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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-03.txt Novell, Inc. 3 July 16, 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. 100 The selection procedure is: 102 1. Determine the attack resistance necessary to satisfy the security 103 requirements of the application. Do this by estimating minimum number 104 of computer operations that the attacker will be forced to do in order 105 to compromise the security of the system and then take the logarithm 106 base two of that number. Call that logarithm value "n". 108 2. Choose a symmetric cipher that has a key with at least n bits and 109 at least that much cryptanalytic strength. 111 3. Choose a key exchange algorithm with a resistance to attack of at 112 least n bits. 114 A fourth consideration might be the public key authentication method 115 used to establish the identity of a user. This might be an RSA digital 116 signature or a DSA digital signature. If the modulus for the 117 authentication method isn't large enough, then the entire basis for 118 trusting the communication might fall apart. The following step 119 is thus added: 121 4. Choose an authentication algorithm with a resistance to attack of at 122 least n bits. 124 1.2 The key exchange algorithms 126 The Diffie-Hellman method uses a group, a generator, and exponents. In 127 today's Internet standards, the group operation is based on modular 128 multiplication. Here, the group is defined by the multiplicative group 129 of an integer, typically a prime p = 2q + 1, where q is a prime, and 130 the arithmetic is done modulo p; the generator (which is often simply 131 2) is denoted by g. 133 In Diffie-Hellman, Alice and Bob first agree (in public or in private) 134 on the values for g and p. Alice chooses a secret large random integer 135 (a), and Bob chooses a secret random large integer (b). Alice sends Bob 136 A, which is g^a mod p; Bob sends Alice B, which is g^b mod p. Next, 137 Alice computes B^a mod p, and Bob computes A^b mod p. These two numbers 138 are equal, and the participants use a simple function of this number as 139 the symmetric key k. 141 Note that Diffie-Hellman key exchange can be done over different kinds 142 of group representations. For instance, elliptic curves defined over 143 finite fields are a particularly efficient way to compute the key 144 exchange [SCH95]. 146 For RSA key exchange, assume that Bob has a public key (m) which is 147 equal to p*q, where p and q are two secret prime numbers, and an 148 encryption exponent e, and a decryption exponent d. For the key 149 exchange, Alice sends Bob E = k^e mod m, where k is the secret 150 symmetric key being exchanged. Bob recovers k by computing E^d mod m, 151 and the two parties use k as their symmetric key. While Bob's 152 encryption exponent e can be quite small (e.g., 17 bits), his 153 decryption exponent d will have as many bits in it as m does. 155 2. Determining the Effort to Factor 157 The RSA public key encryption method is immune to brute force guessing 158 attacks because the modulus will have at least 512 bits, and that is 159 too many possibilities to guess. The Diffie-Hellman exchange is also 160 secure against guessing because the exponents will have at least twice 161 as many bits as the symmetric keys that will be derived from 162 them. However, both methods are susceptible to mathematical attacks 163 that determine the structure of the public keys. 165 Factoring an RSA modulus will result in complete compromise of the 166 security of the private key. Solving the discrete logarithm problem for 167 a Diffie-Hellman modular exponentiation system will similarly destroy 168 the security of all key exchanges using the particular modulus. This 169 document assumes that the difficulty of solving the discrete logarithm 170 problem is equivalent to the difficulty of factoring numbers that are 171 the same size as the modulus. In fact, it is slightly harder because it 172 requires more operations; based on empirical evidence so far, the ratio 173 of difficulty is at least 20, possibly as high as 64. Solving either 174 problem requires a great deal of memory for the last stage of the 175 algorithm, the matrix reduction step. Whether or not this memory 176 requirement will continue to be the limiting factor in solving larger 177 integer problems remains to be seen. At the current time it is not, and 178 there is active research into parallel matrix algorithms that might 179 mitigate the memory requirements for this problem. 181 The number field sieve (NFS) [GOR93] [LEN93] is the best method today 182 for solving the discrete logarithm problem. The formula for estimating 183 the number of simple arithmetic operations needed to factor an integer, 184 n, using the NFS method is: 186 L(n) = k * e^((1.92 + o(1)) * cubrt(ln(n) * (ln(ln(n)))^2)) 188 Many people prefer to discuss the number of MIPS years (MYs) that are 189 needed for large operations such as the number field sieve. For such an 190 estimation, an operation in the L(n) formula is one computer 191 instruction. Empirical evidence indicates that 4 or 5 instructions 192 might be a closer match, but this is a minor factor and this document 193 sticks with one operation/one instruction for this discussion. 195 2.1 Choosing parameters for the equation 197 The expression above has two parameters that can be estimated by 198 empirical means: k and o(1). For the range of numbers we are interested 199 in, there is little distinction between them. 201 One could assume that k is 1 and o(1) is 0. This is reasonably valid if 202 the expression is only used for estimating relative effort (instead of 203 actual effort) and one assumes that the o(1) term is very small over the 204 range of the numbers that are to be factored. 206 Or, one could assume that o(1) is small and roughly constant and thus 207 its value can be folded into k; then estimate k from reported amounts of 208 effort spent factoring large integers in tests. 210 This document uses the second approach in order to get an estimate of 211 the significance of the factor. It appears to be minor, based on the 212 following calculations. 214 Sample values from recent work with the number field sieve, 215 collected by RSA Labs, include: 217 Test name Number of Number of MYs of effort 218 decimal bits 219 digits 220 RSA130 130 430 500 221 RSA140 140 460 2000 222 RSA155 155 512 8000 224 There are no precise measurements of the amount of time used for these 225 factorizations. In most factorization tests, hundreds or thousands of 226 computers are used over a period of several months, but the number of 227 their cycles were used for the factoring project, the precise 228 distribution of processor types, speeds, and so on are not usually 229 reported. However, in all cases, the amount of effort used was far less 230 than the L(n) formula would predict if k was 1 and o(1) was 0. 232 2.2 Choosing k from empirical reports 234 By solving for k from the empirical reports, it appears that k is 235 approximately 0.02. This means that the "effective key strength" of the 236 RSA algorithm is about 5 or 6 bits less than is implied by the naive 237 application of equation L(n) (that is, setting k to 1 and o(1) to 0). 238 These estimates of k are fairly stable over the numbers reported in the 239 table. The estimate is limited to a single significant digit of k 240 because it expresses real uncertainties; however, the effect of 241 additional digits would have make only tiny changes to the recommended 242 key sizes. 244 The factorers of RSA130 used about 1700 MYs, but they felt that this 245 was unrealistically high for prediction purposes; by using more memory 246 on their machines, they could have easily reduced the time to 500 MYs. 247 Thus, the value used in preparing the table above was 500. This story 248 does, however, underscore the difficulty in getting an accurate measure 249 of effort. This document takes the reported effort for factoring RSA155 250 as being the most accurate measure. 252 As a result of examining the empirical data, it appears that the L(n) 253 formula can be used with the o(1) term set to 0 and with k set to 0.02 254 when talking about factoring numbers in the range of 100 to 200 decimal 255 digits. The equation becomes: 257 L(n) = 0.02 * e^(1.92 * cubrt(ln(n) * (ln(ln(n)))^2)) 259 To convert L(n) from simple math instructions to MYs, divide by 260 3*10^13. The equation for the number of MYs needed to factor an integer 261 n then reduces to: 263 MYs = 6 * 10^(-16) * e^(1.92 * cubrt(ln(n) * (ln(ln(n)))^2)) 265 With what confidence can this formula be used for predicting the 266 difficulty of factoring slightly larger numbers? The answer is that it 267 should be a close upper bound, but each factorization effort is usually 268 marked by some improvement in the algorithms or their implementations 269 that makes the running time somewhat shorter than the formula would 270 indicate. 272 2.3 Pollard's rho method 274 In Diffie-Hellman exchanges, there is a second attack, Pollard's rho 275 method [POL78]. The algorithm relies on finding collisions between 276 values computed in a large number space; its success rate is 277 proportional to the square root of the size of the space. Because of 278 Pollard's rho method, the search space in a DH key exchange for the key 279 (the exponent in a g^a term), must be twice as large as the symmetric 280 key. Therefore, to securely derive a key of K bits, an implementation 281 must use an exponent with at least 2*K bits. 283 When the Diffie-Hellman key exchange is done using an elliptic curve 284 method, the NFS methods are of no avail. However, the collision method 285 is still effective, and the need for an exponent (called a multiplier 286 in EC's) with 2*K bits remains. The modulus used for the computation 287 can also be 2*K bits, and this will be substantially smaller than the 288 modulus needed for modular exponentiation methods as the desired 289 security level increases past 64 bits of brute-force attack 290 resistance. 292 One might ask, how can you compare the number of computer instructions 293 really needed for a discrete logarithm attack to the number needed to 294 search the keyspace of a cipher? In comparing the efforts, one should 295 consider what a "basic operation" is. For brute force search of the 296 keyspace of a symmetric encryption algorithm like DES, the basic 297 operation is the time to do a key setup and the time to do one 298 encryption. For discrete logs, the basic operation is a modular 299 squaring. The log of the ratio of these two operations can be used as a 300 "normalizing factor" between the two kinds of computations. However, 301 even for very large moduli (16K bits), this factor amounts to only a few 302 bits of extra effort. 304 2.4 Limits of large memory and many machines 306 Robert Silverman has examined the question of when it will be practical 307 to factor RSA moduli larger than 512 bits. His analysis is based not 308 only on the theoretical number of operations, but it also includes 309 expectations about the availability of actual machines for performing 310 the work (this document is based only on theoretical number of 311 operations). He examines the question of whether or not we can expect 312 there be enough machines, memory, and communication to factor a very 313 large number. 315 The best factoring methods need a lot of random access memory for 316 collecting data relations (sieving) and a critical final step that does 317 a row reduction on a large matrix. The memory requirements are related 318 to the size of the number being factored (or subjected to discrete 319 logarithm solution). Silverman [SILIEEE99][SIL00] has argued that there 320 is a practical limit to the number of machines and the amount of RAM 321 that be brought to bear on a single problem in the foreseeable future. 322 He sees two problems in attacking a 1024-bit RSA modulus: the machines 323 doing the sieving will need 64-bit address spaces and the matrix row 324 reduction machine will need several terabytes of memory. Silverman notes that 325 very few 64-bit machines have been sold, and none of those machines have 326 the 170 gigabytes of memory needed for sieving. Nearly a billion such 327 machines are necessary for the sieving in a reasonable amount of time (a 328 year or two). 330 Silverman's conclusion, based on the history of factoring efforts and 331 Moore's Law, is that 1024-bit RSA moduli will not be factored until 332 about 2037. This implies a much longer lifetime to RSA keys than the 333 theoretical analysis indicates. He argues that predictions about how 334 many machines and memory modules will be available can be with great 335 confidence, based on Moore's Law extrapolations and the recent history 336 of factoring efforts. 338 One should give the practical considerations a great deal of weight, but 339 in a risk analysis, the physical world is less predictable than trend 340 graphs would indicate. In considering how much trust to put into the 341 inability of the computer industry to satisfy the voracious needs of 342 factorers, one must have some insight into economic considerations that 343 are more complicated than the mathematics of factoring. The demand for 344 computer memory is hard to predict because it is based on applications: 345 a "killer app" might come along any day and send the memory industry 346 into a frenzy of sales. The number of processors available on desktops 347 may be limited by the number of desks, but very capable embedded systems 348 account for more processor sales than desktops. As embedded systems 349 absorb networking functions, it is not unimaginable that millions of 350 64-bit processors with at least gigabytes of memory will pervade our 351 environment. 353 The bottom line on this is that the key length recommendations predicted 354 by theory may be overly conservative, but they are what we have used for 355 this document. This question of machine availability is one that should 356 be reconsidered in light of current technology on a regular basis. 358 3. Time to Use the Algorithms 360 This section describes how long it takes to use the algorithms to 361 perform key exchanges. Again, it is important to consider the increased 362 time it takes to exchange symmetric keys when increasing the length of 363 public keys. It is important to avoid choosing unfeasibly long public 364 keys. 366 3.1 Diffie-Hellman Key Exchange 368 A Diffie-Hellman key exchange is done with a group G with a generator g 369 and an exponent x. As noted in the Pollard's rho method section, the 370 exponent has twice as many bits as are needed for the final key. Let 371 the size of the group G be p, let the number of bits in the base 2 372 representation of p be j, and let the number of bits in the exponent be 373 K. 375 In doing the operations that result in a shared key, a generator is 376 raised to a power. The most efficient way to do this involves squaring 377 a number K times and multiplying it several times along the way. Each 378 of the numbers has j/w computer words in it, where w is the number of 379 bits in a computer word (today that will be 32 or 64 bits). A naive 380 assumption is that you will need to do j squarings and j/2 multiplies; 381 fortunately, an efficient implementation will need fewer (NB: for the 382 remainder of this section, n represents j/w). 384 A squaring operation does not need to use quite as many operations as a 385 multiplication; a reasonable estimate is that squaring takes .6 the 386 number of machine instructions of a multiply. If one prepares a table 387 ahead of time with several values of small integer powers of the 388 generator g, then only about one fifth as many multiplies are needed as 389 the naive formula suggests. Therefore, one needs to do the work of 390 approximately .8*K multiplies of n-by-n word numbers. Further, each 391 multiply and squaring must be followed by a modular reduction, and a 392 good assumption is that it is as hard to do a modular reduction as it 393 is to do an n-by-n word multiply. Thus, it takes K reductions for the 394 squarings and .2*K reductions for the multiplies. Summing this, the 395 total effort for a Diffie-Hellman key exchange with K bit exponents and 396 a modulus of n words is approximately 2*K n-by-n-word multiplies. 398 For 32-bit processors, integers that use less than about 30 computer 399 words in their representation require at least n^2 instructions for an 400 n-by-n-word multiply. Larger numbers will use less time, using 401 Karatsuba multiplications, and they will scale as about n^(1.58) for larger 402 n, but that is ignored for the current discussion. Note that 64- 403 bit processors push the "Karatsuba cross-over" number out to even more 404 bits. 406 The basic result is: if you double the size of the Diffie-Hellman 407 modular exponentiation group, you quadruple the number of operations 408 needed for the computation. 410 3.1.1 Diffie-Hellman with elliptic curve groups 412 Note that the ratios for computation effort as a function of 413 modulus size hold even if you are using an elliptic curve 414 (EC) group for Diffie-Hellman. However, for equivalent security, one 415 can use smaller numbers in the case of elliptic curves. Assume that 416 someone has chosen an modular exponentiation group with an 2048 bit 417 modulus as being an appropriate security measure for a Diffie-Hellman 418 application and wants to determine what advantage there would be to 419 using an EC group instead. The calculation is relatively 420 straightforward, if you assume that on the average, it is about 20 421 times more effort to do a squaring or multiplication in an EC group 422 than in a modular exponentiation group. A rough estimate is that an EC 423 group with equivalent security has about 200 bits in its 424 representation. Then, assuming that the time is dominated by n-by-n- 425 word operations, the relative time is computed as: 427 ((2048/200)^2)/20 ~= 5 429 showing that an elliptic curve implementation should be five times as 430 fast as a modular exponentiation implementation. 432 3.2 RSA encryption and decryption 434 Assume that an RSA public key uses a modulus with j bits; its factors 435 are two numbers of about j/2 bits each. The expected computation time 436 for encryption and decryption are different. As before, we denote the 437 number of words in the machine representation of the modulus by the 438 symbol n. 440 Most implementations of RSA use a small exponent for encryption. An 441 encryption may involve as few as 16 squarings and one multiplication, 442 using n-by-n-word operations. Each operation must be followed by a 443 modular reduction, and therefore the time complexity is about 444 16*(.6 + 1) + 1 + 1 ~= 28 n-by-n-word multiplies. 446 RSA decryption must use an exponent that has as many bits as the 447 modulus, j. However, the Chinese Remainder Theorem applies, and all the 448 computations can be done with a modulus of only n/2 words and an 449 exponent of only j/2 bits. The computation must be done twice, once for 450 each factor. The effort is equivalent to 2*(j/2) (n/2 by n/2)-word 451 multiplies. Because multiplying numbers with n/2 words is only 1/4 as 452 difficult as multiplying numbers with n words, the equivalent effort 453 for RSA decryption is j/4 n-by-n-word multiplies. 455 If you double the size of the modulus for RSA, the n-by-n multiplies 456 will take four times as long. Further, the decryption time doubles 457 because the exponent is larger. The overall scaling cost is a factor of 458 4 for encryption, a factor of 8 for decryption. 460 3.3 Real-world examples 462 To make these numbers more real, here are a few examples of software 463 implementations run on current hardware. The examples are included to 464 show rough estimates of reasonable implementations; they are not 465 benchmarks. As with all software, the performance will depend on the 466 exact details of specialization of the code to the problem and the 467 specific hardware. 469 The best time informally reported for a 1024-bit modular exponentiation 470 (the decryption side of 2048-bit RSA), is 0.9 ms (about 450,000 CPU 471 cycles) on a 500 MHz Itanium processor. This shows that newer 472 processors are not losing ground on big number operations; the number of 473 instructions is less than a 32-bit processor uses for a 256-bit modular 474 exponentiation. 476 For less advanced processors timing, the following two tables (computed 477 by Tero Monenen at SSH Communications) for modular exponentiation, such 478 as would be done in a Diffie-Hellman key exchange. 480 Celeron 400 MHz; compiled with GNU C compiler, optimized, some platform 481 specific coding optimizations: 483 group modulus exponent time 484 type size size 485 mod 768 ~150 18 msec 486 mod 1024 ~160 32 msec 487 mod 1536 ~180 82 msec 488 ecn 155 ~150 35 msec 489 ecn 185 ~200 56 msec 491 The group type is from RFC2409 and is either modular exponentiation 492 ("mod") or elliptic curve ("ecn"). All sizes here and in subsequent 493 tables are in bits. 495 Alpha 500 MHz compiled with Digital's C compiler, optimized, no 496 platform specific code: 498 group modulus exponent time 499 type size size 500 mod 768 ~150 12 msec 501 mod 1024 ~160 24 msec 502 mod 1536 ~180 59 msec 503 ecn 155 ~150 20 msec 504 ecn 185 ~200 27 msec 506 The following two tables (computed by Eric Young) were originally 507 for RSA signing operations, using the Chinese Remainder representation. 508 For ease of understanding, the parameters are presented here to show the 509 interior calculations, i.e., the size of the modulus and exponent used 510 by the software. 512 Dual Pentium II-350: 514 equiv equiv equiv 515 modulus exponent time 516 size size 517 256 256 1.5 ms 518 512 512 8.6 ms 519 1024 1024 55.4 ms 520 2048 2048 387 ms 522 Alpha 264 600mhz: 524 equiv equiv equiv 525 modulus exponent time 526 size size 527 512 512 1.4 ms 529 Current chips that accelerate exponentiation can perform 1024-bit 530 exponentiations (1024 bit modulus, 1024 bit exponent) in about 3 531 milliseconds or less. 533 4. Equivalences of Key Sizes 535 In order to determine how strong a public key is needed to protect a 536 particular symmetric key, you first need to determine how much effort 537 is needed to break the symmetric key. Many Internet security protocols 538 require the use of TripleDES for strong symmetric encryption, and it is 539 expected that the Advanced Encryption Standard (AES) will be adopted on 540 the Internet in the coming years. Therefore, these two 541 algorithms are discussed here. 543 If one could simply determine the number of MYs it takes to break 544 TripleDES, the task of computing the public key size of equivalent 545 strength would be easy. Unfortunately, that isn't the case here because 546 there are many examples of DES-specific hardware that encrypt faster 547 than DES in software on a standard CPU. Instead, one must determine the 548 equivalent cost for a system to break TripleDES and a system to break 549 the public key protecting a TripleDES key. 551 In 1998, the Electronic Frontier Foundation (EFF) built a DES-cracking 552 machine [GIL98] for US$130,000 that could test about 1e11 DES keys per 553 second (additional money was spent on the machine's design). The 554 machine's builders fully admit that the machine is not well optimized, 555 and it is estimated that ten times the amount of money could probably 556 create a machine about 50 times as fast. Assuming more optimization by 557 guessing that a system to test TripleDES keys runs about as fast as a 558 system to test DES keys, so approximately US$1 million might test 5e12 559 TripleDES keys per second. 561 In case your adversaries are much richer than EFF, you may want to 562 assume that they have US$1 trillion, enough to test 5e18 keys per 563 second. An exhaustive search of the TripleDES space of 2^112 keys with 564 this quite expensive system would take about 1e15 seconds or about 33 565 million years. (Note that such a system would also need 2^60 bytes of 566 RAM [MH81], which is considered free in this calculation). This seems a 567 needlessly conservative value. However, if computer logic speeds 568 continue to increase in accordance with Moore's Law (doubling in speed 569 every 1.5 years), then one might expect that in about 50 years, the 570 computation could be completed in only one year. For the purposes of 571 illustration, this 50 year resistance against a trillionaire is assumed 572 to be the minimum security requirement for a set of applications. 574 If 112 bits of attack resistance is the system security requirement, 575 then the key exchange system for TripleDES should have equivalent 576 difficulty; that is to say, if the attacker has US$1 trillion, you want 577 him to spend all his money to buy hardware today and to know that he 578 will "crack" the key exchange in not less than 33 million years. 579 (Obviously, a rational attacker would wait for about 45 years before 580 actually spending the money, because he could then get much better 581 hardware, but all attackers benefit from this sort of wait equally.) 583 It is estimated that a typical PC CPU today can generate over 500 MIPs 584 and can be purchased for about US$100 in quantity; thus you get more than 5 585 MIPs/US$. For one trillion US dollars, an attacker can get 5e12 MIP 586 years of computer instructions. This figure is used in the following 587 estimates of equivalent costs for breaking key exchange systems. 589 4.1 Key equivalence against special purpose brute force hardware 591 If the trillionaire attacker is to use conventional CPU's to "crack" a 592 key exchange for a 112 bit key in the same time that the special 593 purpose machine is spending on brute force search for the symmetric 594 key, the key exchange system must use an appropriately large modulus. 595 Assume that the trillionaire performs 5e12 MIPs of instructions per 596 year. Use the following equation to estimate the modulus size to use 597 with RSA encryption or DH key exchange: 599 5*10^33 = (6*10^-16)*e^(1.92*cubrt(ln(n)*(ln(ln(n)))^2)) 601 Solving this approximately for n yields: 603 n = 10^(625) = 2^(2077) 605 Thus, assuming similar logic speeds and the current efficiency of the 606 number field sieve, moduli with about 2100 bits will have about the 607 same resistance against attack as an 112-bit TripleDES key. This 608 indicates that RSA public key encryption should use a modulus with 609 around 2100 bits; for a Diffie-Hellman key exchange, one could use a 610 slightly smaller modulus, but it not a significant difference. 612 4.2 Key equivalence against conventional CPU brute force attack 614 An alternative way of estimating this assumes that the attacker has a 615 less challenging requirement: he must only "crack" the key exchange in 616 less time than a brute force key search against the symmetric key would 617 take with general purpose computers. This is an "apples-to-apples" 618 comparison, because it assumes that the attacker needs only to have 619 computation donated to his effort, not built from a personal or 620 national fortune. The public key modulus will be larger than the one 621 in 4.1, because the symmetric key is going to be viable for a longer 622 period of time. 624 Assume that the number of CPU instructions to encrypt a block of 625 material using TripleDES is 300. The estimated number of computer 626 instructions to break 112 bit TripleDES key: 628 300 * 2^112 629 = 1.6 * 10^(36) 630 = .02*e^(1.92*cubrt(ln(n)*(ln(ln(n)))^2)) 632 Solving this approximately for n yields: 634 n = 10^(734) = 2^(2439) 636 Thus, for general purpose CPU attacks, you can assume that moduli with 637 about 2400 bits will have about the same strength against attack as an 638 112-bit TripleDES key. This indicates that RSA public key encryption 639 should use a modulus with around 2400 bits; for a Diffie-Hellman key 640 exchange, one could use a slightly smaller modulus, but it not a 641 significant difference. 643 Note that some authors assume that the algorithms underlying the number 644 field sieve will continue to get better over time. These authors 645 recommend an even larger modulus, over 4000 bits, for protecting a 112- 646 bit symmetric key for 50 years. This points out the difficulty of 647 long-term cryptographic security: it is all but impossible to predict 648 progress in mathematics and physics over such a long period of time. 650 4.3 A One Year Attack 652 Assuming a trillionaire spends his money today to buy hardware, what 653 size key exchange numbers could he "crack" in one year? He can perform 654 5*e12 MYs of instructions, or 656 3*10^13 * 5*10^12 = .02*e^(1.92*cubrt(ln(n)*(ln(ln(n)))^2)) 658 Solving for an approximation of n yields 660 n = 10^(360) = 2^(1195) 662 This is about as many operations as it would take to crack an 80-bit 663 symmetric key by brute force. 665 Thus, for protecting data that has a secrecy requirement of one year 666 against an incredibly rich attacker, a key exchange modulus with about 667 1200 bits protecting an 80-bit symmetric key is safe even against a 668 nation's resources. 670 4.4 Key equivalence for other ciphers 672 Extending this logic to the AES is straightforward. For purposes of 673 estimation for key searching, one can think of the 128-bit AES as being 674 at least 16 bits stronger than TripleDES but about three times as fast. 675 The time and cost for a brute force attack is approximately 2^(16) more 676 than for TripleDES, and thus, under the assumption that 128 bits of 677 strength is the desired security goal, the recommended key exchange 678 modulus size is about 700 bits longer. 680 If it is possible to design hardware for AES cracking that is 681 considerably more efficient than hardware for DES cracking, then (again 682 under the assumption that the key exchange strength must match the brute 683 force effort) the moduli for protecting the key exchange can be made 684 smaller. However, the existence of such designs is only a matter of 685 speculation at this early moment in the AES lifetime. 687 The AES ciphers have key sizes of 128 bits up to 256 bits. Should a 688 prudent minimum security requirement, and thus the key exchange moduli, 689 have similar strengths? The answer to this depends on whether or not 690 one expect Moore's Law to continue unabated. If it continues, one would 691 expect 128 bit keys to be safe for about 60 years, and 256 bit keys 692 would be safe for another 400 years beyond that, far beyond any 693 imaginable security requirement. But such progress is difficult to 694 predict, as it exceeds the physical capabilities of today's devices and 695 would imply the existence of logic technologies that are unknown or 696 infeasible today. Quantum computing is a candidate, but too little is 697 known today to make confident predictions about its applicability to 698 cryptography (which itself might change over the next 100 years!). 700 If Moore's Law does not continue to hold, if no new computational 701 paradigms emerge, then keys of over 100 bits in length might well be 702 safe "forever". Note, however that others have come up with estimates 703 based on assumptions of new computational paradigms emerging. For 704 example, Lenstra and Verheul's web-based paper "Selecting Cryptographic 705 Key Sizes" chooses a more conservative analysis than the one in this 706 document. 708 4.5 Hash functions for deriving symmetric keys from public key 709 algorithms 711 The Diffie-Hellman algorithm results in a key that is hundreds or 712 thousands of bits long, but ciphers need far fewer bits than that. How 713 can one distill a long key down to a short one without losing strength? 715 Cryptographic one-way hash functions are the building blocks for this, 716 and so long as they use all of the Diffie-Hellman key to derive each 717 block of the symmetric key, they produce keys with sufficient strength. 719 The usual recommendation is to use a good one-way hash function 720 applied to he base material (the result of the key exchange) and to 721 use a subset of the hash function output for the key. However, if the 722 desired key length is greater than the output of the hash function, 723 one might wonder how to reconcile the two. 725 The step of deriving extra key bits must satisfy these requirements: 727 - The bits must not reveal any information about the key exchange secret 729 - The bits must not be correlated with each other 731 - The bits must depend on all the bits of the key exchange secret 733 Any good cryptographic hash function satisfies these three 734 requirements. Note that the number of bits of output of the hash 735 function is not specified. That is because even a hash function with 736 a very short output can be iterated to produce more uncorrelated bits 737 with just a little bit of care. 739 For example, SHA-1 has 160 bits of output. For deriving a key of attack 740 resistance of 160 bits or less, SHA(DHkey) produces a good symmetric 741 key. 743 Suppose one wants a key with attack resistance of 160 bits, but it is to 744 be used with a cipher that uses 192 bit keys. One can iterate SHA-1 as 745 follows: 747 Bits 1-160 of the symmetric key = K1 = SHA(DHkey | 0x00) 748 (that is, concatenate a single octet of value 0x00 to 749 the right side of the DHkey, and then hash) 750 Bits 161-192 of the symmetric key = K2 = 751 select_32_bits(SHA(K1 | 0x01)) 753 But what if one wants 192 bits of strength for the cipher? Then the 754 appropriate calculation is 756 Bits 1-160 of the symmetric key = SHA(0x00 | DHkey) 757 Bits 161-192 of the symmetric key = 758 select_32_bits(SHA(0x01 | DHkey)) 760 (Note that in the description above, instead of concatenating a full 761 octet, concatenating a single bit would also be sufficient.) 763 The important distinction is that in the second case, the DH key is used 764 for each part of the symmetric key. This assures that entropy of the DH 765 key is not lost by iteration of the hash function over the same bits. 767 From an efficiency point of view, if the symmetric key must have a great 768 deal of entropy, it is probably best to use a cryptographic hash 769 function with a large output block (192 bits or more), rather than 770 iterating a smaller one. 772 4.6 Table of effort comparisons 774 In this table it is assumed that attackers use general purpose 775 computers, that the hardware is purchased in the year 2000, and that 776 mathematical knowledge relevant to the problem remains the same as 777 today. This is an pure "apples-to-apples" comparison demonstrating how 778 the time for a key exchange scales with respect to the strength 779 requirement. The subgroup size for DSA is included, if that is being 780 used for supporting authentication as part of the protocol; the DSA 781 modulus must be as long as the DH modulus, but the size of the "q" 782 subgroup is also relevant. 784 Symm. key RSA or DH DSA subgroup 785 size modulus size size 786 (bits) (bits) (bits) 788 70 947 128 789 80 1228 145 790 90 1553 153 791 100 1926 184 792 150 4575 279 793 200 8719 373 794 250 14596 475 796 5. Security Considerations 798 The equations and values given in this document are meant to be as 799 accurate as possible, based on the state of the art in general purpose 800 computers at the time that this document is being written. No 801 predictions can be completely accurate, and the formulas given here are 802 not meant to be definitive statements of fact about cryptographic 803 strengths. For example, some of the empirical results used in 804 calibrating the formulas in this document are probably not completely 805 accurate, and this inaccuracy affects the estimates. It is the authors' 806 hope that the numbers presented here vary from real world experience as 807 little as possible. 809 6. References 811 [DL] B. Dodson, A.K. Lenstra, NFS with four large primes: an explosive 812 experiment, Proceedings Crypto 95, Lecture Notes in Comput. Sci. 963, 813 (1995) 372-385. 815 [GIL98] Cracking DES: Secrets of Encryption Research, Wiretap Politics 816 & Chip Design , Electronic Frontier Foundation, John Gilmore (Ed.), 272 817 pages, May 1998, O'Reilly & Associates; ISBN: 1565925203 819 [GOR93] D. Gordon, "Discrete logarithms in GF(p) using the number field 820 sieve", SIAM Journal on Discrete Mathematics, 6 (1993), 124-138. 822 [LEN93] A. K. Lenstra, H. W. Lenstra, Jr. (eds), The development of the 823 number field sieve, Lecture Notes in Math, 1554, Springer Verlag, 824 Berlin, 1993. 826 [MH81] Merkle, R.C., and Hellman, M., "On the Security of Multiple 827 Encryption", Communications of the ACM, v. 24 n. 7, 1981, pp. 465-467. 829 [ODL95] RSA Labs Cryptobytes, Volume 1, No. 2 - Summer 1995; The Future 830 of Integer Factorization, A. M. Odlyzko 832 [ODL99] A. M. Odlyzko, Discrete logarithms: The past and the future, 833 Designs, Codes, and Cryptography (1999). 835 [POL78] J. Pollard, "Monte Carlo methods for index computation mod p", 836 Mathematics of Computation, 32 (1978), 918-924. 838 [RFC2409] D. Harkens and D. Carrel, "Internet Key Exchange (IKE)", 839 RFC 2409. 841 [SCH95] R. Schroeppel, et al., Fast Key Exchange With Elliptic Curve 842 Systems, In Don Coppersmith, editor, Advances in Cryptology -- CRYPTO 843 '95, volume 963 of Lecture Notes in Computer Science, pages 43-56, 27- 844 31 August 1995. Springer-Verlag 846 [SIL99] R. D. Silverman, RSA Laboratories Bulletin, Number 12 - May 3, 847 1999; An Analysis of Shamir's Factoring Device 849 [SIL00] R. D. Silverman, RSA Laboratories Bulletin, Number 13 - April 2000, 850 A Cost-Based Security Analysis of Symmetric and Asymmetric Key Lengths 852 A. Authors' Addresses 854 Hilarie Orman 855 Volera, Inc. 856 600 E. Timpanogos Circle 857 Orem, UT 84046 858 horman@volera.com 860 Paul Hoffman 861 Internet Mail Consortium and VPN Consortium 862 127 Segre Place 863 Santa Cruz, CA 95060 USA 864 paul.hoffman@imc.org and paul.hoffman@vpnc.org