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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.

Abstract

 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
 usage.
 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
 exchange.

Orman & Hoffman Best Current Practice [Page 1] RFC 3766 Determining Strengths for Public Keys April 2004

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

Orman & Hoffman Best Current Practice [Page 2] RFC 3766 Determining Strengths for Public Keys April 2004

 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
 problems.
 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".

Orman & Hoffman Best Current Practice [Page 3] RFC 3766 Determining Strengths for Public Keys April 2004

    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].

Orman & Hoffman Best Current Practice [Page 4] RFC 3766 Determining Strengths for Public Keys April 2004

 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

Orman & Hoffman Best Current Practice [Page 5] RFC 3766 Determining Strengths for Public Keys April 2004

 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
                  digits
    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
 [DL].

Orman & Hoffman Best Current Practice [Page 6] RFC 3766 Determining Strengths for Public Keys April 2004

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

Orman & Hoffman Best Current Practice [Page 7] RFC 3766 Determining Strengths for Public Keys April 2004

 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

Orman & Hoffman Best Current Practice [Page 8] RFC 3766 Determining Strengths for Public Keys April 2004

 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.

Orman & Hoffman Best Current Practice [Page 9] RFC 3766 Determining Strengths for Public Keys April 2004

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.

Orman & Hoffman Best Current Practice [Page 10] RFC 3766 Determining Strengths for Public Keys April 2004

 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.

Orman & Hoffman Best Current Practice [Page 11] RFC 3766 Determining Strengths for Public Keys April 2004

 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
 exchange.
 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.

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 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

Orman & Hoffman Best Current Practice [Page 13] RFC 3766 Determining Strengths for Public Keys April 2004

 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
 equally.)

Orman & Hoffman Best Current Practice [Page 14] RFC 3766 Determining Strengths for Public Keys April 2004

 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:

Orman & Hoffman Best Current Practice [Page 15] RFC 3766 Determining Strengths for Public Keys April 2004

    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

Orman & Hoffman Best Current Practice [Page 16] RFC 3766 Determining Strengths for Public Keys April 2004

 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

    algorithms
 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
 strength?
 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
 strength.

Orman & Hoffman Best Current Practice [Page 17] RFC 3766 Determining Strengths for Public Keys April 2004

 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:
  1. The bits must not reveal any information about the key exchange

secret

  1. The bits must not be correlated with each other
  1. 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.)

Orman & Hoffman Best Current Practice [Page 18] RFC 3766 Determining Strengths for Public Keys April 2004

 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.

Orman & Hoffman Best Current Practice [Page 19] RFC 3766 Determining Strengths for Public Keys April 2004

 +-------------+-----------+--------------+--------------+
 | 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.

Orman & Hoffman Best Current Practice [Page 20] RFC 3766 Determining Strengths for Public Keys April 2004

 [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.

Orman & Hoffman Best Current Practice [Page 21] RFC 3766 Determining Strengths for Public Keys April 2004

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

Orman & Hoffman Best Current Practice [Page 22] RFC 3766 Determining Strengths for Public Keys April 2004

9. Full Copyright Statement

 Copyright (C) The Internet Society (2004).  This document is subject
 to the rights, licenses and restrictions contained in BCP 78, and
 except as set forth therein, the authors retain all their rights.
 This document and the information contained herein are provided on an
 "AS IS" basis and THE CONTRIBUTOR, THE ORGANIZATION HE/SHE
 REPRESENTS OR IS SPONSORED BY (IF ANY), THE INTERNET SOCIETY AND THE
 INTERNET ENGINEERING TASK FORCE DISCLAIM ALL WARRANTIES, EXPRESS OR
 IMPLIED, INCLUDING BUT NOT LIMITED TO ANY WARRANTY THAT THE USE OF
 THE INFORMATION HEREIN WILL NOT INFRINGE ANY RIGHTS OR ANY IMPLIED
 WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE.

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 Intellectual Property Rights or other rights that might be claimed
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 under such rights might or might not be available; nor does it
 represent that it has made any independent effort to identify any
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 Copies of IPR disclosures made to the IETF Secretariat and any
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 The IETF invites any interested party to bring to its attention
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Acknowledgement

 Funding for the RFC Editor function is currently provided by the
 Internet Society.

Orman & Hoffman Best Current Practice [Page 23]

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