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Run idnits with the --verbose option for more detailed information about the items above. -------------------------------------------------------------------------------- 1 Network Working Group Donald E. Eastlake, 3rd 2 OBSOLETES RFC 1750 Jeffrey I. Schiller 3 Steve Crocker 4 Expires June 2004 December 2003 6 Randomness Requirements for Security 7 ---------- ------------ --- -------- 8 10 Status of This Document 12 This document is intended to become a Best Current Practice. 13 Comments should be sent to the authors. Distribution is unlimited. 15 This document is an Internet-Draft and is in full conformance with 16 all provisions of Section 10 of RFC 2026. Internet-Drafts are 17 working documents of the Internet Engineering Task Force (IETF), its 18 areas, and its working groups. Note that other groups may also 19 distribute working documents as Internet-Drafts. 21 Internet-Drafts are draft documents valid for a maximum of six months 22 and may be updated, replaced, or obsoleted by other documents at any 23 time. It is inappropriate to use Internet-Drafts as reference 24 material or to cite them other than as "work in progress." 26 The list of current Internet-Drafts can be accessed at 27 http://www.ietf.org/ietf/1id-abstracts.txt 29 The list of Internet-Draft Shadow Directories can be accessed at 30 http://www.ietf.org/shadow.html. 32 Abstract 34 Security systems today are built on strong cryptographic algorithms 35 that foil pattern analysis attempts. However, the security of these 36 systems is dependent on generating secret quantities for passwords, 37 cryptographic keys, and similar quantities. The use of pseudo-random 38 processes to generate secret quantities can result in pseudo- 39 security. The sophisticated attacker of these security systems may 40 find it easier to reproduce the environment that produced the secret 41 quantities, searching the resulting small set of possibilities, than 42 to locate the quantities in the whole of the potential number space. 44 Choosing random quantities to foil a resourceful and motivated 45 adversary is surprisingly difficult. This document points out many 46 pitfalls in using traditional pseudo-random number generation 47 techniques for choosing such quantities. It recommends the use of 48 truly random hardware techniques and shows that the existing hardware 49 on many systems can be used for this purpose. It provides 50 suggestions to ameliorate the problem when a hardware solution is not 51 available. And it gives examples of how large such quantities need 52 to be for some applications. 54 Acknowledgements 56 Special thanks to 57 (1) The authors of "Minimal Key Lengths for Symmetric Ciphers to 58 Provide Adequate Commercial Security" which is incorporated as 59 Appendix A. 61 (2) Peter Gutmann who has permitted the incorporation into this 62 replacement for RFC 1750 of material from is paper "Software 63 Generation of Practially Strong Random Numbers". 65 The following other persons (in alphabetic order) contributed to this 66 document: 68 Tony Hansen, Sandy Harris 70 The following persons (in alphabetic order) contributed to RFC 1750, 71 the predeceasor of this document: 73 David M. Balenson, Don T. Davis, Carl Ellison, Marc Horowitz, 74 Christian Huitema, Charlie Kaufman, Steve Kent, Hal Murray, Neil 75 Haller, Richard Pitkin, Tim Redmond, and Doug Tygar. 77 Table of Contents 79 Status of This Document....................................1 81 Abstract...................................................2 82 Acknowledgements...........................................2 84 Table of Contents..........................................3 86 1. Introduction............................................5 88 2. Requirements............................................6 90 3. Traditional Pseudo-Random Sequences.....................8 92 4. Unpredictability.......................................10 93 4.1 Problems with Clocks and Serial Numbers...............10 94 4.2 Timing and Content of External Events.................11 95 4.3 The Fallacy of Complex Manipulation...................11 96 4.4 The Fallacy of Selection from a Large Database........12 98 5. Hardware for Randomness................................13 99 5.1 Volume Required.......................................13 100 5.2 Sensitivity to Skew...................................13 101 5.2.1 Using Stream Parity to De-Skew......................14 102 5.2.2 Using Transition Mappings to De-Skew................15 103 5.2.3 Using FFT to De-Skew................................16 104 5.2.4 Using S-Boxes to De-Skew............................16 105 5.2.5 Using Compression to De-Skew........................17 106 5.3 Existing Hardware Can Be Used For Randomness..........17 107 5.3.1 Using Existing Sound/Video Input....................17 108 5.3.2 Using Existing Disk Drives..........................18 109 5.4 Ring Oscillator Sources...............................18 111 6. Recommended Software Strategy..........................19 112 6.1 Mixing Functions......................................19 113 6.1.1 A Trivial Mixing Function...........................19 114 6.1.2 Stronger Mixing Functions...........................20 115 6.1.3 Diffie-Hellman as a Mixing Function.................21 116 6.1.4 Using a Mixing Function to Stretch Random Bits......22 117 6.1.5 Other Factors in Choosing a Mixing Function.........22 118 6.2 Non-Hardware Sources of Randomness....................23 119 6.3 Cryptographically Strong Sequences....................24 120 6.3.1 Traditional Strong Sequences........................24 121 6.3.2 The Blum Blum Shub Sequence Generator...............25 122 6.3.3 Entropy Pool Techniques.............................26 124 7. Key Generation Standards and Examples..................28 125 7.1 US DoD Recommendations for Password Generation........28 126 7.2 X9.17 Key Generation..................................28 127 7.3 The /dev/random Device under Linux....................29 129 More Table of Contents 131 8. Examples of Randomness Required........................31 132 8.1 Password Generation..................................31 133 8.2 A Very High Security Cryptographic Key................32 134 8.2.1 Effort per Key Trial................................32 135 8.2.2 Meet in the Middle Attacks..........................32 137 9. Conclusion.............................................34 138 10. Security Considerations...............................34 139 Intellectual Property Considerations......................34 141 Appendix: Minimal Secure Key Lengths Study................36 142 A.0 Abstract..............................................36 143 A.1. Encryption Plays an Essential Role in Protecting.....37 144 A.1.1 There is a need for information security............37 145 A.1.2 Encryption to protect confidentiality...............38 146 A.1.3 There are a variety of attackers....................39 147 A.1.4 Strong encryption is not expensive..................40 148 A.2. Brute-Force is becoming easier.......................40 149 A.3. 40-Bit Key Lengths Offer Virtually No Protection.....42 150 A.4. Even DES with 56-Bit Keys Is Increasingly Inadequate.43 151 A.4.1 DES is no panacea today.............................43 152 A.4.2 There are smarter avenues of attack than brute force44 153 A.4.3 Other algorithms are similar........................44 154 A.5. Appropriate Key Lengths for the Future --- A Proposal45 155 A.6 About the Authors.....................................47 156 A.7 Acknowledgement.......................................48 158 Informative References....................................49 160 Authors Addresses.........................................53 161 File Name and Expiration..................................53 163 1. Introduction 165 Software cryptography is coming into wider use and is continuing to 166 spread, although there is a long way to go until it becomes 167 pervasive. 169 Systems like SSH, IPSEC, TLS, S/MIME, PGP, DNSSEC, Kerberos, etc. are 170 maturing and becoming a part of the network landscape [SSH, DNSSEC, 171 IPSEC, MAIL*, TLS]. By comparison, when the previous version of this 172 document [RFC 1750] was issued in 1994, about the only Internet 173 cryptographic security specification in the IETF was the Privacy 174 Enhanced Mail protocol [MAIL PEM]. 176 These systems provide substantial protection against snooping and 177 spoofing. However, there is a potential flaw. At the heart of all 178 cryptographic systems is the generation of secret, unguessable (i.e., 179 random) numbers. 181 For the present, the lack of generally available facilities for 182 generating such unpredictable numbers is an open wound in the design 183 of cryptographic software. For the software developer who wants to 184 build a key or password generation procedure that runs on a wide 185 range of hardware, the only safe strategy so far has been to force 186 the local installation to supply a suitable routine to generate 187 random numbers. To say the least, this is an awkward, error-prone 188 and unpalatable solution. 190 It is important to keep in mind that the requirement is for data that 191 an adversary has a very low probability of guessing or determining. 192 This can easily fail if pseudo-random data is used which only meets 193 traditional statistical tests for randomness or which is based on 194 limited range sources, such as clocks. Frequently such random 195 quantities are determinable by an adversary searching through an 196 embarrassingly small space of possibilities. 198 This Best Current Practice describes techniques for producing random 199 quantities that will be resistant to such attack. It recommends that 200 future systems include hardware random number generation or provide 201 access to existing hardware that can be used for this purpose. It 202 suggests methods for use if such hardware is not available. And it 203 gives some estimates of the number of random bits required for sample 204 applications. 206 2. Requirements 208 A commonly encountered randomness requirement today is the user 209 password. This is usually a simple character string. Obviously, if a 210 password can be guessed, it does not provide security. (For re- 211 usable passwords, it is desirable that users be able to remember the 212 password. This may make it advisable to use pronounceable character 213 strings or phrases composed on ordinary words. But this only affects 214 the format of the password information, not the requirement that the 215 password be very hard to guess.) 217 Many other requirements come from the cryptographic arena. 218 Cryptographic techniques can be used to provide a variety of services 219 including confidentiality and authentication. Such services are 220 based on quantities, traditionally called "keys", that are unknown to 221 and unguessable by an adversary. 223 In some cases, such as the use of symmetric encryption with the one 224 time pads [CRYPTO*] or the US Data Encryption Standard [DES] or 225 Advanced Encryption Standard [AES], the parties who wish to 226 communicate confidentially and/or with authentication must all know 227 the same secret key. In other cases, using what are called 228 asymmetric or "public key" cryptographic techniques, keys come in 229 pairs. One key of the pair is private and must be kept secret by one 230 party, the other is public and can be published to the world. It is 231 computationally infeasible to determine the private key from the 232 public key and knowledge of the public is of no help to an adversary. 233 [ASYMMETRIC, CRYPTO*] 235 The frequency and volume of the requirement for random quantities 236 differs greatly for different cryptographic systems. Using pure RSA 237 [CRYPTO*], random quantities are required when the key pair is 238 generated, but thereafter any number of messages can be signed 239 without a further need for randomness. The public key Digital 240 Signature Algorithm devised by the US National Institute of Standards 241 and Technology (NIST) requires good random numbers for each signature 242 [DSS]. And encrypting with a one time pad, in principle the 243 strongest possible encryption technique, requires a volume of 244 randomness equal to all the messages to be processed [CRYPTO*]. 246 In most of these cases, an adversary can try to determine the 247 "secret" key by trial and error. (This is possible as long as the 248 key is enough smaller than the message that the correct key can be 249 uniquely identified.) The probability of an adversary succeeding at 250 this must be made acceptably low, depending on the particular 251 application. The size of the space the adversary must search is 252 related to the amount of key "information" present in the information 253 theoretic sense [SHANNON]. This depends on the number of different 254 secret values possible and the probability of each value as follows: 256 ----- 257 \ 258 Bits-of-info = \ - p * log ( p ) 259 / i 2 i 260 / 261 ----- 263 where i counts from 1 to the number of possible secret values and p 264 sub i is the probability of the value numbered i. (Since p sub i is 265 less than one, the log will be negative so each term in the sum will 266 be non-negative.) 268 If there are 2^n different values of equal probability, then n bits 269 of information are present and an adversary would, on the average, 270 have to try half of the values, or 2^(n-1) , before guessing the 271 secret quantity. If the probability of different values is unequal, 272 then there is less information present and fewer guesses will, on 273 average, be required by an adversary. In particular, any values that 274 the adversary can know are impossible, or are of low probability, can 275 be initially ignored by an adversary, who will search through the 276 more probable values first. 278 For example, consider a cryptographic system that uses 128 bit keys. 279 If these 128 bit keys are derived by using a fixed pseudo-random 280 number generator that is seeded with an 8 bit seed, then an adversary 281 needs to search through only 256 keys (by running the pseudo-random 282 number generator with every possible seed), not the 2^128 keys that 283 may at first appear to be the case. Only 8 bits of "information" are 284 in these 128 bit keys. 286 3. Traditional Pseudo-Random Sequences 288 Most traditional sources of random numbers use deterministic sources 289 of "pseudo-random" numbers. These typically start with a "seed" 290 quantity and use numeric or logical operations to produce a sequence 291 of values. 293 [KNUTH] has a classic exposition on pseudo-random numbers. 294 Applications he mentions are simulation of natural phenomena, 295 sampling, numerical analysis, testing computer programs, decision 296 making, and games. None of these have the same characteristics as 297 the sort of security uses we are talking about. Only in the last two 298 could there be an adversary trying to find the random quantity. 299 However, in these cases, the adversary normally has only a single 300 chance to use a guessed value. In guessing passwords or attempting 301 to break an encryption scheme, the adversary normally has many, 302 perhaps unlimited, chances at guessing the correct value because they 303 can store the message they are trying to break and repeatedly attack 304 it. They should also be assumed to be aided by a computer. 306 For testing the "randomness" of numbers, Knuth suggests a variety of 307 measures including statistical and spectral. These tests check 308 things like autocorrelation between different parts of a "random" 309 sequence or distribution of its values. But they could be met by a 310 constant stored random sequence, such as the "random" sequence 311 printed in the CRC Standard Mathematical Tables [CRC]. 313 A typical pseudo-random number generation technique, known as a 314 linear congruence pseudo-random number generator, is modular 315 arithmetic where the value numbered N+1 is calculated from the value 316 numbered N by 318 V = ( V * a + b )(Mod c) 319 N+1 N 321 The above technique has a strong relationship to linear shift 322 register pseudo-random number generators, which are well understood 323 cryptographically [SHIFT*]. In such generators bits are introduced 324 at one end of a shift register as the Exclusive Or (binary sum 325 without carry) of bits from selected fixed taps into the register. 326 For example: 328 +----+ +----+ +----+ +----+ 329 | B | <-- | B | <-- | B | <-- . . . . . . <-- | B | <-+ 330 | 0 | | 1 | | 2 | | n | | 331 +----+ +----+ +----+ +----+ | 332 | | | | 333 | | V +-----+ 334 | V +----------------> | | 335 V +-----------------------------> | XOR | 336 +---------------------------------------------------> | | 337 +-----+ 339 V = ( ( V * 2 ) + B .xor. B ... )(Mod 2^n) 340 N+1 N 0 2 342 The goodness of traditional pseudo-random number generator algorithms 343 is measured by statistical tests on such sequences. Carefully chosen 344 values of the initial V and a, b, and c or the placement of shift 345 register tap in the above simple processes can produce excellent 346 statistics. 348 These sequences may be adequate in simulations (Monte Carlo 349 experiments) as long as the sequence is orthogonal to the structure 350 of the space being explored. Even there, subtle patterns may cause 351 problems. However, such sequences are clearly bad for use in 352 security applications. They are fully predictable if the initial 353 state is known. Depending on the form of the pseudo-random number 354 generator, the sequence may be determinable from observation of a 355 short portion of the sequence [CRYPTO*, STERN]. For example, with 356 the generators above, one can determine V(n+1) given knowledge of 357 V(n). In fact, it has been shown that with these techniques, even if 358 only one bit of the pseudo-random values are released, the seed can 359 be determined from short sequences. 361 Not only have linear congruent generators been broken, but techniques 362 are now known for breaking all polynomial congruent generators. 363 [KRAWCZYK] 365 4. Unpredictability 367 Randomness in the traditional sense described in section 3 is NOT the 368 same as the unpredictability required for security use. 370 For example, use of a widely available constant sequence, such as 371 that from the CRC tables, is very weak against an adversary. Once 372 they learn of or guess it, they can easily break all security, future 373 and past, based on the sequence. [CRC] Yet the statistical properties 374 of these tables are good. 376 The following sections describe the limitations of some randomness 377 generation techniques and sources. 379 4.1 Problems with Clocks and Serial Numbers 381 Computer clocks, or similar operating system or hardware values, 382 provide significantly fewer real bits of unpredictability than might 383 appear from their specifications. 385 Tests have been done on clocks on numerous systems and it was found 386 that their behavior can vary widely and in unexpected ways. One 387 version of an operating system running on one set of hardware may 388 actually provide, say, microsecond resolution in a clock while a 389 different configuration of the "same" system may always provide the 390 same lower bits and only count in the upper bits at much lower 391 resolution. This means that successive reads on the clock may 392 produce identical values even if enough time has passed that the 393 value "should" change based on the nominal clock resolution. There 394 are also cases where frequently reading a clock can produce 395 artificial sequential values because of extra code that checks for 396 the clock being unchanged between two reads and increases it by one! 397 Designing portable application code to generate unpredictable numbers 398 based on such system clocks is particularly challenging because the 399 system designer does not always know the properties of the system 400 clocks that the code will execute on. 402 Use of a hardware serial number such as an Ethernet address may also 403 provide fewer bits of uniqueness than one would guess. Such 404 quantities are usually heavily structured and subfields may have only 405 a limited range of possible values or values easily guessable based 406 on approximate date of manufacture or other data. For example, it is 407 likely that a company that manfactures both computers and Ethernet 408 adapters will, at least internally, use its own adapters, which 409 significantly limits the range of built in addresses. 411 Problems such as those described above related to clocks and serial 412 numbers make code to produce unpredictable quantities difficult if 413 the code is to be ported across a variety of computer platforms and 414 systems. 416 4.2 Timing and Content of External Events 418 It is possible to measure the timing and content of mouse movement, 419 key strokes, and similar user events. This is a reasonable source of 420 unguessable data with some qualifications. On some machines, inputs 421 such as key strokes are buffered. Even though the user's inter- 422 keystroke timing may have sufficient variation and unpredictability, 423 there might not be an easy way to access that variation. Another 424 problem is that no standard method exists to sample timing details. 425 This makes it hard to build standard software intended for 426 distribution to a large range of machines based on this technique. 428 The amount of mouse movement or the keys actually hit are usually 429 easier to access than timings but may yield less unpredictability as 430 the user may provide highly repetitive input. 432 Other external events, such as network packet arrival times, can also 433 be used with care. In particular, the possibility of manipulation of 434 such times by an adversary and the lack of history on system start up 435 must be considered. 437 4.3 The Fallacy of Complex Manipulation 439 One strategy which may give a misleading appearance of 440 unpredictability is to take a very complex algorithm (or an excellent 441 traditional pseudo-random number generator with good statistical 442 properties) and calculate a cryptographic key by starting with the 443 current value of a computer system clock as the seed. An adversary 444 who knew roughly when the generator was started would have a 445 relatively small number of seed values to test as they would know 446 likely values of the system clock. Large numbers of pseudo-random 447 bits could be generated but the search space an adversary would need 448 to check could be quite small. 450 Thus very strong and/or complex manipulation of data will not help if 451 the adversary can learn what the manipulation is and there is not 452 enough unpredictability in the starting seed value. Even if they can 453 not learn what the manipulation is, they may be able to use the 454 limited number of results stemming from a limited number of seed 455 values to defeat security. 457 Another serious strategy error is to assume that a very complex 458 pseudo-random number generation algorithm will produce strong random 459 numbers when there has been no theory behind or analysis of the 460 algorithm. There is a excellent example of this fallacy right near 461 the beginning of chapter 3 in [KNUTH] where the author describes a 462 complex algorithm. It was intended that the machine language program 463 corresponding to the algorithm would be so complicated that a person 464 trying to read the code without comments wouldn't know what the 465 program was doing. Unfortunately, actual use of this algorithm 466 showed that it almost immediately converged to a single repeated 467 value in one case and a small cycle of values in another case. 469 Not only does complex manipulation not help you if you have a limited 470 range of seeds but blindly chosen complex manipulation can destroy 471 the randomness in a good seed! 473 4.4 The Fallacy of Selection from a Large Database 475 Another strategy that can give a misleading appearance of 476 unpredictability is selection of a quantity randomly from a database 477 and assume that its strength is related to the total number of bits 478 in the database. For example, typical USENET servers process many 479 megabytes of information per day. Assume a random quantity was 480 selected by fetching 32 bytes of data from a random starting point in 481 this data. This does not yield 32*8 = 256 bits worth of 482 unguessability. Even after allowing that much of the data is human 483 language and probably has no more than 2 or 3 bits of information per 484 byte, it doesn't yield 32*2 = 64 bits of unguessability. For an 485 adversary with access to the same usenet database the unguessability 486 rests only on the starting point of the selection. That is perhaps a 487 little over a couple of dozen bits of unguessability. 489 The same argument applies to selecting sequences from the data on a 490 publicly available CD/DVD recording or any other large public 491 database. If the adversary has access to the same database, this 492 "selection from a large volume of data" step buys very little. 493 However, if a selection can be made from data to which the adversary 494 has no access, such as system buffers on an active multi-user system, 495 it may be of help. 497 5. Hardware for Randomness 499 Is there any hope for true strong portable randomness in the future? 500 There might be. All that's needed is a physical source of 501 unpredictable numbers. 503 A thermal noise (sometimes called Johnson noise in integrated 504 circuits) or radioactive decay source and a fast, free-running 505 oscillator would do the trick directly [GIFFORD]. This is a trivial 506 amount of hardware, and could easily be included as a standard part 507 of a computer system's architecture. Furthermore, any system with a 508 spinning disk or ring oscillator and a stable (crystal) time source 509 or the like has an adequate source of randomness ([DAVIS] and Section 510 5.4). All that's needed is the common perception among computer 511 vendors that this small additional hardware and the software to 512 access it is necessary and useful. 514 5.1 Volume Required 516 How much unpredictability is needed? Is it possible to quantify the 517 requirement in, say, number of random bits per second? 519 The answer is not very much is needed. For AES, the key can be 128 520 bits and, as we show in an example in Section 8, even the highest 521 security system is unlikely to require a keying material of much over 522 200 bits. If a series of keys are needed, they can be generated from 523 a strong random seed using a cryptographically strong sequence as 524 explained in Section 6.3. A few hundred random bits generated at 525 start up or once a day would be enough using such techniques. Even 526 if the random bits are generated as slowly as one per second and it 527 is not possible to overlap the generation process, it should be 528 tolerable in high security applications to wait 200 seconds 529 occasionally. 531 These numbers are trivial to achieve. It could be done by a person 532 repeatedly tossing a coin. Almost any hardware process is likely to 533 be much faster. 535 5.2 Sensitivity to Skew 537 Is there any specific requirement on the shape of the distribution of 538 the random numbers? The good news is the distribution need not be 539 uniform. All that is needed is a conservative estimate of how non- 540 uniform it is to bound performance. Simple techniques to de-skew the 541 bit stream are given below and stronger techniques are mentioned in 542 Section 6.1.2 below. 544 5.2.1 Using Stream Parity to De-Skew 546 Consider taking a sufficiently long string of bits and map the string 547 to "zero" or "one". The mapping will not yield a perfectly uniform 548 distribution, but it can be as close as desired. One mapping that 549 serves the purpose is to take the parity of the string. This has the 550 advantages that it is robust across all degrees of skew up to the 551 estimated maximum skew and is absolutely trivial to implement in 552 hardware. 554 The following analysis gives the number of bits that must be sampled: 556 Suppose the ratio of ones to zeros is 0.5 + e : 0.5 - e, where e is 557 between 0 and 0.5 and is a measure of the "eccentricity" of the 558 distribution. Consider the distribution of the parity function of N 559 bit samples. The probabilities that the parity will be one or zero 560 will be the sum of the odd or even terms in the binomial expansion of 561 (p + q)^N, where p = 0.5 + e, the probability of a one, and q = 0.5 - 562 e, the probability of a zero. 564 These sums can be computed easily as 566 N N 567 1/2 * ( ( p + q ) + ( p - q ) ) 568 and 569 N N 570 1/2 * ( ( p + q ) - ( p - q ) ). 572 (Which one corresponds to the probability the parity will be 1 573 depends on whether N is odd or even.) 575 Since p + q = 1 and p - q = 2e, these expressions reduce to 577 N 578 1/2 * [1 + (2e) ] 579 and 580 N 581 1/2 * [1 - (2e) ]. 583 Neither of these will ever be exactly 0.5 unless e is zero, but we 584 can bring them arbitrarily close to 0.5. If we want the 585 probabilities to be within some delta d of 0.5, i.e. then 587 N 588 ( 0.5 + ( 0.5 * (2e) ) ) < 0.5 + d. 590 Solving for N yields N > log(2d)/log(2e). (Note that 2e is less than 591 1, so its log is negative. Division by a negative number reverses 592 the sense of an inequality.) 593 The following table gives the length of the string which must be 594 sampled for various degrees of skew in order to come within 0.001 of 595 a 50/50 distribution. 597 +---------+--------+-------+ 598 | Prob(1) | e | N | 599 +---------+--------+-------+ 600 | 0.5 | 0.00 | 1 | 601 | 0.6 | 0.10 | 4 | 602 | 0.7 | 0.20 | 7 | 603 | 0.8 | 0.30 | 13 | 604 | 0.9 | 0.40 | 28 | 605 | 0.95 | 0.45 | 59 | 606 | 0.99 | 0.49 | 308 | 607 +---------+--------+-------+ 609 The last entry shows that even if the distribution is skewed 99% in 610 favor of ones, the parity of a string of 308 samples will be within 611 0.001 of a 50/50 distribution. 613 5.2.2 Using Transition Mappings to De-Skew 615 Another technique, originally due to von Neumann [VON NEUMANN], is to 616 examine a bit stream as a sequence of non-overlapping pairs. You 617 could then discard any 00 or 11 pairs found, interpret 01 as a 0 and 618 10 as a 1. Assume the probability of a 1 is 0.5+e and the 619 probability of a 0 is 0.5-e where e is the eccentricity of the source 620 and described in the previous section. Then the probability of each 621 pair is as follows: 623 +------+-----------------------------------------+ 624 | pair | probability | 625 +------+-----------------------------------------+ 626 | 00 | (0.5 - e)^2 = 0.25 - e + e^2 | 627 | 01 | (0.5 - e)*(0.5 + e) = 0.25 - e^2 | 628 | 10 | (0.5 + e)*(0.5 - e) = 0.25 - e^2 | 629 | 11 | (0.5 + e)^2 = 0.25 + e + e^2 | 630 +------+-----------------------------------------+ 632 This technique will completely eliminate any bias but at the expense 633 of taking an indeterminate number of input bits for any particular 634 desired number of output bits. The probability of any particular 635 pair being discarded is 0.5 + 2e^2 so the expected number of input 636 bits to produce X output bits is X/(0.25 - e^2). 638 This technique assumes that the bits are from a stream where each bit 639 has the same probability of being a 0 or 1 as any other bit in the 640 stream and that bits are not correlated, i.e., that the bits are 641 identical independent distributions. If alternate bits were from two 642 correlated sources, for example, the above analysis breaks down. 644 The above technique also provides another illustration of how a 645 simple statistical analysis can mislead if one is not always on the 646 lookout for patterns that could be exploited by an adversary. If the 647 algorithm were mis-read slightly so that overlapping successive bits 648 pairs were used instead of non-overlapping pairs, the statistical 649 analysis given is the same; however, instead of providing an unbiased 650 uncorrelated series of random 1's and 0's, it instead produces a 651 totally predictable sequence of exactly alternating 1's and 0's. 653 5.2.3 Using FFT to De-Skew 655 When real world data consists of strongly biased or correlated bits, 656 it may still contain useful amounts of randomness. This randomness 657 can be extracted through use of the discrete Fourier transform or its 658 optimized variant, the FFT. 660 Using the Fourier transform of the data, strong correlations can be 661 discarded. If adequate data is processed and remaining correlations 662 decay, spectral lines approaching statistical independence and 663 normally distributed randomness can be produced [BRILLINGER]. 665 5.2.4 Using S-Boxes to De-Skew 667 Many modern block encryption functions, including DES and AES, 668 incorporate modules known as S-Boxes (substitution boxes). These 669 produce a smaller number of outputs from a larger number of inputs 670 through a complex non-linear mixing function which would have the 671 effect of concentrating limited entropy in the inputs into the 672 output. 674 S-Boxes sometimes incorporate bent boolean functions which are 675 functions of an even number of bits producing one output bit with 676 maximum non-linearity. Looking at the output for all input pairs 677 differing in any particular bit position, exactly half the outputs 678 are different. 680 An S-Box in which each output bit is produced by a bent function such 681 that any linear combination of these functions is also a bent 682 function is called a "perfect S-Box". Repeated application or 683 cascades of such boxes can be used to de-skew. [SBOX*] 685 5.2.5 Using Compression to De-Skew 687 Reversible compression techniques also provide a crude method of de- 688 skewing a skewed bit stream. This follows directly from the 689 definition of reversible compression and the formula in Section 2 690 above for the amount of information in a sequence. Since the 691 compression is reversible, the same amount of information must be 692 present in the shorter output than was present in the longer input. 693 By the Shannon information equation, this is only possible if, on 694 average, the probabilities of the different shorter sequences are 695 more uniformly distributed than were the probabilities of the longer 696 sequences. Thus the shorter sequences must be de-skewed relative to 697 the input. 699 However, many compression techniques add a somewhat predictable 700 preface to their output stream and may insert such a sequence again 701 periodically in their output or otherwise introduce subtle patterns 702 of their own. They should be considered only a rough technique 703 compared with those described above or in Section 6.1.2. At a 704 minimum, the beginning of the compressed sequence should be skipped 705 and only later bits used for applications requiring random bits. 707 5.3 Existing Hardware Can Be Used For Randomness 709 As described below, many computers come with hardware that can, with 710 care, be used to generate truly random quantities. 712 5.3.1 Using Existing Sound/Video Input 714 Increasingly computers are being built with inputs that digitize some 715 real world analog source, such as sound from a microphone or video 716 input from a camera. Under appropriate circumstances, such input can 717 provide reasonably high quality random bits. The "input" from a 718 sound digitizer with no source plugged in or a camera with the lens 719 cap on, if the system has enough gain to detect anything, is 720 essentially thermal noise. 722 For example, on a SPARCstation, one can read from the /dev/audio 723 device with nothing plugged into the microphone jack. Such data is 724 essentially random noise although it should not be trusted without 725 some checking in case of hardware failure. It will, in any case, 726 need to be de-skewed as described elsewhere. 728 Combining this with compression to de-skew one can, in UNIXese, 729 generate a huge amount of medium quality random data by doing 730 cat /dev/audio | compress - >random-bits-file 732 5.3.2 Using Existing Disk Drives 734 Disk drives have small random fluctuations in their rotational speed 735 due to chaotic air turbulence [DAVIS]. By adding low level disk seek 736 time instrumentation to a system, a series of measurements can be 737 obtained that include this randomness. Such data is usually highly 738 correlated so that significant processing is needed, such as FFT (see 739 section 5.2.3). Nevertheless experimentation has shown that, with 740 such processing, most disk drives easily produce 100 bits a minute or 741 more of excellent random data. 743 Partly offsetting this need for processing is the fact that disk 744 drive failure will normally be rapidly noticed. Thus, problems with 745 this method of random number generation due to hardware failure are 746 unlikely. 748 5.4 Ring Oscillator Sources 750 If an integrated circuit is being designed or field programmed, an 751 odd number of gates can be connected in series to produce a free- 752 running ring oscillator. By sampling a point in the ring at a 753 precise fixed frequency, say one determined by a stable crystal 754 oscialltor, some amount of entropy can be extracted due to slight 755 variations in the free-running osciallator. 757 Such bits will have to be heavily de-skewed as disk rotation timings 758 must be (Section 5.3.2). An engineering study would be needed to 759 determine the amount of entropy being produced depending on the 760 particular design. It may be possible to increase the rate of entropy 761 by xor'ing sampled values from a few ring osciallators with 762 relatively prime lengths or the like. In any case, this can be a 763 good, medium speed source whose cost is a trivial number of gates by 764 modern standards. 766 6. Recommended Software Strategy 768 What is the best overall strategy for meeting the requirement for 769 unguessable random numbers in the absence of a reliable hardware 770 source? It is to obtain random input from a number of uncorrelated 771 sources and to mix them with a strong mixing function. Such a 772 function will preserve the randomness present in any of the sources 773 even if other quantities being combined happen to be fixed or easily 774 guessable. This may be advisable even with a good hardware source, 775 as hardware can also fail, though this should be weighed against any 776 increase in the chance of overall failure due to added software 777 complexity. 779 6.1 Mixing Functions 781 A strong mixing function is one which combines two or more inputs and 782 produces an output where each output bit is a different complex non- 783 linear function of all the input bits. On average, changing any 784 input bit will change about half the output bits. But because the 785 relationship is complex and non-linear, no particular output bit is 786 guaranteed to change when any particular input bit is changed. 788 Consider the problem of converting a stream of bits that is skewed 789 towards 0 or 1 to a shorter stream which is more random, as discussed 790 in Section 5.2 above. This is simply another case where a strong 791 mixing function is desired, mixing the input bits to produce a 792 smaller number of output bits. The technique given in Section 5.2.1 793 of using the parity of a number of bits is simply the result of 794 successively Exclusive Or'ing them which is examined as a trivial 795 mixing function immediately below. Use of stronger mixing functions 796 to extract more of the randomness in a stream of skewed bits is 797 examined in Section 6.1.2. 799 6.1.1 A Trivial Mixing Function 801 A trivial example for single bit inputs is the Exclusive Or function, 802 which is equivalent to addition without carry, as show in the table 803 below. This is a degenerate case in which the one output bit always 804 changes for a change in either input bit. But, despite its 805 simplicity, it will still provide a useful illustration. 807 +-----------+-----------+----------+ 808 | input 1 | input 2 | output | 809 +-----------+-----------+----------+ 810 | 0 | 0 | 0 | 811 | 0 | 1 | 1 | 812 | 1 | 0 | 1 | 813 | 1 | 1 | 0 | 814 +-----------+-----------+----------+ 816 If inputs 1 and 2 are uncorrelated and combined in this fashion then 817 the output will be an even better (less skewed) random bit than the 818 inputs. If we assume an "eccentricity" e as defined in Section 5.2 819 above, then the output eccentricity relates to the input eccentricity 820 as follows: 822 e = 2 * e * e 823 output input 1 input 2 825 Since e is never greater than 1/2, the eccentricity is always 826 improved except in the case where at least one input is a totally 827 skewed constant. This is illustrated in the following table where 828 the top and left side values are the two input eccentricities and the 829 entries are the output eccentricity: 831 +--------+--------+--------+--------+--------+--------+--------+ 832 | e | 0.00 | 0.10 | 0.20 | 0.30 | 0.40 | 0.50 | 833 +--------+--------+--------+--------+--------+--------+--------+ 834 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 835 | 0.10 | 0.00 | 0.02 | 0.04 | 0.06 | 0.08 | 0.10 | 836 | 0.20 | 0.00 | 0.04 | 0.08 | 0.12 | 0.16 | 0.20 | 837 | 0.30 | 0.00 | 0.06 | 0.12 | 0.18 | 0.24 | 0.30 | 838 | 0.40 | 0.00 | 0.08 | 0.16 | 0.24 | 0.32 | 0.40 | 839 | 0.50 | 0.00 | 0.10 | 0.20 | 0.30 | 0.40 | 0.50 | 840 +--------+--------+--------+--------+--------+--------+--------+ 842 However, keep in mind that the above calculations assume that the 843 inputs are not correlated. If the inputs were, say, the parity of 844 the number of minutes from midnight on two clocks accurate to a few 845 seconds, then each might appear random if sampled at random intervals 846 much longer than a minute. Yet if they were both sampled and 847 combined with xor, the result would be zero most of the time. 849 6.1.2 Stronger Mixing Functions 851 The US Government Advanced Encryption Standard [AES] is an example of 852 a strong mixing function for multiple bit quantities. It takes up to 853 384 bits of input (128 bits of "data" and 256 bits of "key") and 854 produces 128 bits of output each of which is dependent on a complex 855 non-linear function of all input bits. Other encryption functions 856 with this characteristic, such as [DES], can also be used by 857 considering them to mix all of their key and data input bits. 859 Another good family of mixing functions are the "message digest" or 860 hashing functions such as The US Government Secure Hash Standards 861 [SHA*] and the MD4, MD5 [MD4, MD5] series. These functions all take 862 an arbitrary amount of input and produce an output mixing all the 863 input bits. The MD* series produce 128 bits of output, SHA-1 produces 864 160 bits, and other SHA functions produce larger numbers of bits. 866 Although the message digest functions are designed for variable 867 amounts of input, AES and other encryption functions can also be used 868 to combine any number of inputs. If 128 bits of output is adequate, 869 the inputs can be packed into a 128 bit data quantity and successive 870 AES keys, padding with zeros if needed, which are then used to 871 successively encrypt using AES in Electronic Codebook Mode [DES 872 MODES]. If more than 128 bits of output are needed, use more complex 873 mixing. For example, if inputs are packed into three quantities, A, 874 B, and C, use AES to encrypt A with B as a key and then with C as a 875 key to produce the 1st part of the output, then encrypt B with C and 876 then A for more output and, if necessary, encrypt C with A and then B 877 for yet more output. Still more output can be produced by reversing 878 the order of the keys given above to stretch things. The same can be 879 done with the hash functions by hashing various subsets of the input 880 data to produce multiple outputs. But keep in mind that it is 881 impossible to get more bits of "randomness" out than are put in. 883 An example of using a strong mixing function would be to reconsider 884 the case of a string of 308 bits each of which is biased 99% towards 885 zero. The parity technique given in Section 5.2.1 above reduced this 886 to one bit with only a 1/1000 deviance from being equally likely a 887 zero or one. But, applying the equation for information given in 888 Section 2, this 308 bit skewed sequence has over 5 bits of 889 information in it. Thus hashing it with SHA-1 and taking the bottom 890 5 bits of the result would yield 5 unbiased random bits as opposed to 891 the single bit given by calculating the parity of the string. 893 6.1.3 Diffie-Hellman as a Mixing Function 895 Diffie-Hellman exponential key exchange is a technique that yields a 896 shared secret between two parties that can be made computationally 897 infeasible for a third party to determine even if they can observe 898 all the messages between the two communicating parties. This shared 899 secret is a mixture of initial quantities generated by each of them 900 [D-H]. If these initial quantities are random, then the shared 901 secret contains the combined randomness of them both, assuming they 902 are uncorrelated. 904 6.1.4 Using a Mixing Function to Stretch Random Bits 906 While it is not necessary for a mixing function to produce the same 907 or fewer bits than its inputs, mixing bits cannot "stretch" the 908 amount of random unpredictability present in the inputs. Thus four 909 inputs of 32 bits each where there is 12 bits worth of 910 unpredicatability (such as 4,096 equally probable values) in each 911 input cannot produce more than 48 bits worth of unpredictable output. 912 The output can be expanded to hundreds or thousands of bits by, for 913 example, mixing with successive integers, but the clever adversary's 914 search space is still 2^48 possibilities. Furthermore, mixing to 915 fewer bits than are input will tend to strengthen the randomness of 916 the output the way using Exclusive Or to produce one bit from two did 917 above. 919 The last table in Section 6.1.1 shows that mixing a random bit with a 920 constant bit with Exclusive Or will produce a random bit. While this 921 is true, it does not provide a way to "stretch" one random bit into 922 more than one. If, for example, a random bit is mixed with a 0 and 923 then with a 1, this produces a two bit sequence but it will always be 924 either 01 or 10. Since there are only two possible values, there is 925 still only the one bit of original randomness. 927 6.1.5 Other Factors in Choosing a Mixing Function 929 For local use, AES has the advantages that it has been widely tested 930 for flaws, is reasonably efficient in software, and is widely 931 documented and implemented with hardware and software implementations 932 available all over the world including open source code. The SHA* 933 family are younger algorithms but there is no particular reason to 934 believe they are flawed. Both SHA* and MD5 were derived from the 935 earlier MD4 algorithm. Some signs of weakness have been found in MD4 936 and MD5. They all have source code available [SHA*, MD*]. 938 AES and SHA* have been vouched for the the US National Security 939 Agency (NSA) on the basis of criteria that primarily remain secret, 940 as was DES. While this has been the cause of much speculation and 941 doubt, investigation of DES over the years has indicated that NSA 942 involvement in modifications to its design, which originated with 943 IBM, was primarily to strengthen it. No concealed or special 944 weakness has been found in DES. It is very likely that the NSA 945 modifications to MD4 to produce the SHA* similarly strengthened these 946 algorithms, possibly against threats not yet known in the public 947 cryptographic community. 949 AES, DES, SHA*, MD4, and MD5 are believed to be royalty free for all 950 purposes. Continued advances in crypography and computing power have 951 cast doubts on MD4 and MD5 so their use is generally not recommended. 953 Another advantage of the SHA* or similar hashing algorithms over 954 encryption algorithms in the past was that they are not subject to 955 the same regulations imposed by the US Government prohibiting the 956 unlicensed export or import of encryption/decryption software and 957 hardware. 959 6.2 Non-Hardware Sources of Randomness 961 The best source of input for mixing would be a hardware randomness 962 such as disk drive timing effected by air turbulence, audio input 963 with thermal noise, or radioactive decay. However, if that is not 964 available there are other possibilities. These include system 965 clocks, system or input/output buffers, user/system/hardware/network 966 serial numbers and/or addresses and timing, and user input. 967 Unfortunately, any of these sources can produce limited or 968 predicatable values under some circumstances. 970 Some of the sources listed above would be quite strong on multi-user 971 systems where, in essence, each user of the system is a source of 972 randomness. However, on a small single user or embedded system, 973 especially at start up, it might be possible for an adversary to 974 assemble a similar configuration. This could give the adversary 975 inputs to the mixing process that were sufficiently correlated to 976 those used originally as to make exhaustive search practical. 978 The use of multiple random inputs with a strong mixing function is 979 recommended and can overcome weakness in any particular input. For 980 example, the timing and content of requested "random" user keystrokes 981 can yield hundreds of random bits but conservative assumptions need 982 to be made. For example, assuming at most a few bits of randomness 983 if the inter-keystroke interval is unique in the sequence up to that 984 point and a similar assumption if the key hit is unique but assuming 985 that no bits of randomness are present in the initial key value or if 986 the timing or key value duplicate previous values. The results of 987 mixing these timings and characters typed could be further combined 988 with clock values and other inputs. 990 This strategy may make practical portable code to produce good random 991 numbers for security even if some of the inputs are very weak on some 992 of the target systems. However, it may still fail against a high 993 grade attack on small single user or embedded systems, especially if 994 the adversary has ever been able to observe the generation process in 995 the past. A hardware based random source is still preferable. 997 6.3 Cryptographically Strong Sequences 999 In cases where a series of random quantities must be generated, an 1000 adversary may learn some values in the sequence. In general, they 1001 should not be able to predict other values from the ones that they 1002 know. 1004 The correct technique is to start with a strong random seed, take 1005 cryptographically strong steps from that seed [CRYPTO2, CRYPTO3], and 1006 do not reveal the complete state of the generator in the sequence 1007 elements. If each value in the sequence can be calculated in a fixed 1008 way from the previous value, then when any value is compromised, all 1009 future values can be determined. This would be the case, for 1010 example, if each value were a constant function of the previously 1011 used values, even if the function were a very strong, non-invertible 1012 message digest function. 1014 (It should be noted that if your technique for generating a sequence 1015 of key values is fast enough, it can trivially be used as the basis 1016 for a confidentiality system. If two parties use the same sequence 1017 generating technique and start with the same seed material, they will 1018 generate identical sequences. These could, for example, be xor'ed at 1019 one end with data being send, encrypting it, and xor'ed with this 1020 data as received, decrypting it due to the reversible properties of 1021 the xor operation.) 1023 6.3.1 Traditional Strong Sequences 1025 A traditional way to achieve a strong sequence has been to have the 1026 values be produced by hashing the quantities produced by 1027 concatenating the seed with successive integers or the like and then 1028 mask the values obtained so as to limit the amount of generator state 1029 available to the adversary. 1031 It may also be possible to use an "encryption" algorithm with a 1032 random key and seed value to encrypt and feedback some or all of the 1033 output encrypted value into the value to be encrypted for the next 1034 iteration. Appropriate feedback techniques will usually be 1035 recommended with the encryption algorithm. An example is shown below 1036 where shifting and masking are used to combine the cypher output 1037 feedback. This type of feedback was recommended by the US Government 1038 in connection with DES [DES MODES] but should be avoided for reasons 1039 described below. 1041 +---------------+ 1042 | V | 1043 | | n |--+ 1044 +--+------------+ | 1045 | | +---------+ 1046 | +---> | | +-----+ 1047 +--+ | Encrypt | <--- | Key | 1048 | +-------- | | +-----+ 1049 | | +---------+ 1050 V V 1051 +------------+--+ 1052 | V | | 1053 | n+1 | 1054 +---------------+ 1056 Note that if a shift of one is used, this is the same as the shift 1057 register technique described in Section 3 above but with the all 1058 important difference that the feedback is determined by a complex 1059 non-linear function of all bits rather than a simple linear or 1060 polynomial combination of output from a few bit position taps. 1062 It has been shown by Donald W. Davies that this sort of shifted 1063 partial output feedback significantly weakens an algorithm compared 1064 will feeding all of the output bits back as input. In particular, 1065 for DES, repeated encrypting a full 64 bit quantity will give an 1066 expected repeat in about 2^63 iterations. Feeding back anything less 1067 than 64 (and more than 0) bits will give an expected repeat in 1068 between 2**31 and 2**32 iterations! 1070 To predict values of a sequence from others when the sequence was 1071 generated by these techniques is equivalent to breaking the 1072 cryptosystem or inverting the "non-invertible" hashing involved with 1073 only partial information available. The less information revealed 1074 each iteration, the harder it will be for an adversary to predict the 1075 sequence. Thus it is best to use only one bit from each value. It 1076 has been shown that in some cases this makes it impossible to break a 1077 system even when the cryptographic system is invertible and can be 1078 broken if all of each generated value was revealed. 1080 6.3.2 The Blum Blum Shub Sequence Generator 1082 Currently the generator which has the strongest public proof of 1083 strength is called the Blum Blum Shub generator after its inventors 1084 [BBS]. It is also very simple and is based on quadratic residues. 1085 It's only disadvantage is that is is computationally intensive 1086 compared with the traditional techniques give in 6.3.1 above. This 1087 is not a major draw back if it is used for moderately infrequent 1088 purposes, such as generating session keys. 1090 Simply choose two large prime numbers, say p and q, which both have 1091 the property that you get a remainder of 3 if you divide them by 4. 1092 Let n = p * q. Then you choose a random number x relatively prime to 1093 n. The initial seed for the generator and the method for calculating 1094 subsequent values are then 1096 2 1097 s = ( x )(Mod n) 1098 0 1100 2 1101 s = ( s )(Mod n) 1102 i+1 i 1104 You must be careful to use only a few bits from the bottom of each s. 1105 It is always safe to use only the lowest order bit. If you use no 1106 more than the 1107 log ( log ( s ) ) 1108 2 2 i 1109 low order bits, then predicting any additional bits from a sequence 1110 generated in this manner is provable as hard as factoring n. As long 1111 as the initial x is secret, you can even make n public if you want. 1113 An intersting characteristic of this generator is that you can 1114 directly calculate any of the s values. In particular 1116 i 1117 ( ( 2 )(Mod (( p - 1 ) * ( q - 1 )) ) ) 1118 s = ( s )(Mod n) 1119 i 0 1121 This means that in applications where many keys are generated in this 1122 fashion, it is not necessary to save them all. Each key can be 1123 effectively indexed and recovered from that small index and the 1124 initial s and n. 1126 6.3.3 Entropy Pool Techniques 1128 Many modern pseudo random number sources utilize the technique of 1129 maintaining a "pool" of bits and providing operations for strongly 1130 mixing input with some randomness into the pool and extracting psuedo 1131 random bits from the pool. This is illustred in the figure below. 1133 +--------+ +------+ +---------+ 1134 --->| Mix In |--->| POOL |--->| Extract |---> 1135 | Bits | | | | Bits | 1136 +--------+ +------+ +---------+ 1137 ^ V 1138 | | 1139 +-----------+ 1141 Bits to be feed into the pool can be any of the various hardware, 1142 environmental, or user input sources discussed above. It is also 1143 common to save the state of the pool on shut down and restore it on 1144 re-starting, if stable storage is available. 1146 In fact, all of the [MD*] and [SHA*] message digest functions are 1147 implemented by internally maintaining a pool substantially larger 1148 than their ultimate output into which the bytes of the message are 1149 mixed and from which the ultimate message digest is extracted. Thus 1150 the figure above can be implemented by using parts of the message 1151 digest code to strongly mix in any new bit supplied and to compute 1152 output bits based on the pool. However, additional code is needed so 1153 that any number of bits can be extracted and appropriate feedback 1154 from the output process is mixed into the pool so as to produce a 1155 strong pseudo-random output stream. 1157 Care must be taken that enough entropy has been added to the pool to 1158 support particular output uses desired. See Section 7.3 for for more 1159 details on an example implementation and [RSA BULL1] for similar 1160 suggestions. 1162 7. Key Generation Standards and Examples 1164 Several public standards and widely deplyed examples are now in place 1165 for the generation of keys without special hardware. Two standards 1166 are described below. Both use DES but any equally strong or stronger 1167 mixing function could be substituted. Then a few widely deployed 1168 examples are described. 1170 7.1 US DoD Recommendations for Password Generation 1172 The United States Department of Defense has specific recommendations 1173 for password generation [DoD]. They suggest using the US Data 1174 Encryption Standard [DES] in Output Feedback Mode [DES MODES] as 1175 follows: 1177 use an initialization vector determined from 1178 the system clock, 1179 system ID, 1180 user ID, and 1181 date and time; 1182 use a key determined from 1183 system interrupt registers, 1184 system status registers, and 1185 system counters; and, 1186 as plain text, use an external randomly generated 64 bit 1187 quantity such as 8 characters typed in by a system 1188 administrator. 1190 The password can then be calculated from the 64 bit "cipher text" 1191 generated in 64-bit Output Feedback Mode. As many bits as are needed 1192 can be taken from these 64 bits and expanded into a pronounceable 1193 word, phrase, or other format if a human being needs to remember the 1194 password. 1196 7.2 X9.17 Key Generation 1198 The American National Standards Institute has specified a method for 1199 generating a sequence of keys as follows: 1201 s is the initial 64 bit seed 1202 0 1204 g is the sequence of generated 64 bit key quantities 1205 n 1207 k is a random key reserved for generating this key sequence 1208 t is the time at which a key is generated to as fine a resolution 1209 as is available (up to 64 bits). 1211 DES ( K, Q ) is the DES encryption of quantity Q with key K 1213 g = DES ( k, DES ( k, t ) .xor. s ) 1214 n n 1216 s = DES ( k, DES ( k, t ) .xor. g ) 1217 n+1 n 1219 If g sub n is to be used as a DES key, then every eighth bit should 1220 be adjusted for parity for that use but the entire 64 bit unmodified 1221 g should be used in calculating the next s. 1223 7.3 The /dev/random Device under Linux 1225 The Linux operating system provides a Kernel resident random number 1226 generator. This generator makes use of events captured by the Kernel 1227 during normal system operation. 1229 The generator consists of a random pool of bytes, by default 512 1230 bytes (represented as 128, 4 byte integers). When an event occurs, 1231 such as a disk drive interrupt, the time of the event is xor'ed into 1232 the pool and the pool is stirred via a primitive polynomial of degree 1233 128. The pool itself is treated as a ring buffer, with new data 1234 being xor'ed (after stirring with the polynomial) across the entire 1235 pool. 1237 Each call that adds entropy to the pool estimates the amount of 1238 likely true entropy the input contains. The pool itself contains a 1239 accumulator that estimates the total over all entropy of the pool. 1241 Input events come from several sources: 1243 1. Keyboard interrupts. The time of the interrupt as well as the scan 1244 code are added to the pool. This in effect adds entropy from the 1245 human operator by measuring inter-keystroke arrival times. 1247 2. Disk completion and other interrupts. A system being used by a 1248 person will likely have a hard to predict pattern of disk 1249 accesses. 1251 3. Mouse motion. The timing as well as mouse position is added in. 1253 When random bytes are required, the pool is hashed with SHA-1 [SHA1] 1254 to yield the returned bytes of randomness. If more bytes are required 1255 than the output of SHA-1 (20 bytes), then the hashed output is 1256 stirred back into the pool and a new hash performed to obtain the 1257 next 20 bytes. As bytes are removed from the pool, the estimate of 1258 entropy is similarly decremented. 1260 To ensure a reasonable random pool upon system startup, the standard 1261 Linux startup scripts (and shutdown scripts) save the pool to a disk 1262 file at shutdown and read this file at system startup. 1264 There are two user exported interfaces. /dev/random returns bytes 1265 from the pool, but blocks when the estimated entropy drops to zero. 1266 As entropy is added to the pool from events, more data becomes 1267 available via /dev/random. Random data obtained /dev/random is 1268 suitable for key generation for long term keys. 1270 /dev/urandom works like /dev/random, however it provides data even 1271 when the entropy estimate for the random pool drops to zero. This 1272 should be fine for session keys. The risk of continuing to take data 1273 even when the pool's entropy estimate is small is that past output 1274 may be computable from current output provided an attacker can 1275 reverse SHA-1. Given that SHA-1 should not be invertible, this is a 1276 reasonable risk. 1278 To obtain random numbers under Linux, all an application needs to do 1279 is open either /dev/random or /dev/urandom and read the desired 1280 number of bytes. 1282 The Linux Random device was written by Theodore Ts'o. It is based 1283 loosely on the random number generator in PGP 2.X and PGP 3.0 (aka 1284 PGP 5.0). 1286 8. Examples of Randomness Required 1288 Below are two examples showing rough calculations of needed 1289 randomness for security. The first is for moderate security 1290 passwords while the second assumes a need for a very high security 1291 cryptographic key. 1293 In addition [ORMAN] and [RSA BULL13] provide information on the 1294 public key lengths that should be used for exchanging symmetric keys. 1296 8.1 Password Generation 1298 Assume that user passwords change once a year and it is desired that 1299 the probability that an adversary could guess the password for a 1300 particular account be less than one in a thousand. Further assume 1301 that sending a password to the system is the only way to try a 1302 password. Then the crucial question is how often an adversary can 1303 try possibilities. Assume that delays have been introduced into a 1304 system so that, at most, an adversary can make one password try every 1305 six seconds. That's 600 per hour or about 15,000 per day or about 1306 5,000,000 tries in a year. Assuming any sort of monitoring, it is 1307 unlikely someone could actually try continuously for a year. In 1308 fact, even if log files are only checked monthly, 500,000 tries is 1309 more plausible before the attack is noticed and steps taken to change 1310 passwords and make it harder to try more passwords. 1312 To have a one in a thousand chance of guessing the password in 1313 500,000 tries implies a universe of at least 500,000,000 passwords or 1314 about 2^29. Thus 29 bits of randomness are needed. This can probably 1315 be achieved using the US DoD recommended inputs for password 1316 generation as it has 8 inputs which probably average over 5 bits of 1317 randomness each (see section 7.1). Using a list of 1000 words, the 1318 password could be expressed as a three word phrase (1,000,000,000 1319 possibilities) or, using case insensitive letters and digits, six 1320 would suffice ((26+10)^6 = 2,176,782,336 possibilities). 1322 For a higher security password, the number of bits required goes up. 1323 To decrease the probability by 1,000 requires increasing the universe 1324 of passwords by the same factor which adds about 10 bits. Thus to 1325 have only a one in a million chance of a password being guessed under 1326 the above scenario would require 39 bits of randomness and a password 1327 that was a four word phrase from a 1000 word list or eight 1328 letters/digits. To go to a one in 10^9 chance, 49 bits of randomness 1329 are needed implying a five word phrase or ten letter/digit password. 1331 In a real system, of course, there are also other factors. For 1332 example, the larger and harder to remember passwords are, the more 1333 likely users are to write them down resulting in an additional risk 1334 of compromise. 1336 8.2 A Very High Security Cryptographic Key 1338 Assume that a very high security key is needed for symmetric 1339 encryption / decryption between two parties. Assume an adversary can 1340 observe communications and knows the algorithm being used. Within 1341 the field of random possibilities, the adversary can try key values 1342 in hopes of finding the one in use. Assume further that brute force 1343 trial of keys is the best the adversary can do. 1345 8.2.1 Effort per Key Trial 1347 How much effort will it take to try each key? For very high security 1348 applications it is best to assume a low value of effort. This 1349 question is considered in detail in Appendix A. It concludes that a 1350 reasonable key length in 1995 for very high security is in the range 1351 of 75 to 90 bits and, since the cost of cryptography does not vary 1352 much with they key size, recommends 90 bits. To update these 1353 recommendations, just add 2/3 of a bit per year for Moore's law 1354 [MOORE]. Thus, in the year 2004, this translates to a determination 1355 that a reasonable key length is in 81 to 96 bit range. 1357 8.2.2 Meet in the Middle Attacks 1359 If chosen or known plain text and the resulting encrypted text are 1360 available, a "meet in the middle" attack is possible if the structure 1361 of the encryption algorithm allows it. (In a known plain text 1362 attack, the adversary knows all or part of the messages being 1363 encrypted, possibly some standard header or trailer fields. In a 1364 chosen plain text attack, the adversary can force some chosen plain 1365 text to be encrypted, possibly by "leaking" an exciting text that 1366 would then be sent by the adversary over an encrypted channel.) 1368 An oversimplified explanation of the meet in the middle attack is as 1369 follows: the adversary can half-encrypt the known or chosen plain 1370 text with all possible first half-keys, sort the output, then half- 1371 decrypt the encoded text with all the second half-keys. If a match 1372 is found, the full key can be assembled from the halves and used to 1373 decrypt other parts of the message or other messages. At its best, 1374 this type of attack can halve the exponent of the work required by 1375 the adversary while adding a large but roughly constant factor of 1376 effort. To be assured of safety against this, a doubling of the 1377 amount of randomness in the very strong key to a minimum of 162 bits 1378 is required for the year 2004 based on the Appendix A analysis. 1380 This amount of randomness is beyond the limit of that in the inputs 1381 recommended by the US DoD for password generation and could require 1382 user typing timing, hardware random number generation, or other 1383 sources. 1385 The meet in the middle attack assumes that the cryptographic 1386 algorithm can be decomposed in this way but we can not rule that out 1387 without a deep knowledge of the algorithm. Even if a basic algorithm 1388 is not subject to a meet in the middle attack, an attempt to produce 1389 a stronger algorithm by applying the basic algorithm twice (or two 1390 different algorithms sequentially) with different keys may gain less 1391 added security than would be expected. Such a composite algorithm 1392 would be subject to a meet in the middle attack. 1394 Enormous resources may be required to mount a meet in the middle 1395 attack but they are probably within the range of the national 1396 security services of a major nation. Essentially all nations spy on 1397 other nations government traffic and several nations are believed to 1398 spy on commercial traffic for economic advantage. 1400 It should be noted that key length calculations such at those above 1401 are controversial and depend on various assumptions about the 1402 cryptographic algorithms in use. In some cases, a professional with 1403 a deep knowledge of code breaking techniques and of the strength of 1404 the algorithm in use could be satisfied with less than half of the 1405 162 bit key size derived above. 1407 9. Conclusion 1409 Generation of unguessable "random" secret quantities for security use 1410 is an essential but difficult task. 1412 Hardware techniques to produce such randomness would be relatively 1413 simple. In particular, the volume and quality would not need to be 1414 high and existing computer hardware, such as disk drives, can be 1415 used. 1417 Computational techniques are available to process low quality random 1418 quantities from multiple sources or a larger quantity of such low 1419 quality input from one source and produce a smaller quantity of 1420 higher quality keying material. In the absence of hardware sources 1421 of randomness, a variety of user and software sources can frequently, 1422 with care, be used instead; however, most modern systems already have 1423 hardware, such as disk drives or audio input, that could be used to 1424 produce high quality randomness. 1426 Once a sufficient quantity of high quality seed key material (a 1427 couple of hundred bits) is available, computational techniques are 1428 available to produce cryptographically strong sequences of 1429 unpredicatable quantities from this seed material. 1431 10. Security Considerations 1433 The entirety of this document concerns techniques and recommendations 1434 for generating unguessable "random" quantities for use as passwords, 1435 cryptographic keys, initialiazation vectors, sequence numbers, and 1436 similar security uses. 1438 Intellectual Property Considerations 1440 The IETF takes no position regarding the validity or scope of any 1441 intellectual property or other rights that might be claimed to 1442 pertain to the implementation or use of the technology described in 1443 this document or the extent to which any license under such rights 1444 might or might not be available; neither does it represent that it 1445 has made any effort to identify any such rights. Information on the 1446 IETF's procedures with respect to rights in standards-track and 1447 standards-related documentation can be found in BCP-11. Copies of 1448 claims of rights made available for publication and any assurances of 1449 licenses to be made available, or the result of an attempt made to 1450 obtain a general license or permission for the use of such 1451 proprietary rights by implementors or users of this specification can 1452 be obtained from the IETF Secretariat. 1454 The IETF invites any interested party to bring to its attention any 1455 copyrights, patents or patent applications, or other proprietary 1456 rights which may cover technology that may be required to practice 1457 this standard. Please address the information to the IETF Executive 1458 Director. 1460 Appendix: Minimal Secure Key Lengths Study 1462 Minimal Key Lengths for Symmetric Ciphers 1463 to Provide Adequate Commercial Security 1465 A Report by an Ad Hoc Group of 1466 Cryptographers and Computer Scientists 1468 Matt Blaze, AT&T Research, mab@research.att.com 1469 Whitfield Diffie, Sun Microsystems, diffie@eng.sun.com 1470 Ronald L. Rivest, MIT LCS, rivest@lcs.mit.edu 1471 Bruce Schneier, Counterpane Systems, schneier@counterpane.com 1472 Tsutomu Shimomura, San Diego Supercomputer Center, tsutomu@sdsc.edu 1473 Eric Thompson Access Data, Inc., eric@accessdata.com 1474 Michael Wiener, Bell Northern Research, wiener@bnr.ca 1476 January 1996 1478 A.0 Abstract 1480 Encryption plays an essential role in protecting the privacy of 1481 electronic information against threats from a variety of potential 1482 attackers. In so doing, modern cryptography employs a combination of 1483 _conventional_ or _symmetric_ cryptographic systems for encrypting 1484 data and _public key_ or _asymmetric_ systems for managing the _keys_ 1485 used by the symmetric systems. Assessing the strength required of 1486 the symmetric cryptographic systems is therefore an essential step in 1487 employing cryptography for computer and communication security. 1489 Technology readily available today (late 1995) makes _brute- 1490 force_ attacks against cryptographic systems considered adequate for 1491 the past several years both fast and cheap. General purpose 1492 computers can be used, but a much more efficient approach is to 1493 employ commercially available _Field Programmable Gate Array (FPGA)_ 1494 technology. For attackers prepared to make a higher initial 1495 investment, custom-made, special-purpose chips make such calculations 1496 much faster and significantly lower the amortized cost per solution. 1498 As a result, cryptosystems with 40-bit keys offer virtually no 1499 protection at this point against brute-force attacks. Even the U.S. 1500 Data Encryption Standard with 56-bit keys is increasingly inadequate. 1501 As cryptosystems often succumb to `smarter' attacks than brute-force 1502 key search, it is also important to remember that the keylengths 1503 discussed here are the minimum needed for security against the 1504 computational threats considered. 1506 Fortunately, the cost of very strong encryption is not 1508 significantly greater than that of weak encryption. Therefore, to 1509 provide adequate protection against the most serious threats --- 1510 well-funded commercial enterprises or government intelligence 1511 agencies --- keys used to protect data today should be at least 75 1512 bits long. To protect information adequately for the next 20 years 1513 in the face of expected advances in computing power, keys in newly- 1514 deployed systems should be at least 90 bits long. 1516 A.1. Encryption Plays an Essential Role in Protecting 1517 the Privacy of Electronic Information" 1519 A.1.1 There is a need for information security 1521 As we write this paper in late 1995, the development of 1522 electronic commerce and the Global Information Infrastructure is at a 1523 critical juncture. The dirt paths of the middle ages only became 1524 highways of business and culture after the security of travelers and 1525 the merchandise they carried could be assured. So too the 1526 information superhighway will be an ill-traveled road unless 1527 information, the goods of the Information Age, can be moved, stored, 1528 bought, and sold securely. Neither corporations nor individuals will 1529 entrust their private business or personal data to computer networks 1530 unless they can assure their information's security. 1532 Today, most forms of information can be stored and processed 1533 electronically. This means a wide variety of information, with 1534 varying economic values and privacy aspects and with a wide variation 1535 in the time over which the information needs to be protected, will be 1536 found on computer networks. Consider the spectrum: 1538 o Electronic Funds Transfers of millions or even billions of 1539 dollars, whose short term security is essential but whose 1540 exposure is brief; 1542 o A company's strategic corporate plans, whose confidentiality 1543 must be preserved for a small number of years; 1545 o A proprietary product (Coke formula, new drug design) that 1546 needs to be protected over its useful life, often decades; 1547 and 1549 o Information private to an individual (medical condition, 1550 employment evaluation) that may need protection for the 1551 lifetime of the individual. 1553 A.1.2 Encryption to protect confidentiality 1555 Encryption Can Provide Strong Confidentiality Protection 1557 Encryption is accomplished by scrambling data using mathematical 1558 procedures that make it extremely difficult and time consuming for 1559 anyone other than authorized recipients --- those with the correct 1560 decryption _keys_ --- to recover the _plain text_. Proper encryption 1561 guarantees that the information will be safe even if it falls into 1562 hostile hands. 1564 Encryption --- and decryption --- can be performed by either 1565 computer software or hardware. Common approaches include writing the 1566 algorithm on a disk for execution by a computer central processor; 1567 placing it in ROM or PROM for execution by a microprocessor; and 1568 isolating storage and execution in a computer accessory device (smart 1569 card or PCMCIA card). 1571 The degree of protection obtained depends on several factors. 1572 These include: the quality of the cryptosystem; the way it is 1573 implemented in software or hardware (especially its reliability and 1574 the manner in which the keys are chosen); and the total number of 1575 possible keys that can be used to encrypt the information. A 1576 cryptographic algorithm is considered strong if: 1578 1. There is no shortcut that allows the opponent to recover the 1579 plain text without using brute force to test keys until the 1580 correct one is found; and 1582 2. The number of possible keys is sufficiently large to make 1583 such an attack infeasible. 1585 The principle here is similar to that of a combination lock on a 1586 safe. If the lock is well designed so that a burglar cannot hear or 1587 feel its inner workings, a person who does not know the combination 1588 can open it only by dialing one set of numbers after another until it 1589 yields. 1591 The sizes of encryption keys are measured in bits and the 1592 difficulty of trying all possible keys grows exponentially with the 1593 number of bits used. Adding one bit to the key doubles the number of 1594 possible keys; adding ten increases it by a factor of more than a 1595 thousand. 1597 There is no definitive way to look at a cipher and determine 1598 whether a shortcut exists. Nonetheless, several encryption 1599 algorithms --- most notably the U.S Data Encryption Standard (DES) 1600 --- have been extensively studied in the public literature and are 1601 widely believed to be of very high quality. An essential element in 1602 cryptographic algorithm design is thus the length of the key, whose 1603 size places an upper bound on the system's strength. 1605 Throughout this paper, we will assume that there are no shortcuts 1606 and treat the length of the key as representative of the 1607 cryptosystem's _workfactor_ --- the minimum amount of effort required 1608 to break the system. It is important to bear in mind, however, that 1609 cryptographers regard this as a rash assumption and many would 1610 recommend keys two or more times as long as needed to resist brute- 1611 force attacks. Prudent cryptographic designs not only employ longer 1612 keys than might appear to be needed, but devote more computation to 1613 encrypting and decrypting. A good example of this is the popular 1614 approach of using _triple-DES_: encrypting the output of DES twice 1615 more, using a total of three distinct keys. 1617 Encryption systems fall into two broad classes. Conventional or 1618 symmetric cryptosystems --- those in which an entity with the ability 1619 to encrypt also has the ability to decrypt and vice versa --- are the 1620 systems under consideration in this paper. The more recent public 1621 key or asymmetric cryptosystems have the property that the ability to 1622 encrypt does not imply the ability to decrypt. In contemporary 1623 cryptography, public-key systems are indispensable for managing the 1624 keys of conventional cryptosystems. All known public key 1625 cryptosystems, however, are subject to shortcut attacks and must 1626 therefore use keys ten or more times the lengths of those discussed 1627 here to achieve the an equivalent level of security. 1629 Although computers permit electronic information to be encrypted 1630 using very large keys, advances in computing power keep pushing up 1631 the size of keys that can be considered large and thus keep making it 1632 easier for individuals and organizations to attack encrypted 1633 information without the expenditure of unreasonable resources. 1635 A.1.3 There are a variety of attackers 1637 There Are Threats from a Variety of Potential Attackers. 1639 Threats to confidentiality of information come from a number of 1640 directions and their forms depend on the resources of the attackers. 1641 `Hackers,' who might be anything from high school students to 1642 commercial programmers, may have access to mainframe computers or 1643 networks of workstations. The same people can readily buy 1644 inexpensive, off-the-shelf, boards, containing _Field Programmable 1645 Gate Array (FPGA)_ chips that function as `programmable hardware' and 1646 vastly increase the effectiveness of a cryptanalytic effort. A 1647 startup company or even a well-heeled individual could afford large 1648 numbers of these chips. A major corporation or organized crime 1649 operation with `serious money' to spend could acquire custom computer 1650 chips specially designed for decryption. An intelligence agency, 1651 engaged in espionage for national economic advantage, could build a 1652 machine employing millions of such chips. 1654 A.1.4 Strong encryption is not expensive 1656 Current Technology Permits Very Strong Encryption for Effectively the 1657 Same Cost As Weaker Encryption. 1659 It is a property of computer encryption that modest increases in 1660 computational cost can produce vast increases in security. 1661 Encrypting information very securely (e.g., with 128-bit keys) 1662 typically requires little more computing than encrypting it weakly 1663 (e.g., with 40-bit keys). In many applications, the cryptography 1664 itself accounts for only a small fraction of the computing costs, 1665 compared to such processes as voice or image compression required to 1666 prepare material for encryption. 1668 One consequence of this uniformity of costs is that there is 1669 rarely any need to tailor the strength of cryptography to the 1670 sensitivity of the information being protected. Even if most of the 1671 information in a system has neither privacy implications nor monetary 1672 value, there is no practical or economic reason to design computer 1673 hardware or software to provide differing levels of encryption for 1674 different messages. It is simplest, most prudent, and thus 1675 fundamentally most economical, to employ a uniformly high level of 1676 encryption: the strongest encryption required for any information 1677 that might be stored or transmitted by a secure system. 1679 A.2. Brute-Force is becoming easier 1681 Readily Available Technology Makes Brute-Force Decryption Attacks 1682 Faster and Cheaper. 1684 The kind of hardware used to mount a brute-force attack against 1685 an encryption algorithm depends on the scale of the cryptanalytic 1686 operation and the total funds available to the attacking enterprise. 1687 In the analysis that follows, we consider three general classes of 1688 technology that are likely to be employed by attackers with differing 1689 resources available to them. Not surprisingly, the cryptanalytic 1690 technologies that require larger up-front investments yield the 1691 lowest cost per recovered key, amortized over the life of the 1692 hardware. 1694 It is the nature of brute-force attacks that they can be 1695 parallelized indefinitely. It is possible to use as many machines as 1696 are available, assigning each to work on a separate part of the 1697 problem. Thus regardless of the technology employed, the search time 1698 can be reduced by adding more equipment; twice as much hardware can 1699 be expected to find the right key in half the time. The total 1700 investment will have doubled, but if the hardware is kept constantly 1701 busy finding keys, the cost per key recovered is unchanged. 1703 At the low end of the technology spectrum is the use of 1704 conventional personal computers or workstations programmed to test 1705 keys. Many people, by virtue of already owning or having access to 1706 the machines, are in a position use such resources at little or no 1707 cost. However, general purpose computers --- laden with such 1708 ancillary equipment as video controllers, keyboards, interfaces, 1709 memory, and disk storage --- make expensive search engines. They are 1710 therefore likely to be employed only by casual attackers who are 1711 unable or unwilling to invest in more specialized equipment. 1713 A more efficient technological approach is to take advantage of 1714 commercially available Field Programmable Gate Arrays. FPGAs 1715 function as programmable hardware and allow faster implementations of 1716 such tasks as encryption and decryption than conventional processors. 1717 FPGAs are a commonly used tool for simple computations that need to 1718 be done very quickly, particularly simulating integrated circuits 1719 during development. 1721 FPGA technology is fast and cheap. The cost of an AT&T ORCA chip 1722 that can test 30 million DES keys per second is $200. This is 1,000 1723 times faster than a PC at about one-tenth the cost! FPGAs are widely 1724 available and, mounted on cards, can be installed in standard PCs 1725 just like sound cards, modems, or extra memory. 1727 FPGA technology may be optimal when the same tool must be used 1728 for attacking a variety of different cryptosystems. Often, as with 1729 DES, a cryptosystem is sufficiently widely used to justify the 1730 construction of more specialized facilities. In these circumstances, 1731 the most cost-effective technology, but the one requiring the largest 1732 initial investment, is the use of _Application-Specific Integrated 1733 Circuits (ASICs)_. A $10 chip can test 200 million keys per second. 1734 This is seven times faster than an FPGA chip at one-twentieth the 1735 cost. 1737 Because ASICs require a far greater engineering investment than 1738 FPGAs and must be fabricated in quantity before they are economical, 1739 this approach is only available to serious, well-funded operations 1740 such as dedicated commercial (or criminal) enterprises and government 1741 intelligence agencies. 1743 A.3. 40-Bit Key Lengths Offer Virtually No Protection 1745 Current U.S. Government policy generally limits exportable mass 1746 market software that incorporates encryption for confidentiality to 1747 using the RC2 or RC4 algorithms with 40-bit keys. A 40-bit key 1748 length means that there are 2^40 possible keys. On average, half of 1749 these (2^39) must be tried to find the correct one. Export of other 1750 algorithms and key lengths must be approved on a case by case basis. 1751 For example, DES with a 56-bit key has been approved for certain 1752 applications such as financial transactions. 1754 The recent successful brute-force attack by two French graduate 1755 students on Netscape's 40-bit RC4 algorithm demonstrates the dangers 1756 of such short keys. These students at the Ecole Polytechnique in 1757 Paris used `idle time' on the school's computers, incurring no cost 1758 to themselves or their school. Even with these limited resources, 1759 they were able to recover the 40-bit key in a few days. 1761 There is no need to have the resources of an institution of 1762 higher education at hand, however. Anyone with a modicum of computer 1763 expertise and a few hundred dollars would be able to attack 40-bit 1764 encryption much faster. An FPGA chip --- costing approximately $400 1765 mounted on a card --- would on average recover a 40-bit key in five 1766 hours. Assuming the FPGA lasts three years and is used continuously 1767 to find keys, the average cost per key is eight cents. 1769 A more determined commercial predator, prepared to spend $10,000 1770 for a set-up with 25 ORCA chips, can find 40-bit keys in an average 1771 of 12 minutes, at the same average eight cent cost. Spending more 1772 money to buy more chips reduces the time accordingly: $300,000 1773 results in a solution in an average of 24 seconds; $10,000,000 1774 results in an average solution in 0.7 seconds. 1776 As already noted, a corporation with substantial resources can 1777 design and commission custom chips that are much faster. By doing 1778 this, a company spending $300,000 could find the right 40-bit key in 1779 an average of 0.18 seconds at 1/10th of a cent per solution; a larger 1780 company or government agency willing to spend $10,000,000 could find 1781 the right key on average in 0.005 seconds (again at 1/10th of a cent 1782 per solution). (Note that the cost per solution remains constant 1783 because we have conservatively assumed constant costs for chip 1784 acquisition --- in fact increasing the quantities purchased of a 1785 custom chip reduces the average chip cost as the initial design and 1786 set-up costs are spread over a greater number of chips.) 1788 These results are summarized in Table I (below). 1790 A.4. Even DES with 56-Bit Keys Is Increasingly Inadequate 1792 A.4.1 DES is no panacea today 1794 The Data Encryption Standard (DES) was developed in the 1970s by 1795 IBM and NSA and adopted by the U.S. Government as a Federal 1796 Information Processing Standard for data encryption. It was intended 1797 to provide strong encryption for the government's sensitive but 1798 unclassified information. It was recognized by many, even at the 1799 time DES was adopted, that technological developments would make 1800 DES's 56-bit key exceedingly vulnerable to attack before the end of 1801 the century. 1803 Today, DES may be the most widely employed encryption algorithm 1804 and continues to be a commonly cited benchmark. Yet DES-like 1805 encryption strength is no panacea. Calculations show that DES is 1806 inadequate against a corporate or government attacker committing 1807 serious resources. The bottom line is that DES is cheaper and easier 1808 to break than many believe. 1810 As explained above, 40-bit encryption provides inadequate 1811 protection against even the most casual of intruders, content to 1812 scavenge time on idle machines or to spend a few hundred dollars. 1813 Against such opponents, using DES with a 56-bit key will provide a 1814 substantial measure of security. At present, it would take a year 1815 and a half for someone using $10,000 worth of FPGA technology to 1816 search out a DES key. In ten years time an investment of this size 1817 would allow one to find a DES key in less than a week. 1819 The real threat to commercial transactions and to privacy on the 1820 Internet is from individuals and organizations willing to invest 1821 substantial time and money. As more and more business and personal 1822 information becomes electronic, the potential rewards to a dedicated 1823 commercial predator also increase significantly and may justify the 1824 commitment of adequate resources. 1826 A serious effort --- on the order of $300,000 --- by a legitimate 1827 or illegitimate business could find a DES key in an average of 19 1828 days using off-the-shelf technology and in only 3 hours using a 1829 custom developed chip. In the latter case, it would cost $38 to find 1830 each key (again assuming a 3 year life to the chip and continual 1831 use). A business or government willing to spend $10,000,000 on 1832 custom chips, could recover DES keys in an average of 6 minutes, for 1833 the same $38 per key. 1835 At the very high end, an organization --- presumably a government 1836 intelligence agency --- willing to spend $300,000,000 could recover 1837 DES keys in 12 seconds each! The investment required is large but 1838 not unheard of in the intelligence community. It is less than the 1839 cost of the Glomar Explorer, built to salvage a single Russian 1840 submarine, and far less than the cost of many spy satellites. Such 1841 an expense might be hard to justify in attacking a single target, but 1842 seems entirely appropriate against a cryptographic algorithm, like 1843 DES, enjoying extensive popularity around the world. 1845 There is ample evidence of the danger presented by government 1846 intelligence agencies seeking to obtain information not only for 1847 military purposes but for commercial advantage. Congressional 1848 hearings in 1993 highlighted instances in which the French and 1849 Japanese governments spied on behalf of their countries' own 1850 businesses. Thus, having to protect commercial information against 1851 such threats is not a hypothetical proposition. 1853 A.4.2 There are smarter avenues of attack than brute force 1855 It is easier to walk around a tree than climb up and down it. 1856 There is no need to break the window of a house to get in if the 1857 front door is unlocked. 1859 Calculations regarding the strength of encryption against brute- 1860 force attack are _worst case_ scenarios. They assume that the 1861 ciphers are in a sense perfect and that attempts to find shortcuts 1862 have failed. One important point is that the crudest approach --- 1863 searching through the keys --- is entirely feasible against many 1864 widely used systems. Another is that the keylengths we discuss are 1865 always minimal. As discussed earlier, prudent designs might use keys 1866 twice or three times as long to provide a margin of safety. 1868 A.4.3 Other algorithms are similar 1870 The Analysis for Other Algorithms Is Roughly Comparable. 1872 The above analysis has focused on the time and money required to 1873 find a key to decrypt information using the RC4 algorithm with a 40- 1874 bit key or the DES algorithm with its 56-bit key, but the results are 1875 not peculiar to these ciphers. Although each algorithm has its own 1876 particular characteristics, the effort required to find the keys of 1877 other ciphers is comparable. There may be some differences as the 1878 result of implementation procedures, but these do not materially 1879 affect the brute-force breakability of algorithms with roughly 1880 comparable key lengths. 1882 Specifically, it has been suggested at times that differences in 1883 set-up procedures, such as the long key-setup process in RC4, result 1884 in some algorithms having effectively longer keys than others. For 1885 the purpose of our analysis, such factors appear to vary the 1886 effective key length by no more than about eight bits. 1888 A.5. Appropriate Key Lengths for the Future --- A Proposal 1890 Table I summarizes the costs of carrying out brute-force attacks 1891 against symmetric cryptosystems with 40-bit and 56-bit keys using 1892 networks of general purpose computers, Field Programmable Gate 1893 Arrays, and special-purpose chips. 1895 It shows that 56 bits provides a level of protection --- about a 1896 year and a half --- that would be adequate for many commercial 1897 purposes against an opponent prepared to invest $10,000. Against an 1898 opponent prepared to invest $300,000, the period of protection has 1899 dropped to the barest minimum of 19 days. Above this, the protection 1900 quickly declines to negligible. A very large, but easily imaginable, 1901 investment by an intelligence agency would clearly allow it to 1902 recover keys in real time. 1904 What workfactor would be required for security today? For an 1905 opponent whose budget lay in the $10 to 300 million range, the time 1906 required to search out keys in a 75-bit keyspace would be between 6 1907 years and 70 days. Although the latter figure may seem comparable to 1908 the `barest minimum' 19 days mentioned earlier, it represents --- 1909 under our amortization assumptions --- a cost of $19 million and a 1910 recovery rate of only five keys a year. The victims of such an 1911 attack would have to be fat targets indeed. 1913 Because many kinds of information must be kept confidential for 1914 long periods of time, assessment cannot be limited to the protection 1915 required today. Equally important, cryptosystems --- especially if 1916 they are standards --- often remain in use for years or even decades. 1917 DES, for example, has been in use for more than 20 years and will 1918 probably continue to be employed for several more. In particular, 1919 the lifetime of a cryptosystem is likely to exceed the lifetime of 1920 any individual product embodying it. 1922 A rough estimate of the minimum strength required as a function 1923 of time can be obtained by applying an empirical rule, popularly 1924 called `Moore's Law,' which holds that the computing power available 1925 for a given cost doubles every 18 months. Taking into account both 1926 the lifetime of cryptographic equipment and the lifetime of the 1927 secrets it protects, we believe it is prudent to require that 1928 encrypted data should still be secure in 20 years. Moore's Law thus 1929 predicts that the keys should be approximately 14 bits longer than 1930 required to protect against an attack today. 1932 *Bearing in mind that the additional computational costs of 1933 stronger encryption are modest, we strongly recommend a minimum key- 1934 length of 90 bits for symmetric cryptosystems.* 1936 It is instructive to compare this recommendation with both 1937 Federal Information Processing Standard 46, The Data Encryption 1938 Standard (DES), and Federal Information Processing Standard 185, The 1939 Escrowed Encryption Standard (EES). DES was proposed 21 years ago 1940 and used a 56-bit key. Applying Moore's Law and adding 14 bits, we 1941 see that the strength of DES when it was proposed in 1975 was 1942 comparable to that of a 70-bit system today. Furthermore, it was 1943 estimated at the time that DES was not strong enough and that keys 1944 could be recovered at a rate of one per day for an investment of 1945 about twenty-million dollars. Our 75-bit estimate today corresponds 1946 to 61 bits in 1975, enough to have moved the cost of key recovery 1947 just out of reach. The Escrowed Encryption Standard, while 1948 unacceptable to many potential users for other reasons, embodies a 1949 notion of appropriate key length that is similar to our own. It uses 1950 80-bit keys, a number that lies between our figures of 75 and 90 1951 bits. 1953 Table I 1955 Time and cost Length Needed 1956 Type of Budget Tool per key recovered for protection 1957 Attacker 40bits 56bits in Late 1995 1959 Pedestrian Hacker 1961 tiny scavenged 1 week infeasible 45 1962 computer 1963 time 1965 $400 FPGA 5 hours 38 years 50 1966 ($0.08) ($5,000) 1968 Small Business 1970 $10,000 FPGA 12 minutes 556 days 55 1971 ($0.08) ($5,000) 1973 Corporate Department 1975 $300K FPGA 24 seconds 19 days 60 1976 or ($0.08) ($5,000) 1977 ASIC .18 seconds 3 hours 1978 ($0.001) ($38) 1980 Big Company 1981 $10M FPGA .7 seconds 13 hours 70 1982 or ($0.08) ($5,000) 1983 ASIC .005 seconds 6 minutes 1984 ($0.001) ($38) 1986 Intellegence Agency 1988 $300M ASIC .0002 seconds 12 seconds 75 1989 ($0.001) ($38) 1991 A.6 About the Authors 1993 *Matt Blaze* is a senior research scientist at AT&T Research in the 1994 area of computer security and cryptography. Recently Blaze 1995 demonstrated weaknesses in the U.S. government's `Clipper Chip' key 1996 escrow encryption system. His current interests include large-scale 1997 trust management and the applications of smartcards. 1999 *Whitfield Diffie* is a distinguished Engineer at Sun Microsystems 2000 specializing in security. In 1976 Diffie and Martin Hellman created 2001 public key cryptography, which solved the problem of sending coded 2002 information between individuals with no prior relationship and is the 2003 basis for widespread encryption in the digital information age. 2005 *Ronald L. Rivest* is a professor of computer science at the 2006 Massachusetts Institute of Technology, and is Associate Director of 2007 MIT's Laboratory for Computer Science. Rivest, together with Leonard 2008 Adleman and Adi Shamir, invented the RSA public-key cryptosystem that 2009 is used widely throughout industry. Ron Rivest is one of the 2010 founders of RSA Data Security Inc. and is the creator of variable key 2011 length symmetric key ciphers (e.g., RC4). 2013 *Bruce Schneier* is president of Counterpane Systems, a consulting 2014 firm specializing in cryptography and computer security. Schneier 2015 writes and speaks frequently on computer security and privacy and is 2016 the author of a leading cryptography textbook, Applied Cryptography, 2017 and is the creator of the symmetric key cipher Blowfish. 2019 *Tsutomu Shimomura* is a computational physicist employed by the San 2020 Diego Supercomputer Center who is an expert in designing software 2021 security tools. Last year, Shimomura was responsible for tracking 2022 down the computer outlaw Kevin Mitnick, who electronically stole and 2023 altered valuable electronic information around the country. 2025 *Eric Thompson* heads AccessData Corporation's cryptanalytic team and 2026 is a frequent lecturer on applied crytography. AccessData 2027 specializes in data recovery and decrypting information utilizing 2028 brute force as well as `smarter' attacks. Regular clients include 2029 the FBI and other law enforcement agencies as well as corporations. 2031 *Michael Wiener* is a cryptographic advisor at Bell-Northern Research 2032 where he focuses on cryptanalysis, security architectures, and 2033 public-key infrastructures. His influential 1993 paper, Efficient 2034 DES Key Search, describes in detail how to construct a machine to 2035 brute force crack DES coded information (and provides cost estimates 2036 as well). 2038 A.7 Acknowledgement 2040 The [Appendix] authors would like to thank the Business Software 2041 Alliance, which provided support for a one-day meeting, held in 2042 Chicago on 20 November 1995. 2044 Informative References 2046 [AES] - "Specification of the Advanced Encryption Standard (AES)", 2047 United States of America, Department of Commerce, National Institute 2048 of Standards and Technology, Federal Information Processing Standard 2049 197, November 2001. 2051 [ASYMMETRIC] - "Secure Communications and Asymmetric Cryptosystems", 2052 edited by Gustavus J. Simmons, AAAS Selected Symposium 69, Westview 2053 Press, Inc. 2055 [BBS] - "A Simple Unpredictable Pseudo-Random Number Generator", SIAM 2056 Journal on Computing, v. 15, n. 2, 1986, L. Blum, M. Blum, & M. Shub. 2058 [BRILLINGER] - "Time Series: Data Analysis and Theory", Holden-Day, 2059 1981, David Brillinger. 2061 [CRC] - "C.R.C. Standard Mathematical Tables", Chemical Rubber 2062 Publishing Company. 2064 [CRYPTO1] - "Cryptography: A Primer", A Wiley-Interscience 2065 Publication, John Wiley & Sons, 1981, Alan G. Konheim. 2067 [CRYPTO2] - "Cryptography: A New Dimension in Computer Data 2068 Security", A Wiley-Interscience Publication, John Wiley & Sons, 1982, 2069 Carl H. Meyer & Stephen M. Matyas. 2071 [CRYPTO3] - "Applied Cryptography: Protocols, Algorithsm, and Source 2072 Code in C", Second Edition, John Wiley & Sons, 1996, Bruce Schneier. 2074 [DAVIS] - "Cryptographic Randomness from Air Turbulence in Disk 2075 Drives", Advances in Cryptology - Crypto '94, Springer-Verlag Lecture 2076 Notes in Computer Science #839, 1984, Don Davis, Ross Ihaka, and 2077 Philip Fenstermacher. 2079 [DES] - "Data Encryption Standard", United States of America, 2080 Department of Commerce, National Institute of Standards and 2081 Technology, Federal Information Processing Standard (FIPS) 46-3, 2082 October 1999. 2083 - "Data Encryption Algorithm", American National Standards Institute, 2084 ANSI X3.92-1981. 2085 (See also FIPS 112, Password Usage, which includes FORTRAN code for 2086 performing DES.) 2088 [DES MODES] - "DES Modes of Operation", United States of America, 2089 Department of Commerce, National Institute of Standards and 2090 Technology, Federal Information Processing Standard (FIPS) 81, 2091 December 1980. 2092 - Data Encryption Algorithm - Modes of Operation, American National 2093 Standards Institute, ANSI X3.106-1983. 2095 [D-H] - "New Directions in Cryptography", IEEE Transactions on 2096 Information Technology, November, 1976, Whitfield Diffie and Martin 2097 E. Hellman. 2099 [DNSSEC] - RFC 2535, "Domain Name System Security Extensions", D. 2100 Eastlake, March 1999. 2102 [DoD] - "Password Management Guideline", United States of America, 2103 Department of Defense, Computer Security Center, CSC-STD-002-85. 2104 (See also FIPS 112, Password Usage, which incorporates CSC-STD-002-85 2105 as one of its appendices.) 2107 [DSS] - "Digital Signature Standard (DSS)", United States of America, 2108 Department of Commerce, National Institute of Standards and 2109 Technoloy, Federal Information Processing Standard (FIPS) 186-2, 2110 January 2000. 2112 [GIFFORD] - "Natural Random Number", MIT/LCS/TM-371, September 1988, 2113 David K. Gifford 2115 [IPSEC] - RFC 2401, "Security Architecture for the Internet 2116 Protocol", S. Kent, R. Atkinson, November 1998 2118 [KNUTH] - "The Art of Computer Programming", Volume 2: Seminumerical 2119 Algorithms, Chapter 3: Random Numbers. Addison Wesley Publishing 2120 Company, Second Edition 1982, Donald E. Knuth. 2122 [KRAWCZYK] - "How to Predict Congruential Generators", Journal of 2123 Algorithms, V. 13, N. 4, December 1992, H. Krawczyk 2125 [MAIL PEM] - RFCs 1421 through 1424: 2126 - RFC 1424, Privacy Enhancement for Internet Electronic Mail: Part 2127 IV: Key Certification and Related Services, 02/10/1993, B. Kaliski 2128 - RFC 1423, Privacy Enhancement for Internet Electronic Mail: Part 2129 III: Algorithms, Modes, and Identifiers, 02/10/1993, D. Balenson 2130 - RFC 1422, Privacy Enhancement for Internet Electronic Mail: Part 2131 II: Certificate-Based Key Management, 02/10/1993, S. Kent 2132 - RFC 1421, Privacy Enhancement for Internet Electronic Mail: Part I: 2133 Message Encryption and Authentication Procedures, 02/10/1993, J. Linn 2135 [MAIL PGP] - RFC 2440, "OpenPGP Message Format", J. Callas, L. 2136 Donnerhacke, H. Finney, R. Thayer", November 1998 2138 [MAIL S/MIME] - RFC 2633, "S/MIME Version 3 Message Specification", 2139 B. Ramsdell, Ed., June 1999. 2141 [MD4] - "The MD4 Message-Digest Algorithm", RFC1320, April 1992, R. 2142 Rivest 2143 [MD5] - "The MD5 Message-Digest Algorithm", RFC1321, April 1992, R. 2144 Rivest 2146 [MOORE] - Moore's Law: the exponential increase the logic density of 2147 silicon circuts. Originally formulated by Gordon Moore in 1964 as a 2148 doubling every year starting in 1962, in the late 1970s the rate fell 2149 to a doubling every 18 months and has remained there through the date 2150 of this document. See "The New Hacker's Dictionary", Third Edition, 2151 MIT Press, ISBN 0-262-18178-9, Eric S. Raymondm 1996. 2153 [ORMAN] - "Determining Strengths For Public Keys Used For Exchanging 2154 Symmetric Keys", draft-orman-public-key-lengths-*.txt, Hilarie Orman, 2155 Paul Hoffman, work in progress. 2157 [RFC 1750] - "Randomness Requirements for Security", D. Eastlake, S. 2158 Crocker, J. Schiller, December 1994. 2160 [RSA BULL1] - "Suggestions for Random Number Generation in Software", 2161 RSA Laboratories Bulletin #1, January 1996. 2163 [RSA BULL13] - "A Cost-Based Security Analysis of Symmetric and 2164 Asymmetric Key Lengths", RSA Laboratories Bulletin #13, Robert 2165 Silverman, April 2000 (revised November 2001). 2167 [SBOX1] - "Practical s-box design", S. Mister, C. Adams, Selected 2168 Areas in Cryptography, 1996. 2169 [SBOX2] - "Perfect Non-linear S-boxes", K. Nyberg, Advances in 2170 Cryptography - Eurocrypt '91 Proceedings, Springer-Verland, 1991. 2172 [SHANNON] - "The Mathematical Theory of Communication", University of 2173 Illinois Press, 1963, Claude E. Shannon. (originally from: Bell 2174 System Technical Journal, July and October 1948) 2176 [SHIFT1] - "Shift Register Sequences", Aegean Park Press, Revised 2177 Edition 1982, Solomon W. Golomb. 2179 [SHIFT2] - "Cryptanalysis of Shift-Register Generated Stream Cypher 2180 Systems", Aegean Park Press, 1984, Wayne G. Barker. 2182 [SHA-1] - "Secure Hash Standard (SHA-1)", United States of American, 2183 National Institute of Science and Technology, Federal Information 2184 Processing Standard (FIPS) 180-1, April 1993. 2185 - RFC 3174, "US Secure Hash Algorithm 1 (SHA1)", D. Eastlake, 2186 P. Jones, September 2001. 2188 [SHA-2] - "Secure Hash Standard", Draft (SHA-2156/384/512), Federal 2189 Information Processing Standard 180-2, not yet issued. 2191 [SSH] - draft-ietf-secsh-*, work in progress. 2193 [STERN] - "Secret Linear Congruential Generators are not 2194 Cryptograhically Secure", Proceedings of IEEE STOC, 1987, J. Stern. 2196 [TLS] - RFC 2246, "The TLS Protocol Version 1.0", T. Dierks, C. 2197 Allen, January 1999. 2199 [VON NEUMANN] - "Various techniques used in connection with random 2200 digits", von Neumann's Collected Works, Vol. 5, Pergamon Press, 1963, 2201 J. von Neumann. 2203 Authors Addresses 2205 Donald E. Eastlake 3rd 2206 Motorola Laboratories 2207 155 Beaver Street 2208 Milford, MA 01757 USA 2210 Telephone: +1 508-786-7554 (w) 2211 +1 508-634-2066 (h) 2212 EMail: Donald.Eastlake@motorola.com 2214 Jeffrey I. Schiller 2215 MIT, Room E40-311 2216 77 Massachusetts Avenue 2217 Cambridge, MA 02139-4307 USA 2219 Telephone: +1 617-253-0161 2220 E-mail: jis@mit.edu 2222 Steve Crocker 2224 EMail: steve@stevecrocker.com 2226 File Name and Expiration 2228 This is file draft-eastlake-randomness2-05.txt. 2230 It expires June 2004.