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Run idnits with the --verbose option for more detailed information about the items above. -------------------------------------------------------------------------------- 1 Internet Draft Hilarie Orman 2 draft-orman-public-key-lengths-02.txt Novell, Inc. 3 March 19, 2001 Paul Hoffman 4 Expires in six months IMC & VPNC 6 Determining Strengths For Public Keys Used 7 For Exchanging Symmetric Keys 9 Status of this memo 11 This document is an Internet-Draft and is in full conformance with all 12 provisions of Section 10 of RFC2026. 14 Internet-Drafts are working documents of the Internet Engineering Task 15 Force (IETF), its areas, and its working groups. Note that other 16 groups may also distribute working documents as Internet-Drafts. 18 Internet-Drafts are draft documents valid for a maximum of six months 19 and may be updated, replaced, or obsoleted by other documents at any 20 time. It is inappropriate to use Internet-Drafts as reference 21 material or to cite them other than as "work in progress." 23 The list of current Internet-Drafts can be accessed at 24 http://www.ietf.org/ietf/1id-abstracts.txt 26 The list of Internet-Draft Shadow Directories can be accessed at 27 http://www.ietf.org/shadow.html. 29 Abstract 31 Implementors of systems that use public key cryptography to exchange 32 symmetric keys need to make the public keys resistant to some 33 predetermined level of attack. That level of attack resistance is the 34 strength of the system, and the symmetric keys that are exchanged must 35 be at least as strong as the system strength requirements. The three 36 quantities, system strength, symmetric key strength, and public key 37 strength, must be consistently matched for any network protocol usage. 39 While it is fairly easy to express the system strength requirements in 40 terms of a symmetric key length and to choose a cipher that has a key 41 length equal to or exceeding that requirement, it is harder to choose a 42 public key that has a cryptographic strength meeting a symmetric key 43 strength requirement. This document explains how to determine the 44 length of an asymmetric key as a function of the length of a symmetric 45 key. Some rules of thumb for estimating equivalent resistance to 46 large-scale attacks on various algorithms are given. The document also 47 addresses how changing the sizes of the underlying large integers 48 (moduli, group sizes, exponents, and so on) changes the time to use the 49 algorithms for key exchange. 51 1. Model of Protecting Symmetric Keys with Public Keys 53 Many books on cryptography and security explain the need to exchange 54 symmetric keys in public as well as the many algorithms that are used 55 for this purpose. However, few of these discussions explain how the 56 strengths of the public keys and the symmetric keys are related. 58 To understand this, picture a house with a strong lock on the front 59 door. Next to the front door is a small lockbox that contains the key 60 to the front door. A would-be burglar who wants to break into the house 61 through the front door has two options: attack the lock on the front 62 door, or attack the lock on the lockbox in order to retrieve the key. 63 Clearly, the burglar is better off attacking the weaker of the two 64 locks. The homeowner in this situation must make sure that adding the 65 second entry option (the lockbox containing the front door key) is at 66 least as strong as the lock on the front door in order not to make the 67 burglar's job easier. 69 An implementor designing a system for exchanging symmetric keys using 70 public key cryptography must make a similar decision. Assume that an 71 attacker wants to learn the contents of a message that is encrypted with a 72 symmetric key, and that the symmetric key was exchanged between the 73 sender and recipient using public key cryptography. The attacker has 74 two options to recover the message: a brute-force attempt to determine 75 the symmetric key by repeated guessing, or mathematical determination 76 of the private key used as the key exchange key. A smart attacker will 77 work on the easier of these two problems. 79 A simple-minded answer to the implementor's problem is to be sure that 80 the key exchange system is always significantly stronger than the 81 symmetric key; this can be done by choosing a very long public key. 82 Such a design is usually not a good idea because the key exchanges 83 become much more expensive in terms of processing time as the length of 84 the public keys go up. Thus, the implementor is faced with the task 85 of trying to match the difficulty of an attack on the symmetric key 86 with the difficulty of an attack on the public key encryption. This 87 analysis is not unnecessary if the key exchange can be performed with 88 extreme security for almost no cost in terms of elapsed time or CPU 89 effort; unfortunately, this is no the case for public key methods today. 91 A third consideration is the minimum security requirement of the user. 92 Assume the user is encrypting with CAST-128 and requires a symmetric 93 key with a resistance time against brute-force attack of 20 years. He 94 might start off by choosing a key with 86 random bits, and then use a 95 one-way function such as SHA-1 to "boost" that to a block of 160 bits, 96 and then take 128 of those bits as the key for CAST-128. In such a 97 case, the key exchange algorithm need only match the difficulty of 86 98 bits, not 128 bits. The selection rule is: 100 1. Determine the number of symmetric key bits matching the security 101 requirement of the application (n). 103 2. Choose a symmetric cipher that has a key with at least n bits, and a 104 cryptanalytic strength of at least that much. 106 3. Choose a key exchange algorithm with a resistance to attack of at 107 least n bits. 109 A fourth consideration might be the public key authentication method 110 used to establish the identity of a user. This might be an RSA digital 111 signature or a DSA digital signature. If the modulus for the 112 authentication method isn't large enough, then the entire basis for 113 trusting the communication might fall apart. The following step 114 is thus added: 116 4. Choose an authentication algorithm with a resistance to attack of at 117 least n bits. 119 1.2 The key exchange algorithms 121 The Diffie-Hellman method uses a group, a generator, and exponents. In 122 today's Internet standards, the group operation is based on modular 123 multiplication. Here, the group is defined by the multiplicative group 124 of an integer, typically a prime p = 2q + 1, where q is a prime, and 125 the arithmetic is done modulo p; the generator (which is often simply 126 2) is denoted by g. 128 In Diffie-Hellman, Alice and Bob first agree (in public or in private) 129 on the values for g and p. Alice chooses a secret large random integer 130 (a), and Bob chooses a secret random large integer (b). Alice sends Bob 131 A, which is g^a mod p; Bob sends Alice B, which is g^b mod p. Next, 132 Alice computes B^a mod p, and Bob computes A^b mod p. These two numbers 133 are equal, and the participants use a simple function of this number as 134 the symmetric key k. 136 Note that Diffie-Hellman key exchange can be done over different kinds 137 of group representations. For instance, elliptic curves defined over 138 finite fields are a particularly efficient way to compute the key 139 exchange [SCH95]. 141 For RSA key exchange, assume that Bob has a public key (m) which is 142 equal to p*q, where p and q are two secret prime numbers, and an 143 encryption exponent e, and a decryption exponent d. For the key 144 exchange, Alice sends Bob E = k^e mod m, where k is the secret 145 symmetric key being exchanged. Bob recovers k by computing E^d mod m, 146 and the two parties use k as their symmetric key. While Bob's 147 encryption exponent e can be quite small (e.g., 17 bits), his 148 decryption exponent d will have as many bits in it as m does. 150 2. Determining the Effort to Factor 152 The RSA public key encryption method is immune to brute force guessing 153 attacks because the modulus will have at least 512 bits. The Diffie- 154 Hellman exchange is also secure against guessing because the exponents 155 will have at least twice as many bits as the symmetric keys that will 156 be derived from them. However, both methods are susceptible to 157 mathematical attacks that determine the structure of the public keys. 159 Factoring an RSA modulus will result in complete compromise of the 160 security of the private key. Solving the discrete logarithm problem for 161 a Diffie-Hellman modular exponentiation system will similarly destroy 162 the security of all key exchanges using the particular modulus. This 163 document assumes that the difficulty of solving the discrete logarithm 164 problem is equivalent to the difficulty of factoring numbers that are 165 the same size as the modulus. In fact, it is slightly harder because it 166 requires more memory; based on empirical evidence so far, the ratio of 167 difficulty is at least 20, possibly as high as 64. Solving either 168 problem requires a great deal of memory for the last stage of the 169 algorithm, the matrix reduction step. Whether or not this memory 170 requirement will be the limiting factor in solving larger integer 171 problems remains to be seen. At the current time it is not, and advances 172 in parallel matrix algorithms are expected to mitigate the memory 173 requirements in the near future. 175 Nearly all cryptographers consider the number field sieve (NFS) [GOR93] 176 [LEN93] the best method today for solving the discrete logarithm 177 problem. The formula for estimating the number of simple arithmetic 178 operations needed to factor an integer, n, using the NFS method is: 180 L(n) = k * e^((1.92 + o(1)) * cubrt(ln(n) * (ln(ln(n)))^2)) 182 Many people prefer to discuss the number of MIPS years (MYs) that are 183 needed for large operations such as the number field sieve. For such an 184 estimation, an operation in the L(n) formula is one computer 185 instruction. Empirical evidence indicates that 4 or 5 instructions 186 might be a closer match, but this is a minor factor and this document 187 sticks with one operation/one instruction for this discussion. 189 2.1 Choosing parameters for the equation 191 The expression above has two parameters that must be estimated by 192 empirical means: k and o(1). Different authors take different 193 approaches to choosing the parameters: 195 - Some authors assume that k is 1 and o(1) is 0. This is reasonably 196 valid if the expression is only used for estimating relative effort 197 (instead of actual effort) and one assumes that the o(1) term is very 198 small over the range of the numbers that are to be factored. 200 - Other authors implicitly assume that o(1) is small and roughly 201 constant and thus its value can be folded into k; they then estimate k 202 from reported amounts of effort spent factoring large integers in 203 tests. 205 This document uses the second approach. 207 Sample values from recent work with the number field sieve, 208 collected by RSA Labs, include: 210 Test name Number of Number of MYs of effort 211 decimal bits 212 digits 213 RSA130 130 430 500 214 RSA140 140 460 2000 215 RSA155 155 512 8000 217 There are no precise measurements of the amount of time used for these 218 factorizations. In most factorization tests, hundreds or thousands of 219 computers are used over a period of several months, but the number of 220 their cycles were used for the factoring project, the precise 221 distribution of processor types, speeds, and so on are not usually 222 reported. However, in all cases, the amount of effort used was far less 223 than the L(n) formula would predict if k was 1 and o(1) was 0. 225 2.2 Choosing k from empirical reports 227 By solving for k from the empirical reports, it appears that k is 228 approximately 0.02. This means that the "effective key strength" of the 229 RSA algorithm is about 5.5 bits less than is implied by the naive 230 application of equation L(n) (that is, setting k to 1 and o(1) to 0). 231 These estimates of k are fairly stable over the numbers reported in the 232 table. The estimate is limited to a single significant digit of k 233 because it expresses real uncertainties; however, the effect of 234 additional digits would have make only tiny changes to the recommended 235 key sizes. 237 The factorers of RSA130 used about 1700 MYs, but they felt that this 238 was unrealistically high for prediction purposes; by using more memory 239 on their machines, they could have easily reduced the time to 500 MYs. 240 Thus, the value used in preparing the table above was 500. This story 241 does, however, underscore the difficulty in getting an accurate measure 242 of effort. This document takes the reported effort for factoring RSA155 243 as being the most accurate measure. 245 As a result of examining the empirical data, it appears that the L(n) 246 formula can be used with the o(1) term set to 0 and with k set to 0.02 247 when talking about factoring numbers in the range of 100 to 200 decimal 248 digits. The equation becomes: 250 L(n) = 0.02 * e^(1.92 * cubrt(ln(n) * (ln(ln(n)))^2)) 252 To convert L(n) from simple math instructions to MYs, divide by 253 3*10^13. The equation for the number of MYs needed to factor an integer 254 n then reduces to: 256 MYs = 6 * 10^(-16) * e^(1.92 * cubrt(ln(n) * (ln(ln(n)))^2)) 258 With what confidence can this formula be used for predicting the 259 difficulty of factoring slightly larger numbers? The answer is that it 260 should be a close upper bound, but each factorization effort is usually 261 marked by some improvement in the algorithms or their implementations 262 that makes the running time somewhat shorter than the formula would 263 indicate. 265 2.3 Pollard's rho method 267 In Diffie-Hellman exchanges, there is a second attack, Pollard's rho 268 method [POL78]. The algorithm relies on finding collisions between 269 values computed in a large number space; its success rate is 270 proportional to the square root of the size of the space. Because of 271 Pollard's rho method, the search space in a DH key exchange for the key 272 (the exponent in a g^a term), must be twice as large as the symmetric 273 key. Therefore, to securely derive a key of K bits, an implementation 274 must use an exponent with at least 2*K bits. 276 When the Diffie-Hellman key exchange is done using an elliptic curve 277 method, the NFS methods are of no avail. However, the collision 278 method is still effective, and the need for an exponent (called a 279 multiplier in EC's) with 2*K bits remains. However, the modulus 280 used for the computation can also be 2*K bits, and this will be 281 substantially smaller than the modulus needed for modular exponentiation 282 methods as the desired security level increases past 64 bits of 283 brute-force attack resistance. 285 2.4 Limits of large memory and many machines 287 Robert Silverman has examined the question of when it will be 288 practical to factor RSA moduli larger than 512 bits. His is based not 289 just on the theoretical number of operations that underlies this 290 document, but it includes expectations about the availability of 291 actual machines for performing the work. He examine the question of 292 whether or not we can expect there be enough machines, memory, and 293 communication to factor a very large number. 295 The best factoring methods need a lot of random access memory for 296 collecting data relations (sieving) and a critical final step that 297 does a row reduction on a large matrix. The memory requirements are 298 related to the size of the number being factored (or subjected to 299 discrete logarithm solution). Silverman [Silv2000] has argued 300 that there is a practical limit to the number of machines and the 301 amount of RAM that be brought to bear on a single problem in the 302 foreseeable future. He sees two problems in attacking a 1024-bit RSA 303 modulus: the machines doing the sieving will need 64-bit address 304 spaces and the matrix row reduction machine will need 6 terabytes of 305 memory. Silverman notes that very few 64-bit machines have been sold, 306 and none of those machines have the 170 gigabytes of memory needed for 307 sieving. Nearly a billion such machines are necessary for the sieving 308 in a reasonable amount of time (a year or two). 310 Silverman's conclusion is that 1024-bit RSA moduli will not be factored 311 until about 2037. This implies a much longer lifetime to RSA keys 312 than the theoretical analysis indicates. He argues that predictions 313 about how many machines and memory modules will be available can be 314 with great confidence, based on Moore's Law extrapolations and the 315 recent history of factoring efforts. 317 One should give the practical considerations a great deal of weight, 318 but in a risk analysis, perhaps the physical world is less 319 predicatable than trend graphs would indicate. In considering how 320 much trust to put into the inability of the computer industry to 321 satisfy the voracious needs of factorers, one must have some insight 322 into economic considerations that are more complicated than the 323 mathematics of factoring. The demand for computer memory is hard to 324 predict because it is based on applications - a "killer app" might 325 come along any day and send the memory industry into a frenzy of 326 sales. The number of processors available on desktops may be limited 327 by the number of desks, but very capable embedded systems account for 328 more processor sales than desktops. As embedded systems absorb 329 networking functions, it is not unimaginable that millions of 64-bit 330 processors with gigabytes of memory will pervade our environment. 332 The bottom line on this is that the key length recommendations 333 predicted by theory may be overly conservative. This question 334 is one that should be reconsidered in light of current technology 335 on a regular basis. 337 2.5 Strong Enough Key, Not Enough Bits 339 If the key exchange or data protection method has a strength matched 340 to the strength of the needed symmetric key, there remains one 341 possible additional step, and that is to securely derive the actual 342 symmetric key. The usual recommendation is to use a good one-way hash 343 function applied to he base material (the result of the key exchange) 344 and to use a subset of the hash function output for the key. However, 345 if the desired key length is greater than the output of the hash 346 function, one might wonder how to reconcile the two. 348 The step of deriving extra key bits must satisfy these requirements: 350 - The bits must not reveal any information about the key exchange secret 352 - The bits must not be correlated with each other 354 - The bits must depend on all the bits of the key exchange secret 356 Any good cryptographic hash function satisfies these three 357 requirements. Note that the number of bits of output of the hash 358 function is not specified. That is because even a hash function with 359 a very short output can be iterated to produce more uncorrelated bits 360 with just a little bit of care. 362 Appendix B of [RFC2409] describes how to derive long keys from short 363 hash functions. Note that this does not increase the key strength at 364 all; it just produces a good, safe, set of bits for a long key. The 365 trick is to make sure the hash function is applied repeatedly to all 366 the bits of the key exchange and one or more additional varying (but 367 not secret) bits. At each stage, the output bits of the hash function 368 can be collected into the output key bits. The RFC 2409 method gets 369 the additional bits from the output of the previous stage: 371 K1 = hash(keyexchangesecret | 0 ) 372 K2 = hash(keyexchangesecret | K1) 373 . . . 374 KN = hash(keyexchangesecret | KN-1) 376 This concatenation of K1 through KN constitutes a good set of 377 key bits. 379 3. Time to Use the Algorithms 381 This section describes how long it takes to use the algorithms to 382 perform key exchanges. Again, it is important to consider the increased 383 time it takes to exchange symmetric keys when increasing the length of 384 public keys in order to not choose unfeasibly long public keys. 386 3.1 Diffie-Hellman Key Exchange 388 A Diffie-Hellman key exchange is done with a group G with a generator g 389 and an exponent x. As noted in the Pollard's rho method section, the 390 exponent has twice as many bits as are needed for the final key. Let 391 the size of the group G be p, let the number of bits in the base 2 392 representation of p be j, and let the number of bits in the exponent be 393 K. 395 In doing the operations that result in a shared key, a generator is 396 raised to a power. The most efficient way to do this involves squaring 397 a number K times and multiplying it several times along the way. Each 398 of the numbers has j/w computer words in it, where w is the number of 399 bits in a computer word (today that will be 32 or 64 bits). 400 A naive assumption is that you will need 401 to do j squarings and j/2 multiplies; fortunately, an efficient 402 implementation will need fewer. For the remainder of this section, 403 n represents j/w. 405 A squaring operation does not need to use quite as many operations as a 406 multiplication; a reasonable estimate is that squaring takes .6 the 407 number of machine instructions of a multiply. If one prepares a table 408 ahead of time with several values of small integer powers of the 409 generator g, then only about one fifth as many multiplies are needed as 410 the naive formula suggests. Therefore, one needs to do the work of 411 approximately .8*K multiplies of n-by-n word numbers. Further, each 412 multiply and squaring must be followed by a modular reduction, and a 413 good assumption is that it is as hard to do a modular reduction as it 414 is to do an n-by-n word multiply. Thus, it takes K reductions for the 415 squarings and .2*K reductions for the multiplies. Summing this, the 416 total effort for a Diffie-Hellman key exchange with K bit exponents and 417 a modulus of n words is approximately 2*K n-by-n-word multiplies. 419 For 32-bit processors, integers that use less than about 30 computer 420 words in their representation require at least n^2 instructions for an 421 n-by-n-word multiply. Larger numbers will use less time, using 422 Karatsuba multiplications, and they will scale as about n^(1.58) for larger 423 n, but that is ignored for the current discussion. Note that 64- 424 bit processors push the "Karatsuba cross-over" number out to even more 425 bits. 427 The basic result is: if you double the size of the Diffie-Hellman 428 modular exponentiation group, you quadruple the number of operations 429 needed for the computation. 431 3.1.1 Diffie-Hellman with elliptic curve groups 433 Note that the ratios for computation effort as a function of 434 modulus size hold even if you are using an elliptic curve 435 (EC) group for Diffie-Hellman. However, for equivalent security, one 436 can use smaller numbers in the case of elliptic curves. Assume that 437 someone has chosen an modular exponentiation group with an 2048 bit 438 modulus as being an appropriate security measure for a Diffie-Hellman 439 application and wants to determine what advantage there would be to 440 using an EC group instead. The calculation is relatively 441 straightforward, if you assume that on the average, it is about 20 442 times more effort to do a squaring or multiplication in an EC group 443 than in a modular exponentiation group. A rough estimate is that an EC 444 group with equivalent security has about 200 bits in its 445 representation. Then, assuming that the time is dominated by n-by-n- 446 word operations, the relative time is computed as: 448 ((2048/200)^2)/20 ~= 5 450 showing that an elliptic curve implementation should be five times as 451 fast as a modular exponentiation implementation. 453 3.2 RSA encryption and decryption 455 Assume that an RSA public key uses a modulus with j bits; its factors 456 are two numbers of about j/2 bits each. The expected computation time 457 for encryption and decryption are different. Denote the number of words 458 in the machine representation of the modulus by the symbol n. 460 Most implementations of RSA use a small exponent for encryption. An 461 encryption may involve as few as 16 squarings and one multiplication, 462 using n-by-n-word operations. Each operation must be followed by a 463 modular reduction, and therefore the time complexity is about 464 16*(.6 + 1) + 1 + 1 ~= 28 n-by-n-word multiplies. 466 RSA decryption must use an exponent that has as many bits as the 467 modulus, j. However, the Chinese Remainder Theorem applies, and all the 468 computations can be done with a modulus of only n/2 words and an 469 exponent of only j/2 bits. The computation must be done twice, once for 470 each factor. The effort is equivalent to 2*(j/2) (n/2 by n/2)-word 471 multiplies. Because multiplying numbers with n/2 words is only 1/4 as 472 difficult as multiplying numbers with n words, the equivalent effort 473 for RSA decryption is j/4 n-by-n-word multiplies. 475 If you double the size of the modulus for RSA, the n-by-n multiplies 476 will take four times as long. Further, the decryption time doubles 477 because the exponent is larger. The overall scaling cost is a factor of 478 4 for encryption, a factor of 8 for decryption. 480 3.3 Real-world examples 482 To make these numbers more real, here are a few examples of software 483 implementations run on current hardware. The examples are included to 484 show rough estimates of reasonable implementations; they are not 485 benchmarks. As with all software, the performance will depend on the 486 exact details of specialization of the code to the problem and the 487 specific hardware. 489 The best time informally reported for a 1024-bit modular exponentiation 490 (the decryption side of 2048-bit RSA), is 0.9 ms (about 450,000 CPU 491 cycles) on a 500 MHz Itanium processor. This shows that newer 492 processors are not losing ground on big number operations; the number of 493 instructions is less than a 32-bit processor uses for a 256-bit modular 494 exponentiation. 496 For less advanced processors timing, the following two tables (computed 497 by Tero Monenen at SSH Communications) for modular exponentiation, such 498 as would be done in a Diffie-Hellman key exchange. 500 Celeron 400 MHz; compiled with GNU C compiler, optimized, some platform 501 specific coding optimizations: 503 group modulus exponent time 504 type size size 505 mod 768 ~150 18 msec 506 mod 1024 ~160 32 msec 507 mod 1536 ~180 82 msec 508 ecn 155 ~150 35 msec 509 ecn 185 ~200 56 msec 511 The group type is from RFC2409 and is either modular exponentiation 512 ("mod") or elliptic curve ("ecn"). All sizes here and in subsequent 513 tables are in bits. 515 Alpha 500 MHz compiled with Digital's C compiler, optimized, no 516 platform specific code: 518 group modulus exponent time 519 type size size 520 mod 768 ~150 12 msec 521 mod 1024 ~160 24 msec 522 mod 1536 ~180 59 msec 523 ecn 155 ~150 20 msec 524 ecn 185 ~200 27 msec 526 The following two tables (computed by Eric Young) were originally 527 for RSA signing operations, using the Chinese Remainder representation. 528 For ease of understanding, the parameters are presented here to show the 529 interior calculations, i.e., the size of the modulus and exponent used 530 by the software. 532 Dual Pentium II-350: 534 equiv equiv equiv 535 modulus exponent time 536 size size 537 256 256 1.5 ms 538 512 512 8.6 ms 539 1024 1024 55.4 ms 540 2048 2048 387 ms 542 Alpha 264 600mhz: 544 equiv equiv equiv 545 modulus exponent time 546 size size 547 512 512 1.4 ms 549 Current chips that accelerate exponentiation can perform 1024-bit 550 exponentiations (1024 bit modulus, 1024 bit exponent) in about 3 551 milliseconds. 553 4. Equivalences of Key Sizes 555 In order to determine how strong a public key is needed to protect a 556 particular symmetric key, you first need to determine how much effort 557 is needed to break the symmetric key. Most Internet security protocols 558 require the use of TripleDES for strong symmetric encryption, and it is 559 expected that the Advanced Encryption Standard (AES) will be adopted on 560 the Internet in the coming years. Therefore, these two 561 algorithms are discussed here. 563 If one could simply determine the number of MYs it takes to break 564 TripleDES, the task of computing the public key size of equivalent 565 strength would be easy. Unfortunately, that isn't the case here because 566 there are many examples of DES-specific hardware that encrypt faster 567 than DES in software on a standard CPU. Instead, one must determine the 568 equivalent cost for a system to break TripleDES and a system to break 569 the public key protecting a TripleDES key. 571 In 1998, the Electronic Frontier Foundation (EFF) built a DES-cracking 572 machine [GIL98] for US$130,000 that could test about 1e11 DES keys per 573 second (additional money was spent on the machine's design). The 574 machine's builders fully admit that the machine is not well optimized, 575 and it is estimated that ten times the amount of money could probably 576 create a machine about 50 times as fast. Assuming more optimization by 577 guessing that a system to test TripleDES keys runs about as fast as a 578 system to test DES keys, so approximately US$1 million might test 5e12 579 TripleDES keys per second. 581 In case your adversaries are much richer than EFF, you may want to 582 assume that they have US$1 trillion, enough to test 5e18 keys per 583 second. An exhaustive search of the TripleDES space of 2^112 keys with 584 this quite expensive system would take about 1e15 seconds or about 33 585 million years. (Note that such a system would also need 2^60 bytes of 586 RAM [MH81], which is considered free in this calculation). This seems a 587 needlessly conservative value. However, if computer logic speeds 588 continue to increase in accordance with Moore's Law (doubling in speed 589 every 1.5 years), then one might expect that in about 50 years, the 590 computation could be completed in only one year. For the purposes of 591 illustration, this 50 year resistance against a trillionaire is assumed 592 to be the minimum security requirement for a set of applications. 594 If 112 bits of attack resistance is the system security requirement, 595 then the key exchange system for TripleDES should have equivalent 596 difficulty; that is to say, if the attacker has US$1 trillion, you want 597 him to spend all his money to buy hardware today and to know that he 598 will "crack" the key exchange in not less than 33 million years. 599 (Obviously, a rational attacker would wait for about 45 years before 600 actually spending the money, because he could then get much better 601 hardware, but all attackers benefit from this sort of wait equally.) 603 It is estimated that a typical PC CPU today can generate about 500 MIPs 604 and can be purchased for about US$100 in quantity; thus you get about 5 605 MIPs/US$. For one trillion US dollars, an attacker can get 5e12 MIP 606 years of computer instructions. This figure is used in the following 607 estimates of equivalent costs for breaking key exchange systems. 609 4.1 Key equivalence against special purpose brute force hardware 611 If the trillionaire attacker is to use conventional CPU's to "crack" a 612 key exchange for a 112 bit key in the same time that the special 613 purpose machine is spending on brute force search for the symmetric 614 key, the key exchange system must use an appropriately large modulus. 615 Assume that the trillionaire performs 5e12 MIPs of instructions per 616 year. Use the following equation to estimate the modulus size to use 617 with RSA encryption or DH key exchange: 619 5*10^33 = (6*10^-16)*e^(1.92*cubrt(ln(n)*(ln(ln(n)))^2)) 621 Solving this approximately for n yields: 623 n = 10^(625) = 2^(2077) 625 Thus, assuming similar logic speeds and the current efficiency of the 626 number field sieve, moduli with about 2100 bits will have about the 627 same resistance against attack as an 112-bit TripleDES key. This 628 indicates that RSA public key encryption should use a modulus with 629 around 2100 bits; for a Diffie-Hellman key exchange, one could use a 630 slightly smaller modulus, but it not a significant difference. 632 4.2 Key equivalence against conventional CPU brute force attack 634 An alternative way of estimating this assumes that the attacker has a 635 less challenging requirement: he must only "crack" the key exchange in 636 less time than a brute force key search against the symmetric key would 637 take with general purpose computers. This is an "apples-to-apples" 638 comparison, because it assumes that the attacker needs only to have 639 computation donated to his effort, not built from a personal or 640 national fortune. The public key modulus will be larger than the one 641 in 4.1, because the symmetric key is going to be viable for a longer 642 period of time. 644 Assume that the number of congenial CPU instructions to encrypt a 645 block of material using TripleDES is 300. The estimated number of 646 computer instructions to break 112 bit TripleDES key: 648 300 * 2^112 649 = 1.6 * 10^(36) 650 = .02*e^(1.92*cubrt(ln(n)*(ln(ln(n)))^2)) 652 Solving this approximately for n yields: 654 n = 10^(734) = 2^(2439) 656 Thus, for general purpose CPU attacks, you can assume that moduli with 657 about 2400 bits will have about the same strength against attack as an 658 112-bit TripleDES key. This indicates that RSA public key encryption 659 should use a modulus with around 2400 bits; for a Diffie-Hellman key 660 exchange, one could use a slightly smaller modulus, but it not a 661 significant difference. 663 Note that some authors assume that the algorithms underlying the number 664 field sieve will continue to get better over time. These authors 665 recommend an even larger modulus, over 4000 bits, for protecting a 112- 666 bit symmetric key for 50 years. This points out the difficulty of 667 long-term cryptographic security: it is all but impossible to predict 668 progress in mathematics and physics over such a long period of time. 670 4.3 A One Year Attack 672 Assuming a trillionaire spends his money today to buy hardware, what 673 size key exchange numbers could he "crack" in one year? He can perform 674 5*e12 MYs of instructions, or 676 3*10^13 * 5*10^12 = .02*e^(1.92*cubrt(ln(n)*(ln(ln(n)))^2)) 678 Solving for an approximation of n yields 680 n = 10^(360) = 2^(1195) 682 This is about as many operations as it would take to crack an 80-bit 683 symmetric key by brute force. 685 Thus, for protecting data that has a secrecy requirement of one year 686 against an incredibly rich attacker, a key exchange modulus with about 687 1200 bits protecting an 80-bit symmetric key is safe even against a 688 nation's resources. 690 4.4 Key equivalence for other ciphers 692 Extending this logic to the AES is straightforward. For purposes of 693 estimation for key searching, one can think of the 128-bit AES as being 694 at least 16 bits stronger than TripleDES but about 5 times as fast. The 695 time and cost for a brute force attack is approximately 2^(14) more than 696 for TripleDES, and thus the recommended key exchange modulus sizes are 697 more than 600 bits longer (based on the difficulty of breaking a 698 discrete log). 700 If it is possible to design hardware for AES cracking that is 701 considerably more efficient than hardware for DES cracking, then the 702 moduli for protecting the key exchange can be made smaller. However, the 703 existence of such designs is only a matter of speculation at this early 704 moment in the AES lifetime. 706 The AES ciphers have key sizes of 128 bits up to 256 bits. Should a 707 prudent minimum security requirement, and thus the key exchange moduli, 708 have similar strengths? The answer to this depends on whether or not 709 one expect Moore's Law to continue unabated. If it continues, one would 710 expect 128 bit keys to be safe for about 60 years, and 256 bit keys 711 would be safe for another 400 years beyond that, far beyond any 712 imaginable security requirement. But such progress is difficult to 713 predict, as it exceeds the physical capabilities of today's devices and 714 would imply the existence of logic technologies that are unknown or 715 infeasible today. Quantum computing is a candidate, but too little is 716 known today to make confident predictions about its applicability to 717 cryptography (which itself might change over the next 100 years!). 719 If Moore's Law does not continue to hold, if no new computational 720 paradigms emerge, then keys of over 100 bits in length might well be 721 safe "forever". Note, however that others have come up with estimates 722 based on assumptions of new computational paradigms emerging. For 723 example, Lenstra and Verheul's web-based paper "Selecting Cryptographic 724 Key Sizes" chooses a more conservative analysis than the one in this 725 document. 727 4.5 Table of effort comparisons 729 In this table it is assumed that attackers use general purpose 730 computers, that the hardware is purchased in the year 2000, and that 731 mathematical knowledge relevant to the problem remains the same as 732 today. This is an pure "apples-to-apples" comparison demonstrating how 733 the time for a key exchange scales with respect to the strength 734 requirement. The subgroup size for DSA is included, if that is being 735 used for supporting authentication as part of the protocol; the DSA 736 modulus must be as long as the DH modulus, but the size of the "q" 737 subgroup is also relevant. 739 Symm. key RSA or DH DSA subgroup 740 size modulus size size 741 (bits) (bits) (bits) 743 70 947 128 744 80 1228 145 745 90 1553 153 746 100 1926 184 747 150 4575 279 748 200 8719 373 749 250 14596 475 751 5. Security Considerations 753 The equations and values given in this document are meant to be as 754 accurate as possible, based on the state of the art in general purpose 755 computers at the time that this document is being written. No 756 predictions can be completely accurate, and the formulas given here are 757 not meant to be definitive statements of fact about cryptographic 758 strengths. For example, some of the empirical results used in 759 calibrating the formulas in this document are probably not completely 760 accurate, and this inaccuracy affects the estimates. It is the authors' 761 hope that the numbers presented here vary from real world experience as 762 little as possible. 764 6. References 766 [DL] B. Dodson, A.K. Lenstra, NFS with four large primes: an explosive 767 experiment, Proceedings Crypto 95, Lecture Notes in Comput. Sci. 963, 768 (1995) 372-385. 770 [GIL98] Cracking DES: Secrets of Encryption Research, Wiretap Politics 771 & Chip Design , Electronic Frontier Foundation, John Gilmore (Ed.), 272 772 pages, May 1998, O'Reilly & Associates; ISBN: 1565925203 774 [GOR93] D. Gordon, "Discrete logarithms in GF(p) using the number field 775 sieve", SIAM Journal on Discrete Mathematics, 6 (1993), 124-138. 777 [LEN93] A. K. Lenstra, H. W. Lenstra, Jr. (eds), The development of the 778 number field sieve, Lecture Notes in Math, 1554, Springer Verlag, 779 Berlin, 1993. 781 [MH81] Merkle, R.C., and Hellman, M., "On the Security of Multiple 782 Encryption", Communications of the ACM, v. 24 n. 7, 1981, pp. 465-467. 784 [ODL95] RSA Labs Cryptobytes, Volume 1, No. 2 - Summer 1995; The Future 785 of Integer Factorization, A. M. Odlyzko 787 [ODL99] A. M. Odlyzko, Discrete logarithms: The past and the future, 788 Designs, Codes, and Cryptography (1999). To appear. 790 [POL78] J. Pollard, "Monte Carlo methods for index computation mod p", 791 Mathematics of Computation, 32 (1978), 918-924. 793 [RFC2409] D. Harkens and D. Carrel, "Internet Key Exchange (IKE)", 794 RFC 2409. 796 [SCH95] R. Schroeppel, et al., Fast Key Exchange With Elliptic Curve 797 Systems, In Don Coppersmith, editor, Advances in Cryptology -- CRYPTO 798 '95, volume 963 of Lecture Notes in Computer Science, pages 43-56, 27- 799 31 August 1995. Springer-Verlag 801 [SIL99] R. D. Silverman, RSA Laboratories Bulletin, Number 12 - May 3, 802 1999; An Analysis of Shamir's Factoring Device 804 [SIL00] R. D. Silverman, RSA Laboratories Bulletin, Number 13 - April 2000, 805 A Cost-Based Security Analysis of Symmetric and Asymmetric Key Lengths 807 A. Authors' Addresses 809 Hilarie Orman 810 Novell, Inc. 811 1800 South Novell Place 812 Provo, UT 84606 813 horman@novell.com 815 Paul Hoffman 816 Internet Mail Consortium and VPN Consortium 817 127 Segre Place 818 Santa Cruz, CA 95060 USA 819 paul.hoffman@imc.org and paul.hoffman@vpnc.org