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Network Working Group                                           H. Orman
Request for Comments: 3766                            Purple Streak Dev.
BCP: 86                                                       P. Hoffman
Category: Best Current Practice                           VPN Consortium
                                                              April 2004

               Determining Strengths For Public Keys Used
                     For Exchanging Symmetric Keys

Status of this Memo

   This document specifies an Internet Best Current Practices for the
   Internet Community, and requests discussion and suggestions for
   improvements.  Distribution of this memo is unlimited.

Copyright Notice

   Copyright (C) The Internet Society (2004).  All Rights Reserved.


   Implementors of systems that use public key cryptography to exchange
   symmetric keys need to make the public keys resistant to some
   predetermined level of attack.  That level of attack resistance is
   the strength of the system, and the symmetric keys that are exchanged
   must be at least as strong as the system strength requirements.  The
   three quantities, system strength, symmetric key strength, and public
   key strength, must be consistently matched for any network protocol

   While it is fairly easy to express the system strength requirements
   in terms of a symmetric key length and to choose a cipher that has a
   key length equal to or exceeding that requirement, it is harder to
   choose a public key that has a cryptographic strength meeting a
   symmetric key strength requirement.  This document explains how to
   determine the length of an asymmetric key as a function of a
   symmetric key strength requirement.  Some rules of thumb for
   estimating equivalent resistance to large-scale attacks on various
   algorithms are given.  The document also addresses how changing the
   sizes of the underlying large integers (moduli, group sizes,
   exponents, and so on) changes the time to use the algorithms for key

Table of Contents

   1.  Model of Protecting Symmetric Keys with Public Keys. . . . . .  2
       1.1. The key exchange algorithms . . . . . . . . . . . . . . .  4
   2.  Determining the Effort to Factor . . . . . . . . . . . . . . .  5
       2.1. Choosing parameters for the equation. . . . . . . . . . .  6
       2.2. Choosing k from empirical reports . . . . . . . . . . . .  7
       2.3. Pollard's rho method. . . . . . . . . . . . . . . . . . .  7
       2.4. Limits of large memory and many machines. . . . . . . . .  8
       2.5. Special purpose machines. . . . . . . . . . . . . . . . .  9
   3.  Compute Time for the Algorithms. . . . . . . . . . . . . . . . 10
       3.1. Diffie-Hellman Key Exchange . . . . . . . . . . . . . . . 10
            3.1.1. Diffie-Hellman with elliptic curve groups. . . . . 11
       3.2. RSA encryption and decryption . . . . . . . . . . . . . . 11
       3.3. Real-world examples . . . . . . . . . . . . . . . . . . . 12
   4.  Equivalences of Key Sizes. . . . . . . . . . . . . . . . . . . 13
       4.1. Key equivalence against special purpose brute force
            hardware. . . . . . . . . . . . . . . . . . . . . . . . . 15
       4.2. Key equivalence against conventional CPU brute force
            attack. . . . . . . . . . . . . . . . . . . . . . . . . . 15
       4.3. A One Year Attack: 80 bits of strength. . . . . . . . . . 16
       4.4. Key equivalence for other ciphers . . . . . . . . . . . . 16
       4.5. Hash functions for deriving symmetric keys from public
            key algorithms. . . . . . . . . . . . . . . . . . . . . . 17
       4.6. Importance of randomness. . . . . . . . . . . . . . . . . 19
   5.  Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 19
       5.1. TWIRL Correction. . . . . . . . . . . . . . . . . . . . . 20
   6.  Security Considerations. . . . . . . . . . . . . . . . . . . . 20
   7.  References . . . . . . . . . . . . . . . . . . . . . . . . . . 20
       7.1. Informational References. . . . . . . . . . . . . . . . . 20
   8.  Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . 22
   9.  Full Copyright Statement . . . . . . . . . . . . . . . . . . . 23

1.  Model of Protecting Symmetric Keys with Public Keys

   Many books on cryptography and security explain the need to exchange
   symmetric keys in public as well as the many algorithms that are used
   for this purpose.  However, few of these discussions explain how the
   strengths of the public keys and the symmetric keys are related.

   To understand this, picture a house with a strong lock on the front
   door.  Next to the front door is a small lockbox that contains the
   key to the front door.  A would-be burglar who wants to break into
   the house through the front door has two options: attack the lock on
   the front door, or attack the lock on the lockbox in order to
   retrieve the key.  Clearly, the burglar is better off attacking the
   weaker of the two locks.  The homeowner in this situation must make

   sure that adding the second entry option (the lockbox containing the
   front door key) is at least as strong as the lock on the front door,
   in order not to make the burglar's job easier.

   An implementor designing a system for exchanging symmetric keys using
   public key cryptography must make a similar decision.  Assume that an
   attacker wants to learn the contents of a message that is encrypted
   with a symmetric key, and that the symmetric key was exchanged
   between the sender and recipient using public key cryptography.  The
   attacker has two options to recover the message: a brute-force
   attempt to determine the symmetric key by repeated guessing, or
   mathematical determination of the private key used as the key
   exchange key.  A smart attacker will work on the easier of these two

   A simple-minded answer to the implementor's problem is to be sure
   that the key exchange system is always significantly stronger than
   the symmetric key; this can be done by choosing a very long public
   key.  Such a design is usually not a good idea because the key
   exchanges become much more expensive in terms of processing time as
   the length of the public keys go up.  Thus, the implementor is faced
   with the task of trying to match the difficulty of an attack on the
   symmetric key with the difficulty of an attack on the public key
   encryption.  This analysis is not necessary if the key exchange can
   be performed with extreme security for almost no cost in terms of
   elapsed time or CPU effort; unfortunately, this is not the case for
   public key methods today.

   A third consideration is the minimum security requirement of the
   user.  Assume the user is encrypting with CAST-128 and requires a
   symmetric key with a resistance time against brute-force attack of 20
   years.  He might start off by choosing a key with 86 random bits, and
   then use a one-way function such as SHA-1 to "boost" that to a block
   of 160 bits, and then take 128 of those bits as the key for CAST-128.
   In such a case, the key exchange algorithm need only match the
   difficulty of 86 bits, not 128 bits.

   The selection procedure is:

   1. Determine the attack resistance necessary to satisfy the security
      requirements of the application.  Do this by estimating the
      minimum number of computer operations that the attacker will be
      forced to do in order to compromise the security of the system and
      then take the logarithm base two of that number.  Call that
      logarithm value "n".

      A 1996 report recommended 90 bits as a good all-around choice for
      system security.  The 90 bit number should be increased by about
      2/3 bit/year, or about 96 bits in 2005.

   2. Choose a symmetric cipher that has a key with at least n bits and
      at least that much cryptanalytic strength.

   3. Choose a key exchange algorithm with a resistance to attack of at
      least n bits.

   A fourth consideration might be the public key authentication method
   used to establish the identity of a user.  This might be an RSA
   digital signature or a DSA digital signature.  If the modulus for the
   authentication method isn't large enough, then the entire basis for
   trusting the communication might fall apart.  The following step is
   thus added:

   4. Choose an authentication algorithm with a resistance to attack of
      at least n bits.  This ensures that a similar key exchanged cannot
      be forged between the two parties during the secrecy lifetime of
      the encrypted material.  This may not be strictly necessary if the
      authentication keys are changed frequently and they have a well-
      understood usage lifetime, but in lieu of this, the n bit guidance
      is sound.

1.1.  The key exchange algorithms

   The Diffie-Hellman method uses a group, a generator, and exponents.
   In today's Internet standards, the group operation is based on
   modular multiplication.  Here, the group is defined by the
   multiplicative group of an integer, typically a prime p = 2q + 1,
   where q is a prime, and the arithmetic is done modulo p; the
   generator (which is often simply 2) is denoted by g.

   In Diffie-Hellman, Alice and Bob first agree (in public or in
   private) on the values for g and p.  Alice chooses a secret large
   random integer (a), and Bob chooses a secret random large integer
   (b).  Alice sends Bob A, which is g^a mod p; Bob sends Alice B, which
   is g^b mod p.  Next, Alice computes B^a mod p, and Bob computes A^b
   mod p.  These two numbers are equal, and the participants use a
   simple function of this number as the symmetric key k.

   Note that Diffie-Hellman key exchange can be done over different
   kinds of group representations.  For instance, elliptic curves
   defined over finite fields are a particularly efficient way to
   compute the key exchange [SCH95].

   For RSA key exchange, assume that Bob has a public key (m) which is
   equal to p*q, where p and q are two secret prime numbers, and an
   encryption exponent e, and a decryption exponent d.  For the key
   exchange, Alice sends Bob E = k^e mod m, where k is the secret
   symmetric key being exchanged.  Bob recovers k by computing E^d mod
   m, and the two parties use k as their symmetric key.  While Bob's
   encryption exponent e can be quite small (e.g., 17 bits), his
   decryption exponent d will have as many bits in it as m does.

2.  Determining the Effort to Factor

   The RSA public key encryption method is immune to brute force
   guessing attacks because the modulus (and thus, the secret exponent
   d) will have at least 512 bits, and that is too many possibilities to
   guess.  The Diffie-Hellman exchange is also secure against guessing
   because the exponents will have at least twice as many bits as the
   symmetric keys that will be derived from them.  However, both methods
   are susceptible to mathematical attacks that determine the structure
   of the public keys.

   Factoring an RSA modulus will result in complete compromise of the
   security of the private key.  Solving the discrete logarithm problem
   for a Diffie-Hellman modular exponentiation system will similarly
   destroy the security of all key exchanges using the particular
   modulus.  This document assumes that the difficulty of solving the
   discrete logarithm problem is equivalent to the difficulty of
   factoring numbers that are the same size as the modulus.  In fact, it
   is slightly harder because it requires more operations; based on
   empirical evidence so far, the ratio of difficulty is at least 20,
   possibly as high as 64.  Solving either problem requires a great deal
   of memory for the last stage of the algorithm, the matrix reduction
   step.  Whether or not this memory requirement will continue to be the
   limiting factor in solving larger integer problems remains to be
   seen.  At the current time it is not, and there is active research
   into parallel matrix algorithms that might mitigate the memory
   requirements for this problem.

   The number field sieve (NFS) [GOR93] [LEN93] is the best method today
   for solving the discrete logarithm problem.  The formula for
   estimating the number of simple arithmetic operations needed to
   factor an integer, n, using the NFS method is:

      L(n) = k * e^((1.92 + o(1)) * cubrt(ln(n) * (ln(ln(n)))^2))

   Many people prefer to discuss the number of MIPS years (MYs) that are
   needed for large operations such as the number field sieve.  For such
   an estimation, an operation in the L(n) formula is one computer

   instruction.  Empirical evidence indicates that 4 or 5 instructions
   might be a closer match, but this is a minor factor and this document
   sticks with one operation/one instruction for this discussion.

2.1.  Choosing parameters for the equation

   The expression above has two parameters that can be estimated by
   empirical means: k and o(1).  For the range of numbers we are
   interested in, there is little distinction between them.

   One could assume that k is 1 and o(1) is 0.  This is reasonably valid
   if the expression is only used for estimating relative effort
   (instead of actual effort) and one assumes that the o(1) term is very
   small over the range of the numbers that are to be factored.

   Or, one could assume that o(1) is small and roughly constant and thus
   its value can be folded into k; then estimate k from reported amounts
   of effort spent factoring large integers in tests.

   This document uses the second approach in order to get an estimate of
   the significance of the factor.  It appears to be minor, based on the
   following calculations.

   Sample values from recent work with the number field sieve include:

      Test name   Number of   Number of   MYs of effort
                    decimal      bits
      RSA130         130         430            500
      RSA140         140         460           2000
      RSA155         155         512           8000
      RSA160         160         528           3000

   There are few precise measurements of the amount of time used for
   these factorizations.  In most factorization tests, hundreds or
   thousands of computers are used over a period of several months, but
   the number of their cycles were used for the factoring project, the
   precise distribution of processor types, speeds, and so on are not
   usually reported.  However, in all the above cases, the amount of
   effort used was far less than the L(n) formula would predict if k was
   1 and o(1) was 0.

   A similar estimate of effort, done in 1995, is in [ODL95].

   Results indicating that for the Number Field Sieve factoring method,
   the actual number of operations is less than expected, are found in

2.2.  Choosing k from empirical reports

   By solving for k from the empirical reports, it appears that k is
   approximately 0.02.  This means that the "effective key strength" of
   the RSA algorithm is about 5 or 6 bits less than is implied by the
   naive application of equation L(n) (that is, setting k to 1 and o(1)
   to 0). These estimates of k are fairly stable over the numbers
   reported in the table.  The estimate is limited to a single
   significant digit of k because it expresses real uncertainties;
   however, the effect of additional digits would have make only tiny
   changes to the recommended key sizes.

   The factorers of RSA130 used about 1700 MYs, but they felt that this
   was unrealistically high for prediction purposes; by using more
   memory on their machines, they could have easily reduced the time to
   500 MYs.  Thus, the value used in preparing the table above was 500.
   This story does, however, underscore the difficulty in getting an
   accurate measure of effort.  This document takes the reported effort
   for factoring RSA155 as being the most accurate measure.

   As a result of examining the empirical data, it appears that the L(n)
   formula can be used with the o(1) term set to 0 and with k set to
   0.02 when talking about factoring numbers in the range of 100 to 200
   decimal digits.  The equation becomes:

      L(n) =  0.02 * e^(1.92 * cubrt(ln(n) * (ln(ln(n)))^2))

   To convert L(n) from simple math instructions to MYs, divide by
   3*10^13.  The equation for the number of MYs needed to factor an
   integer n then reduces to:

      MYs = 6 * 10^(-16) * e^(1.92 * cubrt(ln(n) * (ln(ln(n)))^2))

   With what confidence can this formula be used for predicting the
   difficulty of factoring slightly larger numbers?  The answer is that
   it should be a close upper bound, but each factorization effort is
   usually marked by some improvement in the algorithms or their
   implementations that makes the running time somewhat shorter than the
   formula would indicate.

2.3.  Pollard's rho method

   In Diffie-Hellman exchanges, there is a second attack, Pollard's rho
   method [POL78].  The algorithm relies on finding collisions between
   values computed in a large number space; its success rate is
   proportional to the square root of the size of the space.  Because of
   Pollard's rho method, the search space in a DH key exchange for the
   key (the exponent in a g^a term), must be twice as large as the

   symmetric key.  Therefore, to securely derive a key of K bits, an
   implementation must use an exponent with at least 2*K bits.  See
   [ODL99] for more detail.

   When the Diffie-Hellman key exchange is done using an elliptic curve
   method, the NFS methods are of no avail.  However, the collision
   method is still effective, and the need for an exponent (called a
   multiplier in EC's) with 2*K bits remains.  The modulus used for the
   computation can also be 2*K bits, and this will be substantially
   smaller than the modulus needed for modular exponentiation methods as
   the desired security level increases past 64 bits of brute-force
   attack resistance.

   One might ask, how can you compare the number of computer
   instructions really needed for a discrete logarithm attack to the
   number needed to search the keyspace of a cipher? In comparing the
   efforts, one should consider what a "basic operation" is.  For brute
   force search of the keyspace of a symmetric encryption algorithm like
   DES, the basic operation is the time to do a key setup and the time
   to do one encryption.  For discrete logs, the basic operation is a
   modular squaring.  The log of the ratio of these two operations can
   be used as a "normalizing factor" between the two kinds of
   computations.  However, even for very large moduli (16K bits), this
   factor amounts to only a few bits of extra effort.

2.4.  Limits of large memory and many machines

   Robert Silverman has examined the question of when it will be
   practical to factor RSA moduli larger than 512 bits.  His analysis is
   based not only on the theoretical number of operations, but it also
   includes expectations about the availability of actual machines for
   performing the work (this document is based only on theoretical
   number of operations).  He examines the question of whether or not we
   can expect there be enough machines, memory, and communication to
   factor a very large number.

   The best factoring methods need a lot of random access memory for
   collecting data relations (sieving) and a critical final step that
   does a row reduction on a large matrix.  The memory requirements are
   related to the size of the number being factored (or subjected to
   discrete logarithm solution).  Silverman [SILIEEE99] [SIL00] has
   argued that there is a practical limit to the number of machines and
   the amount of RAM that can be brought to bear on a single problem in
   the foreseeable future.  He sees two problems in attacking a 1024-bit
   RSA modulus: the machines doing the sieving will need 64-bit address
   spaces and the matrix row reduction machine will need several
   terabytes of memory. Silverman notes that very few 64-bit machines

   that have the 170 gigabytes of memory needed for sieving have been
   sold.  Nearly a billion such machines are necessary for the sieving
   in a reasonable amount of time (a year or two).

   Silverman's conclusion, based on the history of factoring efforts and
   Moore's Law, is that 1024-bit RSA moduli will not be factored until
   about 2037.  This implies a much longer lifetime to RSA keys than the
   theoretical analysis indicates.  He argues that predictions about how
   many machines and memory modules will be available can be with great
   confidence, based on Moore's Law extrapolations and the recent
   history of factoring efforts.

   One should give the practical considerations a great deal of weight,
   but in a risk analysis, the physical world is less predictable than
   trend graphs would indicate.  In considering how much trust to put
   into the inability of the computer industry to satisfy the voracious
   needs of factorers, one must have some insight into economic
   considerations that are more complicated than the mathematics of
   factoring.  The demand for computer memory is hard to predict because
   it is based on applications:  a "killer app" might come along any day
   and send the memory industry into a frenzy of sales.  The number of
   processors available on desktops may be limited by the number of
   desks, but very capable embedded systems account for more processor
   sales than desktops.  As embedded systems absorb networking
   functions, it is not unimaginable that millions of 64-bit processors
   with at least gigabytes of memory will pervade our environment.

   The bottom line on this is that the key length recommendations
   predicted by theory may be overly conservative, but they are what we
   have used for this document.  This question of machine availability
   is one that should be reconsidered in light of current technology on
   a regular basis.

2.5.  Special purpose machines

   In August of 2003, a design for a special-purpose "sieving machine"
   (TWIRL) surfaced [Shamir2003], and it substantially changed the cost
   estimates for factoring numbers up to 1024 bits in size.  By applying
   many high-speed VLSI components in parallel, such a machine might be
   able to carry out the sieving of 512-bit numbers in 10 minutes at a
   cost of $10K for the hardware.  A larger version could sieve a 1024-
   bit number in one year for a cost of $10M.  The work cites some
   advances in approaches to the row reduction step in concluding that
   the security of 1024-bit RSA moduli is doubtful.

   The estimates for the time and cost for factoring 512-bit and 1024-
   bit numbers correspond to a speed-up factor of about 2 million over
   what can be achieved with commodity processors of a few years ago.

3.  Compute Time for the Algorithms

   This section describes how long it takes to use the algorithms to
   perform key exchanges.  Again, it is important to consider the
   increased time it takes to exchange symmetric keys when increasing
   the length of public keys.  It is important to avoid choosing
   unfeasibly long public keys.

3.1.  Diffie-Hellman Key Exchange

   A Diffie-Hellman key exchange is done with a finite cyclic group G
   with a generator g and an exponent x.  As noted in the Pollard's rho
   method section, the exponent has twice as many bits as are needed for
   the final key.  Let the size of the group G be p, let the number of
   bits in the base 2 representation of p be j, and let the number of
   bits in the exponent be K.

   In doing the operations that result in a shared key, a generator is
   raised to a power.  The most efficient way to do this involves
   squaring a number K times and multiplying it several times along the
   way.  Each of the numbers has j/w computer words in it, where w is
   the number of bits in a computer word (today that will be 32 or 64
   bits).  A naive assumption is that you will need to do j squarings
   and j/2 multiplies; fortunately, an efficient implementation will
   need fewer (NB: for the remainder of this section, n represents j/w).

   A squaring operation does not need to use quite as many operations as
   a multiplication; a reasonable estimate is that squaring takes .6 the
   number of machine instructions of a multiply.  If one prepares a
   table ahead of time with several values of small integer powers of
   the generator g, then only about one fifth as many multiplies are
   needed as the naive formula suggests.  Therefore, one needs to do the
   work of approximately .8*K multiplies of n-by-n word numbers.
   Further, each multiply and squaring must be followed by a modular
   reduction, and a good assumption is that it is as hard to do a
   modular reduction as it is to do an n-by-n word multiply.  Thus, it
   takes K reductions for the squarings and .2*K reductions for the
   multiplies.  Summing this, the total effort for a Diffie-Hellman key
   exchange with K bit exponents and a modulus of n words is
   approximately 2*K n-by-n-word multiplies.

   For 32-bit processors, integers that use less than about 30 computer
   words in their representation require at least n^2 instructions for
   an n-by-n-word multiply.  Larger numbers will use less time, using
   Karatsuba multiplications, and they will scale as about n^(1.58) for
   larger n, but that is ignored for the current discussion.  Note that
   64-bit processors push the "Karatsuba cross-over" number out to even
   more bits.

   The basic result is: if you double the size of the Diffie-Hellman
   modular exponentiation group, you quadruple the number of operations
   needed for the computation.

3.1.1.  Diffie-Hellman with elliptic curve groups

   Note that the ratios for computation effort as a function of modulus
   size hold even if you are using an elliptic curve (EC) group for
   Diffie-Hellman.  However, for equivalent security, one can use
   smaller numbers in the case of elliptic curves.  Assume that someone
   has chosen an modular exponentiation group with an 2048 bit modulus
   as being an appropriate security measure for a Diffie-Hellman
   application and wants to determine what advantage there would be to
   using an EC group instead.  The calculation is relatively
   straightforward, if you assume that on the average, it is about 20
   times more effort to do a squaring or multiplication in an EC group
   than in a modular exponentiation group.  A rough estimate is that an
   EC group with equivalent security has about 200 bits in its
   representation.  Then, assuming that the time is dominated by n-by-n-
   word operations, the relative time is computed as:

      ((2048/200)^2)/20 ~= 5

   showing that an elliptic curve implementation should be five times as
   fast as a modular exponentiation implementation.

3.2.  RSA encryption and decryption

   Assume that an RSA public key uses a modulus with j bits; its factors
   are two numbers of about j/2 bits each.  The expected computation
   time for encryption and decryption are different.  As before, we
   denote the number of words in the machine representation of the
   modulus by the symbol n.

   Most implementations of RSA use a small exponent for encryption.  An
   encryption may involve as few as 16 squarings and one multiplication,
   using n-by-n-word operations.  Each operation must be followed by a
   modular reduction, and therefore the time complexity is about 16*(.6
   + 1) + 1 + 1 ~= 28 n-by-n-word multiplies.

   RSA decryption must use an exponent that has as many bits as the
   modulus, j.  However, the Chinese Remainder Theorem applies, and all
   the computations can be done with a modulus of only n/2 words and an
   exponent of only j/2 bits.  The computation must be done twice, once
   for each factor.  The effort is equivalent to  2*(j/2) (n/2 by n/2)-
   word multiplies.  Because multiplying numbers with n/2 words is only
   1/4 as difficult as multiplying numbers with n words, the equivalent
   effort for RSA decryption is j/4 n-by-n-word multiplies.

   If you double the size of the modulus for RSA, the n-by-n multiplies
   will take four times as long.  Further, the decryption time doubles
   because the exponent is larger.  The overall scaling cost is a factor
   of 4 for encryption, a factor of 8 for decryption.

3.3.  Real-world examples

   To make these numbers more real, here are a few examples of software
   implementations run on hardware that was current as of a few years
   before the publication of this document.  The examples are included
   to show rough estimates of reasonable implementations; they are not
   benchmarks.  As with all software, the performance will depend on the
   exact details of specialization of the code to the problem and the
   specific hardware.

   The best time informally reported for a 1024-bit modular
   exponentiation (the decryption side of 2048-bit RSA), is 0.9 ms
   (about 450,000 CPU cycles) on a 500 MHz Itanium processor.  This
   shows that newer processors are not losing ground on big number
   operations; the number of instructions is less than a 32-bit
   processor uses for a 256-bit modular exponentiation.

   For less advanced processors timing, the following two tables
   (computed by Tero Monenen at SSH Communications) for modular
   exponentiation, such as would be done in a Diffie-Hellman key

   Celeron 400 MHz; compiled with GNU C compiler, optimized, some
   platform specific coding optimizations:

      group  modulus   exponent    time
      type    size       size
       mod    768       ~150       18 msec
       mod   1024       ~160       32 msec
       mod   1536       ~180       82 msec
       ecn    155       ~150       35 msec
       ecn    185       ~200       56 msec

   The group type is from [RFC2409] and is either modular exponentiation
   ("mod") or elliptic curve ("ecn").  All sizes here and in subsequent
   tables are in bits.

   Alpha 500 MHz compiled with Digital's C compiler, optimized, no
   platform specific code:

      group  modulus    exponent       time
      type    size       size
       mod    768       ~150          12 msec
       mod   1024       ~160          24 msec
       mod   1536       ~180          59 msec
       ecn    155       ~150          20 msec
       ecn    185       ~200          27 msec

   The following two tables (computed by Eric Young) were originally for
   RSA signing operations, using the Chinese Remainder representation.
   For ease of understanding, the parameters are presented here to show
   the interior calculations, i.e., the size of the modulus and exponent
   used by the software.

   Dual Pentium II-350:

       equiv      equiv         equiv
      modulus    exponent       time
       size        size
        256        256         1.5 ms
        512        512         8.6 ms
       1024       1024        55.4 ms
       2048       2048       387   ms

   Alpha 264 600mhz:

       equiv       equiv        equiv
      modulus     exponent      time
       size        size
       512         512         1.4 ms

   Recent chips that accelerate exponentiation can perform 1024-bit
   exponentiations (1024 bit modulus, 1024 bit exponent) in about 3
   milliseconds or less.

4.  Equivalences of Key Sizes

   In order to determine how strong a public key is needed to protect a
   particular symmetric key, you first need to determine how much effort
   is needed to break the symmetric key.  Many Internet security
   protocols require the use of TripleDES for strong symmetric
   encryption, and it is expected that the Advanced Encryption Standard
   (AES) will be adopted on the Internet in the coming years.
   Therefore, these two algorithms are discussed here.  In this section,
   for illustrative purposes, we will implicitly assume that the system

   security requirement is 112 bits; this doesn't mean that 112 bits is
   recommended.  In fact, 112 bits is arguably too strong for any
   practical purpose.  It is used for illustration simply because that
   is the upper bound on the strength of TripleDES.

   If one could simply determine the number of MYs it takes to break
   TripleDES, the task of computing the public key size of equivalent
   strength would be easy.  Unfortunately, that isn't the case here
   because there are many examples of DES-specific hardware that encrypt
   faster than DES in software on a standard CPU.  Instead, one must
   determine the equivalent cost for a system to break TripleDES and a
   system to break the public key protecting a TripleDES key.

   In 1998, the Electronic Frontier Foundation (EFF) built a DES-
   cracking machine [GIL98] for US$130,000 that could test about 1e11
   DES keys per second (additional money was spent on the machine's
   design).  The machine's builders fully admit that the machine is not
   well optimized, and it is estimated that ten times the amount of
   money could probably create a machine about 50 times as fast.
   Assuming more optimization by guessing that a system to test
   TripleDES keys runs about as fast as a system to test DES keys, so
   approximately US$1 million might test 5e12 TripleDES keys per second.

   In case your adversaries are much richer than EFF, you may want to
   assume that they have US$1 trillion, enough to test 5e18 keys per
   second.  An exhaustive search of the effective TripleDES space of
   2^112 keys with this quite expensive system would take about 1e15
   seconds or about 33 million years.  (Note that such a system would
   also need 2^60 bytes of RAM [MH81], which is considered free in this
   calculation).  This seems a needlessly conservative value.  However,
   if computer logic speeds continue to increase in accordance with
   Moore's Law (doubling in speed every 1.5 years), then one might
   expect that in about 50 years, the computation could be completed in
   only one year.  For the purposes of illustration, this 50 year
   resistance against a trillionaire is assumed to be the minimum
   security requirement for a set of applications.

   If 112 bits of attack resistance is the system security requirement,
   then the key exchange system for TripleDES should have equivalent
   difficulty; that is to say, if the attacker has US$1 trillion, you
   want him to spend all his money to buy hardware today and to know
   that he will "crack" the key exchange in not less than 33 million
   years.  (Obviously, a rational attacker would wait for about 45 years
   before actually spending the money, because he could then get much
   better hardware, but all attackers benefit from this sort of wait

   It is estimated that a typical PC CPU of just a few years ago can
   generate over 500 MIPs and could be purchased for about US$100 in
   quantity; thus you get more than 5 MIPs/US$.  Again, this number
   doubles about every 18 months.  For one trillion US dollars, an
   attacker can get 5e12 MIP years of computer instructions on that
   recent-vintage hardware.  This figure is used in the following
   estimates of equivalent costs for breaking key exchange systems.

4.1.  Key equivalence against special purpose brute force hardware

   If the trillionaire attacker is to use conventional CPU's to "crack"
   a key exchange for a 112 bit key in the same time that the special
   purpose machine is spending on brute force search for the symmetric
   key, the key exchange system must use an appropriately large modulus.
   Assume that the trillionaire performs 5e12 MIPs of instructions per
   year.  Use the following equation to estimate the modulus size to use
   with RSA encryption or DH key exchange:

      5*10^33 = (6*10^-16)*e^(1.92*cubrt(ln(n)*(ln(ln(n)))^2))

   Solving this approximately for n yields:

      n = 10^(625) = 2^(2077)

   Thus, assuming similar logic speeds and the current efficiency of the
   number field sieve, moduli with about 2100 bits will have about the
   same resistance against attack as an 112-bit TripleDES key.  This
   indicates that RSA public key encryption should use a modulus with
   around 2100 bits; for a Diffie-Hellman key exchange, one could use a
   slightly smaller modulus, but it is not a significant difference.

4.2 Key equivalence against conventional CPU brute force attack

   An alternative way of estimating this assumes that the attacker has a
   less challenging requirement: he must only "crack" the key exchange
   in less time than a brute force key search against the symmetric key
   would take with general purpose computers.  This is an "apples-to-
   apples" comparison, because it assumes that the attacker needs only
   to have computation donated to his effort, not built from a personal
   or national fortune.  The public key modulus will be larger than the
   one in 4.1, because the symmetric key is going to be viable for a
   longer period of time.

   Assume that the number of CPU instructions to encrypt a block of
   material using TripleDES is 300.  The estimated number of computer
   instructions to break 112 bit TripleDES key:

      300 * 2^112
      = 1.6 * 10^(36)
      = .02*e^(1.92*cubrt(ln(n)*(ln(ln(n)))^2))

   Solving this approximately for n yields:

      n = 10^(734) = 2^(2439)

   Thus, for general purpose CPU attacks, you can assume that moduli
   with about 2400 bits will have about the same strength against attack
   as an 112-bit TripleDES key.  This indicates that RSA public key
   encryption should use a modulus with around 2400 bits; for a Diffie-
   Hellman key exchange, one could use a slightly smaller modulus, but
   it not a significant difference.

   Note that some authors assume that the algorithms underlying the
   number field sieve will continue to get better over time.  These
   authors recommend an even larger modulus, over 4000 bits, for
   protecting a 112-bit symmetric key for 50 years.  This points out the
   difficulty of long-term cryptographic security: it is all but
   impossible to predict progress in mathematics and physics over such a
   long period of time.

4.3.  A One Year Attack: 80 bits of strength

   Assuming a trillionaire spends his money today to buy hardware, what
   size key exchange numbers could he "crack" in one year?  He can
   perform 5*e12 MYs of instructions, or

      3*10^13 * 5*10^12 = .02*e^(1.92*cubrt(ln(n)*(ln(ln(n)))^2))

   Solving for an approximation of n yields

      n = 10^(360) = 2^(1195)

   This is about as many operations as it would take to crack an 80-bit
   symmetric key by brute force.

   Thus, for protecting data that has a secrecy requirement of one year
   against an incredibly rich attacker, a key exchange modulus with
   about 1200 bits protecting an 80-bit symmetric key is safe even
   against a nation's resources.

4.4.  Key equivalence for other ciphers

   Extending this logic to the AES is straightforward.  For purposes of
   estimation for key searching, one can think of the 128-bit AES as
   being at least 16 bits stronger than TripleDES but about three times

   as fast.  The time and cost for a brute force attack is approximately
   2^(16) more than for TripleDES, and thus, under the assumption that
   128 bits of strength is the desired security goal, the recommended
   key exchange modulus size is about 700 bits longer.

   If it is possible to design hardware for AES cracking that is
   considerably more efficient than hardware for DES cracking, then
   (again under the assumption that the key exchange strength must match
   the brute force effort) the moduli for protecting the key exchange
   can be made smaller.  However, the existence of such designs is only
   a matter of speculation at this early moment in the AES lifetime.

   The AES ciphers have key sizes of 128 bits up to 256 bits.  Should a
   prudent minimum security requirement, and thus the key exchange
   moduli, have similar strengths? The answer to this depends on whether
   or not one expect Moore's Law to continue unabated.  If it continues,
   one would expect 128 bit keys to be safe for about 60 years, and 256
   bit keys would be safe for another 400 years beyond that, far beyond
   any imaginable security requirement.  But such progress is difficult
   to predict, as it exceeds the physical capabilities of today's
   devices and would imply the existence of logic technologies that are
   unknown or infeasible today.  Quantum computing is a candidate, but
   too little is known today to make confident predictions about its
   applicability to cryptography (which itself might change over the
   next 100 years!).

   If Moore's Law does not continue to hold, if no new computational
   paradigms emerge, then keys of over 100 bits in length might well be
   safe "forever".  Note, however that others have come up with
   estimates based on assumptions of new computational paradigms
   emerging.  For example, Lenstra and Verheul's web-based paper
   "Selecting Cryptographic Key Sizes" chooses a more conservative
   analysis than the one in this document.

4.5.  Hash functions for deriving symmetric keys from public key

   The Diffie-Hellman algorithm results in a key that is hundreds or
   thousands of bits long, but ciphers need far fewer bits than that.
   How can one distill a long key down to a short one without losing

   Cryptographic one-way hash functions are the building blocks for
   this, and so long as they use all of the Diffie-Hellman key to derive
   each block of the symmetric key, they produce keys with sufficient

   The usual recommendation is to use a good one-way hash function
   applied to he base material (the result of the key exchange) and to
   use a subset of the hash function output for the key.  However, if
   the desired key length is greater than the output of the hash
   function, one might wonder how to reconcile the two.

   The step of deriving extra key bits must satisfy these requirements:

   -  The bits must not reveal any information about the key exchange

   -  The bits must not be correlated with each other

   -  The bits must depend on all the bits of the key exchange secret

   Any good cryptographic hash function satisfies these three
   requirements.  Note that the number of bits of output of the hash
   function is not specified.  That is because even a hash function with
   a very short output can be iterated to produce more uncorrelated bits
   with just a little bit of care.

   For example, SHA-1 has 160 bits of output.  For deriving a key of
   attack resistance of 160 bits or less, SHA(DHkey) produces a good
   symmetric key.

   Suppose one wants a key with attack resistance of 160 bits, but it is
   to be used with a cipher that uses 192 bit keys.  One can iterate
   SHA-1 as follows:

      Bits 1-160   of the symmetric key = K1 = SHA(DHkey | 0x00)
                   (that is, concatenate a single octet of value 0x00 to
                   the right side of the DHkey, and then hash)
      Bits 161-192 of the symmetric key = K2 =
                   select_32_bits(SHA(K1 | 0x01))

   But what if one wants 192 bits of strength for the cipher?  Then the
   appropriate calculation is

      Bits 1-160   of the symmetric key = SHA(0x00 | DHkey)
      Bits 161-192 of the symmetric key =
                   select_32_bits(SHA(0x01 | DHkey))

   (Note that in the description above, instead of concatenating a full
   octet, concatenating a single bit would also be sufficient.)

   The important distinction is that in the second case, the DH key is
   used for each part of the symmetric key.  This assures that entropy
   of the DH key is not lost by iteration of the hash function over the
   same bits.

   From an efficiency point of view, if the symmetric key must have a
   great deal of entropy, it is probably best to use a cryptographic
   hash function with a large output block (192 bits or more), rather
   than iterating a smaller one.

   Newer hash algorithms with longer output (such as SHA-256, SHA-384,
   and SHA-512) can be used with the same level of security as the
   stretching algorithm described above.

4.6.  Importance of randomness

   Some of the calculations described in this document require random
   inputs; for example, the secret Diffie-Hellman exponents must be
   chosen based on n truly random bits (where n is the system security
   requirement).  The number of truly random bits is extremely important
   to determining the strength of the output of the calculations.  Using
   truly random numbers is often overlooked, and many security
   applications have been significantly weakened by using insufficient
   random inputs.  A much more complete description of the importance of
   random numbers can be found in [ECS].

5.  Conclusion

   In this table it is assumed that attackers use general purpose
   computers, that the hardware is purchased in the year 2000, and that
   mathematical knowledge relevant to the problem remains the same as
   today.  This is an pure "apples-to-apples" comparison demonstrating
   how the time for a key exchange scales with respect to the strength
   requirement.  The subgroup size for DSA is included, if that is being
   used for supporting authentication as part of the protocol; the DSA
   modulus must be as long as the DH modulus, but the size of the "q"
   subgroup is also relevant.

   | System      |           |              |              |
   | requirement | Symmetric | RSA or DH    | DSA subgroup |
   | for attack  | key size  | modulus size | size         |
   | resistance  | (bits)    | (bits)       | (bits)       |
   | (bits)      |           |              |              |
   |     70      |     70    |      947     |     129      |
   |     80      |     80    |     1228     |     148      |
   |     90      |     90    |     1553     |     167      |
   |    100      |    100    |     1926     |     186      |
   |    150      |    150    |     4575     |     284      |
   |    200      |    200    |     8719     |     383      |
   |    250      |    250    |    14596     |     482      |

5.1.  TWIRL Correction

   If the TWIRL machine becomes a reality, and if there are advances in
   parallelism for row reduction in factoring, then conservative
   estimates would subtract about 11 bits from the system security
   column of the table.  Thus, in order to get 89 bits of security, one
   would need an RSA modulus of about 1900 bits.

6.  Security Considerations

   The equations and values given in this document are meant to be as
   accurate as possible, based on the state of the art in general
   purpose computers at the time that this document is being written.
   No predictions can be completely accurate, and the formulas given
   here are not meant to be definitive statements of fact about
   cryptographic strengths.  For example, some of the empirical results
   used in calibrating the formulas in this document are probably not
   completely accurate, and this inaccuracy affects the estimates.  It
   is the authors' hope that the numbers presented here vary from real
   world experience as little as possible.

7.  References

7.1.  Informational References

   [DL]        Dodson, B. and A. K. Lenstra, NFS with four large primes:
               an explosive experiment, Proceedings Crypto 95, Lecture
               Notes in Comput. Sci. 963, (1995) 372-385.

   [ECS]       Eastlake, D., Crocker, S. and J. Schiller, "Randomness
               Recommendations for Security", RFC 1750, December 1994.

   [GIL98]     Cracking DES: Secrets of Encryption Research, Wiretap
               Politics & Chip Design , Electronic Frontier Foundation,
               John Gilmore (Ed.), 272 pages, May 1998, O'Reilly &
               Associates; ISBN: 1565925203

   [GOR93]     Gordon, D., "Discrete logarithms in GF(p) using the
               number field sieve", SIAM Journal on Discrete
               Mathematics, 6 (1993), 124-138.

   [LEN93]     Lenstra, A. K. and H. W. Lenstra, Jr. (eds), The
               development of the number field sieve, Lecture Notes in
               Math, 1554, Springer Verlag, Berlin, 1993.

   [MH81]      Merkle, R.C., and Hellman, M., "On the Security of
               Multiple Encryption", Communications of the ACM, v. 24 n.
               7, 1981, pp. 465-467.

   [ODL95]     RSA Labs Cryptobytes, Volume 1, No. 2 - Summer 1995; The
               Future of Integer Factorization, A. M. Odlyzko

   [ODL99]     A. M. Odlyzko, Discrete logarithms: The past and the
               future, Designs, Codes, and Cryptography (1999).

   [POL78]     J. Pollard, "Monte Carlo methods for index computation
               mod p", Mathematics of Computation, 32 (1978), 918-924.

   [RFC2409]   Harkins, D. and D. Carrel, "The Internet Key Exchange
               (IKE)", RFC 2409, November 1998.

   [SCH95]     R. Schroeppel, et al., Fast Key Exchange With Elliptic
               Curve Systems, In Don Coppersmith, editor, Advances in
               Cryptology -- CRYPTO 31 August 1995. Springer-Verlag

   [SHAMIR03]  Shamir, Adi and Eran Tromer, "Factoring Large Numbers
               with the TWIRL Device", Advances in Cryptology - CRYPTO
               2003, Springer, Lecture Notes in Computer Science 2729.

   [SIL00]     R. D. Silverman, RSA Laboratories Bulletin, Number 13 -
               April 2000, A Cost-Based Security Analysis of Symmetric
               and Asymmetric Key Lengths

   [SILIEEE99] R. D. Silverman, "The Mythical MIPS Year", IEEE Computer,
               August 1999.

8. Authors' Addresses

   Hilarie Orman
   Purple Streak Development
   500 S. Maple Dr.
   Salem, UT 84653

   EMail: hilarie@purplestreak.com and ho@alum.mit.edu

   Paul Hoffman
   VPN Consortium
   127 Segre Place
   Santa Cruz, CA  95060 USA

   EMail: paul.hoffman@vpnc.org

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