP (s, k).

      3.  Output the signature S.

8.1.2.  Signature Verification Operation

   RSASSA-PSS-VERIFY ((n, e), M, S)

   Input:

      (n, e)  signer's RSA public key
      M       message whose signature is to be verified, an octet string
      S       signature to be verified, an octet string of length k,
              where k is the length in octets of the RSA modulus n

   Output:  "valid signature" or "invalid signature"

   Steps:

      1.  Length checking: If the length of the signature S is not k
          octets, output "invalid signature" and stop.

      2.  RSA verification:

          a.  Convert the signature S to an integer signature
              representative s (see Section 4.2):

                 s = OS2IP (S).

          b.  Apply the RSAVP1 verification primitive (Section 5.2.2) to
              the RSA public key (n, e) and the signature representative
              s to produce an integer message representative m:

                 m = RSAVP1 ((n, e), s).

              If RSAVP1 output "signature representative out of range",
              output "invalid signature" and stop.

          c.  Convert the message representative m to an encoded message
              EM of length emLen = \ceil ((modBits - 1)/8) octets, where
              modBits is the length in bits of the RSA modulus n (see
              Section 4.1):

                 EM = I2OSP (m, emLen).

              Note that emLen will be one less than k if modBits - 1 is
              divisible by 8 and equal to k otherwise.  If I2OSP outputs
              "integer too large", output "invalid signature" and stop.

      3.  EMSA-PSS verification: Apply the EMSA-PSS verification
          operation (Section 9.1.2) to the message M and the encoded
          message EM to determine whether they are consistent:

             Result = EMSA-PSS-VERIFY (M, EM, modBits - 1).

      4.  If Result = "consistent", output "valid signature".
          Otherwise, output "invalid signature".

8.2.  RSASSA-PKCS1-v1_5

   RSASSA-PKCS1-v1_5 combines the RSASP1 and RSAVP1 primitives with the
   EMSA-PKCS1-v1_5 encoding method.  It is compatible with the IFSSA
   scheme defined in IEEE 1363 [IEEE1363], where the signature and
   verification primitives are IFSP-RSA1 and IFVP-RSA1, and the message
   encoding method is EMSA-PKCS1-v1_5 (which is not defined in IEEE 1363
   but is in IEEE 1363a [IEEE1363A]).

   The length of messages on which RSASSA-PKCS1-v1_5 can operate is
   either unrestricted or constrained by a very large number, depending
   on the hash function underlying the EMSA-PKCS1-v1_5 method.

   Assuming that computing e-th roots modulo n is infeasible and the
   hash function in EMSA-PKCS1-v1_5 has appropriate properties,
   RSASSA-PKCS1-v1_5 is conjectured to provide secure signatures.  More
   precisely, forging signatures without knowing the RSA private key is
   conjectured to be computationally infeasible.  Also, in the encoding
   method EMSA-PKCS1-v1_5, a hash function identifier is embedded in the
   encoding.  Because of this feature, an adversary trying to find a
   message with the same signature as a previously signed message must
   find collisions of the particular hash function being used; attacking
   a different hash function than the one selected by the signer is not
   useful to the adversary.  See [HASHID] for further discussion.

   Note: As noted in PKCS #1 v1.5, the EMSA-PKCS1-v1_5 encoding method
   has the property that the encoded message, converted to an integer
   message representative, is guaranteed to be large and at least
   somewhat "random".  This prevents attacks of the kind proposed by
   Desmedt and Odlyzko [CHOSEN] where multiplicative relationships
   between message representatives are developed by factoring the
   message representatives into a set of small values (e.g., a set of
   small primes).  Coron, Naccache, and Stern [PADDING] showed that a
   stronger form of this type of attack could be quite effective against
   some instances of the ISO/IEC 9796-2 signature scheme.  They also
   analyzed the complexity of this type of attack against the
   EMSA-PKCS1-v1_5 encoding method and concluded that an attack would be
   impractical, requiring more operations than a collision search on the
   underlying hash function (i.e., more than 2^80 operations).
   Coppersmith, Halevi, and Jutla [FORGERY] subsequently extended Coron
   et al.'s attack to break the ISO/IEC 9796-1 signature scheme with
   message recovery.  The various attacks illustrate the importance of
   carefully constructing the input to the RSA signature primitive,
   particularly in a signature scheme with message recovery.
   Accordingly, the EMSA-PKCS-v1_5 encoding method explicitly includes a
   hash operation and is not intended for signature schemes with message
   recovery.  Moreover, while no attack is known against the
   EMSA-PKCS-v1_5 encoding method, a gradual transition to EMSA-PSS is
   recommended as a precaution against future developments.

8.2.1.  Signature Generation Operation

   RSASSA-PKCS1-V1_5-SIGN (K, M)

   Input:

      K        signer's RSA private key
      M        message to be signed, an octet string

   Output:

      S        signature, an octet string of length k, where k is the
               length in octets of the RSA modulus n

   Errors:  "message too long"; "RSA modulus too short"

   Steps:

      1.  EMSA-PKCS1-v1_5 encoding: Apply the EMSA-PKCS1-v1_5 encoding
          operation (Section 9.2) to the message M to produce an encoded
          message EM of length k octets:

             EM = EMSA-PKCS1-V1_5-ENCODE (M, k).

          If the encoding operation outputs "message too long", output
          "message too long" and stop.  If the encoding operation
          outputs "intended encoded message length too short", output
          "RSA modulus too short" and stop.

      2.  RSA signature:

          a.  Convert the encoded message EM to an integer message
              representative m (see Section 4.2):

                 m = OS2IP (EM).

          b.  Apply the RSASP1 signature primitive (Section 5.2.1) to
              the RSA private key K and the message representative m to
              produce an integer signature representative s:

                 s = RSASP1 (K, m).

          c.  Convert the signature representative s to a signature S of
              length k octets (see Section 4.1):

                 S = I2OSP (s, k).

      3.  Output the signature S.

8.2.2.  Signature Verification Operation

   RSASSA-PKCS1-V1_5-VERIFY ((n, e), M, S)

   Input:

      (n, e)  signer's RSA public key
      M       message whose signature is to be verified, an octet string

      S       signature to be verified, an octet string of length k,
              where k is the length in octets of the RSA modulus n

   Output  "valid signature" or "invalid signature"

   Errors:  "message too long"; "RSA modulus too short"

   Steps:

      1.  Length checking: If the length of the signature S is not k
          octets, output "invalid signature" and stop.

      2.  RSA verification:

          a.  Convert the signature S to an integer signature
              representative s (see Section 4.2):

                 s = OS2IP (S).

          b.  Apply the RSAVP1 verification primitive (Section 5.2.2) to
              the RSA public key (n, e) and the signature representative
              s to produce an integer message representative m:

                 m = RSAVP1 ((n, e), s).

              If RSAVP1 outputs "signature representative out of range",
              output "invalid signature" and stop.

          c.  Convert the message representative m to an encoded message
              EM of length k octets (see Section 4.1):

                 EM = I2OSP (m, k).

              If I2OSP outputs "integer too large", output "invalid
              signature" and stop.

      3.  EMSA-PKCS1-v1_5 encoding: Apply the EMSA-PKCS1-v1_5 encoding
          operation (Section 9.2) to the message M to produce a second
          encoded message EM' of length k octets:

             EM' = EMSA-PKCS1-V1_5-ENCODE (M, k).

          If the encoding operation outputs "message too long", output
          "message too long" and stop.  If the encoding operation
          outputs "intended encoded message length too short", output
          "RSA modulus too short" and stop.

      4.  Compare the encoded message EM and the second encoded message
          EM'.  If they are the same, output "valid signature";
          otherwise, output "invalid signature".

      Note: Another way to implement the signature verification
      operation is to apply a "decoding" operation (not specified in
      this document) to the encoded message to recover the underlying
      hash value, and then compare it to a newly computed hash value.
      This has the advantage that it requires less intermediate storage
      (two hash values rather than two encoded messages), but the
      disadvantage that it requires additional code.

9.  Encoding Methods for Signatures with Appendix

   Encoding methods consist of operations that map between octet string
   messages and octet-string-encoded messages, which are converted to
   and from integer message representatives in the schemes.  The integer
   message representatives are processed via the primitives.  The
   encoding methods thus provide the connection between the schemes,
   which process messages, and the primitives.

   An encoding method for signatures with appendix, for the purposes of
   this document, consists of an encoding operation and optionally a
   verification operation.  An encoding operation maps a message M to an
   encoded message EM of a specified length.  A verification operation
   determines whether a message M and an encoded message EM are
   consistent, i.e., whether the encoded message EM is a valid encoding
   of the message M.

   The encoding operation may introduce some randomness, so that
   different applications of the encoding operation to the same message
   will produce different encoded messages, which has benefits for
   provable security.  For such an encoding method, both an encoding and
   a verification operation are needed unless the verifier can reproduce
   the randomness (e.g., by obtaining the salt value from the signer).
   For a deterministic encoding method, only an encoding operation is
   needed.

   Two encoding methods for signatures with appendix are employed in the
   signature schemes and are specified here: EMSA-PSS and
   EMSA-PKCS1-v1_5.

9.1.  EMSA-PSS

   This encoding method is parameterized by the choice of hash function,
   mask generation function, and salt length.  These options should be
   fixed for a given RSA key, except that the salt length can be
   variable (see [JONSSON] for discussion).  Suggested hash and mask
   generation functions are given in Appendix B.  The encoding method is
   based on Bellare and Rogaway's Probabilistic Signature Scheme (PSS)
   [RSARABIN][PSS].  It is randomized and has an encoding operation and
   a verification operation.

   Figure 2 illustrates the encoding operation.

      __________________________________________________________________

                                     +-----------+
                                     |     M     |
                                     +-----------+
                                           |
                                           V
                                         Hash
                                           |
                                           V
                             +--------+----------+----------+
                        M' = |Padding1|  mHash   |   salt   |
                             +--------+----------+----------+
                                            |
                  +--------+----------+     V
            DB =  |Padding2|   salt   |   Hash
                  +--------+----------+     |
                            |               |
                            V               |
                           xor <--- MGF <---|
                            |               |
                            |               |
                            V               V
                  +-------------------+----------+--+
            EM =  |    maskedDB       |     H    |bc|
                  +-------------------+----------+--+
      __________________________________________________________________

   Figure 2: EMSA-PSS Encoding Operation

   Note that the verification operation follows reverse steps to recover
   salt and then forward steps to recompute and compare H.

   Notes:

   1.  The encoding method defined here differs from the one in Bellare
       and Rogaway's submission to IEEE 1363a [PSS] in three respects:

       *  It applies a hash function rather than a mask generation
          function to the message.  Even though the mask generation
          function is based on a hash function, it seems more natural to
          apply a hash function directly.

       *  The value that is hashed together with the salt value is the
          string (0x)00 00 00 00 00 00 00 00 || mHash rather than the
          message M itself.  Here, mHash is the hash of M.  Note that
          the hash function is the same in both steps.  See Note 3 below
          for further discussion.  (Also, the name "salt" is used
          instead of "seed", as it is more reflective of the value's
          role.)

       *  The encoded message in EMSA-PSS has nine fixed bits; the first
          bit is 0 and the last eight bits form a "trailer field", the
          octet 0xbc.  In the original scheme, only the first bit is
          fixed.  The rationale for the trailer field is for
          compatibility with the Integer Factorization Signature
          Primitive using Rabin-Williams (IFSP-RW) in IEEE 1363
          [IEEE1363] and the corresponding primitive in ISO/IEC
          9796-2:2010 [ISO9796].

   2.  Assuming that the mask generation function is based on a hash
       function, it is RECOMMENDED that the hash function be the same as
       the one that is applied to the message; see Section 8.1 for
       further discussion.

   3.  Without compromising the security proof for RSASSA-PSS, one may
       perform Steps 1 and 2 of EMSA-PSS-ENCODE and EMSA-PSS-VERIFY (the
       application of the hash function to the message) outside the
       module that computes the rest of the signature operation, so that
       mHash rather than the message M itself is input to the module.
       In other words, the security proof for RSASSA-PSS still holds
       even if an opponent can control the value of mHash.  This is
       convenient if the module has limited I/O bandwidth, e.g., a smart
       card.  Note that previous versions of PSS [RSARABIN][PSS] did not
       have this property.  Of course, it may be desirable for other
       security reasons to have the module process the full message.
       For instance, the module may need to "see" what it is signing if
       it does not trust the component that computes the hash value.

   4.  Typical salt lengths in octets are hLen (the length of the output
       of the hash function Hash) and 0.  In both cases, the security of
       RSASSA-PSS can be closely related to the hardness of inverting
       RSAVP1.  Bellare and Rogaway [RSARABIN] give a tight lower bound
       for the security of the original RSA-PSS scheme, which
       corresponds roughly to the former case, while Coron [FDH] gives a
       lower bound for the related Full Domain Hashing scheme, which
       corresponds roughly to the latter case.  In [PSSPROOF], Coron
       provides a general treatment with various salt lengths ranging
       from 0 to hLen; see [IEEE1363A] for discussion.  See also
       [JONSSON], which adapts the security proofs in [RSARABIN]
       [PSSPROOF] to address the differences between the original and
       the present version of RSA-PSS as listed in Note 1 above.

   5.  As noted in IEEE 1363a [IEEE1363A], the use of randomization in
       signature schemes -- such as the salt value in EMSA-PSS -- may
       provide a "covert channel" for transmitting information other
       than the message being signed.  For more on covert channels, see
       [SIMMONS].

9.1.1.  Encoding Operation

   EMSA-PSS-ENCODE (M, emBits)

   Options:

      Hash     hash function (hLen denotes the length in octets of
               the hash function output)
      MGF      mask generation function
      sLen     intended length in octets of the salt

   Input:

      M        message to be encoded, an octet string
      emBits   maximal bit length of the integer OS2IP (EM) (see Section
               4.2), at least 8hLen + 8sLen + 9

   Output:

      EM       encoded message, an octet string of length emLen = \ceil
               (emBits/8)

   Errors:  "Encoding error"; "message too long"

   Steps:

      1.   If the length of M is greater than the input limitation for
           the hash function (2^61 - 1 octets for SHA-1), output
           "message too long" and stop.

      2.   Let mHash = Hash(M), an octet string of length hLen.

      3.   If emLen < hLen + sLen + 2, output "encoding error" and stop.

      4.   Generate a random octet string salt of length sLen; if sLen =
           0, then salt is the empty string.

      5.   Let

              M' = (0x)00 00 00 00 00 00 00 00 || mHash || salt;

           M' is an octet string of length 8 + hLen + sLen with eight
           initial zero octets.

      6.   Let H = Hash(M'), an octet string of length hLen.

      7.   Generate an octet string PS consisting of emLen - sLen - hLen
           - 2 zero octets.  The length of PS may be 0.

      8.   Let DB = PS || 0x01 || salt; DB is an octet string of length
           emLen - hLen - 1.

      9.   Let dbMask = MGF(H, emLen - hLen - 1).

      10.  Let maskedDB = DB \xor dbMask.

      11.  Set the leftmost 8emLen - emBits bits of the leftmost octet
           in maskedDB to zero.

      12.  Let EM = maskedDB || H || 0xbc.

      13.  Output EM.

9.1.2.  Verification Operation

   EMSA-PSS-VERIFY (M, EM, emBits)

   Options:

      Hash     hash function (hLen denotes the length in octets of
               the hash function output)
      MGF      mask generation function
      sLen     intended length in octets of the salt

   Input:

      M        message to be verified, an octet string
      EM       encoded message, an octet string of length emLen = \ceil
               (emBits/8)
      emBits   maximal bit length of the integer OS2IP (EM) (see Section
               4.2), at least 8hLen + 8sLen + 9

   Output:  "consistent" or "inconsistent"

   Steps:

      1.   If the length of M is greater than the input limitation for
           the hash function (2^61 - 1 octets for SHA-1), output
           "inconsistent" and stop.

      2.   Let mHash = Hash(M), an octet string of length hLen.

      3.   If emLen < hLen + sLen + 2, output "inconsistent" and stop.

      4.   If the rightmost octet of EM does not have hexadecimal value
           0xbc, output "inconsistent" and stop.

      5.   Let maskedDB be the leftmost emLen - hLen - 1 octets of EM,
           and let H be the next hLen octets.

      6.   If the leftmost 8emLen - emBits bits of the leftmost octet in
           maskedDB are not all equal to zero, output "inconsistent" and
           stop.

      7.   Let dbMask = MGF(H, emLen - hLen - 1).

      8.   Let DB = maskedDB \xor dbMask.

      9.   Set the leftmost 8emLen - emBits bits of the leftmost octet
           in DB to zero.

      10.  If the emLen - hLen - sLen - 2 leftmost octets of DB are not
           zero or if the octet at position emLen - hLen - sLen - 1 (the
           leftmost position is "position 1") does not have hexadecimal
           value 0x01, output "inconsistent" and stop.

      11.  Let salt be the last sLen octets of DB.

      12.  Let

              M' = (0x)00 00 00 00 00 00 00 00 || mHash || salt ;

           M' is an octet string of length 8 + hLen + sLen with eight
           initial zero octets.

      13.  Let H' = Hash(M'), an octet string of length hLen.

      14.  If H = H', output "consistent".  Otherwise, output
           "inconsistent".

9.2.  EMSA-PKCS1-v1_5

   This encoding method is deterministic and only has an encoding
   operation.

   EMSA-PKCS1-v1_5-ENCODE (M, emLen)

   Option:

      Hash     hash function (hLen denotes the length in octets of
               the hash function output)

   Input:

      M        message to be encoded
      emLen    intended length in octets of the encoded message, at
               least tLen + 11, where tLen is the octet length of the
               Distinguished Encoding Rules (DER) encoding T of
               a certain value computed during the encoding operation

   Output:

      EM       encoded message, an octet string of length emLen

   Errors:  "message too long"; "intended encoded message length too
      short"

   Steps:

      1.  Apply the hash function to the message M to produce a hash
          value H:

             H = Hash(M).

          If the hash function outputs "message too long", output
          "message too long" and stop.

      2.  Encode the algorithm ID for the hash function and the hash
          value into an ASN.1 value of type DigestInfo (see
          Appendix A.2.4) with the DER, where the type DigestInfo has
          the syntax

               DigestInfo ::= SEQUENCE {
                   digestAlgorithm AlgorithmIdentifier,
                   digest OCTET STRING
               }

          The first field identifies the hash function and the second
          contains the hash value.  Let T be the DER encoding of the
          DigestInfo value (see the notes below), and let tLen be the
          length in octets of T.

      3.  If emLen < tLen + 11, output "intended encoded message length
          too short" and stop.

      4.  Generate an octet string PS consisting of emLen - tLen - 3
          octets with hexadecimal value 0xff.  The length of PS will be
          at least 8 octets.

      5.  Concatenate PS, the DER encoding T, and other padding to form
          the encoded message EM as

             EM = 0x00 || 0x01 || PS || 0x00 || T.

      6.  Output EM.

   Notes:

   1.  For the nine hash functions mentioned in Appendix B.1, the DER
       encoding T of the DigestInfo value is equal to the following:

         MD2:     (0x)30 20 30 0c 06 08 2a 86 48 86 f7 0d 02 02 05 00 04
                      10 || H.
         MD5:     (0x)30 20 30 0c 06 08 2a 86 48 86 f7 0d 02 05 05 00 04
                      10 || H.
         SHA-1:   (0x)30 21 30 09 06 05 2b 0e 03 02 1a 05 00 04 14 || H.
         SHA-224:  (0x)30 2d 30 0d 06 09 60 86 48 01 65 03 04 02 04
                      05 00 04 1c || H.
         SHA-256: (0x)30 31 30 0d 06 09 60 86 48 01 65 03 04 02 01 05 00
                      04 20 || H.
         SHA-384: (0x)30 41 30 0d 06 09 60 86 48 01 65 03 04 02 02 05 00
                      04 30 || H.
         SHA-512: (0x)30 51 30 0d 06 09 60 86 48 01 65 03 04 02 03 05 00
                      04 40 || H.
         SHA-512/224:  (0x)30 2d 30 0d 06 09 60 86 48 01 65 03 04 02 05
                           05 00 04 1c || H.
         SHA-512/256:  (0x)30 31 30 0d 06 09 60 86 48 01 65 03 04 02 06
                           05 00 04 20 || H.

   2.  In version 1.5 of this document, T was defined as the BER
       encoding, rather than the DER encoding, of the DigestInfo value.
       In particular, it is possible -- at least in theory -- that the
       verification operation defined in this document (as well as in
       version 2.0) rejects a signature that is valid with respect to
       the specification given in PKCS #1 v1.5.  This occurs if other
       rules than DER are applied to DigestInfo (e.g., an indefinite
       length encoding of the underlying SEQUENCE type).  While this is
       unlikely to be a concern in practice, a cautious implementor may
       choose to employ a verification operation based on a BER decoding
       operation as specified in PKCS #1 v1.5.  In this manner,
       compatibility with any valid implementation based on PKCS #1 v1.5
       is obtained.  Such a verification operation should indicate
       whether the underlying BER encoding is a DER encoding and hence
       whether the signature is valid with respect to the specification
       given in this document.

10.  Security Considerations

   Security considerations are discussed throughout this memo.

11.  References

11.1.  Normative References

   [GARNER]   Garner, H., "The Residue Number System", IRE Transactions
              on Electronic Computers, Volume EC-8, Issue 2, pp.
              140-147, DOI 10.1109/TEC.1959.5219515, June 1959.

   [RFC2119]  Bradner, S., "Key words for use in RFCs to Indicate
              Requirement Levels", BCP 14, RFC 2119,
              DOI 10.17487/RFC2119, March 1997,
              <http://www.rfc-editor.org/info/rfc2119>.

   [RSA]      Rivest, R., Shamir, A., and L. Adleman, "A Method for
              Obtaining Digital Signatures and Public-Key
              Cryptosystems", Communications of the ACM, Volume 21,
              Issue 2, pp. 120-126, DOI 10.1145/359340.359342, February
              1978.

11.2.  Informative References

   [ANSIX944] ANSI, "Key Establishment Using Integer Factorization
              Cryptography", ANSI X9.44-2007, August 2007.

   [BKS]      Bleichenbacher, D., Kaliski, B., and J. Staddon, "Recent
              Results on PKCS #1: RSA Encryption Standard", RSA
              Laboratories, Bulletin No. 7, June 1998.

   [BLEICHENBACHER]
              Bleichenbacher, D., "Chosen Ciphertext Attacks Against
              Protocols Based on the RSA Encryption Standard PKCS #1",
              Lecture Notes in Computer Science, Volume 1462, pp. 1-12,
              1998.

   [CHOSEN]   Desmedt, Y. and A. Odlyzko, "A Chosen Text Attack on the
              RSA Cryptosystem and Some Discrete Logarithm Schemes",
              Lecture Notes in Computer Science, Volume 218, pp.
              516-522, 1985.

   [COCHRAN]  Cochran, M., "Notes on the Wang et al. 2^63 SHA-1
              Differential Path", Cryptology ePrint Archive: Report
              2007/474, August 2008, <http://eprint.iacr.org/2007/474>.

   [FASTDEC]  Quisquater, J. and C. Couvreur, "Fast Decipherment
              Algorithm for RSA Public-Key Cryptosystem", Electronic
              Letters, Volume 18, Issue 21, pp. 905-907,
              DOI 10.1049/el:19820617, October 1982.

   [FDH]      Coron, J., "On the Exact Security of Full Domain Hash",
              Lecture Notes in Computer Science, Volume 1880, pp.
              229-235, 2000.

   [FOPS]     Fujisaki, E., Okamoto, T., Pointcheval, D., and J. Stern,
              "RSA-OAEP is Secure under the RSA Assumption", Lecture
              Notes in Computer Science, Volume 2139, pp. 260-274,
              August 2001.

   [FORGERY]  Coppersmith, D., Halevi, S., and C. Jutla, "ISO 9796-1 and
              the new forgery strategy", rump session of Crypto, August
              1999.

   [HAASTAD]  Haastad, J., "Solving Simultaneous Modular Equations of
              Low Degree", SIAM Journal on Computing, Volume 17,
              Issue 2, pp. 336-341, DOI 10.1137/0217019, April 1988.

   [HANDBOOK] Menezes, A., van Oorschot, P., and S. Vanstone, "Handbook
              of Applied Cryptography", CRC Press, ISBN: 0849385237,
              1996.

   [HASHID]   Kaliski, B., "On Hash Function Firewalls in Signature
              Schemes", Lecture Notes in Computer Science, Volume 2271,
              pp. 1-16, DOI 10.1007/3-540-45760-7_1, February 2002.

   [IEEE1363] IEEE, "Standard Specifications for Public Key
              Cryptography", IEEE Std 1363-2000,
              DOI 10.1109/IEEESTD.2000.92292, August 2000,
              <http://ieeexplore.ieee.org/document/891000/>.

   [IEEE1363A]
              IEEE, "Standard Specifications for Public Key Cryptography
              - Amendment 1: Additional Techniques", IEEE Std 1363a-
              2004, DOI 10.1109/IEEESTD.2004.94612, September 2004,
              <http://ieeexplore.ieee.org/document/1335427/>.

   [ISO18033] International Organization for Standardization,
              "Information technology -- Security techniques --
              Encryption algorithms - Part 2: Asymmetric ciphers", ISO/
              IEC 18033-2:2006, May 2006.

   [ISO9594]  International Organization for Standardization,
              "Information technology - Open Systems Interconnection -
              The Directory: Public-key and attribute certificate
              frameworks", ISO/IEC 9594-8:2008, December 2008.

   [ISO9796]  International Organization for Standardization,
              "Information technology - Security techniques - Digital
              signature schemes giving message recovery - Part 2:
              Integer factorization based mechanisms",
              ISO/IEC 9796-2:2010, December 2010.

   [JONSSON]  Jonsson, J., "Security Proofs for the RSA-PSS Signature
              Scheme and Its Variants", Cryptology ePrint
              Archive: Report 2001/053, March 2002,
              <http://eprint.iacr.org/2001/053>.

   [LOWEXP]   Coppersmith, D., Franklin, M., Patarin, J., and M. Reiter,
              "Low-Exponent RSA with Related Messages", Lecture Notes in
              Computer Science, Volume 1070, pp. 1-9, 1996.

   [MANGER]   Manger, J., "A Chosen Ciphertext Attack on RSA Optimal
              Asymmetric Encryption Padding (OAEP) as Standardized in
              PKCS #1 v2.0", Lecture Notes in Computer Science, Volume
              2139, pp. 230-238, DOI 10.1007/3-540-44647-8_14, 2001.

   [MD4]      Dobbertin, H., "Cryptanalysis of MD4", Lecture Notes in
              Computer Science, Volume 1039, pp. 53-69,
              DOI 10.1007/3-540-60865-6_43, 1996.

   [MD4FIRST] Dobbertin, H., "The First Two Rounds of MD4 are Not One-
              Way", Lecture Notes in Computer Science, Volume 1372, pp.
              284-292, DOI 10.1007/3-540-69710-1_19, March 1998.

   [MD4LAST]  den Boer, B. and A. Bosselaers, "An Attack on the Last Two
              Rounds of MD4", Lecture Notes in Computer Science, Volume
              576, pp. 194-203, DOI 10.1007/3-540-46766-1_14, 1992.

   [NEWATTACK]
              Coron, J., Joye, M., Naccache, D., and P. Paillier, "New
              Attacks on PKCS #1 v1.5 Encryption", Lecture Notes in
              Computer Science, Volume 1807, pp. 369-381,
              DOI 10.1007/3-540-45539-6_25, May 2000.

   [OAEP]     Bellare, M. and P. Rogaway, "Optimal Asymmetric Encryption
              - How to Encrypt with RSA", Lecture Notes in Computer
              Science, Volume 950, pp. 92-111, November 1995.

   [PA98]     Bellare, M., Desai, A., Pointcheval, D., and P. Rogaway,
              "Relations Among Notions of Security for Public-Key
              Encryption Schemes", Lecture Notes in Computer
              Science, Volume 1462, pp. 26-45, DOI 10.1007/BFb0055718,
              1998.

   [PADDING]  Coron, J., Naccache, D., and J. Stern, "On the Security of
              RSA Padding", Lecture Notes in Computer Science, Volume
              1666, pp. 1-18, DOI 10.1007/3-540-48405-1_1, December
              1999.

   [PKCS1_22] RSA Laboratories, "PKCS #1: RSA Cryptography Standard
              Version 2.2", October 2012.

   [PREFIX]   Stevens, M., Lenstra, A., and B. de Weger, "Chosen-prefix
              collisions for MD5 and applications", International
              Journal of Applied Cryptography, Volume 2, No. 4, pp.
              322-359, July 2012.

   [PSS]      Bellare, M. and P. Rogaway, "PSS: Provably Secure Encoding
              Method for Digital Signatures", Submission to IEEE P1363a,
              August 1998, <http://grouper.ieee.org/groups/1363/
              P1363a/contributions/pss-submission.pdf>.

   [PSSPROOF] Coron, J., "Optimal Security Proofs for PSS and Other
              Signature Schemes", Lecture Notes in Computer
              Science, Volume 2332, pp. 272-287,
              DOI 10.1007/3-540-46035-7_18, 2002.

   [RFC1319]  Kaliski, B., "The MD2 Message-Digest Algorithm", RFC 1319,
              DOI 10.17487/RFC1319, April 1992,
              <http://www.rfc-editor.org/info/rfc1319>.

   [RFC1321]  Rivest, R., "The MD5 Message-Digest Algorithm", RFC 1321,
              DOI 10.17487/RFC1321, April 1992,
              <http://www.rfc-editor.org/info/rfc1321>.

   [RFC2313]  Kaliski, B., "PKCS #1: RSA Encryption Version 1.5",
              RFC 2313, DOI 10.17487/RFC2313, March 1998,
              <http://www.rfc-editor.org/info/rfc2313>.

   [RFC2315]  Kaliski, B., "PKCS #7: Cryptographic Message Syntax
              Version 1.5", RFC 2315, DOI 10.17487/RFC2315, March 1998,
              <http://www.rfc-editor.org/info/rfc2315>.

   [RFC2437]  Kaliski, B. and J. Staddon, "PKCS #1: RSA Cryptography
              Specifications Version 2.0", RFC 2437,
              DOI 10.17487/RFC2437, October 1998,
              <http://www.rfc-editor.org/info/rfc2437>.

   [RFC3447]  Jonsson, J. and B. Kaliski, "Public-Key Cryptography
              Standards (PKCS) #1: RSA Cryptography Specifications
              Version 2.1", RFC 3447, DOI 10.17487/RFC3447, February
              2003, <http://www.rfc-editor.org/info/rfc3447>.

   [RFC5246]  Dierks, T. and E. Rescorla, "The Transport Layer Security
              (TLS) Protocol Version 1.2", RFC 5246,
              DOI 10.17487/RFC5246, August 2008,
              <http://www.rfc-editor.org/info/rfc5246>.

   [RFC5652]  Housley, R., "Cryptographic Message Syntax (CMS)", STD 70,
              RFC 5652, DOI 10.17487/RFC5652, September 2009,
              <http://www.rfc-editor.org/info/rfc5652>.

   [RFC5958]  Turner, S., "Asymmetric Key Packages", RFC 5958,
              DOI 10.17487/RFC5958, August 2010,
              <http://www.rfc-editor.org/info/rfc5958>.

   [RFC6149]  Turner, S. and L. Chen, "MD2 to Historic Status",
              RFC 6149, DOI 10.17487/RFC6149, March 2011,
              <http://www.rfc-editor.org/info/rfc6149>.

   [RFC7292]  Moriarty, K., Ed., Nystrom, M., Parkinson, S., Rusch, A.,
              and M. Scott, "PKCS #12: Personal Information Exchange
              Syntax v1.1", RFC 7292, DOI 10.17487/RFC7292, July 2014,
              <http://www.rfc-editor.org/info/rfc7292>.

   [RSARABIN] Bellare, M. and P. Rogaway, "The Exact Security of Digital
              Signatures - How to Sign with RSA and Rabin", Lecture
              Notes in Computer Science, Volume 1070, pp. 399-416,
              DOI 10.1007/3-540-68339-9_34, 1996.

   [RSATLS]   Jonsson, J. and B. Kaliski, "On the Security of RSA
              Encryption in TLS", Lecture Notes in Computer
              Science, Volume 2442, pp. 127-142,
              DOI 10.1007/3-540-45708-9_9, 2002.

   [SHA1CRYPT]
              Wang, X., Yao, A., and F. Yao, "Cryptanalysis on SHA-1",
              Lecture Notes in Computer Science, Volume 2442, pp.
              127-142, February 2005,
              <http://csrc.nist.gov/groups/ST/hash/documents/
              Wang_SHA1-New-Result.pdf>.

   [SHOUP]    Shoup, V., "OAEP Reconsidered (Extended Abstract)",
              Lecture Notes in Computer Science, Volume 2139, pp.
              239-259, DOI 10.1007/3-540-44647-8_15, 2001.

   [SHS]      National Institute of Standards and Technology, "Secure
              Hash Standard (SHS)", FIPS PUB 180-4, August 2015,
              <http://dx.doi.org/10.6028/NIST.FIPS.180-4>.

   [SILVERMAN]
              Silverman, R., "A Cost-Based Security Analysis of
              Symmetric and Asymmetric Key Lengths", RSA
              Laboratories, Bulletin No. 13, 2000.

   [SIMMONS]  Simmons, G., "Subliminal Communication is Easy Using the
              DSA", Lecture Notes in Computer Science, Volume 765, pp.
              218-232, DOI 10.1007/3-540-48285-7_18, 1994.

Appendix A.  ASN.1 Syntax

A.1.  RSA Key Representation

   This section defines ASN.1 object identifiers for RSA public and
   private keys and defines the types RSAPublicKey and RSAPrivateKey.
   The intended application of these definitions includes X.509
   certificates, PKCS #8 [RFC5958], and PKCS #12 [RFC7292].

   The object identifier rsaEncryption identifies RSA public and private
   keys as defined in Appendices A.1.1 and A.1.2.  The parameters field
   has associated with this OID in a value of type AlgorithmIdentifier
   SHALL have a value of type NULL.

      rsaEncryption    OBJECT IDENTIFIER ::= { pkcs-1 1 }

   The definitions in this section have been extended to support multi-
   prime RSA, but they are backward compatible with previous versions.

A.1.1.  RSA Public Key Syntax

   An RSA public key should be represented with the ASN.1 type
   RSAPublicKey:

         RSAPublicKey ::= SEQUENCE {
             modulus           INTEGER,  -- n
             publicExponent    INTEGER   -- e
         }

   The fields of type RSAPublicKey have the following meanings:

   o  modulus is the RSA modulus n.

   o  publicExponent is the RSA public exponent e.

A.1.2.  RSA Private Key Syntax

   An RSA private key should be represented with the ASN.1 type
   RSAPrivateKey:

         RSAPrivateKey ::= SEQUENCE {
             version           Version,
             modulus           INTEGER,  -- n
             publicExponent    INTEGER,  -- e
             privateExponent   INTEGER,  -- d
             prime1            INTEGER,  -- p
             prime2            INTEGER,  -- q
             exponent1         INTEGER,  -- d mod (p-1)
             exponent2         INTEGER,  -- d mod (q-1)
             coefficient       INTEGER,  -- (inverse of q) mod p
             otherPrimeInfos   OtherPrimeInfos OPTIONAL
         }

   The fields of type RSAPrivateKey have the following meanings:

   o  version is the version number, for compatibility with future
      revisions of this document.  It SHALL be 0 for this version of the
      document, unless multi-prime is used; in which case, it SHALL be
      1.

            Version ::= INTEGER { two-prime(0), multi(1) }
               (CONSTRAINED BY
               {-- version must be multi if otherPrimeInfos present --})

   o  modulus is the RSA modulus n.

   o  publicExponent is the RSA public exponent e.

   o  privateExponent is the RSA private exponent d.

   o  prime1 is the prime factor p of n.

   o  prime2 is the prime factor q of n.

   o  exponent1 is d mod (p - 1).

   o  exponent2 is d mod (q - 1).

   o  coefficient is the CRT coefficient q^(-1) mod p.

   o  otherPrimeInfos contains the information for the additional primes
      r_3, ..., r_u, in order.  It SHALL be omitted if version is 0 and
      SHALL contain at least one instance of OtherPrimeInfo if version
      is 1.

            OtherPrimeInfos ::= SEQUENCE SIZE(1..MAX) OF OtherPrimeInfo

            OtherPrimeInfo ::= SEQUENCE {
                prime             INTEGER,  -- ri
                exponent          INTEGER,  -- di
                coefficient       INTEGER   -- ti
            }

   The fields of type OtherPrimeInfo have the following meanings:

   o  prime is a prime factor r_i of n, where i >= 3.

   o  exponent is d_i = d mod (r_i - 1).

   o  coefficient is the CRT coefficient t_i = (r_1 * r_2 * ... *
      r_(i-1))^(-1) mod r_i.

   Note: It is important to protect the RSA private key against both
   disclosure and modification.  Techniques for such protection are
   outside the scope of this document.  Methods for storing and
   distributing private keys and other cryptographic data are described
   in PKCS #12 and #15.

A.2.  Scheme Identification

   This section defines object identifiers for the encryption and
   signature schemes.  The schemes compatible with PKCS #1 v1.5 have the
   same definitions as in PKCS #1 v1.5.  The intended application of
   these definitions includes X.509 certificates and PKCS #7.

   Here are type identifier definitions for the PKCS #1 OIDs:

   PKCS1Algorithms    ALGORITHM-IDENTIFIER ::= {
       { OID rsaEncryption                PARAMETERS NULL } |
       { OID md2WithRSAEncryption         PARAMETERS NULL } |
       { OID md5WithRSAEncryption         PARAMETERS NULL } |
       { OID sha1WithRSAEncryption        PARAMETERS NULL } |
       { OID sha224WithRSAEncryption      PARAMETERS NULL } |
       { OID sha256WithRSAEncryption      PARAMETERS NULL } |
       { OID sha384WithRSAEncryption      PARAMETERS NULL } |
       { OID sha512WithRSAEncryption      PARAMETERS NULL } |
       { OID sha512-224WithRSAEncryption  PARAMETERS NULL } |
       { OID sha512-256WithRSAEncryption  PARAMETERS NULL } |
       { OID id-RSAES-OAEP   PARAMETERS RSAES-OAEP-params } |
       PKCS1PSourceAlgorithms                               |
       { OID id-RSASSA-PSS   PARAMETERS RSASSA-PSS-params },
       ...  -- Allows for future expansion --
   }

A.2.1.  RSAES-OAEP

   The object identifier id-RSAES-OAEP identifies the RSAES-OAEP
   encryption scheme.

       id-RSAES-OAEP    OBJECT IDENTIFIER ::= { pkcs-1 7 }

   The parameters field associated with this OID in a value of type
   AlgorithmIdentifier SHALL have a value of type RSAES-OAEP-params:

   RSAES-OAEP-params ::= SEQUENCE {
       hashAlgorithm      [0] HashAlgorithm     DEFAULT sha1,
       maskGenAlgorithm   [1] MaskGenAlgorithm  DEFAULT mgf1SHA1,
       pSourceAlgorithm   [2] PSourceAlgorithm  DEFAULT pSpecifiedEmpty
   }

   The fields of type RSAES-OAEP-params have the following meanings:

   o  hashAlgorithm identifies the hash function.  It SHALL be an
      algorithm ID with an OID in the set OAEP-PSSDigestAlgorithms.  For
      a discussion of supported hash functions, see Appendix B.1.

       HashAlgorithm ::= AlgorithmIdentifier {
          {OAEP-PSSDigestAlgorithms}
       }

       OAEP-PSSDigestAlgorithms    ALGORITHM-IDENTIFIER ::= {
           { OID id-sha1       PARAMETERS NULL }|
           { OID id-sha224     PARAMETERS NULL }|
           { OID id-sha256     PARAMETERS NULL }|
           { OID id-sha384     PARAMETERS NULL }|
           { OID id-sha512     PARAMETERS NULL }|
           { OID id-sha512-224 PARAMETERS NULL }|
           { OID id-sha512-256 PARAMETERS NULL },
           ...  -- Allows for future expansion --
       }

   The default hash function is SHA-1:

       sha1    HashAlgorithm ::= {
           algorithm   id-sha1,
           parameters  SHA1Parameters : NULL
       }

       SHA1Parameters ::= NULL

   o  maskGenAlgorithm identifies the mask generation function.  It
      SHALL be an algorithm ID with an OID in the set
      PKCS1MGFAlgorithms, which for this version SHALL consist of
      id-mgf1, identifying the MGF1 mask generation function (see
      Appendix B.2.1).  The parameters field associated with id-mgf1
      SHALL be an algorithm ID with an OID in the set
      OAEP-PSSDigestAlgorithms, identifying the hash function on which
      MGF1 is based.

       MaskGenAlgorithm ::= AlgorithmIdentifier { {PKCS1MGFAlgorithms} }

       PKCS1MGFAlgorithms    ALGORITHM-IDENTIFIER ::= {
           { OID id-mgf1 PARAMETERS HashAlgorithm },
           ...  -- Allows for future expansion --
       }

   o  The default mask generation function is MGF1 with SHA-1:

       mgf1SHA1    MaskGenAlgorithm ::= {
           algorithm   id-mgf1,
           parameters  HashAlgorithm : sha1
       }

   o  pSourceAlgorithm identifies the source (and possibly the value) of
      the label L.  It SHALL be an algorithm ID with an OID in the set
      PKCS1PSourceAlgorithms, which for this version SHALL consist of
      id-pSpecified, indicating that the label is specified explicitly.
      The parameters field associated with id-pSpecified SHALL have a
      value of type OCTET STRING, containing the label.  In previous
      versions of this specification, the term "encoding parameters" was
      used rather than "label", hence the name of the type below.

       PSourceAlgorithm ::= AlgorithmIdentifier {
          {PKCS1PSourceAlgorithms}
       }

       PKCS1PSourceAlgorithms    ALGORITHM-IDENTIFIER ::= {
           { OID id-pSpecified PARAMETERS EncodingParameters },
           ...  -- Allows for future expansion --
       }

       id-pSpecified    OBJECT IDENTIFIER ::= { pkcs-1 9 }

       EncodingParameters ::= OCTET STRING(SIZE(0..MAX))

   o  The default label is an empty string (so that lHash will contain
      the hash of the empty string):

       pSpecifiedEmpty    PSourceAlgorithm ::= {
           algorithm   id-pSpecified,
           parameters  EncodingParameters : emptyString
       }

       emptyString    EncodingParameters ::= ''H

   If all of the default values of the fields in RSAES-OAEP-params are
   used, then the algorithm identifier will have the following value:

       rSAES-OAEP-Default-Identifier    RSAES-AlgorithmIdentifier ::= {
           algorithm   id-RSAES-OAEP,
           parameters  RSAES-OAEP-params : {
               hashAlgorithm       sha1,
               maskGenAlgorithm    mgf1SHA1,
               pSourceAlgorithm    pSpecifiedEmpty
           }
       }

       RSAES-AlgorithmIdentifier ::= AlgorithmIdentifier  {
           {PKCS1Algorithms}
       }

A.2.2.  RSAES-PKCS-v1_5

   The object identifier rsaEncryption (see Appendix A.1) identifies the
   RSAES-PKCS1-v1_5 encryption scheme.  The parameters field associated
   with this OID in a value of type AlgorithmIdentifier SHALL have a
   value of type NULL.  This is the same as in PKCS #1 v1.5.

       rsaEncryption    OBJECT IDENTIFIER ::= { pkcs-1 1 }

A.2.3.  RSASSA-PSS

   The object identifier id-RSASSA-PSS identifies the RSASSA-PSS
   encryption scheme.

       id-RSASSA-PSS    OBJECT IDENTIFIER ::= { pkcs-1 10 }

   The parameters field associated with this OID in a value of type
   AlgorithmIdentifier SHALL have a value of type RSASSA-PSS-params:

   RSASSA-PSS-params ::= SEQUENCE {
       hashAlgorithm      [0] HashAlgorithm      DEFAULT sha1,
       maskGenAlgorithm   [1] MaskGenAlgorithm   DEFAULT mgf1SHA1,
       saltLength         [2] INTEGER            DEFAULT 20,
       trailerField       [3] TrailerField       DEFAULT trailerFieldBC
   }

   The fields of type RSASSA-PSS-params have the following meanings:

   o  hashAlgorithm identifies the hash function.  It SHALL be an
      algorithm ID with an OID in the set OAEP-PSSDigestAlgorithms (see
      Appendix A.2.1).  The default hash function is SHA-1.

   o  maskGenAlgorithm identifies the mask generation function.  It
      SHALL be an algorithm ID with an OID in the set PKCS1MGFAlgorithms
      (see Appendix A.2.1).  The default mask generation function is
      MGF1 with SHA-1.  For MGF1 (and more generally, for other mask
      generation functions based on a hash function), it is RECOMMENDED
      that the underlying hash function be the same as the one
      identified by hashAlgorithm; see Note 2 in Section 9.1 for further
      comments.

   o  saltLength is the octet length of the salt.  It SHALL be an
      integer.  For a given hashAlgorithm, the default value of
      saltLength is the octet length of the hash value.  Unlike the
      other fields of type RSASSA-PSS-params, saltLength does not need
      to be fixed for a given RSA key pair.

   o  trailerField is the trailer field number, for compatibility with
      IEEE 1363a [IEEE1363A].  It SHALL be 1 for this version of the
      document, which represents the trailer field with hexadecimal
      value 0xbc.  Other trailer fields (including the trailer field
      HashID || 0xcc in IEEE 1363a) are not supported in this document.

       TrailerField ::= INTEGER { trailerFieldBC(1) }

   If the default values of the hashAlgorithm, maskGenAlgorithm, and
   trailerField fields of RSASSA-PSS-params are used, then the algorithm
   identifier will have the following value:

       rSASSA-PSS-Default-Identifier    RSASSA-AlgorithmIdentifier ::= {
           algorithm   id-RSASSA-PSS,
           parameters  RSASSA-PSS-params : {
               hashAlgorithm       sha1,
               maskGenAlgorithm    mgf1SHA1,
               saltLength          20,
               trailerField        trailerFieldBC
           }
       }

       RSASSA-AlgorithmIdentifier ::= AlgorithmIdentifier {
           {PKCS1Algorithms}
       }

   Note: In some applications, the hash function underlying a signature
   scheme is identified separately from the rest of the operations in
   the signature scheme.  For instance, in PKCS #7 [RFC2315], a hash
   function identifier is placed before the message and a "digest
   encryption" algorithm identifier (indicating the rest of the
   operations) is carried with the signature.  In order for PKCS #7 to
   support the RSASSA-PSS signature scheme, an object identifier would
   need to be defined for the operations in RSASSA-PSS after the hash
   function (analogous to the RSAEncryption OID for the
   RSASSA-PKCS1-v1_5 scheme).  S/MIME Cryptographic Message Syntax (CMS)
   [RFC5652] takes a different approach.  Although a hash function
   identifier is placed before the message, an algorithm identifier for
   the full signature scheme may be carried with a CMS signature (this
   is done for DSA signatures).  Following this convention, the
   id-RSASSA-PSS OID can be used to identify RSASSA-PSS signatures in
   CMS.  Since CMS is considered the successor to PKCS #7 and new
   developments such as the addition of support for RSASSA-PSS will be
   pursued with respect to CMS rather than PKCS #7, an OID for the "rest
   of" RSASSA-PSS is not defined in this version of PKCS #1.

A.2.4.  RSASSA-PKCS-v1_5

   The object identifier for RSASSA-PKCS1-v1_5 SHALL be one of the
   following.  The choice of OID depends on the choice of hash
   algorithm: MD2, MD5, SHA-1, SHA-224, SHA-256, SHA-384, SHA-512,
   SHA-512/224, or SHA-512/256.  Note that if either MD2 or MD5 is used,
   then the OID is just as in PKCS #1 v1.5.  For each OID, the
   parameters field associated with this OID in a value of type
   AlgorithmIdentifier SHALL have a value of type NULL.  The OID should
   be chosen in accordance with the following table:

         Hash algorithm   OID
         ------------------------------------------------------------
         MD2              md2WithRSAEncryption        ::= {pkcs-1 2}
         MD5              md5WithRSAEncryption        ::= {pkcs-1 4}
         SHA-1            sha1WithRSAEncryption       ::= {pkcs-1 5}
         SHA-256          sha224WithRSAEncryption     ::= {pkcs-1 14}
         SHA-256          sha256WithRSAEncryption     ::= {pkcs-1 11}
         SHA-384          sha384WithRSAEncryption     ::= {pkcs-1 12}
         SHA-512          sha512WithRSAEncryption     ::= {pkcs-1 13}
         SHA-512/224      sha512-224WithRSAEncryption ::= {pkcs-1 15}
         SHA-512/256      sha512-256WithRSAEncryption ::= {pkcs-1 16}

   The EMSA-PKCS1-v1_5 encoding method includes an ASN.1 value of type
   DigestInfo, where the type DigestInfo has the syntax

       DigestInfo ::= SEQUENCE {
           digestAlgorithm DigestAlgorithm,
           digest OCTET STRING
       }

   digestAlgorithm identifies the hash function and SHALL be an
   algorithm ID with an OID in the set PKCS1-v1-5DigestAlgorithms.  For
   a discussion of supported hash functions, see Appendix B.1.

       DigestAlgorithm ::= AlgorithmIdentifier {
          {PKCS1-v1-5DigestAlgorithms}
       }

       PKCS1-v1-5DigestAlgorithms    ALGORITHM-IDENTIFIER ::= {
           { OID id-md2        PARAMETERS NULL }|
           { OID id-md5        PARAMETERS NULL }|
           { OID id-sha1       PARAMETERS NULL }|
           { OID id-sha224     PARAMETERS NULL }|
           { OID id-sha256     PARAMETERS NULL }|
           { OID id-sha384     PARAMETERS NULL }|
           { OID id-sha512     PARAMETERS NULL }|
           { OID id-sha512-224 PARAMETERS NULL }|
           { OID id-sha512-256 PARAMETERS NULL }
       }

Appendix B.  Supporting Techniques

   This section gives several examples of underlying functions
   supporting the encryption schemes in Section 7 and the encoding
   methods in Section 9.  A range of techniques is given here to allow
   compatibility with existing applications as well as migration to new
   techniques.  While these supporting techniques are appropriate for
   applications to implement, none of them is required to be
   implemented.  It is expected that profiles for PKCS #1 v2.2 will be
   developed that specify particular supporting techniques.

   This section also gives object identifiers for the supporting
   techniques.

B.1.  Hash Functions

   Hash functions are used in the operations contained in Sections 7 and
   9.  Hash functions are deterministic, meaning that the output is
   completely determined by the input.  Hash functions take octet
   strings of variable length and generate fixed-length octet strings.
   The hash functions used in the operations contained in Sections 7 and
   9 should generally be collision-resistant.  This means that it is
   infeasible to find two distinct inputs to the hash function that
   produce the same output.  A collision-resistant hash function also
   has the desirable property of being one-way; this means that given an
   output, it is infeasible to find an input whose hash is the specified
   output.  In addition to the requirements, the hash function should
   yield a mask generation function (Appendix B.2) with pseudorandom
   output.

   Nine hash functions are given as examples for the encoding methods in
   this document: MD2 [RFC1319] (which was retired by [RFC6149]), MD5
   [RFC1321], SHA-1, SHA-224, SHA-256, SHA-384, SHA-512, SHA-512/224,
   and SHA-512/256 [SHS].  For the RSAES-OAEP encryption scheme and
   EMSA-PSS encoding method, only SHA-1, SHA-224, SHA-256, SHA-384, SHA-
   512, SHA-512/224, and SHA-512/256 are RECOMMENDED.  For the EMSA-
   PKCS1-v1_5 encoding method, SHA-224, SHA-256, SHA-384, SHA-512, SHA-
   512/224, and SHA-512/256 are RECOMMENDED for new applications.  MD2,
   MD5, and SHA-1 are recommended only for compatibility with existing
   applications based on PKCS #1 v1.5.

   The object identifiers id-md2, id-md5, id-sha1, id-sha224, id-sha256,
   id-sha384, id-sha512, id-sha512/224, and id-sha512/256 identify the
   respective hash functions:

       id-md2      OBJECT IDENTIFIER ::= {
           iso (1) member-body (2) us (840) rsadsi (113549)
           digestAlgorithm (2) 2
       }

       id-md5      OBJECT IDENTIFIER ::= {
           iso (1) member-body (2) us (840) rsadsi (113549)
           digestAlgorithm (2) 5
       }

       id-sha1    OBJECT IDENTIFIER ::= {
           iso(1) identified-organization(3) oiw(14) secsig(3)
            algorithms(2) 26
       }

       id-sha224    OBJECT IDENTIFIER ::= {
           joint-iso-itu-t (2) country (16) us (840) organization (1)
           gov (101) csor (3) nistalgorithm (4) hashalgs (2) 4
       }

       id-sha256    OBJECT IDENTIFIER ::= {
           joint-iso-itu-t (2) country (16) us (840) organization (1)
           gov (101) csor (3) nistalgorithm (4) hashalgs (2) 1
       }

       id-sha384    OBJECT IDENTIFIER ::= {
           joint-iso-itu-t (2) country (16) us (840) organization (1)
           gov (101) csor (3) nistalgorithm (4) hashalgs (2) 2
       }

       id-sha512    OBJECT IDENTIFIER ::= {
           joint-iso-itu-t (2) country (16) us (840) organization (1)
           gov (101) csor (3) nistalgorithm (4) hashalgs (2) 3
       }

       id-sha512-224    OBJECT IDENTIFIER ::= {
           joint-iso-itu-t (2) country (16) us (840) organization (1)
           gov (101) csor (3) nistalgorithm (4) hashalgs (2) 5
       }

       id-sha512-256    OBJECT IDENTIFIER ::= {
           joint-iso-itu-t (2) country (16) us (840) organization (1)
           gov (101) csor (3) nistalgorithm (4) hashalgs (2) 6
       }

   The parameters field associated with these OIDs in a value of type
   AlgorithmIdentifier SHALL have a value of type NULL.

   The parameters field associated with id-md2 and id-md5 in a value of
   type AlgorithmIdentifier shall have a value of type NULL.

   The parameters field associated with id-sha1, id-sha224, id-sha256,
   id-sha384, id-sha512, id-sha512/224, and id-sha512/256 should
   generally be omitted, but if present, it shall have a value of type
   NULL.

   This is to align with the definitions originally promulgated by NIST.
   For the SHA algorithms, implementations MUST accept
   AlgorithmIdentifier values both without parameters and with NULL
   parameters.

   Exception: When formatting the DigestInfoValue in EMSA-PKCS1-v1_5
   (see Section 9.2), the parameters field associated with id-sha1,
   id-sha224, id-sha256, id-sha384, id-sha512, id-sha512/224, and
   id-sha512/256 shall have a value of type NULL.  This is to maintain
   compatibility with existing implementations and with the numeric
   information values already published for EMSA-PKCS1-v1_5, which are
   also reflected in IEEE 1363a [IEEE1363A].

   Note: Version 1.5 of PKCS #1 also allowed for the use of MD4 in
   signature schemes.  The cryptanalysis of MD4 has progressed
   significantly in the intervening years.  For example, Dobbertin [MD4]
   demonstrated how to find collisions for MD4 and that the first two
   rounds of MD4 are not one-way [MD4FIRST].  Because of these results
   and others (e.g., [MD4LAST]), MD4 is NOT RECOMMENDED.

   Further advances have been made in the cryptanalysis of MD2 and MD5,
   especially after the findings of Stevens et al.  [PREFIX] on chosen-

   prefix collisions on MD5.  MD2 and MD5 should be considered
   cryptographically broken and removed from existing applications.
   This version of the standard supports MD2 and MD5 just for backwards-
   compatibility reasons.

   There have also been advances in the cryptanalysis of SHA-1.
   Particularly, the results of Wang et al.  [SHA1CRYPT] (which have
   been independently verified by M.  Cochran in his analysis [COCHRAN])
   on using a differential path to find collisions in SHA-1, which
   conclude that the security strength of the SHA-1 hashing algorithm is
   significantly reduced.  However, this reduction is not significant
   enough to warrant the removal of SHA-1 from existing applications,
   but its usage is only recommended for backwards-compatibility
   reasons.

   To address these concerns, only SHA-224, SHA-256, SHA-384, SHA-512,
   SHA-512/224, and SHA-512/256 are RECOMMENDED for new applications.
   As of today, the best (known) collision attacks against these hash
   functions are generic attacks with complexity 2L/2, where L is the
   bit length of the hash output.  For the signature schemes in this
   document, a collision attack is easily translated into a signature
   forgery.  Therefore, the value L / 2 should be at least equal to the
   desired security level in bits of the signature scheme (a security
   level of B bits means that the best attack has complexity 2B).  The
   same rule of thumb can be applied to RSAES-OAEP; it is RECOMMENDED
   that the bit length of the seed (which is equal to the bit length of
   the hash output) be twice the desired security level in bits.

B.2.  Mask Generation Functions

   A mask generation function takes an octet string of variable length
   and a desired output length as input and outputs an octet string of
   the desired length.  There may be restrictions on the length of the
   input and output octet strings, but such bounds are generally very
   large.  Mask generation functions are deterministic; the octet string
   output is completely determined by the input octet string.  The
   output of a mask generation function should be pseudorandom: Given
   one part of the output but not the input, it should be infeasible to
   predict another part of the output.  The provable security of
   RSAES-OAEP and RSASSA-PSS relies on the random nature of the output
   of the mask generation function, which in turn relies on the random
   nature of the underlying hash.

   One mask generation function is given here: MGF1, which is based on a
   hash function.  MGF1 coincides with the mask generation functions
   defined in IEEE 1363 [IEEE1363] and ANSI X9.44 [ANSIX944].  Future
   versions of this document may define other mask generation functions.

B.2.1.  MGF1

   MGF1 is a mask generation function based on a hash function.

   MGF1 (mgfSeed, maskLen)

   Options:

      Hash     hash function (hLen denotes the length in octets of
               the hash function output)

   Input:

      mgfSeed  seed from which mask is generated, an octet string
      maskLen  intended length in octets of the mask, at most 2^32 hLen

   Output:

      mask     mask, an octet string of length maskLen

   Error: "mask too long"

   Steps:

   1.  If maskLen > 2^32 hLen, output "mask too long" and stop.

   2.  Let T be the empty octet string.

   3.  For counter from 0 to \ceil (maskLen / hLen) - 1, do the
       following:

       A.  Convert counter to an octet string C of length 4 octets (see
           Section 4.1):

              C = I2OSP (counter, 4) .

       B.  Concatenate the hash of the seed mgfSeed and C to the octet
           string T:

              T = T || Hash(mgfSeed || C) .

   4.  Output the leading maskLen octets of T as the octet string mask.

   The object identifier id-mgf1 identifies the MGF1 mask generation
   function:

      id-mgf1    OBJECT IDENTIFIER ::= { pkcs-1 8 }

   The parameters field associated with this OID in a value of type
   AlgorithmIdentifier shall have a value of type hashAlgorithm,
   identifying the hash function on which MGF1 is based.

Appendix C.  ASN.1 Module

   -- PKCS #1 v2.2 ASN.1 Module
   -- Revised October 27, 2012

   -- This module has been checked for conformance with the
   -- ASN.1 standard by the OSS ASN.1 Tools

   PKCS-1 {
       iso(1) member-body(2) us(840) rsadsi(113549) pkcs(1) pkcs-1(1)
       modules(0) pkcs-1(1)
   }

   DEFINITIONS EXPLICIT TAGS ::=

   BEGIN

   -- EXPORTS ALL
   -- All types and values defined in this module are exported for use
   -- in other ASN.1 modules.

   IMPORTS

   id-sha224, id-sha256, id-sha384, id-sha512, id-sha512-224,
   id-sha512-256
       FROM NIST-SHA2 {
           joint-iso-itu-t(2) country(16) us(840) organization(1)
           gov(101) csor(3) nistalgorithm(4) hashAlgs(2)
       };

   -- ============================
   --   Basic object identifiers
   -- ============================

   -- The DER encoding of this in hexadecimal is:
   -- (0x)06 08
   --        2A 86 48 86 F7 0D 01 01
   --
   pkcs-1    OBJECT IDENTIFIER ::= {
       iso(1) member-body(2) us(840) rsadsi(113549) pkcs(1) 1
   }

   --
   -- When rsaEncryption is used in an AlgorithmIdentifier,

   -- the parameters MUST be present and MUST be NULL.
   --
   rsaEncryption    OBJECT IDENTIFIER ::= { pkcs-1 1 }

   --
   -- When id-RSAES-OAEP is used in an AlgorithmIdentifier, the
   -- parameters MUST be present and MUST be RSAES-OAEP-params.
   --
   id-RSAES-OAEP    OBJECT IDENTIFIER ::= { pkcs-1 7 }

   --
   -- When id-pSpecified is used in an AlgorithmIdentifier, the
   -- parameters MUST be an OCTET STRING.
   --
   id-pSpecified    OBJECT IDENTIFIER ::= { pkcs-1 9 }

   --
   -- When id-RSASSA-PSS is used in an AlgorithmIdentifier, the
   -- parameters MUST be present and MUST be RSASSA-PSS-params.
   --
   id-RSASSA-PSS    OBJECT IDENTIFIER ::= { pkcs-1 10 }

   --
   -- When the following OIDs are used in an AlgorithmIdentifier,
   -- the parameters MUST be present and MUST be NULL.
   --
   md2WithRSAEncryption         OBJECT IDENTIFIER ::= { pkcs-1 2 }
   md5WithRSAEncryption         OBJECT IDENTIFIER ::= { pkcs-1 4 }
   sha1WithRSAEncryption        OBJECT IDENTIFIER ::= { pkcs-1 5 }
   sha224WithRSAEncryption      OBJECT IDENTIFIER ::= { pkcs-1 14 }
   sha256WithRSAEncryption      OBJECT IDENTIFIER ::= { pkcs-1 11 }
   sha384WithRSAEncryption      OBJECT IDENTIFIER ::= { pkcs-1 12 }
   sha512WithRSAEncryption      OBJECT IDENTIFIER ::= { pkcs-1 13 }
   sha512-224WithRSAEncryption  OBJECT IDENTIFIER ::= { pkcs-1 15 }
   sha512-256WithRSAEncryption  OBJECT IDENTIFIER ::= { pkcs-1 16 }

   --
   -- This OID really belongs in a module with the secsig OIDs.
   --
   id-sha1    OBJECT IDENTIFIER ::= {
       iso(1) identified-organization(3) oiw(14) secsig(3) algorithms(2)
       26
   }

   --
   -- OIDs for MD2 and MD5, allowed only in EMSA-PKCS1-v1_5.
   --
   id-md2 OBJECT IDENTIFIER ::= {

       iso(1) member-body(2) us(840) rsadsi(113549) digestAlgorithm(2) 2
   }

   id-md5 OBJECT IDENTIFIER ::= {
       iso(1) member-body(2) us(840) rsadsi(113549) digestAlgorithm(2) 5
   }

   --
   -- When id-mgf1 is used in an AlgorithmIdentifier, the parameters
   -- MUST be present and MUST be a HashAlgorithm, for example, sha1.
   --
   id-mgf1    OBJECT IDENTIFIER ::= { pkcs-1 8 }

   -- ================
   --   Useful types
   -- ================

   ALGORITHM-IDENTIFIER ::= CLASS {
       &id    OBJECT IDENTIFIER  UNIQUE,
       &Type  OPTIONAL
   }
       WITH SYNTAX { OID &id [PARAMETERS &Type] }

   -- Note: the parameter InfoObjectSet in the following definitions
   -- allows a distinct information object set to be specified for sets
   -- of algorithms such as:
   -- DigestAlgorithms    ALGORITHM-IDENTIFIER ::= {
   --     { OID id-md2  PARAMETERS NULL }|
   --     { OID id-md5  PARAMETERS NULL }|
   --     { OID id-sha1 PARAMETERS NULL }
   -- }
   --

   AlgorithmIdentifier { ALGORITHM-IDENTIFIER:InfoObjectSet } ::=
       SEQUENCE {
         algorithm
             ALGORITHM-IDENTIFIER.&id({InfoObjectSet}),
         parameters
             ALGORITHM-IDENTIFIER.&Type({InfoObjectSet}{@.algorithm})
               OPTIONAL
   }

   -- ==============
   --   Algorithms
   -- ==============

   --
   -- Allowed EME-OAEP and EMSA-PSS digest algorithms.

   --
   OAEP-PSSDigestAlgorithms    ALGORITHM-IDENTIFIER ::= {
       { OID id-sha1       PARAMETERS NULL }|
       { OID id-sha224     PARAMETERS NULL }|
       { OID id-sha256     PARAMETERS NULL }|
       { OID id-sha384     PARAMETERS NULL }|
       { OID id-sha512     PARAMETERS NULL }|
       { OID id-sha512-224 PARAMETERS NULL }|
       { OID id-sha512-256 PARAMETERS NULL },
       ...  -- Allows for future expansion --
   }

   --
   -- Allowed EMSA-PKCS1-v1_5 digest algorithms.
   --
   PKCS1-v1-5DigestAlgorithms    ALGORITHM-IDENTIFIER ::= {
       { OID id-md2        PARAMETERS NULL }|
       { OID id-md5        PARAMETERS NULL }|
       { OID id-sha1       PARAMETERS NULL }|
       { OID id-sha224     PARAMETERS NULL }|
       { OID id-sha256     PARAMETERS NULL }|
       { OID id-sha384     PARAMETERS NULL }|
       { OID id-sha512     PARAMETERS NULL }|
       { OID id-sha512-224 PARAMETERS NULL }|
       { OID id-sha512-256 PARAMETERS NULL }
   }

   -- When id-md2 and id-md5 are used in an AlgorithmIdentifier, the
   -- parameters field shall have a value of type NULL.

   -- When id-sha1, id-sha224, id-sha256, id-sha384, id-sha512,
   -- id-sha512-224, and id-sha512-256 are used in an
   -- AlgorithmIdentifier, the parameters (which are optional) SHOULD be
   -- omitted, but if present, they SHALL have a value of type NULL.
   -- However, implementations MUST accept AlgorithmIdentifier values
   -- both without parameters and with NULL parameters.

   -- Exception: When formatting the DigestInfoValue in EMSA-PKCS1-v1_5
   -- (see Section 9.2), the parameters field associated with id-sha1,
   -- id-sha224, id-sha256, id-sha384, id-sha512, id-sha512-224, and
   -- id-sha512-256 SHALL have a value of type NULL.  This is to
   -- maintain compatibility with existing implementations and with the
   -- numeric information values already published for EMSA-PKCS1-v1_5,
   -- which are also reflected in IEEE 1363a.

   sha1    HashAlgorithm ::= {
       algorithm   id-sha1,
       parameters  SHA1Parameters : NULL

   }

   HashAlgorithm ::= AlgorithmIdentifier { {OAEP-PSSDigestAlgorithms} }

   SHA1Parameters ::= NULL

   --
   -- Allowed mask generation function algorithms.
   -- If the identifier is id-mgf1, the parameters are a HashAlgorithm.
   --
   PKCS1MGFAlgorithms    ALGORITHM-IDENTIFIER ::= {
       { OID id-mgf1 PARAMETERS HashAlgorithm },
       ...  -- Allows for future expansion --
   }

   --
   -- Default AlgorithmIdentifier for id-RSAES-OAEP.maskGenAlgorithm and
   -- id-RSASSA-PSS.maskGenAlgorithm.
   --
   mgf1SHA1    MaskGenAlgorithm ::= {
       algorithm   id-mgf1,
       parameters  HashAlgorithm : sha1
   }

   MaskGenAlgorithm ::= AlgorithmIdentifier { {PKCS1MGFAlgorithms} }

   --
   -- Allowed algorithms for pSourceAlgorithm.
   --
   PKCS1PSourceAlgorithms    ALGORITHM-IDENTIFIER ::= {
       { OID id-pSpecified PARAMETERS EncodingParameters },
       ...  -- Allows for future expansion --
   }

   EncodingParameters ::= OCTET STRING(SIZE(0..MAX))

   --
   -- This identifier means that the label L is an empty string, so the
   -- digest of the empty string appears in the RSA block before
   -- masking.
   --

   pSpecifiedEmpty    PSourceAlgorithm ::= {
       algorithm   id-pSpecified,
       parameters  EncodingParameters : emptyString
   }

   PSourceAlgorithm ::= AlgorithmIdentifier { {PKCS1PSourceAlgorithms} }

   emptyString    EncodingParameters ::= ''H

   --
   -- Type identifier definitions for the PKCS #1 OIDs.
   --
   PKCS1Algorithms    ALGORITHM-IDENTIFIER ::= {
       { OID rsaEncryption                PARAMETERS NULL } |
       { OID md2WithRSAEncryption         PARAMETERS NULL } |
       { OID md5WithRSAEncryption         PARAMETERS NULL } |
       { OID sha1WithRSAEncryption        PARAMETERS NULL } |
       { OID sha224WithRSAEncryption      PARAMETERS NULL } |
       { OID sha256WithRSAEncryption      PARAMETERS NULL } |
       { OID sha384WithRSAEncryption      PARAMETERS NULL } |
       { OID sha512WithRSAEncryption      PARAMETERS NULL } |
       { OID sha512-224WithRSAEncryption  PARAMETERS NULL } |
       { OID sha512-256WithRSAEncryption  PARAMETERS NULL } |
       { OID id-RSAES-OAEP   PARAMETERS RSAES-OAEP-params } |
       PKCS1PSourceAlgorithms                               |
       { OID id-RSASSA-PSS   PARAMETERS RSASSA-PSS-params },
       ...  -- Allows for future expansion --
   }

   -- ===================
   --   Main structures
   -- ===================

   RSAPublicKey ::= SEQUENCE {
       modulus           INTEGER,  -- n
       publicExponent    INTEGER   -- e
   }

   --
   -- Representation of RSA private key with information for the CRT
   -- algorithm.
   --
   RSAPrivateKey ::= SEQUENCE {
       version           Version,
       modulus           INTEGER,  -- n
       publicExponent    INTEGER,  -- e
       privateExponent   INTEGER,  -- d
       prime1            INTEGER,  -- p
       prime2            INTEGER,  -- q
       exponent1         INTEGER,  -- d mod (p-1)
       exponent2         INTEGER,  -- d mod (q-1)
       coefficient       INTEGER,  -- (inverse of q) mod p
       otherPrimeInfos   OtherPrimeInfos OPTIONAL
   }

   Version ::= INTEGER { two-prime(0), multi(1) }
       (CONSTRAINED BY
         {-- version MUST
    be multi if otherPrimeInfos present --})

   OtherPrimeInfos ::= SEQUENCE SIZE(1..MAX) OF OtherPrimeInfo

   OtherPrimeInfo ::= SEQUENCE {
       prime             INTEGER,  -- ri
       exponent          INTEGER,  -- di
       coefficient       INTEGER   -- ti
   }

   --
   -- AlgorithmIdentifier.parameters for id-RSAES-OAEP.
   -- Note that the tags in this Sequence are explicit.
   --
   RSAES-OAEP-params ::= SEQUENCE {
       hashAlgorithm      [0] HashAlgorithm     DEFAULT sha1,
       maskGenAlgorithm   [1] MaskGenAlgorithm  DEFAULT mgf1SHA1,
       pSourceAlgorithm   [2] PSourceAlgorithm  DEFAULT pSpecifiedEmpty
   }

   --
   -- Identifier for default RSAES-OAEP algorithm identifier.
   -- The DER encoding of this is in hexadecimal:
   -- (0x)30 0D
   --        06 09
   --           2A 86 48 86 F7 0D 01 01 07
   --        30 00
   -- Notice that the DER encoding of default values is "empty".
   --

   rSAES-OAEP-Default-Identifier    RSAES-AlgorithmIdentifier ::= {
       algorithm   id-RSAES-OAEP,
       parameters  RSAES-OAEP-params : {
           hashAlgorithm       sha1,
           maskGenAlgorithm    mgf1SHA1,
           pSourceAlgorithm    pSpecifiedEmpty
       }
   }

   RSAES-AlgorithmIdentifier ::= AlgorithmIdentifier {
       {PKCS1Algorithms}
   }

   --

   -- AlgorithmIdentifier.parameters for id-RSASSA-PSS.
   -- Note that the tags in this Sequence are explicit.
   --
   RSASSA-PSS-params ::= SEQUENCE {
       hashAlgorithm      [0] HashAlgorithm      DEFAULT sha1,
       maskGenAlgorithm   [1] MaskGenAlgorithm   DEFAULT mgf1SHA1,
       saltLength         [2] INTEGER            DEFAULT 20,
       trailerField       [3] TrailerField       DEFAULT trailerFieldBC
   }

   TrailerField ::= INTEGER { trailerFieldBC(1) }

   --
   -- Identifier for default RSASSA-PSS algorithm identifier
   -- The DER encoding of this is in hexadecimal:
   -- (0x)30 0D
   --        06 09
   --           2A 86 48 86 F7 0D 01 01 0A
   --        30 00
   -- Notice that the DER encoding of default values is "empty".
   --
   rSASSA-PSS-Default-Identifier    RSASSA-AlgorithmIdentifier ::= {
       algorithm   id-RSASSA-PSS,
       parameters  RSASSA-PSS-params : {
           hashAlgorithm       sha1,
           maskGenAlgorithm    mgf1SHA1,
           saltLength          20,
           trailerField        trailerFieldBC
       }
   }

   RSASSA-AlgorithmIdentifier ::= AlgorithmIdentifier {
       {PKCS1Algorithms}
   }

   --
   -- Syntax for the EMSA-PKCS1-v1_5 hash identifier.
   --
   DigestInfo ::= SEQUENCE {
       digestAlgorithm DigestAlgorithm,
       digest OCTET STRING
   }

   DigestAlgorithm ::= AlgorithmIdentifier {
       {PKCS1-v1-5DigestAlgorithms}
   }

   END

Appendix D.  Revision History of PKCS #1

   Versions 1.0 - 1.5:

      Versions 1.0 - 1.3 were distributed to participants in RSA Data
      Security, Inc.'s Public-Key Cryptography Standards meetings in
      February and March 1991.

      Version 1.4 was part of the June 3, 1991 initial public release of
      PKCS.  Version 1.4 was published as NIST/OSI Implementors'
      Workshop document SEC-SIG-91-18.

      Version 1.5 incorporated several editorial changes, including
      updates to the references and the addition of a revision history.
      The following substantive changes were made:

      *  Section 10: "MD4 with RSA" signature and verification processes
         were added.

      *  Section 11: md4WithRSAEncryption object identifier was added.

      Version 1.5 was republished as [RFC2313] (which was later
      obsoleted by [RFC2437]).

   Version 2.0:

      Version 2.0 incorporated major editorial changes in terms of the
      document structure and introduced the RSAES-OAEP encryption
      scheme.  This version continued to support the encryption and
      signature processes in version 1.5, although the hash algorithm
      MD4 was no longer allowed due to cryptanalytic advances in the
      intervening years.  Version 2.0 was republished as [RFC2437]
      (which was later obsoleted by [RFC3447]).

   Version 2.1:

      Version 2.1 introduced multi-prime RSA and the RSASSA-PSS
      signature scheme with appendix along with several editorial
      improvements.  This version continued to support the schemes in
      version 2.0.  Version 2.1 was republished as [RFC3447].

   Version 2.2:

      Version 2.2 updates the list of allowed hashing algorithms to
      align them with FIPS 180-4 [SHS], therefore adding SHA-224,
      SHA-512/224, and SHA-512/256.  The following substantive changes
      were made:

      *  Object identifiers for sha224WithRSAEncryption,
         sha512-224WithRSAEncryption, and sha512-256WithRSAEncryption
         were added.

      *  This version continues to support the schemes in version 2.1.

Appendix E.  About PKCS

   The Public-Key Cryptography Standards are specifications produced by
   RSA Laboratories in cooperation with secure systems developers
   worldwide for the purpose of accelerating the deployment of public-
   key cryptography.  First published in 1991 as a result of meetings
   with a small group of early adopters of public-key technology, the
   PKCS documents have become widely referenced and implemented.
   Contributions from the PKCS series have become part of many formal
   and de facto standards, including ANSI X9 and IEEE P1363 documents,
   PKIX, Secure Electronic Transaction (SET), S/MIME, SSL/TLS, and
   Wireless Application Protocol (WAP) / WAP Transport Layer Security
   (WTLS).

   Further development of most PKCS documents occurs through the IETF.
   Suggestions for improvement are welcome.

Acknowledgements

   This document is based on a contribution of RSA Laboratories, the
   research center of RSA Security Inc.

Authors' Addresses

   Kathleen M. Moriarty (editor)
   EMC Corporation
   176 South Street
   Hopkinton, MA  01748
   United States of America

   Email: kathleen.moriarty@emc.com

   Burt Kaliski
   Verisign
   12061 Bluemont Way
   Reston, VA  20190
   United States of America

   Email: bkaliski@verisign.com
   URI:   http://verisignlabs.com

   Jakob Jonsson
   Subset AB
   Munkbrogtan 4
   Stockholm  SE-11127
   Sweden

   Phone: +46 8 428 687 43
   Email: jakob.jonsson@subset.se

   Andreas Rusch
   RSA
   345 Queen Street
   Brisbane, QLD  4000
   Australia

   Email: andreas.rusch@rsa.com

 

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