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RFC 6508 - Sakai-Kasahara Key Encryption (SAKKE)


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Internet Engineering Task Force (IETF)                         M. Groves
Request for Comments: 6508                                          CESG
Category: Informational                                    February 2012
ISSN: 2070-1721

                 Sakai-Kasahara Key Encryption (SAKKE)

Abstract

   In this document, the Sakai-Kasahara Key Encryption (SAKKE) algorithm
   is described.  This uses Identity-Based Encryption to exchange a
   shared secret from a Sender to a Receiver.

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 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 5741.

   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/rfc6508.

Copyright Notice

   Copyright (c) 2012 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 ....................................................2
      1.1. Requirements Terminology ...................................3
   2. Notation and Definitions ........................................3
      2.1. Notation ...................................................3
      2.2. Definitions ................................................5
      2.3. Parameters to Be Defined or Negotiated .....................6
   3. Elliptic Curves and Pairings ....................................7
      3.1. E(F_p^2) and the Distortion Map ............................7
      3.2. The Tate-Lichtenbaum Pairing ...............................7
   4. Representation of Values ........................................9
   5. Supporting Algorithms ..........................................10
      5.1. Hashing to an Integer Range ...............................10
   6. The SAKKE Cryptosystem .........................................11
      6.1. Setup .....................................................11
           6.1.1. Secret Key Extraction ..............................11
           6.1.2. User Provisioning ..................................11
      6.2. Key Exchange ..............................................12
           6.2.1. Sender .............................................12
           6.2.2. Receiver ...........................................12
      6.3. Group Communications ......................................13
   7. Security Considerations ........................................13
   8. References .....................................................15
      8.1. Normative References ......................................15
      8.2. Informative References ....................................15
   Appendix A. Test Data..............................................17

1.  Introduction

   This document defines an efficient use of Identity-Based Encryption
   (IBE) based on bilinear pairings.  The Sakai-Kasahara IBE
   cryptosystem [S-K] is described for establishment of a shared secret
   value.  This document adds to the IBE options available in [RFC5091],
   providing an efficient primitive and an additional family of curves.

   This document is restricted to a particular family of curves (see
   Section 2.1) that have the benefit of a simple and efficient method
   of calculating the pairing on which the Sakai-Kasahara IBE
   cryptosystem is based.

   IBE schemes allow public and private keys to be derived from
   Identifiers.  In fact, the Identifier can itself be viewed as
   corresponding to a public key or certificate in a traditional public
   key system.  However, in IBE, the Identifier can be formed by both
   Sender and Receiver, which obviates the necessity of providing public
   keys through a third party or of transmitting certified public keys

   during each session establishment.  Furthermore, in an IBE system,
   calculation of keys can occur as needed, and indeed, messages can be
   sent to users who are yet to enroll.

   The Sakai-Kasahara primitive described in this document supports
   simplex transmission of messages from a Sender to a Receiver.  The
   choice of elliptic curve pairing on which the primitive is based
   allows simple and efficient implementations.

   The Sakai-Kasahara Key Encryption scheme described in this document
   is drawn from the Sakai-Kasahara Key Encapsulation Mechanism (SK-KEM)
   scheme (as modified to support multi-party communications) submitted
   to the IEEE P1363 Working Group in [SK-KEM].

1.1.  Requirements Terminology

   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 and Definitions

2.1.  Notation

   n      A security parameter; the size of symmetric keys in bits to be
          exchanged by SAKKE.

   p      A prime, which is the order of the finite field F_p.  In this
          document, p is always congruent to 3 modulo 4.

   F_p    The finite field of order p.

   F*     The multiplicative group of the non-zero elements in the field
          F; e.g., (F_p)* is the multiplicative group of the finite
          field F_p.

   q      An odd prime that divides p + 1.  To provide the desired level
          of security, lg(q) MUST be greater than 2*n.

   E      An elliptic curve defined over F_p, having a subgroup of order
          q.  In this document, we use supersingular curves with
          equation y^2 = x^3 - 3 * x modulo p.  This curve is chosen
          because of the efficiency and simplicity advantages it offers.
          The choice of -3 for the coefficient of x provides advantages
          for elliptic curve arithmetic that are explained in [P1363].
          A further reason for this choice of curve is that Barreto's
          trick [Barreto] of eliminating the computation of the
          denominators when calculating the pairing applies.

   E(F)   The additive group of points of affine coordinates (x,y) with
          x, y in the field F, that satisfy the curve equation for E.

   P      A point of E(F_p) that generates the cyclic subgroup of order
          q.  The coordinates of P are given by P = (P_x,P_y).  These
          coordinates are in F_p, and they satisfy the curve equation.

   0      The null element of any additive group of points on an
          elliptic curve, also called the point at infinity.

   F_p^2  The extension field of degree 2 of the field F_p.  In this
          document, we use a particular instantiation of this field;
          F_p^2 = F_p[i], where i^2 + 1 = 0.

   PF_p   The projectivization of F_p.  We define this to be
          (F_p^2)*/(F_p)*.  Note that PF_p is cyclic and has order
          p + 1, which is divisible by q.

   G[q]   The q-torsion of a group G.  This is the subgroup generated by
          points of order q in G.

   < , >  A version of the Tate-Lichtenbaum pairing.  In this document,
          this is a bilinear map from E(F_p)[q] x E(F_p)[q] onto the
          subgroup of order q in PF_p.  A full definition is given in
          Section 3.2.

   Hash   A cryptographic hash function.

   lg(x)  The base 2 logarithm of the real value x.

   The following conventions are assumed for curve operations:

      Point addition - If A and B are two points on a curve E, their sum
         is denoted as A + B.

      Scalar multiplication - If A is a point on a curve, and k an
         integer, the result of adding A to itself a total of k times is
         denoted [k]A.

   We assume that the following concrete representations of mathematical
   objects are used:

      Elements of F_p - The p elements of F_p are represented directly
         using the integers from 0 to p-1.

      Elements of F_p^2 - The elements of F_p^2 = F_p[i] are represented
         as x_1 + i * x_2, where x_1 and x_2 are elements of F_p.

      Elements of PF_p - Elements of PF_p are cosets of (F_p)* in
         (F_p^2)*.  Every element of F_p^2 can be written unambiguously
         in the form x_1 + i * x_2, where x_1 and x_2 are elements of
         F_p.  Thus, elements of PF_p (except the unique element of
         order 2) can be represented unambiguously by x_2/x_1 in F_p.
         Since q is odd, every element of PF_p[q] can be represented by
         an element of F_p in this manner.

   This representation of elements in PF_p[q] allows efficient
   implementation of PF_p[q] group operations, as these can be defined
   using arithmetic in F_p.  If a and b are elements of F_p representing
   elements A and B of PF_p[q], respectively, then A * B in PF_p[q] is
   represented by (a + b)/(1 - a * b) in F_p.

2.2.  Definitions

   Identifier - Each user of an IBE system MUST have a unique,
      unambiguous identifying string that can be easily derived by all
      valid communicants.  This string is the user's Identifier.  An
      Identifier is an integer in the range 2 to q-1.  The method by
      which Identifiers are formed MUST be defined for each application.

   Key Management Service (KMS) - The Key Management Service is a
      trusted third party for the IBE system.  It derives system secrets
      and distributes key material to those authorized to obtain it.
      Applications MAY support mutual communication between the users of
      multiple KMSs.  We denote KMSs by KMS_T, KMS_S, etc.

   Public parameters - The public parameters are a set of parameters
      that are held by all users of an IBE system.  Such a system MAY
      contain multiple KMSs.  Each application of SAKKE MUST define the
      set of public parameters to be used.  The parameters needed are p,
      q, E, P, g=<P,P>, Hash, and n.

   Master Secret (z_T) - The Master Secret z_T is the master key
      generated and privately kept by KMS_T and is used by KMS_T to
      generate the private keys of the users that it provisions; it is
      an integer in the range 2 to q-1.

   KMS Public Key: Z_T = [z_T]P - The KMS Public Key Z_T is used to form
      Public Key Establishment Keys for all users provisioned by KMS_T;
      it is a point of order q in E(F_p).  It MUST be provisioned by
      KMS_T to all who are authorized to send messages to users of the
      IBE system.

   Receiver Secret Key (RSK) - Each user enrolled in an IBE system is
      provisioned with a Receiver Secret Key by its KMS.  The RSK
      provided to a user with Identifier 'a' by KMS_T is denoted
      K_(a,T).  In SAKKE, the RSK is a point of order q in E(F_p).

   Shared Secret Value (SSV) - The aim of the SAKKE scheme is for the
      Sender to securely transmit a shared secret value to the Receiver.
      The SSV is an integer in the range 0 to (2^n) - 1.

   Encapsulated Data - The Encapsulated Data are used to transmit secret
      information securely to the Receiver.  They can be computed
      directly from the Receiver's Identifier, the public parameters,
      the KMS Public Key, and the SSV to be transmitted.  In SAKKE, the
      Encapsulated Data are a point of order q in E(F_p) and an integer
      in the range 0 to (2^n) - 1.  They are formatted as described in
      Section 4.

2.3.  Parameters to Be Defined or Negotiated

   In order for an application to make use of the SAKKE algorithm, the
   communicating hosts MUST agree on values for several of the
   parameters described above.  The curve equation (E) and the pairing
   (< , >) are constant and used for all applications.

   For the following parameters, each application MUST either define an
   application-specific constant value or define a mechanism for hosts
   to negotiate a value:

      * n

      * p

      * q

      * P = (P_x,P_y)

      * g = <P,P>

      * Hash

3.  Elliptic Curves and Pairings

   E is a supersingular elliptic curve (of j-invariant 1728).  E(F_p)
   contains a cyclic subgroup of order q, denoted E(F_p)[q], whereas the
   larger object E(F_p^2) contains the direct product of two cyclic
   subgroups of order q, denoted E(F_p^2)[q].

   P is a generator of E(F_p)[q].  It is specified by the (affine)
   coordinates (P_x,P_y) in F_p, satisfying the curve equation.

   Routines for point addition and doubling on E(F_p) can be found in
   Appendix A.10 of [P1363].

3.1.  E(F_p^2) and the Distortion Map

   If (Q_x,Q_y) are (affine) coordinates in F_p for some point (denoted
   Q) on E(F_p)[q], then (-Q_x,iQ_y) are (affine) coordinates in F_p^2
   for some point on E(F_p^2)[q].  This latter point is denoted [i]Q, by
   analogy with the definition for scalar multiplication.  The two
   points P and [i]P together generate E(F_p^2)[q].  The map [i]: E(F_p)
   -> E(F_p^2) is sometimes termed the distortion map.

3.2.  The Tate-Lichtenbaum Pairing

   We proceed to describe the pairing < , > to be used in SAKKE.  We
   will need to evaluate polynomials f_R that depend on points on
   E(F_p)[q].  Miller's algorithm [Miller] provides a method for
   evaluation of f_R(X), where X is some element of E(F_p^2)[q] and R is
   some element of E(F_p)[q] and f_R is some polynomial over F_p whose
   divisor is (q)(R) - (q)(0).  Note that f_R is defined only up to
   scalars of F_p.

   The version of the Tate-Lichtenbaum pairing used in this document is
   given by <R,Q> = f_R([i]Q)^c / (F_p)*.  It satisfies the bilinear
   relation <[x]R,Q> = <R,[x]Q> = <R,Q>^x for all Q, R in E(F_p)[q], for
   all integers x.  Note that the domain of definition is restricted to
   E(F_p)[q] x E(F_p)[q] so that certain optimizations are natural.

   We provide pseudocode for computing <R,Q>, with elliptic curve
   arithmetic expressed in affine coordinates.  We make use of Barreto's
   trick [Barreto] for avoiding the calculation of denominators.  Note
   that this section does not fully describe the most efficient way of
   computing the pairing; it is possible to compute the pairing without
   any explicit reference to the extension field F_p^2.  This reduces
   the number and complexity of the operations needed to compute the
   pairing.

   <CODE BEGINS>

   /*
   Copyright (c) 2012 IETF Trust and the persons identified as
   authors of the code.  All rights reserved.

   Redistribution and use in source and binary forms, with or without
   modification, is permitted pursuant to, and subject to the license
   terms contained in, the Simplified BSD License set forth in
   Section 4.c of the IETF Trust's Legal Provisions Relating to
   IETF Documents (http://trustee.ietf.org/license-info).
   */

       Routine for computing the pairing <R,Q>:

         Input R, Q points on E(F_p)[q];

         Initialize variables:

           v = (F_p)*;    // An element of PF_p[q]
           C = R;         // An element of E(F_p)[q]
           c = (p+1)/q;   // An integer

         for bits of q-1, starting with the second most significant
         bit, ending with the least significant bit, do

           // gradient of line through C, C, [-2]C.
           l = 3*( C_x^2 - 1 ) / ( 2*C_y );

           //accumulate line evaluated at [i]Q into v
           v = v^2 * ( l*( Q_x + C_x ) + ( i*Q_y - C_y ) );

           C = [2]C;

           if bit is 1, then

             // gradient of line through C, R, -C-R.
             l = ( C_y - R_y )/( C_x - R_x );

             //accumulate line evaluated at [i]Q into v
             v = v * ( l*( Q_x + C_x ) + ( i*Q_y - C_y ) );

             C = C+R;

           end if;
         end for;

         t = v^c;

         return representative in F_p of t;

       End of routine;

       Routine for computing representative in F_p of elements of PF_p:

         Input t, in F_p^2, representing an element of PF_p;

         Represent t as a + i*b, with a,b in F_p;
         return b/a;

       End of routine;

   <CODE ENDS>

4.  Representation of Values

   This section provides canonical representations of values that MUST
   be used to ensure interoperability of implementations.  The following
   representations MUST be used for input into hash functions and for
   transmission.

   Integers           Integers MUST be represented as an octet string,
                      with bit length a multiple of 8.  To achieve this,
                      the integer is represented most significant bit
                      first, and padded with zero bits on the left until
                      an octet string of the necessary length is
                      obtained.  This is the octet string representation
                      described in Section 6 of [RFC6090].

   F_p elements       Elements of F_p MUST be represented as integers in
                      the range 0 to p-1 using the octet string
                      representation defined above.  Such octet strings
                      MUST have length L = Ceiling(lg(p)/8).

   PF_p elements      Elements of PF_p MUST be represented as an element
                      of F_p using the algorithm in Section 3.2.  They
                      are therefore represented as octet strings as
                      defined above and are L octets in length.
                      Representation of the unique element of order 2 in
                      PF_p will not be required.

   Points on E        Elliptic curve points MUST be represented in
                      uncompressed form as defined in Section 2.2 of
                      [RFC5480].  For an elliptic curve point (x,y) with
                      x and y in F_p, this representation is given by

                      0x04 || x' || y', where x' is the octet string
                      representing x, y' is the octet string
                      representing y, and || denotes concatenation.  The
                      representation is 2*L+1 octets in length.

   Encapsulated Data  The Encapsulated Data MUST be represented as an
                      elliptic curve point concatenated with an integer
                      in the range 0 to (2 ^ n) - 1.  Since the length
                      of the representation of elements of F_p is well
                      defined given p, these data can be unambiguously
                      parsed to retrieve their components.  The
                      Encapsulated Data is 2*L + n + 1 octets in length.

5.  Supporting Algorithms

5.1.  Hashing to an Integer Range

   We use the function HashToIntegerRange( s, n, hashfn ) to hash
   strings to an integer range.  Given a string (s), a hash function
   (hashfn), and an integer (n), this function returns a value between 0
   and n - 1.

   Input:

      * an octet string, s

      * an integer, n <= (2^hashlen)^hashlen

      * a hash function, hashfn, with output length hashlen bits

   Output:

      * an integer, v, in the range 0 to n-1

   Method:

      1) Let A = hashfn( s )

      2) Let h_0 = 00...00, a string of null bits of length hashlen bits

      3) Let l = Ceiling(lg(n)/hashlen)

      4) For each i in 1 to l, do:

         a) Let h_i = hashfn(h_(i - 1))

         b) Let v_i = hashfn(h_i || A), where || denotes concatenation

      5) Let v' = v_1 || ...  || v_l

      6) Let v = v' mod n

6.  The SAKKE Cryptosystem

   This section describes the Sakai-Kasahara Key Encryption algorithm.
   It draws from the cryptosystem first described in [S-K].

6.1.  Setup

   All users share a set of public parameters with a KMS.  In most
   circumstances, it is expected that a system will only use a single
   KMS.  However, it is possible for users provisioned by different KMSs
   to interoperate, provided that they use a common set of public
   parameters and that they each possess the necessary KMS Public Keys.
   In order to facilitate this interoperation, it is anticipated that
   parameters will be published in application-specific standards.

   KMS_T chooses its KMS Master Secret, z_T.  It MUST randomly select a
   value in the range 2 to q-1, and assigns this value to z_T.  It MUST
   derive its KMS Public Key, Z_T, by performing the calculation Z_T =
   [z_T]P.

6.1.1.  Secret Key Extraction

   The KMS derives each RSK from an Identifier and its KMS Master
   Secret.  It MUST derive a RSK for each user that it provisions.

   For Identifier 'a', the RSK K_(a,T) provided by KMS_T MUST be derived
   by KMS_T as K_(a,T) = [(a + z_T)^-1]P, where 'a' is interpreted as an
   integer, and the inversion is performed modulo q.

6.1.2.  User Provisioning

   The KMS MUST provide its KMS Public Key to all users through an
   authenticated channel.  RSKs MUST be supplied to all users through a
   channel that provides confidentiality and mutual authentication.  The
   mechanisms that provide security for these channels are beyond the
   scope of this document: they are application specific.

   Upon receipt of key material, each user MUST verify its RSK.  For
   Identifier 'a', RSKs from KMS_T are verified by checking that the
   following equation holds: < [a]P + Z, K_(a,T) > = g, where 'a' is
   interpreted as an integer.

6.2.  Key Exchange

   A Sender forms Encapsulated Data and sends it to the Receiver, who
   processes it.  The result is a shared secret that can be used as
   keying material for securing further communications.  We denote the
   Sender A with Identifier 'a'; we denote the Receiver B with
   Identifier 'b'; Identifiers are to be interpreted as integers in the
   algorithms below.  Let A be provisioned by KMS_T and B be provisioned
   by KMS_S.

6.2.1.  Sender

   In order to form Encapsulated Data to send to device B who is
   provisioned by KMS_S, A needs to hold Z_S.  It is anticipated that
   this will have been provided to A by KMS_T along with its User
   Private Keys.  The Sender MUST carry out the following steps:

      1) Select a random ephemeral integer value for the SSV in the
         range 0 to 2^n - 1;

      2) Compute r = HashToIntegerRange( SSV || b, q, Hash );

      3) Compute R_(b,S) = [r]([b]P + Z_S) in E(F_p);

      4) Compute the Hint, H;

         a) Compute g^r.  Note that g is an element of PF_p[q]
            represented by an element of F_p.  Thus, in order to
            calculate g^r, the operation defined in Section 2.1 for
            calculation of A * B in PF_p[q] is to be used as part of a
            square and multiply (or similar) exponentiation algorithm,
            rather than the regular F_p operations;

         b) Compute H := SSV XOR HashToIntegerRange( g^r, 2^n, Hash );

      5) Form the Encapsulated Data ( R_(b,S), H ), and transmit it
         to B;

      6) Output SSV for use to derive key material for the application
         to be keyed.

6.2.2.  Receiver

   Device B receives Encapsulated Data from device A.  In order to
   process this, it requires its RSK, K_(b,S), which will have been
   provisioned in advance by KMS_S.  The method by which keys are
   provisioned by the KMS is application specific.  The Receiver MUST
   carry out the following steps to derive and verify the SSV:

      1) Parse the Encapsulated Data ( R_(b,S), H ), and extract R_(b,S)
         and H;

      2) Compute w := < R_(b,S), K_(b,S) >.  Note that by bilinearity,
         w = g^r;

      3) Compute SSV = H XOR HashToIntegerRange( w, 2^n, Hash );

      4) Compute r = HashToIntegerRange( SSV || b, q, Hash );

      5) Compute TEST = [r]([b]P + Z_S) in E(F_p).  If TEST does not
         equal R_(b,S), then B MUST NOT use the SSV to derive key
         material;

      6) Output SSV for use to derive key material for the application
         to be keyed.

6.3.  Group Communications

   The SAKKE scheme can be used to exchange SSVs for group
   communications.  To provide a shared secret to multiple Receivers, a
   Sender MUST form Encapsulated Data for each of their Identifiers and
   transmit the appropriate data to each Receiver.  Any party possessing
   the group SSV MAY extend the group by forming Encapsulated Data for a
   new group member.

   While the Sender needs to form multiple Encapsulated Data, the fact
   that the sending operation avoids pairings means that the extension
   to multiple Receivers can be carried out more efficiently than for
   alternative IBE schemes that require the Sender to compute a pairing.

7.  Security Considerations

   This document describes the SAKKE cryptographic algorithm.  We assume
   that the security provided by this algorithm depends entirely on the
   secrecy of the secret keys it uses, and that for an adversary to
   defeat this security, he will need to perform computationally
   intensive cryptanalytic attacks to recover a secret key.  Note that a
   security proof exists for SAKKE in the Random Oracle Model [SK-KEM].

   When defining public parameters, guidance on parameter sizes from
   [SP800-57] SHOULD be followed.  Note that the size of the F_p^2
   discrete logarithm on which the security rests is 2*lg(p).  Table 1
   shows bits of security afforded by various sizes of p.  If k bits of
   security are needed, then lg(q) SHOULD be chosen to be at least 2*k.
   Similarly, if k bits of security are needed, then a hash with output
   size at least 2*k SHOULD be chosen.

         Bits of Security | lg(p)
         -------------------------
         80               |   512
         112              |  1024
         128              |  1536
         192              |  3840
         256              |  7680

      Table 1: Comparable Strengths, Taken from Table 2 of [SP800-57]

   The KMS Master Secret provides the security for each device
   provisioned by the KMS.  It MUST NOT be revealed to any other entity.
   Each user's RSK protects the SAKKE communications it receives.  This
   key MUST NOT be revealed to any entity other than the trusted KMS and
   the authorized user.

   In order to ensure that the RSK is received only by an authorized
   device, it MUST be provided through a secure channel.  The security
   offered by this system is no greater than the security provided by
   this delivery channel.

   Note that IBE systems have different properties than other asymmetric
   cryptographic schemes with regard to key recovery.  The KMS (and
   hence any administrator with appropriate privileges) can create RSKs
   for arbitrary Identifiers, and procedures to monitor the creation of
   RSKs, such as logging of administrator actions, SHOULD be defined by
   any functioning implementation of SAKKE.

   Identifiers MUST be defined unambiguously by each application of
   SAKKE.  Note that it is not necessary to hash the data in a format
   for Identifiers (except in the case where its size would be greater
   than that of q).  In this way, any weaknesses that might be caused by
   collisions in hash functions can be avoided without reliance on the
   structure of the Identifier format.  Applications of SAKKE MAY
   include a time/date component in their Identifier format to ensure
   that Identifiers (and hence RSKs) are only valid for a fixed period
   of time.

   The randomness of values stipulated to be selected at random in
   SAKKE, as described in this document, is essential to the security
   provided by SAKKE.  If the ephemeral value r selected by the Sender
   is not chosen at random, then the SSV, which is used to provide key
   material for further communications, could be predictable.  Guidance
   on the generation of random values for security can be found in
   [RFC4086].

8.  References

8.1.  Normative References

   [RFC2119]   Bradner, S., "Key words for use in RFCs to Indicate
               Requirement Levels", BCP 14, RFC 2119, March 1997.

   [RFC5480]   Turner, S., Brown, D., Yiu, K., Housley, R., and T. Polk,
               "Elliptic Curve Cryptography Subject Public Key
               Information", RFC 5480, March 2009.

   [RFC6090]   McGrew, D., Igoe, K., and M. Salter, "Fundamental
               Elliptic Curve Cryptography Algorithms", RFC 6090,
               February 2011.

   [S-K]       Sakai, R., Ohgishi, K., and M. Kasahara, "ID based
               cryptosystem based on pairing on elliptic curves",
               Symposium on Cryptography and Information Security -
               SCIS, 2001.

   [SK-KEM]    Barbosa, M., Chen, L., Cheng, Z., Chimley, M., Dent, A.,
               Farshim, P., Harrison, K., Malone-Lee, J., Smart, N., and
               F. Vercauteren, "SK-KEM: An Identity-Based KEM",
               submission for IEEE P1363.3, June 2006,
               (http://grouper.ieee.org/groups/1363/IBC/
               submissions/Barbosa-SK-KEM-2006-06.pdf).

   [SP800-57]  Barker, E., Barker, W., Burr, W., Polk, W., and M. Smid,
               "Recommendation for Key Management - Part 1: General
               (Revised)", NIST Special Publication 800-57, March 2007.

8.2.  Informative References

   [Barreto]   Barreto, P., Kim, H., Lynn, B., and M. Scott, "Efficient
               Algorithms for Pairing-Based Cryptosystems", Advances in
               Cryptology - Crypto 2002, LNCS 2442, Springer-Verlag
               (2002), pp. 354-369.

   [Miller]    Miller, V., "The Weil pairing, and its efficient
               calculation", J. Cryptology 17 (2004), 235-261.

   [P1363]     IEEE P1363-2000, "Standard Specifications for Public-Key
               Cryptography", 2001.

   [RFC4086]   Eastlake 3rd, D., Schiller, J., and S. Crocker,
               "Randomness Requirements for Security", BCP 106,
               RFC 4086, June 2005.

   [RFC5091]   Boyen, X. and L. Martin, "Identity-Based Cryptography
               Standard (IBCS) #1: Supersingular Curve Implementations
               of the BF and BB1 Cryptosystems", RFC 5091,
               December 2007.

   [RFC6509]   Groves, M., "MIKEY-SAKKE: Sakai-Kasahara Key Encryption
               in Multimedia Internet KEYing (MIKEY)", RFC 6509,
               February 2012.

Appendix A.  Test Data

   This appendix provides test data for SAKKE with the public parameters
   defined in Appendix A of [RFC6509].  'b' represents the Identifier of
   the Responder.  The value "mask" is the value used to mask the SSV
   and is defined to be
   HashToIntegerRange( g^r, 2^n, Hash ).

      // --------------------------------------------------------
      // The KMS generates:

        z      = AFF429D3 5F84B110 D094803B 3595A6E2 998BC99F

        Zx     = 5958EF1B 1679BF09 9B3A030D F255AA6A
                 23C1D8F1 43D4D23F 753E69BD 27A832F3
                 8CB4AD53 DDEF4260 B0FE8BB4 5C4C1FF5
                 10EFFE30 0367A37B 61F701D9 14AEF097
                 24825FA0 707D61A6 DFF4FBD7 273566CD
                 DE352A0B 04B7C16A 78309BE6 40697DE7
                 47613A5F C195E8B9 F328852A 579DB8F9
                 9B1D0034 479EA9C5 595F47C4 B2F54FF2

        Zy     = 1508D375 14DCF7A8 E143A605 8C09A6BF
                 2C9858CA 37C25806 5AE6BF75 32BC8B5B
                 63383866 E0753C5A C0E72709 F8445F2E
                 6178E065 857E0EDA 10F68206 B63505ED
                 87E534FB 2831FF95 7FB7DC61 9DAE6130
                 1EEACC2F DA3680EA 4999258A 833CEA8F
                 C67C6D19 487FB449 059F26CC 8AAB655A
                 B58B7CC7 96E24E9A 39409575 4F5F8BAE

      // --------------------------------------------------------
      // Creating Encapsulated Data

        b      = 3230 31312D30 32007465 6C3A2B34
                 34373730 30393030 31323300

        SSV    = 12345678 9ABCDEF0 12345678 9ABCDEF0

        r      = HashToIntegerRange(
                 12345678 9ABCDEF0 12345678 9ABCDEF0
                 32303131 2D303200 74656C3A 2B343437
                 37303039 30303132 3300, q, SHA-256 )

               = 13EE3E1B 8DAC5DB1 68B1CEB3 2F0566A4
                 C273693F 78BAFFA2 A2EE6A68 6E6BD90F
                 8206CCAB 84E7F42E D39BD4FB 131012EC
                 CA2ECD21 19414560 C17CAB46 B956A80F
                 58A3302E B3E2C9A2 28FBA7ED 34D8ACA2
                 392DA1FF B0B17B23 20AE09AA EDFD0235
                 F6FE0EB6 5337A63F 9CC97728 B8E5AD04
                 60FADE14 4369AA5B 21662132 47712096

        Rbx    = 44E8AD44 AB8592A6 A5A3DDCA 5CF896C7
                 18043606 A01D650D EF37A01F 37C228C3
                 32FC3173 54E2C274 D4DAF8AD 001054C7
                 6CE57971 C6F4486D 57230432 61C506EB
                 F5BE438F 53DE04F0 67C776E0 DD3B71A6
                 29013328 3725A532 F21AF145 126DC1D7
                 77ECC27B E50835BD 28098B8A 73D9F801
                 D893793A 41FF5C49 B87E79F2 BE4D56CE

        Rby    = 557E134A D85BB1D4 B9CE4F8B E4B08A12
                 BABF55B1 D6F1D7A6 38019EA2 8E15AB1C
                 9F76375F DD1210D4 F4351B9A 009486B7
                 F3ED46C9 65DED2D8 0DADE4F3 8C6721D5
                 2C3AD103 A10EBD29 59248B4E F006836B
                 F097448E 6107C9ED EE9FB704 823DF199
                 F832C905 AE45F8A2 47A072D8 EF729EAB
                 C5E27574 B07739B3 4BE74A53 2F747B86

        g^r    = 7D2A8438 E6291C64 9B6579EB 3B79EAE9
                 48B1DE9E 5F7D1F40 70A08F8D B6B3C515
                 6F2201AF FBB5CB9D 82AA3EC0 D0398B89
                 ABC78A13 A760C0BF 3F77E63D 0DF3F1A3
                 41A41B88 11DF197F D6CD0F00 3125606F
                 4F109F40 0F7292A1 0D255E3C 0EBCCB42
                 53FB182C 68F09CF6 CD9C4A53 DA6C74AD
                 007AF36B 8BCA979D 5895E282 F483FCD6

        mask   = HashToIntegerRange(
                 7D2A8438 E6291C64 9B6579EB 3B79EAE9
                 48B1DE9E 5F7D1F40 70A08F8D B6B3C515
                 6F2201AF FBB5CB9D 82AA3EC0 D0398B89
                 ABC78A13 A760C0BF 3F77E63D 0DF3F1A3
                 41A41B88 11DF197F D6CD0F00 3125606F
                 4F109F40 0F7292A1 0D255E3C 0EBCCB42
                 53FB182C 68F09CF6 CD9C4A53 DA6C74AD
                 007AF36B 8BCA979D 5895E282 F483FCD6, 2^128, SHA-256 )

               = 9BD4EA1E 801D37E6 2AD2FAB0 D4F5BBF7

        H      = 89E0BC66 1AA1E916 38E6ACC8 4E496507

      // --------------------------------------------------------
      // Receiver processing

      // Device receives Kb from the KMS

        Kbx    = 93AF67E5 007BA6E6 A80DA793 DA300FA4
                 B52D0A74 E25E6E7B 2B3D6EE9 D18A9B5C
                 5023597B D82D8062 D3401956 3BA1D25C
                 0DC56B7B 979D74AA 50F29FBF 11CC2C93
                 F5DFCA61 5E609279 F6175CEA DB00B58C
                 6BEE1E7A 2A47C4F0 C456F052 59A6FA94
                 A634A40D AE1DF593 D4FECF68 8D5FC678
                 BE7EFC6D F3D68353 25B83B2C 6E69036B

        Kby    = 155F0A27 241094B0 4BFB0BDF AC6C670A
                 65C325D3 9A069F03 659D44CA 27D3BE8D
                 F311172B 55416018 1CBE94A2 A783320C
                 ED590BC4 2644702C F371271E 496BF20F
                 588B78A1 BC01ECBB 6559934B DD2FB65D
                 2884318A 33D1A42A DF5E33CC 5800280B
                 28356497 F87135BA B9612A17 26042440
                 9AC15FEE 996B744C 33215123 5DECB0F5

      // Device processes Encapsulated Data

        w      = 7D2A8438 E6291C64 9B6579EB 3B79EAE9
                 48B1DE9E 5F7D1F40 70A08F8D B6B3C515
                 6F2201AF FBB5CB9D 82AA3EC0 D0398B89
                 ABC78A13 A760C0BF 3F77E63D 0DF3F1A3
                 41A41B88 11DF197F D6CD0F00 3125606F
                 4F109F40 0F7292A1 0D255E3C 0EBCCB42
                 53FB182C 68F09CF6 CD9C4A53 DA6C74AD
                 007AF36B 8BCA979D 5895E282 F483FCD6

        SSV    = 12345678 9ABCDEF0 12345678 9ABCDEF0

        r      = 13EE3E1B 8DAC5DB1 68B1CEB3 2F0566A4
                 C273693F 78BAFFA2 A2EE6A68 6E6BD90F
                 8206CCAB 84E7F42E D39BD4FB 131012EC
                 CA2ECD21 19414560 C17CAB46 B956A80F
                 58A3302E B3E2C9A2 28FBA7ED 34D8ACA2
                 392DA1FF B0B17B23 20AE09AA EDFD0235
                 F6FE0EB6 5337A63F 9CC97728 B8E5AD04
                 60FADE14 4369AA5B 21662132 47712096

        TESTx  = 44E8AD44 AB8592A6 A5A3DDCA 5CF896C7
                 18043606 A01D650D EF37A01F 37C228C3
                 32FC3173 54E2C274 D4DAF8AD 001054C7
                 6CE57971 C6F4486D 57230432 61C506EB
                 F5BE438F 53DE04F0 67C776E0 DD3B71A6
                 29013328 3725A532 F21AF145 126DC1D7
                 77ECC27B E50835BD 28098B8A 73D9F801
                 D893793A 41FF5C49 B87E79F2 BE4D56CE

        TESTy  = 557E134A D85BB1D4 B9CE4F8B E4B08A12
                 BABF55B1 D6F1D7A6 38019EA2 8E15AB1C
                 9F76375F DD1210D4 F4351B9A 009486B7
                 F3ED46C9 65DED2D8 0DADE4F3 8C6721D5
                 2C3AD103 A10EBD29 59248B4E F006836B
                 F097448E 6107C9ED EE9FB704 823DF199
                 F832C905 AE45F8A2 47A072D8 EF729EAB
                 C5E27574 B07739B3 4BE74A53 2F747B86

        TEST == Rb

      // --------------------------------------------------------
      // HashToIntegerRange( M, q, SHA-256 ) example

        M      = 12345678 9ABCDEF0 12345678 9ABCDEF0
                 32303131 2D303200 74656C3A 2B343437
                 37303039 30303132 3300

        A      = E04D4EF6 9DF86893 22B39AE3 80284617
                 4A93BEDB 1E3D2A2C 5F2C7EA0 05513EBA

        h0     = 00000000 00000000 00000000 00000000
                 00000000 00000000 00000000 00000000

        h1     = 66687AAD F862BD77 6C8FC18B 8E9F8E20
                 08971485 6EE233B3 902A591D 0D5F2925

        h2     = 2B32DB6C 2C0A6235 FB1397E8 225EA85E
                 0F0E6E8C 7B126D00 16CCBDE0 E667151E

        h3     = 12771355 E46CD47C 71ED1721 FD5319B3
                 83CCA3A1 F9FCE3AA 1C8CD3BD 37AF20D7

        h4     = FE15C0D3 EBE314FA D720A08B 839A004C
                 2E6386F5 AECC19EC 74807D19 20CB6AEB

        v1     = FA2656CA 1D2DBD79 015AE918 773DFEDC
                 24957C91 E3C9C335 40D6BF6D 7C3C0055

        v2     = F016CD67 59620AD7 87669E3A DD887DF6
                 25895A91 0CEE1486 91A06735 B2F0A248

        v3     = AC45C6F9 7F83BCE0 A2BBD0A1 4CF4D7F4
                 CB3590FB FAF93AE7 1C64E426 185710B5

        v4     = E65D50BD 551A54EF 981F535E 072DE98D
                 2223ACAD 4621E026 3B0A61EA C56DB078

       v mod q = 13EE3E1B 8DAC5DB1 68B1CEB3 2F0566A4
                 C273693F 78BAFFA2 A2EE6A68 6E6BD90F
                 8206CCAB 84E7F42E D39BD4FB 131012EC
                 CA2ECD21 19414560 C17CAB46 B956A80F
                 58A3302E B3E2C9A2 28FBA7ED 34D8ACA2
                 392DA1FF B0B17B23 20AE09AA EDFD0235
                 F6FE0EB6 5337A63F 9CC97728 B8E5AD04
                 60FADE14 4369AA5B 21662132 47712096

      // --------------------------------------------------------

Author's Address

   Michael Groves
   CESG
   Hubble Road
   Cheltenham
   GL51 8HJ
   UK

   EMail: Michael.Groves@cesg.gsi.gov.uk

 

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