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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 February 2004 January 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 alpahbetic 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 Diff-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..................................28 128 More Table of Contents 130 7.3 The /dev/random Device under Linux....................29 132 8. Examples of Randomness Required........................31 133 8.1 Password Generation..................................31 134 8.2 A Very High Security Cryptographic Key................32 135 8.2.1 Effort per Key Trial................................32 136 8.2.2 Meet in the Middle Attacks..........................32 138 9. Conclusion.............................................34 139 10. Security Considerations...............................34 140 Intellectual Property Considerations......................34 142 Appendix: Minimal Secure Key Lengths Study................36 143 A.0 Abstract..............................................36 144 A.1. Encryption Plays an Essential Role in Protecting.....37 145 A.1.1 There is a need for information security............37 146 A.1.2 Encryption to protect confidentiality...............38 147 A.1.3 There are a variety of attackers....................39 148 A.1.4 Strong encryption is not expensive..................40 149 A.2. Brute-Forece is becoming easier......................40 150 A.3. 40-Bit Key Lengths Offer Virtually No Protection.....42 151 A.4. Even DES with 56-Bit Keys Is Increasingly Inadequate.43 152 A.4.1 DES is no panacea today.............................43 153 A.4.2 There are smarter avenues of attack than brute 154 force...............................................44 155 A.4.3 Other algorithms are similar........................44 156 A.5. Appropriate Key Lengths for the Future --- A 157 Proposal.............................................45 158 A.6 About the Authors.....................................47 159 A.7 Acknowledgement.......................................48 161 Informative References....................................49 163 Authors Addresses.........................................53 164 File Name and Expiration..................................53 166 1. Introduction 168 Software cryptography is coming into wider use and is continuing to 169 spread, although there is a long way to go until it becomes 170 pervasive. 172 Systems like SSH, IPSEC, TLS, S/MIME, PGP, DNSSEC, Kerberos, etc. are 173 maturing and becoming a part of the network landscape [SSH, DNSSEC, 174 IPSEC, MAIL*, TLS]. By comparison, when the previous version of this 175 document [RFC 1750] was issued in 1994, about the only Internet 176 cryptographic security specification in the IETF was the Privacy 177 Enhanced Mail protocol [MAIL PEM]. 179 These systems provide substantial protection against snooping and 180 spoofing. However, there is a potential flaw. At the heart of all 181 cryptographic systems is the generation of secret, unguessable (i.e., 182 random) numbers. 184 For the present, the lack of generally available facilities for 185 generating such unpredictable numbers is an open wound in the design 186 of cryptographic software. For the software developer who wants to 187 build a key or password generation procedure that runs on a wide 188 range of hardware, the only safe strategy so far has been to force 189 the local installation to supply a suitable routine to generate 190 random numbers. To say the least, this is an awkward, error-prone 191 and unpalatable solution. 193 It is important to keep in mind that the requirement is for data that 194 an adversary has a very low probability of guessing or determining. 195 This can easily fail if pseudo-random data is used which only meets 196 traditional statistical tests for randomness or which is based on 197 limited range sources, such as clocks. Frequently such random 198 quantities are determinable by an adversary searching through an 199 embarrassingly small space of possibilities. 201 This Best Current Practice describes techniques for producing random 202 quantities that will be resistant to such attack. It recommends that 203 future systems include hardware random number generation or provide 204 access to existing hardware that can be used for this purpose. It 205 suggests methods for use if such hardware is not available. And it 206 gives some estimates of the number of random bits required for sample 207 applications. 209 2. Requirements 211 A commonly encountered randomness requirement today is the user 212 password. This is usually a simple character string. Obviously, if a 213 password can be guessed, it does not provide security. (For re- 214 usable passwords, it is desirable that users be able to remember the 215 password. This may make it advisable to use pronounceable character 216 strings or phrases composed on ordinary words. But this only affects 217 the format of the password information, not the requirement that the 218 password be very hard to guess.) 220 Many other requirements come from the cryptographic arena. 221 Cryptographic techniques can be used to provide a variety of services 222 including confidentiality and authentication. Such services are 223 based on quantities, traditionally called "keys", that are unknown to 224 and unguessable by an adversary. 226 In some cases, such as the use of symmetric encryption with the one 227 time pads [CRYPTO*] or the US Data Encryption Standard [DES] or 228 Advanced Encryption Standard [AES], the parties who wish to 229 communicate confidentially and/or with authentication must all know 230 the same secret key. In other cases, using what are called 231 asymmetric or "public key" cryptographic techniques, keys come in 232 pairs. One key of the pair is private and must be kept secret by one 233 party, the other is public and can be published to the world. It is 234 computationally infeasible to determine the private key from the 235 public key and knowledge of the public is of no help to an adversary. 236 [ASYMMETRIC, CRYPTO*] 238 The frequency and volume of the requirement for random quantities 239 differs greatly for different cryptographic systems. Using pure RSA 240 [CRYPTO*], random quantities are required when the key pair is 241 generated, but thereafter any number of messages can be signed 242 without a further need for randomness. The public key Digital 243 Signature Algorithm devised by the US National Institute of Standards 244 and Technology (NIST) requires good random numbers for each signature 245 [DSS]. And encrypting with a one time pad, in principle the 246 strongest possible encryption technique, requires a volume of 247 randomness equal to all the messages to be processed [CRYPTO*]. 249 In most of these cases, an adversary can try to determine the 250 "secret" key by trial and error. (This is possible as long as the 251 key is enough smaller than the message that the correct key can be 252 uniquely identified.) The probability of an adversary succeeding at 253 this must be made acceptably low, depending on the particular 254 application. The size of the space the adversary must search is 255 related to the amount of key "information" present in the information 256 theoretic sense [SHANNON]. This depends on the number of different 257 secret values possible and the probability of each value as follows: 259 ----- 260 \ 261 Bits-of-info = \ - p * log ( p ) 262 / i 2 i 263 / 264 ----- 266 where i counts from 1 to the number of possible secret values and p 267 sub i is the probability of the value numbered i. (Since p sub i is 268 less than one, the log will be negative so each term in the sum will 269 be non-negative.) 271 If there are 2^n different values of equal probability, then n bits 272 of information are present and an adversary would, on the average, 273 have to try half of the values, or 2^(n-1) , before guessing the 274 secret quantity. If the probability of different values is unequal, 275 then there is less information present and fewer guesses will, on 276 average, be required by an adversary. In particular, any values that 277 the adversary can know are impossible, or are of low probability, can 278 be initially ignored by an adversary, who will search through the 279 more probable values first. 281 For example, consider a cryptographic system that uses 128 bit keys. 282 If these 128 bit keys are derived by using a fixed pseudo-random 283 number generator that is seeded with an 8 bit seed, then an adversary 284 needs to search through only 256 keys (by running the pseudo-random 285 number generator with every possible seed), not the 2^128 keys that 286 may at first appear to be the case. Only 8 bits of "information" are 287 in these 128 bit keys. 289 3. Traditional Pseudo-Random Sequences 291 Most traditional sources of random numbers use deterministic sources 292 of "pseudo-random" numbers. These typically start with a "seed" 293 quantity and use numeric or logical operations to produce a sequence 294 of values. 296 [KNUTH] has a classic exposition on pseudo-random numbers. 297 Applications he mentions are simulation of natural phenomena, 298 sampling, numerical analysis, testing computer programs, decision 299 making, and games. None of these have the same characteristics as 300 the sort of security uses we are talking about. Only in the last two 301 could there be an adversary trying to find the random quantity. 302 However, in these cases, the adversary normally has only a single 303 chance to use a guessed value. In guessing passwords or attempting 304 to break an encryption scheme, the adversary normally has many, 305 perhaps unlimited, chances at guessing the correct value because they 306 can store the message they are trying to break and repeatedly attack 307 it. They should also be assumed to be aided by a computer. 309 For testing the "randomness" of numbers, Knuth suggests a variety of 310 measures including statistical and spectral. These tests check 311 things like autocorrelation between different parts of a "random" 312 sequence or distribution of its values. But they could be met by a 313 constant stored random sequence, such as the "random" sequence 314 printed in the CRC Standard Mathematical Tables [CRC]. 316 A typical pseudo-random number generation technique, known as a 317 linear congruence pseudo-random number generator, is modular 318 arithmetic where the value numbered N+1 is calculated from the value 319 numbered N by 321 V = ( V * a + b )(Mod c) 322 N+1 N 324 The above technique has a strong relationship to linear shift 325 register pseudo-random number generators, which are well understood 326 cryptographically [SHIFT*]. In such generators bits are introduced 327 at one end of a shift register as the Exclusive Or (binary sum 328 without carry) of bits from selected fixed taps into the register. 329 For example: 331 +----+ +----+ +----+ +----+ 332 | B | <-- | B | <-- | B | <-- . . . . . . <-- | B | <-+ 333 | 0 | | 1 | | 2 | | n | | 334 +----+ +----+ +----+ +----+ | 335 | | | | 336 | | V +-----+ 337 | V +----------------> | | 338 V +-----------------------------> | XOR | 339 +---------------------------------------------------> | | 340 +-----+ 342 V = ( ( V * 2 ) + B .xor. B ... )(Mod 2^n) 343 N+1 N 0 2 345 The goodness of traditional pseudo-random number generator algorithms 346 is measured by statistical tests on such sequences. Carefully chosen 347 values of the initial V and a, b, and c or the placement of shift 348 register tap in the above simple processes can produce excellent 349 statistics. 351 These sequences may be adequate in simulations (Monte Carlo 352 experiments) as long as the sequence is orthogonal to the structure 353 of the space being explored. Even there, subtle patterns may cause 354 problems. However, such sequences are clearly bad for use in 355 security applications. They are fully predictable if the initial 356 state is known. Depending on the form of the pseudo-random number 357 generator, the sequence may be determinable from observation of a 358 short portion of the sequence [CRYPTO*, STERN]. For example, with 359 the generators above, one can determine V(n+1) given knowledge of 360 V(n). In fact, it has been shown that with these techniques, even if 361 only one bit of the pseudo-random values are released, the seed can 362 be determined from short sequences. 364 Not only have linear congruent generators been broken, but techniques 365 are now known for breaking all polynomial congruent generators. 366 [KRAWCZYK] 368 4. Unpredictability 370 Randomness in the traditional sense described in section 3 is NOT the 371 same as the unpredictability required for security use. 373 For example, use of a widely available constant sequence, such as 374 that from the CRC tables, is very weak against an adversary. Once 375 they learn of or guess it, they can easily break all security, future 376 and past, based on the sequence. [CRC] Yet the statistical properties 377 of these tables are good. 379 The following sections describe the limitations of some randomness 380 generation techniques and sources. 382 4.1 Problems with Clocks and Serial Numbers 384 Computer clocks, or similar operating system or hardware values, 385 provide significantly fewer real bits of unpredictability than might 386 appear from their specifications. 388 Tests have been done on clocks on numerous systems and it was found 389 that their behavior can vary widely and in unexpected ways. One 390 version of an operating system running on one set of hardware may 391 actually provide, say, microsecond resolution in a clock while a 392 different configuration of the "same" system may always provide the 393 same lower bits and only count in the upper bits at much lower 394 resolution. This means that successive reads on the clock may 395 produce identical values even if enough time has passed that the 396 value "should" change based on the nominal clock resolution. There 397 are also cases where frequently reading a clock can produce 398 artificial sequential values because of extra code that checks for 399 the clock being unchanged between two reads and increases it by one! 400 Designing portable application code to generate unpredictable numbers 401 based on such system clocks is particularly challenging because the 402 system designer does not always know the properties of the system 403 clocks that the code will execute on. 405 Use of a hardware serial number such as an Ethernet address may also 406 provide fewer bits of uniqueness than one would guess. Such 407 quantities are usually heavily structured and subfields may have only 408 a limited range of possible values or values easily guessable based 409 on approximate date of manufacture or other data. For example, it is 410 likely that a company that manfactures both computers and Ethernet 411 adapters will, at least internally, use its own adapters, which 412 significantly limits the range of built in addresses. 414 Problems such as those described above related to clocks and serial 415 numbers make code to produce unpredictable quantities difficult if 416 the code is to be ported across a variety of computer platforms and 417 systems. 419 4.2 Timing and Content of External Events 421 It is possible to measure the timing and content of mouse movement, 422 key strokes, and similar user events. This is a reasonable source of 423 unguessable data with some qualifications. On some machines, inputs 424 such as key strokes are buffered. Even though the user's inter- 425 keystroke timing may have sufficient variation and unpredictability, 426 there might not be an easy way to access that variation. Another 427 problem is that no standard method exists to sample timing details. 428 This makes it hard to build standard software intended for 429 distribution to a large range of machines based on this technique. 431 The amount of mouse movement or the keys actually hit are usually 432 easier to access than timings but may yield less unpredictability as 433 the user may provide highly repetitive input. 435 Other external events, such as network packet arrival times, can also 436 be used with care. In particular, the possibility of manipulation of 437 such times by an adversary and the lack of history on system start up 438 must be considered. 440 4.3 The Fallacy of Complex Manipulation 442 One strategy which may give a misleading appearance of 443 unpredictability is to take a very complex algorithm (or an excellent 444 traditional pseudo-random number generator with good statistical 445 properties) and calculate a cryptographic key by starting with the 446 current value of a computer system clock as the seed. An adversary 447 who knew roughly when the generator was started would have a 448 relatively small number of seed values to test as they would know 449 likely values of the system clock. Large numbers of pseudo-random 450 bits could be generated but the search space an adversary would need 451 to check could be quite small. 453 Thus very strong and/or complex manipulation of data will not help if 454 the adversary can learn what the manipulation is and there is not 455 enough unpredictability in the starting seed value. Even if they can 456 not learn what the manipulation is, they may be able to use the 457 limited number of results stemming from a limited number of seed 458 values to defeat security. 460 Another serious strategy error is to assume that a very complex 461 pseudo-random number generation algorithm will produce strong random 462 numbers when there has been no theory behind or analysis of the 463 algorithm. There is a excellent example of this fallacy right near 464 the beginning of chapter 3 in [KNUTH] where the author describes a 465 complex algorithm. It was intended that the machine language program 466 corresponding to the algorithm would be so complicated that a person 467 trying to read the code without comments wouldn't know what the 468 program was doing. Unfortunately, actual use of this algorithm 469 showed that it almost immediately converged to a single repeated 470 value in one case and a small cycle of values in another case. 472 Not only does complex manipulation not help you if you have a limited 473 range of seeds but blindly chosen complex manipulation can destroy 474 the randomness in a good seed! 476 4.4 The Fallacy of Selection from a Large Database 478 Another strategy that can give a misleading appearance of 479 unpredictability is selection of a quantity randomly from a database 480 and assume that its strength is related to the total number of bits 481 in the database. For example, typical USENET servers process many 482 megabytes of information per day. Assume a random quantity was 483 selected by fetching 32 bytes of data from a random starting point in 484 this data. This does not yield 32*8 = 256 bits worth of 485 unguessability. Even after allowing that much of the data is human 486 language and probably has no more than 2 or 3 bits of information per 487 byte, it doesn't yield 32*2 = 64 bits of unguessability. For an 488 adversary with access to the same usenet database the unguessability 489 rests only on the starting point of the selection. That is perhaps a 490 little over a couple of dozen bits of unguessability. 492 The same argument applies to selecting sequences from the data on a 493 publicly available CD/DVD recording or any other large public 494 database. If the adversary has access to the same database, this 495 "selection from a large volume of data" step buys very little. 496 However, if a selection can be made from data to which the adversary 497 has no access, such as system buffers on an active multi-user system, 498 it may be of help. 500 5. Hardware for Randomness 502 Is there any hope for true strong portable randomness in the future? 503 There might be. All that's needed is a physical source of 504 unpredictable numbers. 506 A thermal noise (sometimes called Johnson noise in integrated 507 circuits) or radioactive decay source and a fast, free-running 508 oscillator would do the trick directly [GIFFORD]. This is a trivial 509 amount of hardware, and could easily be included as a standard part 510 of a computer system's architecture. Furthermore, any system with a 511 spinning disk or ring oscillator and a stable (crystal) time source 512 or the like has an adequate source of randomness ([DAVIS] and Section 513 5.4). All that's needed is the common perception among computer 514 vendors that this small additional hardware and the software to 515 access it is necessary and useful. 517 5.1 Volume Required 519 How much unpredictability is needed? Is it possible to quantify the 520 requirement in, say, number of random bits per second? 522 The answer is not very much is needed. For AES, the key can be 128 523 bits and, as we show in an example in Section 8, even the highest 524 security system is unlikely to require a keying material of much over 525 200 bits. If a series of keys are needed, they can be generated from 526 a strong random seed using a cryptographically strong sequence as 527 explained in Section 6.3. A few hundred random bits generated at 528 start up or once a day would be enough using such techniques. Even 529 if the random bits are generated as slowly as one per second and it 530 is not possible to overlap the generation process, it should be 531 tolerable in high security applications to wait 200 seconds 532 occasionally. 534 These numbers are trivial to achieve. It could be done by a person 535 repeatedly tossing a coin. Almost any hardware process is likely to 536 be much faster. 538 5.2 Sensitivity to Skew 540 Is there any specific requirement on the shape of the distribution of 541 the random numbers? The good news is the distribution need not be 542 uniform. All that is needed is a conservative estimate of how non- 543 uniform it is to bound performance. Simple techniques to de-skew the 544 bit stream are given below and stronger techniques are mentioned in 545 Section 6.1.2 below. 547 5.2.1 Using Stream Parity to De-Skew 549 Consider taking a sufficiently long string of bits and map the string 550 to "zero" or "one". The mapping will not yield a perfectly uniform 551 distribution, but it can be as close as desired. One mapping that 552 serves the purpose is to take the parity of the string. This has the 553 advantages that it is robust across all degrees of skew up to the 554 estimated maximum skew and is absolutely trivial to implement in 555 hardware. 557 The following analysis gives the number of bits that must be sampled: 559 Suppose the ratio of ones to zeros is 0.5 + e : 0.5 - e, where e is 560 between 0 and 0.5 and is a measure of the "eccentricity" of the 561 distribution. Consider the distribution of the parity function of N 562 bit samples. The probabilities that the parity will be one or zero 563 will be the sum of the odd or even terms in the binomial expansion of 564 (p + q)^N, where p = 0.5 + e, the probability of a one, and q = 0.5 - 565 e, the probability of a zero. 567 These sums can be computed easily as 569 N N 570 1/2 * ( ( p + q ) + ( p - q ) ) 571 and 572 N N 573 1/2 * ( ( p + q ) - ( p - q ) ). 575 (Which one corresponds to the probability the parity will be 1 576 depends on whether N is odd or even.) 578 Since p + q = 1 and p - q = 2e, these expressions reduce to 580 N 581 1/2 * [1 + (2e) ] 582 and 583 N 584 1/2 * [1 - (2e) ]. 586 Neither of these will ever be exactly 0.5 unless e is zero, but we 587 can bring them arbitrarily close to 0.5. If we want the 588 probabilities to be within some delta d of 0.5, i.e. then 590 N 591 ( 0.5 + ( 0.5 * (2e) ) ) < 0.5 + d. 593 Solving for N yields N > log(2d)/log(2e). (Note that 2e is less than 594 1, so its log is negative. Division by a negative number reverses 595 the sense of an inequality.) 596 The following table gives the length of the string which must be 597 sampled for various degrees of skew in order to come within 0.001 of 598 a 50/50 distribution. 600 +---------+--------+-------+ 601 | Prob(1) | e | N | 602 +---------+--------+-------+ 603 | 0.5 | 0.00 | 1 | 604 | 0.6 | 0.10 | 4 | 605 | 0.7 | 0.20 | 7 | 606 | 0.8 | 0.30 | 13 | 607 | 0.9 | 0.40 | 28 | 608 | 0.95 | 0.45 | 59 | 609 | 0.99 | 0.49 | 308 | 610 +---------+--------+-------+ 612 The last entry shows that even if the distribution is skewed 99% in 613 favor of ones, the parity of a string of 308 samples will be within 614 0.001 of a 50/50 distribution. 616 5.2.2 Using Transition Mappings to De-Skew 618 Another technique, originally due to von Neumann [VON NEUMANN], is to 619 examine a bit stream as a sequence of non-overlapping pairs. You 620 could then discard any 00 or 11 pairs found, interpret 01 as a 0 and 621 10 as a 1. Assume the probability of a 1 is 0.5+e and the 622 probability of a 0 is 0.5-e where e is the eccentricity of the source 623 and described in the previous section. Then the probability of each 624 pair is as follows: 626 +------+-----------------------------------------+ 627 | pair | probability | 628 +------+-----------------------------------------+ 629 | 00 | (0.5 - e)^2 = 0.25 - e + e^2 | 630 | 01 | (0.5 - e)*(0.5 + e) = 0.25 - e^2 | 631 | 10 | (0.5 + e)*(0.5 - e) = 0.25 - e^2 | 632 | 11 | (0.5 + e)^2 = 0.25 + e + e^2 | 633 +------+-----------------------------------------+ 635 This technique will completely eliminate any bias but at the expense 636 of taking an indeterminate number of input bits for any particular 637 desired number of output bits. The probability of any particular 638 pair being discarded is 0.5 + 2e^2 so the expected number of input 639 bits to produce X output bits is X/(0.25 - e^2). 641 This technique assumes that the bits are from a stream where each bit 642 has the same probability of being a 0 or 1 as any other bit in the 643 stream and that bits are not correlated, i.e., that the bits are 644 identical independent distributions. If alternate bits were from two 645 correlated sources, for example, the above analysis breaks down. 647 The above technique also provides another illustration of how a 648 simple statistical analysis can mislead if one is not always on the 649 lookout for patterns that could be exploited by an adversary. If the 650 algorithm were mis-read slightly so that overlapping successive bits 651 pairs were used instead of non-overlapping pairs, the statistical 652 analysis given is the same; however, instead of providing an unbiased 653 uncorrelated series of random 1's and 0's, it instead produces a 654 totally predictable sequence of exactly alternating 1's and 0's. 656 5.2.3 Using FFT to De-Skew 658 When real world data consists of strongly biased or correlated bits, 659 it may still contain useful amounts of randomness. This randomness 660 can be extracted through use of the discrete Fourier transform or its 661 optimized variant, the FFT. 663 Using the Fourier transform of the data, strong correlations can be 664 discarded. If adequate data is processed and remaining correlations 665 decay, spectral lines approaching statistical independence and 666 normally distributed randomness can be produced [BRILLINGER]. 668 5.2.4 Using S-Boxes to De-Skew 670 Many modern block encryption functions, including DES and AES, 671 incorporate modules known as S-Boxes (substitution boxes). These 672 produce a smaller number of outputs from a larger number of inputs 673 through a complex non-linear mixing function which would have the 674 effect of concentrating limited entropy in the inputs into the 675 output. 677 S-Boxes sometimes incorporate bent boolean functions which are 678 functions of an even number of bits producing one output bit with 679 maximum non-linearity. Looking at the output for all input pairs 680 differing in any particular bit position, exactly half the outputs 681 are different. 683 An S-Box in which each output bit is produced by a bent function such 684 that any linear combination of these functions is also a bent 685 function is called a "perfect S-Box". Repeated application or 686 cascades of such boxes can be used to de-skew. [SBOX*] 688 5.2.5 Using Compression to De-Skew 690 Reversible compression techniques also provide a crude method of de- 691 skewing a skewed bit stream. This follows directly from the 692 definition of reversible compression and the formula in Section 2 693 above for the amount of information in a sequence. Since the 694 compression is reversible, the same amount of information must be 695 present in the shorter output than was present in the longer input. 696 By the Shannon information equation, this is only possible if, on 697 average, the probabilities of the different shorter sequences are 698 more uniformly distributed than were the probabilities of the longer 699 sequences. Thus the shorter sequences must be de-skewed relative to 700 the input. 702 However, many compression techniques add a somewhat predictable 703 preface to their output stream and may insert such a sequence again 704 periodically in their output or otherwise introduce subtle patterns 705 of their own. They should be considered only a rough technique 706 compared with those described above or in Section 6.1.2. At a 707 minimum, the beginning of the compressed sequence should be skipped 708 and only later bits used for applications requiring random bits. 710 5.3 Existing Hardware Can Be Used For Randomness 712 As described below, many computers come with hardware that can, with 713 care, be used to generate truly random quantities. 715 5.3.1 Using Existing Sound/Video Input 717 Increasingly computers are being built with inputs that digitize some 718 real world analog source, such as sound from a microphone or video 719 input from a camera. Under appropriate circumstances, such input can 720 provide reasonably high quality random bits. The "input" from a 721 sound digitizer with no source plugged in or a camera with the lens 722 cap on, if the system has enough gain to detect anything, is 723 essentially thermal noise. 725 For example, on a SPARCstation, one can read from the /dev/audio 726 device with nothing plugged into the microphone jack. Such data is 727 essentially random noise although it should not be trusted without 728 some checking in case of hardware failure. It will, in any case, 729 need to be de-skewed as described elsewhere. 731 Combining this with compression to de-skew one can, in UNIXese, 732 generate a huge amount of medium quality random data by doing 733 cat /dev/audio | compress - >random-bits-file 735 5.3.2 Using Existing Disk Drives 737 Disk drives have small random fluctuations in their rotational speed 738 due to chaotic air turbulence [DAVIS]. By adding low level disk seek 739 time instrumentation to a system, a series of measurements can be 740 obtained that include this randomness. Such data is usually highly 741 correlated so that significant processing is needed, such as FFT (see 742 section 5.2.3). Nevertheless experimentation has shown that, with 743 such processing, most disk drives easily produce 100 bits a minute or 744 more of excellent random data. 746 Partly offsetting this need for processing is the fact that disk 747 drive failure will normally be rapidly noticed. Thus, problems with 748 this method of random number generation due to hardware failure are 749 unlikely. 751 5.4 Ring Oscillator Sources 753 If an integrated circuit is being designed or field programmed, an 754 odd number of gates can be connected in series to produce a free- 755 running ring oscillator. By sampling a point in the ring at a 756 precise fixed frequency, say one determined by a stable crystal 757 oscialltor, some amount of entropy can be extracted due to slight 758 variations in the free-running osciallator. 760 Such bits will have to be heavily de-skewed as disk rotation timings 761 must be (Section 5.3.2). An engineering study would be needed to 762 determine the amount of entropy being produced depending on the 763 particular design. It may be possible to increase the rate of entropy 764 by xor'ing sampled values from a few ring osciallators with 765 relatively prime lengths or the like. In any case, this can be a 766 good, medium speed source whose cost is a trivial number of gates by 767 modern standards. 769 6. Recommended Software Strategy 771 What is the best overall strategy for meeting the requirement for 772 unguessable random numbers in the absence of a reliable hardware 773 source? It is to obtain random input from a number of uncorrelated 774 sources and to mix them with a strong mixing function. Such a 775 function will preserve the randomness present in any of the sources 776 even if other quantities being combined happen to be fixed or easily 777 guessable. This may be advisable even with a good hardware source, 778 as hardware can also fail, though this should be weighed against any 779 increase in the chance of overall failure due to added software 780 complexity. 782 6.1 Mixing Functions 784 A strong mixing function is one which combines two or more inputs and 785 produces an output where each output bit is a different complex non- 786 linear function of all the input bits. On average, changing any 787 input bit will change about half the output bits. But because the 788 relationship is complex and non-linear, no particular output bit is 789 guaranteed to change when any particular input bit is changed. 791 Consider the problem of converting a stream of bits that is skewed 792 towards 0 or 1 to a shorter stream which is more random, as discussed 793 in Section 5.2 above. This is simply another case where a strong 794 mixing function is desired, mixing the input bits to produce a 795 smaller number of output bits. The technique given in Section 5.2.1 796 of using the parity of a number of bits is simply the result of 797 successively Exclusive Or'ing them which is examined as a trivial 798 mixing function immediately below. Use of stronger mixing functions 799 to extract more of the randomness in a stream of skewed bits is 800 examined in Section 6.1.2. 802 6.1.1 A Trivial Mixing Function 804 A trivial example for single bit inputs is the Exclusive Or function, 805 which is equivalent to addition without carry, as show in the table 806 below. This is a degenerate case in which the one output bit always 807 changes for a change in either input bit. But, despite its 808 simplicity, it will still provide a useful illustration. 810 +-----------+-----------+----------+ 811 | input 1 | input 2 | output | 812 +-----------+-----------+----------+ 813 | 0 | 0 | 0 | 814 | 0 | 1 | 1 | 815 | 1 | 0 | 1 | 816 | 1 | 1 | 0 | 817 +-----------+-----------+----------+ 819 If inputs 1 and 2 are uncorrelated and combined in this fashion then 820 the output will be an even better (less skewed) random bit than the 821 inputs. If we assume an "eccentricity" e as defined in Section 5.2 822 above, then the output eccentricity relates to the input eccentricity 823 as follows: 825 e = 2 * e * e 826 output input 1 input 2 828 Since e is never greater than 1/2, the eccentricity is always 829 improved except in the case where at least one input is a totally 830 skewed constant. This is illustrated in the following table where 831 the top and left side values are the two input eccentricities and the 832 entries are the output eccentricity: 834 +--------+--------+--------+--------+--------+--------+--------+ 835 | e | 0.00 | 0.10 | 0.20 | 0.30 | 0.40 | 0.50 | 836 +--------+--------+--------+--------+--------+--------+--------+ 837 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 838 | 0.10 | 0.00 | 0.02 | 0.04 | 0.06 | 0.08 | 0.10 | 839 | 0.20 | 0.00 | 0.04 | 0.08 | 0.12 | 0.16 | 0.20 | 840 | 0.30 | 0.00 | 0.06 | 0.12 | 0.18 | 0.24 | 0.30 | 841 | 0.40 | 0.00 | 0.08 | 0.16 | 0.24 | 0.32 | 0.40 | 842 | 0.50 | 0.00 | 0.10 | 0.20 | 0.30 | 0.40 | 0.50 | 843 +--------+--------+--------+--------+--------+--------+--------+ 845 However, keep in mind that the above calculations assume that the 846 inputs are not correlated. If the inputs were, say, the parity of 847 the number of minutes from midnight on two clocks accurate to a few 848 seconds, then each might appear random if sampled at random intervals 849 much longer than a minute. Yet if they were both sampled and 850 combined with xor, the result would be zero most of the time. 852 6.1.2 Stronger Mixing Functions 854 The US Government Advanced Encryption Standard [AES] is an example of 855 a strong mixing function for multiple bit quantities. It takes up to 856 384 bits of input (128 bits of "data" and 256 bits of "key") and 857 produces 128 bits of output each of which is dependent on a complex 858 non-linear function of all input bits. Other encryption functions 859 with this characteristic, such as [DES], can also be used by 860 considering them to mix all of their key and data input bits. 862 Another good family of mixing functions are the "message digest" or 863 hashing functions such as The US Government Secure Hash Standards 864 [SHA*] and the MD4, MD5 [MD4, MD5] series. These functions all take 865 an arbitrary amount of input and produce an output mixing all the 866 input bits. The MD* series produce 128 bits of output, SHA-1 produces 867 160 bits, and other SHA functions produce larger numbers of bits. 869 Although the message digest functions are designed for variable 870 amounts of input, AES and other encryption functions can also be used 871 to combine any number of inputs. If 128 bits of output is adequate, 872 the inputs can be packed into a 128 bit data quantity and successive 873 AES keys, padding with zeros if needed, which are then used to 874 successively encrypt using AES in Electronic Codebook Mode [DES 875 MODES]. If more than 128 bits of output are needed, use more complex 876 mixing. For example, if inputs are packed into three quantities, A, 877 B, and C, use AES to encrypt A with B as a key and then with C as a 878 key to produce the 1st part of the output, then encrypt B with C and 879 then A for more output and, if necessary, encrypt C with A and then B 880 for yet more output. Still more output can be produced by reversing 881 the order of the keys given above to stretch things. The same can be 882 done with the hash functions by hashing various subsets of the input 883 data to produce multiple outputs. But keep in mind that it is 884 impossible to get more bits of "randomness" out than are put in. 886 An example of using a strong mixing function would be to reconsider 887 the case of a string of 308 bits each of which is biased 99% towards 888 zero. The parity technique given in Section 5.2.1 above reduced this 889 to one bit with only a 1/1000 deviance from being equally likely a 890 zero or one. But, applying the equation for information given in 891 Section 2, this 308 bit skewed sequence has over 5 bits of 892 information in it. Thus hashing it with SHA-1 and taking the bottom 893 5 bits of the result would yield 5 unbiased random bits as opposed to 894 the single bit given by calculating the parity of the string. 896 6.1.3 Diff-Hellman as a Mixing Function 898 Diffie-Hellman exponential key exchange is a technique that yields a 899 shared secret between two parties that can be made computationally 900 infeasible for a third party to determine even if they can observe 901 all the messages between the two communicating parties. This shared 902 secret is a mixture of initial quantities generated by each of them 903 [D-H]. If these initial quantities are random, then the shared 904 secret contains the combined randomness of them both, assuming they 905 are uncorrelated. 907 6.1.4 Using a Mixing Function to Stretch Random Bits 909 While it is not necessary for a mixing function to produce the same 910 or fewer bits than its inputs, mixing bits cannot "stretch" the 911 amount of random unpredictability present in the inputs. Thus four 912 inputs of 32 bits each where there is 12 bits worth of 913 unpredicatability (such as 4,096 equally probable values) in each 914 input cannot produce more than 48 bits worth of unpredictable output. 915 The output can be expanded to hundreds or thousands of bits by, for 916 example, mixing with successive integers, but the clever adversary's 917 search space is still 2^48 possibilities. Furthermore, mixing to 918 fewer bits than are input will tend to strengthen the randomness of 919 the output the way using Exclusive Or to produce one bit from two did 920 above. 922 The last table in Section 6.1.1 shows that mixing a random bit with a 923 constant bit with Exclusive Or will produce a random bit. While this 924 is true, it does not provide a way to "stretch" one random bit into 925 more than one. If, for example, a random bit is mixed with a 0 and 926 then with a 1, this produces a two bit sequence but it will always be 927 either 01 or 10. Since there are only two possible values, there is 928 still only the one bit of original randomness. 930 6.1.5 Other Factors in Choosing a Mixing Function 932 For local use, AES has the advantages that it has been widely tested 933 for flaws, is reasonably efficient in software, and is widely 934 documented and implemented with hardware and software implementations 935 available all over the world including open source code. The SHA* 936 family are younger algorithms but there is no particular reason to 937 believe they are flawed. Both SHA* and MD5 were derived from the 938 earlier MD4 algorithm. Some signs of weakness have been found in MD4 939 and MD5. They all have source code available [SHA*, MD*]. 941 AES and SHA* have been vouched for the the US National Security 942 Agency (NSA) on the basis of criteria that primarily remain secret, 943 as was DES. While this has been the cause of much speculation and 944 doubt, investigation of DES over the years has indicated that NSA 945 involvement in modifications to its design, which originated with 946 IBM, was primarily to strengthen it. No concealed or special 947 weakness has been found in DES. It is very likely that the NSA 948 modifications to MD4 to produce the SHA* similarly strengthened these 949 algorithms, possibly against threats not yet known in the public 950 cryptographic community. 952 AES, DES, SHA*, MD4, and MD5 are believed to be royalty free for all 953 purposes. Continued advances in crypography and computing power have 954 cast doubts on MD4 and MD5 so their use is generally not recommended. 956 Another advantage of the SHA* or similar hashing algorithms over 957 encryption algorithms in the past was that they are not subject to 958 the same regulations imposed by the US Government prohibiting the 959 unlicensed export or import of encryption/decryption software and 960 hardware. 962 6.2 Non-Hardware Sources of Randomness 964 The best source of input for mixing would be a hardware randomness 965 such as disk drive timing effected by air turbulence, audio input 966 with thermal noise, or radioactive decay. However, if that is not 967 available there are other possibilities. These include system 968 clocks, system or input/output buffers, user/system/hardware/network 969 serial numbers and/or addresses and timing, and user input. 970 Unfortunately, any of these sources can produce limited or 971 predicatable values under some circumstances. 973 Some of the sources listed above would be quite strong on multi-user 974 systems where, in essence, each user of the system is a source of 975 randomness. However, on a small single user or embedded system, 976 especially at start up, it might be possible for an adversary to 977 assemble a similar configuration. This could give the adversary 978 inputs to the mixing process that were sufficiently correlated to 979 those used originally as to make exhaustive search practical. 981 The use of multiple random inputs with a strong mixing function is 982 recommended and can overcome weakness in any particular input. For 983 example, the timing and content of requested "random" user keystrokes 984 can yield hundreds of random bits but conservative assumptions need 985 to be made. For example, assuming at most a few bits of randomness 986 if the inter-keystroke interval is unique in the sequence up to that 987 point and a similar assumption if the key hit is unique but assuming 988 that no bits of randomness are present in the initial key value or if 989 the timing or key value duplicate previous values. The results of 990 mixing these timings and characters typed could be further combined 991 with clock values and other inputs. 993 This strategy may make practical portable code to produce good random 994 numbers for security even if some of the inputs are very weak on some 995 of the target systems. However, it may still fail against a high 996 grade attack on small single user or embedded systems, especially if 997 the adversary has ever been able to observe the generation process in 998 the past. A hardware based random source is still preferable. 1000 6.3 Cryptographically Strong Sequences 1002 In cases where a series of random quantities must be generated, an 1003 adversary may learn some values in the sequence. In general, they 1004 should not be able to predict other values from the ones that they 1005 know. 1007 The correct technique is to start with a strong random seed, take 1008 cryptographically strong steps from that seed [CRYPTO2, CRYPTO3], and 1009 do not reveal the complete state of the generator in the sequence 1010 elements. If each value in the sequence can be calculated in a fixed 1011 way from the previous value, then when any value is compromised, all 1012 future values can be determined. This would be the case, for 1013 example, if each value were a constant function of the previously 1014 used values, even if the function were a very strong, non-invertible 1015 message digest function. 1017 (It should be noted that if your technique for generating a sequence 1018 of key values is fast enough, it can trivially be used as the basis 1019 for a confidentiality system. If two parties use the same sequence 1020 generating technique and start with the same seed material, they will 1021 generate identical sequences. These could, for example, be xor'ed at 1022 one end with data being send, encrypting it, and xor'ed with this 1023 data as received, decrypting it due to the reversible properties of 1024 the xor operation.) 1026 6.3.1 Traditional Strong Sequences 1028 A traditional way to achieve a strong sequence has been to have the 1029 values be produced by hashing the quantities produced by 1030 concatenating the seed with successive integers or the like and then 1031 mask the values obtained so as to limit the amount of generator state 1032 available to the adversary. 1034 It may also be possible to use an "encryption" algorithm with a 1035 random key and seed value to encrypt and feedback some or all of the 1036 output encrypted value into the value to be encrypted for the next 1037 iteration. Appropriate feedback techniques will usually be 1038 recommended with the encryption algorithm. An example is shown below 1039 where shifting and masking are used to combine the cypher output 1040 feedback. This type of feedback was recommended by the US Government 1041 in connection with DES [DES MODES] but should be avoided for reasons 1042 described below. 1044 +---------------+ 1045 | V | 1046 | | n |--+ 1047 +--+------------+ | 1048 | | +---------+ 1049 | +---> | | +-----+ 1050 +--+ | Encrypt | <--- | Key | 1051 | +-------- | | +-----+ 1052 | | +---------+ 1053 V V 1054 +------------+--+ 1055 | V | | 1056 | n+1 | 1057 +---------------+ 1059 Note that if a shift of one is used, this is the same as the shift 1060 register technique described in Section 3 above but with the all 1061 important difference that the feedback is determined by a complex 1062 non-linear function of all bits rather than a simple linear or 1063 polynomial combination of output from a few bit position taps. 1065 It has been shown by Donald W. Davies that this sort of shifted 1066 partial output feedback significantly weakens an algorithm compared 1067 will feeding all of the output bits back as input. In particular, 1068 for DES, repeated encrypting a full 64 bit quantity will give an 1069 expected repeat in about 2^63 iterations. Feeding back anything less 1070 than 64 (and more than 0) bits will give an expected repeat in 1071 between 2**31 and 2**32 iterations! 1073 To predict values of a sequence from others when the sequence was 1074 generated by these techniques is equivalent to breaking the 1075 cryptosystem or inverting the "non-invertible" hashing involved with 1076 only partial information available. The less information revealed 1077 each iteration, the harder it will be for an adversary to predict the 1078 sequence. Thus it is best to use only one bit from each value. It 1079 has been shown that in some cases this makes it impossible to break a 1080 system even when the cryptographic system is invertible and can be 1081 broken if all of each generated value was revealed. 1083 6.3.2 The Blum Blum Shub Sequence Generator 1085 Currently the generator which has the strongest public proof of 1086 strength is called the Blum Blum Shub generator after its inventors 1087 [BBS]. It is also very simple and is based on quadratic residues. 1088 It's only disadvantage is that is is computationally intensive 1089 compared with the traditional techniques give in 6.3.1 above. This 1090 is not a major draw back if it is used for moderately infrequent 1091 purposes, such as generating session keys. 1093 Simply choose two large prime numbers, say p and q, which both have 1094 the property that you get a remainder of 3 if you divide them by 4. 1095 Let n = p * q. Then you choose a random number x relatively prime to 1096 n. The initial seed for the generator and the method for calculating 1097 subsequent values are then 1099 2 1100 s = ( x )(Mod n) 1101 0 1103 2 1104 s = ( s )(Mod n) 1105 i+1 i 1107 You must be careful to use only a few bits from the bottom of each s. 1108 It is always safe to use only the lowest order bit. If you use no 1109 more than the 1110 log ( log ( s ) ) 1111 2 2 i 1112 low order bits, then predicting any additional bits from a sequence 1113 generated in this manner is provable as hard as factoring n. As long 1114 as the initial x is secret, you can even make n public if you want. 1116 An intersting characteristic of this generator is that you can 1117 directly calculate any of the s values. In particular 1119 i 1120 ( ( 2 )(Mod (( p - 1 ) * ( q - 1 )) ) ) 1121 s = ( s )(Mod n) 1122 i 0 1124 This means that in applications where many keys are generated in this 1125 fashion, it is not necessary to save them all. Each key can be 1126 effectively indexed and recovered from that small index and the 1127 initial s and n. 1129 6.3.3 Entropy Pool Techniques 1131 Many modern pseudo random number sources utilize the technique of 1132 maintaining a "pool" of bits and providing operations for strongly 1133 mixing input with some randomness into the pool and extracting psuedo 1134 random bits from the pool. This is illustred in the figure below. 1136 +--------+ +------+ +---------+ 1137 --->| Mix In |--->| POOL |--->| Extract |---> 1138 | Bits | | | | Bits | 1139 +--------+ +------+ +---------+ 1140 ^ V 1141 | | 1142 +-----------+ 1144 Bits to be feed into the pool can be any of the various hardware, 1145 environmental, or user input sources discussed above. It is also 1146 common to save the state of the pool on shut down and restore it on 1147 re-starting, if stable storage is available. 1149 In fact, all of the [MD*] and [SHA*] message digest functions are 1150 implemented by internally maintaining a pool substantially larger 1151 than their ultimate output into which the bytes of the message are 1152 mixed and from which the ultimate message digest is extracted. Thus 1153 the figure above can be implemented by using parts of the message 1154 digest code to strongly mix in any new bit supplied and to compute 1155 output bits based on the pool. However, additional code is needed so 1156 that any number of bits can be extracted and appropriate feedback 1157 from the output process is mixed into the pool so as to produce a 1158 strong pseudo-random output stream. 1160 Care must be taken that enough entropy has been added to the pool to 1161 support particular output uses desired. See Section 7.3 for for more 1162 details on an example implementation and [RSA BULL1] for similar 1163 suggestions. 1165 7. Key Generation Standards and Examples 1167 Several public standards and widely deplyed examples are now in place 1168 for the generation of keys without special hardware. Two standards 1169 are described below. Both use DES but any equally strong or stronger 1170 mixing function could be substituted. Then a few widely deployed 1171 examples are described. 1173 7.1 US DoD Recommendations for Password Generation 1175 The United States Department of Defense has specific recommendations 1176 for password generation [DoD]. They suggest using the US Data 1177 Encryption Standard [DES] in Output Feedback Mode [DES MODES] as 1178 follows: 1180 use an initialization vector determined from 1181 the system clock, 1182 system ID, 1183 user ID, and 1184 date and time; 1185 use a key determined from 1186 system interrupt registers, 1187 system status registers, and 1188 system counters; and, 1189 as plain text, use an external randomly generated 64 bit 1190 quantity such as 8 characters typed in by a system 1191 administrator. 1193 The password can then be calculated from the 64 bit "cipher text" 1194 generated in 64-bit Output Feedback Mode. As many bits as are needed 1195 can be taken from these 64 bits and expanded into a pronounceable 1196 word, phrase, or other format if a human being needs to remember the 1197 password. 1199 7.2 X9.17 Key Generation 1201 The American National Standards Institute has specified a method for 1202 generating a sequence of keys as follows: 1204 s is the initial 64 bit seed 1205 0 1207 g is the sequence of generated 64 bit key quantities 1208 n 1210 k is a random key reserved for generating this key sequence 1211 t is the time at which a key is generated to as fine a resolution 1212 as is available (up to 64 bits). 1214 DES ( K, Q ) is the DES encryption of quantity Q with key K 1216 g = DES ( k, DES ( k, t ) .xor. s ) 1217 n n 1219 s = DES ( k, DES ( k, t ) .xor. g ) 1220 n+1 n 1222 If g sub n is to be used as a DES key, then every eighth bit should 1223 be adjusted for parity for that use but the entire 64 bit unmodified 1224 g should be used in calculating the next s. 1226 7.3 The /dev/random Device under Linux 1228 The Linux operating system provides a Kernel resident random number 1229 generator. This generator makes use of events captured by the Kernel 1230 during normal system operation. 1232 The generator consists of a random pool of bytes, by default 512 1233 bytes (represented as 128, 4 byte integers). When an event occurs, 1234 such as a disk drive interrupt, the time of the event is xor'ed into 1235 the pool and the pool is stirred via a primitive polynomial of degree 1236 128. The pool itself is treated as a ring buffer, with new data 1237 being xor'ed (after stirring with the polynomial) across the entire 1238 pool. 1240 Each call that adds entropy to the pool estimates the amount of 1241 likely true entropy the input contains. The pool itself contains a 1242 accumulator that estimates the total over all entropy of the pool. 1244 Input events come from several sources: 1246 1. Keyboard interrupts. The time of the interrupt as well as the scan 1247 code are added to the pool. This in effect adds entropy from the 1248 human operator by measuring inter-keystroke arrival times. 1250 2. Disk completion and other interrupts. A system being used by a 1251 person will likely have a hard to predict pattern of disk 1252 accesses. 1254 3. Mouse motion. The timing as well as mouse position is added in. 1256 When random bytes are required, the pool is hashed with SHA-1 [SHA1] 1257 to yield the returned bytes of randomness. If more bytes are required 1258 than the output of SHA-1 (20 bytes), then the hashed output is 1259 stirred back into the pool and a new hash performed to obtain the 1260 next 20 bytes. As bytes are removed from the pool, the estimate of 1261 entropy is similarly decremented. 1263 To ensure a reasonable random pool upon system startup, the standard 1264 Linux startup scripts (and shutdown scripts) save the pool to a disk 1265 file at shutdown and read this file at system startup. 1267 There are two user exported interfaces. /dev/random returns bytes 1268 from the pool, but blocks when the estimated entropy drops to zero. 1269 As entropy is added to the pool from events, more data becomes 1270 available via /dev/random. Random data obtained /dev/random is 1271 suitable for key generation for long term keys. 1273 /dev/urandom works like /dev/random, however it provides data even 1274 when the entropy estimate for the random pool drops to zero. This 1275 should be fine for session keys. The risk of continuing to take data 1276 even when the pools entropy estimate is small is that past output may 1277 be computable from current output provided an attacker can reverse 1278 SHA-1. Given that SHA-1 should not be invertible, this is a 1279 reasonable risk. 1281 To obtain random numbers under Linux, all an application needs to do 1282 is open either /dev/random or /dev/urandom and read the desired 1283 number of bytes. 1285 The Linux Random device was written by Theodore Ts'o. It is based 1286 loosely on the random number generator in PGP 2.X and PGP 3.0 (aka 1287 PGP 5.0). 1289 8. Examples of Randomness Required 1291 Below are two examples showing rough calculations of needed 1292 randomness for security. The first is for moderate security 1293 passwords while the second assumes a need for a very high security 1294 cryptographic key. 1296 In addition [ORMAN] and [RSA BULL13] provide information on the 1297 public key lengths that should be used for exchanging symmetric keys. 1299 8.1 Password Generation 1301 Assume that user passwords change once a year and it is desired that 1302 the probability that an adversary could guess the password for a 1303 particular account be less than one in a thousand. Further assume 1304 that sending a password to the system is the only way to try a 1305 password. Then the crucial question is how often an adversary can 1306 try possibilities. Assume that delays have been introduced into a 1307 system so that, at most, an adversary can make one password try every 1308 six seconds. That's 600 per hour or about 15,000 per day or about 1309 5,000,000 tries in a year. Assuming any sort of monitoring, it is 1310 unlikely someone could actually try continuously for a year. In 1311 fact, even if log files are only checked monthly, 500,000 tries is 1312 more plausible before the attack is noticed and steps taken to change 1313 passwords and make it harder to try more passwords. 1315 To have a one in a thousand chance of guessing the password in 1316 500,000 tries implies a universe of at least 500,000,000 passwords or 1317 about 2^29. Thus 29 bits of randomness are needed. This can probably 1318 be achieved using the US DoD recommended inputs for password 1319 generation as it has 8 inputs which probably average over 5 bits of 1320 randomness each (see section 7.1). Using a list of 1000 words, the 1321 password could be expressed as a three word phrase (1,000,000,000 1322 possibilities) or, using case insensitive letters and digits, six 1323 would suffice ((26+10)^6 = 2,176,782,336 possibilities). 1325 For a higher security password, the number of bits required goes up. 1326 To decrease the probability by 1,000 requires increasing the universe 1327 of passwords by the same factor which adds about 10 bits. Thus to 1328 have only a one in a million chance of a password being guessed under 1329 the above scenario would require 39 bits of randomness and a password 1330 that was a four word phrase from a 1000 word list or eight 1331 letters/digits. To go to a one in 10^9 chance, 49 bits of randomness 1332 are needed implying a five word phrase or ten letter/digit password. 1334 In a real system, of course, there are also other factors. For 1335 example, the larger and harder to remember passwords are, the more 1336 likely users are to write them down resulting in an additional risk 1337 of compromise. 1339 8.2 A Very High Security Cryptographic Key 1341 Assume that a very high security key is needed for symmetric 1342 encryption / decryption between two parties. Assume an adversary can 1343 observe communications and knows the algorithm being used. Within 1344 the field of random possibilities, the adversary can try key values 1345 in hopes of finding the one in use. Assume further that brute force 1346 trial of keys is the best the adversary can do. 1348 8.2.1 Effort per Key Trial 1350 How much effort will it take to try each key? For very high security 1351 applications it is best to assume a low value of effort. This 1352 question is considered in detail in Appendix A. It concludes that a 1353 reasonable key length in 1995 for very high security is in the range 1354 of 75 to 90 bits and, since the cost of cryptography does not vary 1355 much with they key size, recommends 90 bits. To update these 1356 recommendations, just add 2/3 of a bit per year for Moore's law 1357 [MOORE]. Thus, in the year 2004, this translates to a determination 1358 that a reasonable key length is in 81 to 96 bit range. 1360 8.2.2 Meet in the Middle Attacks 1362 If chosen or known plain text and the resulting encrypted text are 1363 available, a "meet in the middle" attack is possible if the structure 1364 of the encryption algorithm allows it. (In a known plain text 1365 attack, the adversary knows all or part of the messages being 1366 encrypted, possibly some standard header or trailer fields. In a 1367 chosen plain text attack, the adversary can force some chosen plain 1368 text to be encrypted, possibly by "leaking" an exciting text that 1369 would then be sent by the adversary over an encrypted channel.) 1371 An oversimplified explanation of the meet in the middle attack is as 1372 follows: the adversary can half-encrypt the known or chosen plain 1373 text with all possible first half-keys, sort the output, then half- 1374 decrypt the encoded text with all the second half-keys. If a match 1375 is found, the full key can be assembled from the halves and used to 1376 decrypt other parts of the message or other messages. At its best, 1377 this type of attack can halve the exponent of the work required by 1378 the adversary while adding a large but roughly constant factor of 1379 effort. To be assured of safety against this, a doubling of the 1380 amount of randomness in the very strong key to a minimum of 162 bits 1381 is required for the year 2004 based on the Appendix A analysis. 1383 This amount of randomness is beyond the limit of that in the inputs 1384 recommended by the US DoD for password generation and could require 1385 user typing timing, hardware random number generation, or other 1386 sources. 1388 The meet in the middle attack assumes that the cryptographic 1389 algorithm can be decomposed in this way but we can not rule that out 1390 without a deep knowledge of the algorithm. Even if a basic algorithm 1391 is not subject to a meet in the middle attack, an attempt to produce 1392 a stronger algorithm by applying the basic algorithm twice (or two 1393 different algorithms sequentially) with different keys may gain less 1394 added security than would be expected. Such a composite algorithm 1395 would be subject to a meet in the middle attack. 1397 Enormous resources may be required to mount a meet in the middle 1398 attack but they are probably within the range of the national 1399 security services of a major nation. Essentially all nations spy on 1400 other nations government traffic and several nations are believed to 1401 spy on commercial traffic for economic advantage. 1403 It should be noted that key length calculations such at those above 1404 are controversial and depend on various assumptions about the 1405 cryptographic algorithms in use. In some cases, a professional with 1406 a deep knowledge of code breaking techniques and of the strength of 1407 the algorithm in use could be satisfied with less than half of the 1408 162 bit key size derived above. 1410 9. Conclusion 1412 Generation of unguessable "random" secret quantities for security use 1413 is an essential but difficult task. 1415 Hardware techniques to produce such randomness would be relatively 1416 simple. In particular, the volume and quality would not need to be 1417 high and existing computer hardware, such as disk drives, can be 1418 used. 1420 Computational techniques are available to process low quality random 1421 quantities from multiple sources or a larger quantity of such low 1422 quality input from one source and produce a smaller quantity of 1423 higher quality keying material. In the absence of hardware sources 1424 of randomness, a variety of user and software sources can frequently, 1425 with care, be used instead; however, most modern systems already have 1426 hardware, such as disk drives or audio input, that could be used to 1427 produce high quality randomness. 1429 Once a sufficient quantity of high quality seed key material (a 1430 couple of hundred bits) is available, computational techniques are 1431 available to produce cryptographically strong sequences of 1432 unpredicatable quantities from this seed material. 1434 10. Security Considerations 1436 The entirety of this document concerns techniques and recommendations 1437 for generating unguessable "random" quantities for use as passwords, 1438 cryptographic keys, initialiazation vectors, sequence numbers, and 1439 similar security uses. 1441 Intellectual Property Considerations 1443 The IETF takes no position regarding the validity or scope of any 1444 intellectual property or other rights that might be claimed to 1445 pertain to the implementation or use of the technology described in 1446 this document or the extent to which any license under such rights 1447 might or might not be available; neither does it represent that it 1448 has made any effort to identify any such rights. Information on the 1449 IETF's procedures with respect to rights in standards-track and 1450 standards-related documentation can be found in BCP-11. Copies of 1451 claims of rights made available for publication and any assurances of 1452 licenses to be made available, or the result of an attempt made to 1453 obtain a general license or permission for the use of such 1454 proprietary rights by implementors or users of this specification can 1455 be obtained from the IETF Secretariat. 1457 The IETF invites any interested party to bring to its attention any 1458 copyrights, patents or patent applications, or other proprietary 1459 rights which may cover technology that may be required to practice 1460 this standard. Please address the information to the IETF Executive 1461 Director. 1463 Appendix: Minimal Secure Key Lengths Study 1465 Minimal Key Lengths for Symmetric Ciphers 1466 to Provide Adequate Commercial Security 1468 A Report by an Ad Hoc Group of 1469 Cryptographers and Computer Scientists 1471 Matt Blaze, AT&T Research, mab@research.att.com 1472 Whitfield Diffie, Sun Microsystems, diffie@eng.sun.com 1473 Ronald L. Rivest, MIT LCS, rivest@lcs.mit.edu 1474 Bruce Schneier, Counterpane Systems, schneier@counterpane.com 1475 Tsutomu Shimomura, San Diego Supercomputer Center, tsutomu@sdsc.edu 1476 Eric Thompson Access Data, Inc., eric@accessdata.com 1477 Michael Wiener, Bell Northern Research, wiener@bnr.ca 1479 January 1996 1481 A.0 Abstract 1483 Encryption plays an essential role in protecting the privacy of 1484 electronic information against threats from a variety of potential 1485 attackers. In so doing, modern cryptography employs a combination of 1486 _conventional_ or _symmetric_ cryptographic systems for encrypting 1487 data and _public key_ or _asymmetric_ systems for managing the _keys_ 1488 used by the symmetric systems. Assessing the strength required of 1489 the symmetric cryptographic systems is therefore an essential step in 1490 employing cryptography for computer and communication security. 1492 Technology readily available today (late 1995) makes _brute- 1493 force_ attacks against cryptographic systems considered adequate for 1494 the past several years both fast and cheap. General purpose 1495 computers can be used, but a much more efficient approach is to 1496 employ commercially available _Field Programmable Gate Array (FPGA)_ 1497 technology. For attackers prepared to make a higher initial 1498 investment, custom-made, special-purpose chips make such calculations 1499 much faster and significantly lower the amortized cost per solution. 1501 As a result, cryptosystems with 40-bit keys offer virtually no 1502 protection at this point against brute-force attacks. Even the U.S. 1503 Data Encryption Standard with 56-bit keys is increasingly inadequate. 1504 As cryptosystems often succumb to `smarter' attacks than brute-force 1505 key search, it is also important to remember that the keylengths 1506 discussed here are the minimum needed for security against the 1507 computational threats considered. 1509 Fortunately, the cost of very strong encryption is not 1511 significantly greater than that of weak encryption. Therefore, to 1512 provide adequate protection against the most serious threats --- 1513 well-funded commercial enterprises or government intelligence 1514 agencies --- keys used to protect data today should be at least 75 1515 bits long. To protect information adequately for the next 20 years 1516 in the face of expected advances in computing power, keys in newly- 1517 deployed systems should be at least 90 bits long. 1519 A.1. Encryption Plays an Essential Role in Protecting 1520 the Privacy of Electronic Information" 1522 A.1.1 There is a need for information security 1524 As we write this paper in late 1995, the development of 1525 electronic commerce and the Global Information Infrastructure is at a 1526 critical juncture. The dirt paths of the middle ages only became 1527 highways of business and culture after the security of travelers and 1528 the merchandise they carried could be assured. So too the 1529 information superhighway will be an ill-traveled road unless 1530 information, the goods of the Information Age, can be moved, stored, 1531 bought, and sold securely. Neither corporations nor individuals will 1532 entrust their private business or personal data to computer networks 1533 unless they can assure their information's security. 1535 Today, most forms of information can be stored and processed 1536 electronically. This means a wide variety of information, with 1537 varying economic values and privacy aspects and with a wide variation 1538 in the time over which the information needs to be protected, will be 1539 found on computer networks. Consider the spectrum: 1541 o Electronic Funds Transfers of millions or even billions of 1542 dollars, whose short term security is essential but whose 1543 exposure is brief; 1545 o A company's strategic corporate plans, whose confidentiality 1546 must be preserved for a small number of years; 1548 o A proprietary product (Coke formula, new drug design) that 1549 needs to be protected over its useful life, often decades; 1550 and 1552 o Information private to an individual (medical condition, 1553 employment evaluation) that may need protection for the 1554 lifetime of the individual. 1556 A.1.2 Encryption to protect confidentiality 1558 Encryption Can Provide Strong Confidentiality Protection 1560 Encryption is accomplished by scrambling data using mathematical 1561 procedures that make it extremely difficult and time consuming for 1562 anyone other than authorized recipients --- those with the correct 1563 decryption _keys_ --- to recover the _plain text_. Proper encryption 1564 guarantees that the information will be safe even if it falls into 1565 hostile hands. 1567 Encryption --- and decryption --- can be performed by either 1568 computer software or hardware. Common approaches include writing the 1569 algorithm on a disk for execution by a computer central processor; 1570 placing it in ROM or PROM for execution by a microprocessor; and 1571 isolating storage and execution in a computer accessory device (smart 1572 card or PCMCIA card). 1574 The degree of protection obtained depends on several factors. 1575 These include: the quality of the cryptosystem; the way it is 1576 implemented in software or hardware (especially its reliability and 1577 the manner in which the keys are chosen); and the total number of 1578 possible keys that can be used to encrypt the information. A 1579 cryptographic algorithm is considered strong if: 1581 1. There is no shortcut that allows the opponent to recover the 1582 plain text without using brute force to test keys until the 1583 correct one is found; and 1585 2. The number of possible keys is sufficiently large to make 1586 such an attack infeasible. 1588 The principle here is similar to that of a combination lock on a 1589 safe. If the lock is well designed so that a burglar cannot hear or 1590 feel its inner workings, a person who does not know the combination 1591 can open it only by dialing one set of numbers after another until it 1592 yields. 1594 The sizes of encryption keys are measured in bits and the 1595 difficulty of trying all possible keys grows exponentially with the 1596 number of bits used. Adding one bit to the key doubles the number of 1597 possible keys; adding ten increases it by a factor of more than a 1598 thousand. 1600 There is no definitive way to look at a cipher and determine 1601 whether a shortcut exists. Nonetheless, several encryption 1602 algorithms --- most notably the U.S Data Encryption Standard (DES) 1603 --- have been extensively studied in the public literature and are 1604 widely believed to be of very high quality. An essential element in 1605 cryptographic algorithm design is thus the length of the key, whose 1606 size places an upper bound on the system's strength. 1608 Throughout this paper, we will assume that there are no shortcuts 1609 and treat the length of the key as representative of the 1610 cryptosystem's _workfactor_ --- the minimum amount of effort required 1611 to break the system. It is important to bear in mind, however, that 1612 cryptographers regard this as a rash assumption and many would 1613 recommend keys two or more times as long as needed to resist brute- 1614 force attacks. Prudent cryptographic designs not only employ longer 1615 keys than might appear to be needed, but devote more computation to 1616 encrypting and decrypting. A good example of this is the popular 1617 approach of using _triple-DES_: encrypting the output of DES twice 1618 more, using a total of three distinct keys. 1620 Encryption systems fall into two broad classes. Conventional or 1621 symmetric cryptosystems --- those in which an entity with the ability 1622 to encrypt also has the ability to decrypt and vice versa --- are the 1623 systems under consideration in this paper. The more recent public 1624 key or asymmetric cryptosystems have the property that the ability to 1625 encrypt does not imply the ability to decrypt. In contemporary 1626 cryptography, public-key systems are indispensable for managing the 1627 keys of conventional cryptosystems. All known public key 1628 cryptosystems, however, are subject to shortcut attacks and must 1629 therefore use keys ten or more times the lengths of those discussed 1630 here to achieve the an equivalent level of security. 1632 Although computers permit electronic information to be encrypted 1633 using very large keys, advances in computing power keep pushing up 1634 the size of keys that can be considered large and thus keep making it 1635 easier for individuals and organizations to attack encrypted 1636 information without the expenditure of unreasonable resources. 1638 A.1.3 There are a variety of attackers 1640 There Are Threats from a Variety of Potential Attackers. 1642 Threats to confidentiality of information come from a number of 1643 directions and their forms depend on the resources of the attackers. 1644 `Hackers,' who might be anything from high school students to 1645 commercial programmers, may have access to mainframe computers or 1646 networks of workstations. The same people can readily buy 1647 inexpensive, off-the-shelf, boards, containing _Field Programmable 1648 Gate Array (FPGA)_ chips that function as `programmable hardware' and 1649 vastly increase the effectiveness of a cryptanalytic effort. A 1650 startup company or even a well-heeled individual could afford large 1651 numbers of these chips. A major corporation or organized crime 1652 operation with `serious money' to spend could acquire custom computer 1653 chips specially designed for decryption. An intelligence agency, 1654 engaged in espionage for national economic advantage, could build a 1655 machine employing millions of such chips. 1657 A.1.4 Strong encryption is not expensive 1659 Current Technology Permits Very Strong Encryption for Effectively the 1660 Same Cost As Weaker Encryption. 1662 It is a property of computer encryption that modest increases in 1663 computational cost can produce vast increases in security. 1664 Encrypting information very securely (e.g., with 128-bit keys) 1665 typically requires little more computing than encrypting it weakly 1666 (e.g., with 40-bit keys). In many applications, the cryptography 1667 itself accounts for only a small fraction of the computing costs, 1668 compared to such processes as voice or image compression required to 1669 prepare material for encryption. 1671 One consequence of this uniformity of costs is that there is 1672 rarely any need to tailor the strength of cryptography to the 1673 sensitivity of the information being protected. Even if most of the 1674 information in a system has neither privacy implications nor monetary 1675 value, there is no practical or economic reason to design computer 1676 hardware or software to provide differing levels of encryption for 1677 different messages. It is simplest, most prudent, and thus 1678 fundamentally most economical, to employ a uniformly high level of 1679 encryption: the strongest encryption required for any information 1680 that might be stored or transmitted by a secure system. 1682 A.2. Brute-Forece is becoming easier 1684 Readily Available Technology Makes Brute-Force Decryption Attacks 1685 Faster and Cheaper. 1687 The kind of hardware used to mount a brute-force attack against 1688 an encryption algorithm depends on the scale of the cryptanalytic 1689 operation and the total funds available to the attacking enterprise. 1690 In the analysis that follows, we consider three general classes of 1691 technology that are likely to be employed by attackers with differing 1692 resources available to them. Not surprisingly, the cryptanalytic 1693 technologies that require larger up-front investments yield the 1694 lowest cost per recovered key, amortized over the life of the 1695 hardware. 1697 It is the nature of brute-force attacks that they can be 1698 parallelized indefinitely. It is possible to use as many machines as 1699 are available, assigning each to work on a separate part of the 1700 problem. Thus regardless of the technology employed, the search time 1701 can be reduced by adding more equipment; twice as much hardware can 1702 be expected to find the right key in half the time. The total 1703 investment will have doubled, but if the hardware is kept constantly 1704 busy finding keys, the cost per key recovered is unchanged. 1706 At the low end of the technology spectrum is the use of 1707 conventional personal computers or workstations programmed to test 1708 keys. Many people, by virtue of already owning or having access to 1709 the machines, are in a position use such resources at little or no 1710 cost. However, general purpose computers --- laden with such 1711 ancillary equipment as video controllers, keyboards, interfaces, 1712 memory, and disk storage --- make expensive search engines. They are 1713 therefore likely to be employed only by casual attackers who are 1714 unable or unwilling to invest in more specialized equipment. 1716 A more efficient technological approach is to take advantage of 1717 commercially available Field Programmable Gate Arrays. FPGAs 1718 function as programmable hardware and allow faster implementations of 1719 such tasks as encryption and decryption than conventional processors. 1720 FPGAs are a commonly used tool for simple computations that need to 1721 be done very quickly, particularly simulating integrated circuits 1722 during development. 1724 FPGA technology is fast and cheap. The cost of an AT&T ORCA chip 1725 that can test 30 million DES keys per second is $200. This is 1,000 1726 times faster than a PC at about one-tenth the cost! FPGAs are widely 1727 available and, mounted on cards, can be installed in standard PCs 1728 just like sound cards, modems, or extra memory. 1730 FPGA technology may be optimal when the same tool must be used 1731 for attacking a variety of different cryptosystems. Often, as with 1732 DES, a cryptosystem is sufficiently widely used to justify the 1733 construction of more specialized facilities. In these circumstances, 1734 the most cost-effective technology, but the one requiring the largest 1735 initial investment, is the use of _Application-Specific Integrated 1736 Circuits (ASICs)_. A $10 chip can test 200 million keys per second. 1737 This is seven times faster than an FPGA chip at one-twentieth the 1738 cost. 1740 Because ASICs require a far greater engineering investment than 1741 FPGAs and must be fabricated in quantity before they are economical, 1742 this approach is only available to serious, well-funded operations 1743 such as dedicated commercial (or criminal) enterprises and government 1744 intelligence agencies. 1746 A.3. 40-Bit Key Lengths Offer Virtually No Protection 1748 Current U.S. Government policy generally limits exportable mass 1749 market software that incorporates encryption for confidentiality to 1750 using the RC2 or RC4 algorithms with 40-bit keys. A 40-bit key 1751 length means that there are 2^40 possible keys. On average, half of 1752 these (2^39) must be tried to find the correct one. Export of other 1753 algorithms and key lengths must be approved on a case by case basis. 1754 For example, DES with a 56-bit key has been approved for certain 1755 applications such as financial transactions. 1757 The recent successful brute-force attack by two French graduate 1758 students on Netscape's 40-bit RC4 algorithm demonstrates the dangers 1759 of such short keys. These students at the Ecole Polytechnique in 1760 Paris used `idle time' on the school's computers, incurring no cost 1761 to themselves or their school. Even with these limited resources, 1762 they were able to recover the 40-bit key in a few days. 1764 There is no need to have the resources of an institution of 1765 higher education at hand, however. Anyone with a modicum of computer 1766 expertise and a few hundred dollars would be able to attack 40-bit 1767 encryption much faster. An FPGA chip --- costing approximately $400 1768 mounted on a card --- would on average recover a 40-bit key in five 1769 hours. Assuming the FPGA lasts three years and is used continuously 1770 to find keys, the average cost per key is eight cents. 1772 A more determined commercial predator, prepared to spend $10,000 1773 for a set-up with 25 ORCA chips, can find 40-bit keys in an average 1774 of 12 minutes, at the same average eight cent cost. Spending more 1775 money to buy more chips reduces the time accordingly: $300,000 1776 results in a solution in an average of 24 seconds; $10,000,000 1777 results in an average solution in 0.7 seconds. 1779 As already noted, a corporation with substantial resources can 1780 design and commission custom chips that are much faster. By doing 1781 this, a company spending $300,000 could find the right 40-bit key in 1782 an average of 0.18 seconds at 1/10th of a cent per solution; a larger 1783 company or government agency willing to spend $10,000,000 could find 1784 the right key on average in 0.005 seconds (again at 1/10th of a cent 1785 per solution). (Note that the cost per solution remains constant 1786 because we have conservatively assumed constant costs for chip 1787 acquisition --- in fact increasing the quantities purchased of a 1788 custom chip reduces the average chip cost as the initial design and 1789 set-up costs are spread over a greater number of chips.) 1791 These results are summarized in Table I (below). 1793 A.4. Even DES with 56-Bit Keys Is Increasingly Inadequate 1795 A.4.1 DES is no panacea today 1797 The Data Encryption Standard (DES) was developed in the 1970s by 1798 IBM and NSA and adopted by the U.S. Government as a Federal 1799 Information Processing Standard for data encryption. It was intended 1800 to provide strong encryption for the government's sensitive but 1801 unclassified information. It was recognized by many, even at the 1802 time DES was adopted, that technological developments would make 1803 DES's 56-bit key exceedingly vulnerable to attack before the end of 1804 the century. 1806 Today, DES may be the most widely employed encryption algorithm 1807 and continues to be a commonly cited benchmark. Yet DES-like 1808 encryption strength is no panacea. Calculations show that DES is 1809 inadequate against a corporate or government attacker committing 1810 serious resources. The bottom line is that DES is cheaper and easier 1811 to break than many believe. 1813 As explained above, 40-bit encryption provides inadequate 1814 protection against even the most casual of intruders, content to 1815 scavenge time on idle machines or to spend a few hundred dollars. 1816 Against such opponents, using DES with a 56-bit key will provide a 1817 substantial measure of security. At present, it would take a year 1818 and a half for someone using $10,000 worth of FPGA technology to 1819 search out a DES key. In ten years time an investment of this size 1820 would allow one to find a DES key in less than a week. 1822 The real threat to commercial transactions and to privacy on the 1823 Internet is from individuals and organizations willing to invest 1824 substantial time and money. As more and more business and personal 1825 information becomes electronic, the potential rewards to a dedicated 1826 commercial predator also increase significantly and may justify the 1827 commitment of adequate resources. 1829 A serious effort --- on the order of $300,000 --- by a legitimate 1830 or illegitimate business could find a DES key in an average of 19 1831 days using off-the-shelf technology and in only 3 hours using a 1832 custom developed chip. In the latter case, it would cost $38 to find 1833 each key (again assuming a 3 year life to the chip and continual 1834 use). A business or government willing to spend $10,000,000 on 1835 custom chips, could recover DES keys in an average of 6 minutes, for 1836 the same $38 per key. 1838 At the very high end, an organization --- presumably a government 1839 intelligence agency --- willing to spend $300,000,000 could recover 1840 DES keys in 12 seconds each! The investment required is large but 1841 not unheard of in the intelligence community. It is less than the 1842 cost of the Glomar Explorer, built to salvage a single Russian 1843 submarine, and far less than the cost of many spy satellites. Such 1844 an expense might be hard to justify in attacking a single target, but 1845 seems entirely appropriate against a cryptographic algorithm, like 1846 DES, enjoying extensive popularity around the world. 1848 There is ample evidence of the danger presented by government 1849 intelligence agencies seeking to obtain information not only for 1850 military purposes but for commercial advantage. Congressional 1851 hearings in 1993 highlighted instances in which the French and 1852 Japanese governments spied on behalf of their countries' own 1853 businesses. Thus, having to protect commercial information against 1854 such threats is not a hypothetical proposition. 1856 A.4.2 There are smarter avenues of attack than brute force 1858 It is easier to walk around a tree than climb up and down it. 1859 There is no need to break the window of a house to get in if the 1860 front door is unlocked. 1862 Calculations regarding the strength of encryption against brute- 1863 force attack are _worst case_ scenarios. They assume that the 1864 ciphers are in a sense perfect and that attempts to find shortcuts 1865 have failed. One important point is that the crudest approach --- 1866 searching through the keys --- is entirely feasible against many 1867 widely used systems. Another is that the keylengths we discuss are 1868 always minimal. As discussed earlier, prudent designs might use keys 1869 twice or three times as long to provide a margin of safety. 1871 A.4.3 Other algorithms are similar 1873 The Analysis for Other Algorithms Is Roughly Comparable. 1875 The above analysis has focused on the time and money required to 1876 find a key to decrypt information using the RC4 algorithm with a 1877 40-bit key or the DES algorithm with its 56-bit key, but the results 1878 are not peculiar to these ciphers. Although each algorithm has its 1879 own particular characteristics, the effort required to find the keys 1880 of other ciphers is comparable. There may be some differences as the 1881 result of implementation procedures, but these do not materially 1882 affect the brute-force breakability of algorithms with roughly 1883 comparable key lengths. 1885 Specifically, it has been suggested at times that differences in 1886 set-up procedures, such as the long key-setup process in RC4, result 1887 in some algorithms having effectively longer keys than others. For 1888 the purpose of our analysis, such factors appear to vary the 1889 effective key length by no more than about eight bits. 1891 A.5. Appropriate Key Lengths for the Future --- A Proposal 1893 Table I summarizes the costs of carrying out brute-force attacks 1894 against symmetric cryptosystems with 40-bit and 56-bit keys using 1895 networks of general purpose computers, Field Programmable Gate 1896 Arrays, and special-purpose chips. 1898 It shows that 56 bits provides a level of protection --- about a 1899 year and a half --- that would be adequate for many commercial 1900 purposes against an opponent prepared to invest $10,000. Against an 1901 opponent prepared to invest $300,000, the period of protection has 1902 dropped to the barest minimum of 19 days. Above this, the protection 1903 quickly declines to negligible. A very large, but easily imaginable, 1904 investment by an intelligence agency would clearly allow it to 1905 recover keys in real time. 1907 What workfactor would be required for security today? For an 1908 opponent whose budget lay in the $10 to 300 million range, the time 1909 required to search out keys in a 75-bit keyspace would be between 6 1910 years and 70 days. Although the latter figure may seem comparable to 1911 the `barest minimum' 19 days mentioned earlier, it represents --- 1912 under our amortization assumptions --- a cost of $19 million and a 1913 recovery rate of only five keys a year. The victims of such an 1914 attack would have to be fat targets indeed. 1916 Because many kinds of information must be kept confidential for 1917 long periods of time, assessment cannot be limited to the protection 1918 required today. Equally important, cryptosystems --- especially if 1919 they are standards --- often remain in use for years or even decades. 1920 DES, for example, has been in use for more than 20 years and will 1921 probably continue to be employed for several more. In particular, 1922 the lifetime of a cryptosystem is likely to exceed the lifetime of 1923 any individual product embodying it. 1925 A rough estimate of the minimum strength required as a function 1926 of time can be obtained by applying an empirical rule, popularly 1927 called `Moore's Law,' which holds that the computing power available 1928 for a given cost doubles every 18 months. Taking into account both 1929 the lifetime of cryptographic equipment and the lifetime of the 1930 secrets it protects, we believe it is prudent to require that 1931 encrypted data should still be secure in 20 years. Moore's Law thus 1932 predicts that the keys should be approximately 14 bits longer than 1933 required to protect against an attack today. 1935 *Bearing in mind that the additional computational costs of 1936 stronger encryption are modest, we strongly recommend a minimum key- 1937 length of 90 bits for symmetric cryptosystems.* 1939 It is instructive to compare this recommendation with both 1940 Federal Information Processing Standard 46, The Data Encryption 1941 Standard (DES), and Federal Information Processing Standard 185, The 1942 Escrowed Encryption Standard (EES). DES was proposed 21 years ago 1943 and used a 56-bit key. Applying Moore's Law and adding 14 bits, we 1944 see that the strength of DES when it was proposed in 1975 was 1945 comparable to that of a 70-bit system today. Furthermore, it was 1946 estimated at the time that DES was not strong enough and that keys 1947 could be recovered at a rate of one per day for an investment of 1948 about twenty-million dollars. Our 75-bit estimate today corresponds 1949 to 61 bits in 1975, enough to have moved the cost of key recovery 1950 just out of reach. The Escrowed Encryption Standard, while 1951 unacceptable to many potential users for other reasons, embodies a 1952 notion of appropriate key length that is similar to our own. It uses 1953 80-bit keys, a number that lies between our figures of 75 and 90 1954 bits. 1956 Table I 1958 Time and cost Length Needed 1959 Type of Budget Tool per key recovered for protection 1960 Attacker 40bits 56bits in Late 1995 1962 Pedestrian Hacker 1964 tiny scavenged 1 week infeasible 45 1965 computer 1966 time 1968 $400 FPGA 5 hours 38 years 50 1969 ($0.08) ($5,000) 1971 Small Business 1973 $10,000 FPGA 12 minutes 556 days 55 1974 ($0.08) ($5,000) 1976 Corporate Department 1978 $300K FPGA 24 seconds 19 days 60 1979 or ($0.08) ($5,000) 1980 ASIC .18 seconds 3 hours 1981 ($0.001) ($38) 1983 Big Company 1984 $10M FPGA .7 seconds 13 hours 70 1985 or ($0.08) ($5,000) 1986 ASIC .005 seconds 6 minutes 1987 ($0.001) ($38) 1989 Intellegence Agency 1991 $300M ASIC .0002 seconds 12 seconds 75 1992 ($0.001) ($38) 1994 A.6 About the Authors 1996 *Matt Blaze* is a senior research scientist at AT&T Research in the 1997 area of computer security and cryptography. Recently Blaze 1998 demonstrated weaknesses in the U.S. government's `Clipper Chip' key 1999 escrow encryption system. His current interests include large-scale 2000 trust management and the applications of smartcards. 2002 *Whitfield Diffie* is a distinguished Engineer at Sun Microsystems 2003 specializing in security. In 1976 Diffie and Martin Hellman created 2004 public key cryptography, which solved the problem of sending coded 2005 information between individuals with no prior relationship and is the 2006 basis for widespread encryption in the digital information age. 2008 *Ronald L. Rivest* is a professor of computer science at the 2009 Massachusetts Institute of Technology, and is Associate Director of 2010 MIT's Laboratory for Computer Science. Rivest, together with Leonard 2011 Adleman and Adi Shamir, invented the RSA public-key cryptosystem that 2012 is used widely throughout industry. Ron Rivest is one of the 2013 founders of RSA Data Security Inc. and is the creator of variable key 2014 length symmetric key ciphers (e.g., RC4). 2016 *Bruce Schneier* is president of Counterpane Systems, a consulting 2017 firm specializing in cryptography and computer security. Schneier 2018 writes and speaks frequently on computer security and privacy and is 2019 the author of a leading cryptography textbook, Applied Cryptography, 2020 and is the creator of the symmetric key cipher Blowfish. 2022 *Tsutomu Shimomura* is a computational physicist employed by the San 2023 Diego Supercomputer Center who is an expert in designing software 2024 security tools. Last year, Shimomura was responsible for tracking 2025 down the computer outlaw Kevin Mitnick, who electronically stole and 2026 altered valuable electronic information around the country. 2028 *Eric Thompson* heads AccessData Corporation's cryptanalytic team and 2029 is a frequent lecturer on applied crytography. AccessData 2030 specializes in data recovery and decrypting information utilizing 2031 brute force as well as `smarter' attacks. Regular clients include 2032 the FBI and other law enforcement agencies as well as corporations. 2034 *Michael Wiener* is a cryptographic advisor at Bell-Northern Research 2035 where he focuses on cryptanalysis, security architectures, and 2036 public-key infrastructures. His influential 1993 paper, Efficient 2037 DES Key Search, describes in detail how to construct a machine to 2038 brute force crack DES coded information (and provides cost estimates 2039 as well). 2041 A.7 Acknowledgement 2043 The [Appendix] authors would like to thank the Business Software 2044 Alliance, which provided support for a one-day meeting, held in 2045 Chicago on 20 November 1995. 2047 Informative References 2049 [AES] - "Specification of theAdvanced Encryption Standard (AES)", 2050 United States of America, Department of Commerce, National Institute 2051 of Standards and Technology, Federal Information Processing Standard 2052 197, November 2001. 2054 [ASYMMETRIC] - "Secure Communications and Asymmetric Cryptosystems", 2055 edited by Gustavus J. Simmons, AAAS Selected Symposium 69, Westview 2056 Press, Inc. 2058 [BBS] - "A Simple Unpredictable Pseudo-Random Number Generator", SIAM 2059 Journal on Computing, v. 15, n. 2, 1986, L. Blum, M. Blum, & M. Shub. 2061 [BRILLINGER] - "Time Series: Data Analysis and Theory", Holden-Day, 2062 1981, David Brillinger. 2064 [CRC] - "C.R.C. Standard Mathematical Tables", Chemical Rubber 2065 Publishing Company. 2067 [CRYPTO1] - "Cryptography: A Primer", A Wiley-Interscience 2068 Publication, John Wiley & Sons, 1981, Alan G. Konheim. 2070 [CRYPTO2] - "Cryptography: A New Dimension in Computer Data 2071 Security", A Wiley-Interscience Publication, John Wiley & Sons, 1982, 2072 Carl H. Meyer & Stephen M. Matyas. 2074 [CRYPTO3] - "Applied Cryptography: Protocols, Algorithsm, and Source 2075 Code in C", Second Edition, John Wiley & Sons, 1996, Bruce Schneier. 2077 [DAVIS] - "Cryptographic Randomness from Air Turbulence in Disk 2078 Drives", Advances in Cryptology - Crypto '94, Springer-Verlag Lecture 2079 Notes in Computer Science #839, 1984, Don Davis, Ross Ihaka, and 2080 Philip Fenstermacher. 2082 [DES] - "Data Encryption Standard", United States of America, 2083 Department of Commerce, National Institute of Standards and 2084 Technology, Federal Information Processing Standard (FIPS) 46-3, 2085 October 1999. 2086 - "Data Encryption Algorithm", American National Standards Institute, 2087 ANSI X3.92-1981. 2088 (See also FIPS 112, Password Usage, which includes FORTRAN code for 2089 performing DES.) 2091 [DES MODES] - "DES Modes of Operation", United States of America, 2092 Department of Commerce, National Institute of Standards and 2093 Technology, Federal Information Processing Standard (FIPS) 81, 2094 December 1980. 2095 - Data Encryption Algorithm - Modes of Operation, American National 2096 Standards Institute, ANSI X3.106-1983. 2098 [D-H] - "New Directions in Cryptography", IEEE Transactions on 2099 Information Technology, November, 1976, Whitfield Diffie and Martin 2100 E. Hellman. 2102 [DNSSEC] - RFC 2535, "Domain Name System Security Extensions", D. 2103 Eastlake, March 1999. 2105 [DoD] - "Password Management Guideline", United States of America, 2106 Department of Defense, Computer Security Center, CSC-STD-002-85. 2107 (See also FIPS 112, Password Usage, which incorporates CSC-STD-002-85 2108 as one of its appendices.) 2110 [DSS] - "Digital Signature Standard (DSS)", United States of America, 2111 Department of Commerce, National Institute of Standards and 2112 Technoloy, Federal Information Processing Standard (FIPS) 186-2, 2113 January 2000. 2115 [GIFFORD] - "Natural Random Number", MIT/LCS/TM-371, September 1988, 2116 David K. Gifford 2118 [IPSEC] - RFC 2401, "Security Architecture for the Internet 2119 Protocol", S. Kent, R. Atkinson, November 1998 2121 [KNUTH] - "The Art of Computer Programming", Volume 2: Seminumerical 2122 Algorithms, Chapter 3: Random Numbers. Addison Wesley Publishing 2123 Company, Second Edition 1982, Donald E. Knuth. 2125 [KRAWCZYK] - "How to Predict Congruential Generators", Journal of 2126 Algorithms, V. 13, N. 4, December 1992, H. Krawczyk 2128 [MAIL PEM] - RFCs 1421 through 1424: 2129 - RFC 1424, Privacy Enhancement for Internet Electronic Mail: Part 2130 IV: Key Certification and Related Services, 02/10/1993, B. Kaliski 2131 - RFC 1423, Privacy Enhancement for Internet Electronic Mail: Part 2132 III: Algorithms, Modes, and Identifiers, 02/10/1993, D. Balenson 2133 - RFC 1422, Privacy Enhancement for Internet Electronic Mail: Part 2134 II: Certificate-Based Key Management, 02/10/1993, S. Kent 2135 - RFC 1421, Privacy Enhancement for Internet Electronic Mail: Part I: 2136 Message Encryption and Authentication Procedures, 02/10/1993, J. Linn 2138 [MAIL PGP] - RFC 2440, "OpenPGP Message Format", J. Callas, L. 2139 Donnerhacke, H. Finney, R. Thayer", November 1998 2141 [MAIL S/MIME] - RFC 2633, "S/MIME Version 3 Message Specification", 2142 B. Ramsdell, Ed., June 1999. 2144 [MD4] - "The MD4 Message-Digest Algorithm", RFC1320, April 1992, R. 2145 Rivest 2146 [MD5] - "The MD5 Message-Digest Algorithm", RFC1321, April 1992, R. 2147 Rivest 2149 [MOORE] - Moore's Law: the exponential increase the logic density of 2150 silicon circuts. Originally formulated by Gordon Moore in 1964 as a 2151 doubling every year starting in 1962, in the late 1970s the rate fell 2152 to a doubling every 18 months and has remained there through the date 2153 of this document. See "The New Hacker's Dictionary", Third Edition, 2154 MIT Press, ISBN 0-262-18178-9, Eric S. Raymondm 1996. 2156 [ORMAN] - "Determining Strengths For Public Keys Used For Exchanging 2157 Symmetric Keys", draft-orman-public-key-lengths-*.txt, Hilarie Orman, 2158 Paul Hoffman, work in progress. 2160 [RFC 1750] - "Randomness Requirements for Security", D. Eastlake, S. 2161 Crocker, J. Schiller, December 1994. 2163 [RSA BULL1] - "Suggestions for Random Number Generation in Software", 2164 RSA Laboratories Bulletin #1, January 1996. 2166 [RSA BULL13] - "A Cost-Based Security Analysis of Symmetric and 2167 Asymmetric Key Lengths", RSA Laboratories Bulletin #13, Robert 2168 Silverman, April 2000 (revised November 2001). 2170 [SBOX1] - "Practical s-box design", S. Mister, C. Adams, Selected 2171 Areas in Cryptography, 1996. 2172 [SBOX2] - "Perfect Non-linear S-boxes", K. Nyberg, Advances in 2173 Cryptography - Eurocrypt '91 Proceedings, Springer-Verland, 1991. 2175 [SHANNON] - "The Mathematical Theory of Communication", University of 2176 Illinois Press, 1963, Claude E. Shannon. (originally from: Bell 2177 System Technical Journal, July and October 1948) 2179 [SHIFT1] - "Shift Register Sequences", Aegean Park Press, Revised 2180 Edition 1982, Solomon W. Golomb. 2182 [SHIFT2] - "Cryptanalysis of Shift-Register Generated Stream Cypher 2183 Systems", Aegean Park Press, 1984, Wayne G. Barker. 2185 [SHA-1] - "Secure Hash Standard (SHA-1)", United States of American, 2186 National Institute of Science and Technology, Federal Information 2187 Processing Standard (FIPS) 180-1, April 1993. 2188 - RFC 3174, "US Secure Hash Algorithm 1 (SHA1)", D. Eastlake, 2189 P. Jones, September 2001. 2191 [SHA-2] - "Secure Hash Standard", Draft (SHA-2156/384/512), Federal 2192 Information Processing Standard 180-2, not yet issued. 2194 [SSH] - draft-ietf-secsh-*, work in progress. 2196 [STERN] - "Secret Linear Congruential Generators are not 2197 Cryptograhically Secure", Proceedings of IEEE STOC, 1987, J. Stern. 2199 [TLS] - RFC 2246, "The TLS Protocol Version 1.0", T. Dierks, C. 2200 Allen, January 1999. 2202 [VON NEUMANN] - "Various techniques used in connection with random 2203 digits", von Neumann's Collected Works, Vol. 5, Pergamon Press, 1963, 2204 J. von Neumann. 2206 Authors Addresses 2208 Donald E. Eastlake 3rd 2209 Motorola Laboratories 2210 155 Beaver Street 2211 Milford, MA 01757 USA 2213 Telephone: +1 508-851-8280 (w) 2214 +1 508-634-2066 (h) 2215 EMail: Donald.Eastlake@motorola.com 2217 Jeffrey I. Schiller 2218 MIT, Room E40-311 2219 77 Massachusetts Avenue 2220 Cambridge, MA 02139-4307 USA 2222 Telephone: +1 617-253-0161 2223 E-mail: jis@mit.edu 2225 Steve Crocker 2227 EMail: steve@stevecrocker.com 2229 File Name and Expiration 2231 This is file draft-eastlake-randomness2-04.txt. 2233 It expires February 2004.