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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 January 2003 July 2002 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..................................28 127 7.3 The /dev/random Device under Linux....................29 129 8. Examples of Randomness Required........................31 130 8.1 Password Generation..................................31 131 8.2 A Very High Security Cryptographic Key................32 132 8.2.1 Effort per Key Trial................................32 133 8.2.2 Meet in the Middle Attacks..........................32 135 9. Conclusion.............................................34 136 10. Security Considerations...............................34 138 Appendix: Minimal Secure Key Lengths Study................35 139 A.0 Abstract..............................................35 140 A.1. Encryption Plays an Essential Role in Protecting.....36 141 A.1.1 There is a need for information security............36 142 A.1.2 Encryption to protect confidentiality...............37 143 A.1.3 There are a variety of attackers....................38 144 A.1.4 Strong encryption is not expensive..................39 145 A.2. Brute-Forece is becoming easier......................39 146 A.3. 40-Bit Key Lengths Offer Virtually No Protection.....41 147 A.4. Even DES with 56-Bit Keys Is Increasingly Inadequate.42 148 A.4.1 DES is no panacea today.............................42 149 A.4.2 There are smarter attacks than brute force..........43 150 A.4.3 Other algorithms are similar........................43 151 A.5. Appropriate Key Lengths for the Future - A Proposal..44 152 A.6 About the Authors.....................................46 153 A.7 Acknowledgement.......................................47 155 References................................................48 157 Authors Addresses.........................................52 158 File Name and Expiration..................................52 160 1. Introduction 162 Software cryptography is coming into wider use and is continuing to 163 spread, although there is a long way to go until it becomes 164 pervasive. 166 Systems like SSH, IPSEC, TLS, S/MIME, PGP, DNSSEC, Kerberos, etc. are 167 maturing and becoming a part of the network landscape [SSH, DNSSEC, 168 IPSEC, MAIL*, TLS]. By comparison, when the previous version of this 169 document [RFC 1750] was issued in 1994, about the only cryptographic 170 security specification in the IETF was the Privacy Enhanced Mail 171 protocol [MAIL PEM]. 173 These systems provide substantial protection against snooping and 174 spoofing. However, there is a potential flaw. At the heart of all 175 cryptographic systems is the generation of secret, unguessable (i.e., 176 random) numbers. 178 For the present, the lack of generally available facilities for 179 generating such unpredictable numbers is an open wound in the design 180 of cryptographic software. For the software developer who wants to 181 build a key or password generation procedure that runs on a wide 182 range of hardware, the only safe strategy so far has been to force 183 the local installation to supply a suitable routine to generate 184 random numbers. To say the least, this is an awkward, error-prone 185 and unpalatable solution. 187 It is important to keep in mind that the requirement is for data that 188 an adversary has a very low probability of guessing or determining. 189 This can easily fail if pseudo-random data is used which only meets 190 traditional statistical tests for randomness or which is based on 191 limited range sources, such as clocks. Frequently such random 192 quantities are determinable by an adversary searching through an 193 embarrassingly small space of possibilities. 195 This Best Current Practice describes techniques for producing random 196 quantities that will be resistant to such attack. It recommends that 197 future systems include hardware random number generation or provide 198 access to existing hardware that can be used for this purpose. It 199 suggests methods for use if such hardware is not available. And it 200 gives some estimates of the number of random bits required for sample 201 applications. 203 2. Requirements 205 A commonly encountered randomness requirement today is the user 206 password. This is usually a simple character string. Obviously, if a 207 password can be guessed, it does not provide security. (For re- 208 usable passwords, it is desirable that users be able to remember the 209 password. This may make it advisable to use pronounceable character 210 strings or phrases composed on ordinary words. But this only affects 211 the format of the password information, not the requirement that the 212 password be very hard to guess.) 214 Many other requirements come from the cryptographic arena. 215 Cryptographic techniques can be used to provide a variety of services 216 including confidentiality and authentication. Such services are 217 based on quantities, traditionally called "keys", that are unknown to 218 and unguessable by an adversary. 220 In some cases, such as the use of symmetric encryption with the one 221 time pads [CRYPTO*] or the US Data Encryption Standard [DES] or 222 Advanced Encryption Standard [AES], the parties who wish to 223 communicate confidentially and/or with authentication must all know 224 the same secret key. In other cases, using what are called 225 asymmetric or "public key" cryptographic techniques, keys come in 226 pairs. One key of the pair is private and must be kept secret by one 227 party, the other is public and can be published to the world. It is 228 computationally infeasible to determine the private key from the 229 public key and knowledge of the public is of no help to an adversary. 230 [ASYMMETRIC, CRYPTO*] 232 The frequency and volume of the requirement for random quantities 233 differs greatly for different cryptographic systems. Using pure RSA 234 [CRYPTO*], random quantities are required when the key pair is 235 generated, but thereafter any number of messages can be signed 236 without a further need for randomness. The public key Digital 237 Signature Algorithm devised by the US National Institute of Standards 238 and Technology (NIST) requires good random numbers for each signature 239 [DSS]. And encrypting with a one time pad, in principle the 240 strongest possible encryption technique, requires a volume of 241 randomness equal to all the messages to be processed [CRYPTO*]. 243 In most of these cases, an adversary can try to determine the 244 "secret" key by trial and error. (This is possible as long as the 245 key is enough smaller than the message that the correct key can be 246 uniquely identified.) The probability of an adversary succeeding at 247 this must be made acceptably low, depending on the particular 248 application. The size of the space the adversary must search is 249 related to the amount of key "information" present in the information 250 theoretic sense [SHANNON]. This depends on the number of different 251 secret values possible and the probability of each value as follows: 253 ----- 254 \ 255 Bits-of-info = \ - p * log ( p ) 256 / i 2 i 257 / 258 ----- 260 where i counts from 1 to the number of possible secret values and p 261 sub i is the probability of the value numbered i. (Since p sub i is 262 less than one, the log will be negative so each term in the sum will 263 be non-negative.) 265 If there are 2^n different values of equal probability, then n bits 266 of information are present and an adversary would, on the average, 267 have to try half of the values, or 2^(n-1) , before guessing the 268 secret quantity. If the probability of different values is unequal, 269 then there is less information present and fewer guesses will, on 270 average, be required by an adversary. In particular, any values that 271 the adversary can know are impossible, or are of low probability, can 272 be initially ignored by an adversary, who will search through the 273 more probable values first. 275 For example, consider a cryptographic system that uses 128 bit keys. 276 If these 128 bit keys are derived by using a fixed pseudo-random 277 number generator that is seeded with an 8 bit seed, then an adversary 278 needs to search through only 256 keys (by running the pseudo-random 279 number generator with every possible seed), not the 2^128 keys that 280 may at first appear to be the case. Only 8 bits of "information" are 281 in these 128 bit keys. 283 3. Traditional Pseudo-Random Sequences 285 Most traditional sources of random numbers use deterministic sources 286 of "pseudo-random" numbers. These typically start with a "seed" 287 quantity and use numeric or logical operations to produce a sequence 288 of values. 290 [KNUTH] has a classic exposition on pseudo-random numbers. 291 Applications he mentions are simulation of natural phenomena, 292 sampling, numerical analysis, testing computer programs, decision 293 making, and games. None of these have the same characteristics as 294 the sort of security uses we are talking about. Only in the last two 295 could there be an adversary trying to find the random quantity. 296 However, in these cases, the adversary normally has only a single 297 chance to use a guessed value. In guessing passwords or attempting 298 to break an encryption scheme, the adversary normally has many, 299 perhaps unlimited, chances at guessing the correct value because they 300 can store the message they are trying to break and repeatedly attack 301 it. They should also be assumed to be aided by a computer. 303 For testing the "randomness" of numbers, Knuth suggests a variety of 304 measures including statistical and spectral. These tests check 305 things like autocorrelation between different parts of a "random" 306 sequence or distribution of its values. But they could be met by a 307 constant stored random sequence, such as the "random" sequence 308 printed in the CRC Standard Mathematical Tables [CRC]. 310 A typical pseudo-random number generation technique, known as a 311 linear congruence pseudo-random number generator, is modular 312 arithmetic where the value numbered N+1 is calculated from the value 313 numbered N by 315 V = ( V * a + b )(Mod c) 316 N+1 N 318 The above technique has a strong relationship to linear shift 319 register pseudo-random number generators, which are well understood 320 cryptographically [SHIFT*]. In such generators bits are introduced 321 at one end of a shift register as the Exclusive Or (binary sum 322 without carry) of bits from selected fixed taps into the register. 323 For example: 325 +----+ +----+ +----+ +----+ 326 | B | <-- | B | <-- | B | <-- . . . . . . <-- | B | <-+ 327 | 0 | | 1 | | 2 | | n | | 328 +----+ +----+ +----+ +----+ | 329 | | | | 330 | | V +-----+ 331 | V +----------------> | | 332 V +-----------------------------> | XOR | 333 +---------------------------------------------------> | | 334 +-----+ 336 V = ( ( V * 2 ) + B .xor. B ... )(Mod 2^n) 337 N+1 N 0 2 339 The goodness of traditional pseudo-random number generator algorithms 340 is measured by statistical tests on such sequences. Carefully chosen 341 values of the initial V and a, b, and c or the placement of shift 342 register tap in the above simple processes can produce excellent 343 statistics. 345 These sequences may be adequate in simulations (Monte Carlo 346 experiments) as long as the sequence is orthogonal to the structure 347 of the space being explored. Even there, subtle patterns may cause 348 problems. However, such sequences are clearly bad for use in 349 security applications. They are fully predictable if the initial 350 state is known. Depending on the form of the pseudo-random number 351 generator, the sequence may be determinable from observation of a 352 short portion of the sequence [CRYPTO*, STERN]. For example, with 353 the generators above, one can determine V(n+1) given knowledge of 354 V(n). In fact, it has been shown that with these techniques, even if 355 only one bit of the pseudo-random values are released, the seed can 356 be determined from short sequences. 358 Not only have linear congruent generators been broken, but techniques 359 are now known for breaking all polynomial congruent generators. 360 [KRAWCZYK] 362 4. Unpredictability 364 Randomness in the traditional sense described in section 3 is NOT the 365 same as the unpredictability required for security use. 367 For example, use of a widely available constant sequence, such as 368 that from the CRC tables, is very weak against an adversary. Once 369 they learn of or guess it, they can easily break all security, future 370 and past, based on the sequence. [CRC] Yet the statistical properties 371 of these tables are good. 373 The following sections describe the limitations of some randomness 374 generation techniques and sources. 376 4.1 Problems with Clocks and Serial Numbers 378 Computer clocks, or similar operating system or hardware values, 379 provide significantly fewer real bits of unpredictability than might 380 appear from their specifications. 382 Tests have been done on clocks on numerous systems and it was found 383 that their behavior can vary widely and in unexpected ways. One 384 version of an operating system running on one set of hardware may 385 actually provide, say, microsecond resolution in a clock while a 386 different configuration of the "same" system may always provide the 387 same lower bits and only count in the upper bits at much lower 388 resolution. This means that successive reads on the clock may 389 produce identical values even if enough time has passed that the 390 value "should" change based on the nominal clock resolution. There 391 are also cases where frequently reading a clock can produce 392 artificial sequential values because of extra code that checks for 393 the clock being unchanged between two reads and increases it by one! 394 Designing portable application code to generate unpredictable numbers 395 based on such system clocks is particularly challenging because the 396 system designer does not always know the properties of the system 397 clocks that the code will execute on. 399 Use of a hardware serial number such as an Ethernet address may also 400 provide fewer bits of uniqueness than one would guess. Such 401 quantities are usually heavily structured and subfields may have only 402 a limited range of possible values or values easily guessable based 403 on approximate date of manufacture or other data. For example, it is 404 likely that a company that manfactures both computers and Ethernet 405 adapters will, at least internally, use its own adapters, which 406 significantly limits the range of built in addresses. 408 Problems such as those described above related to clocks and serial 409 numbers make code to produce unpredictable quantities difficult if 410 the code is to be ported across a variety of computer platforms and 411 systems. 413 4.2 Timing and Content of External Events 415 It is possible to measure the timing and content of mouse movement, 416 key strokes, and similar user events. This is a reasonable source of 417 unguessable data with some qualifications. On some machines, inputs 418 such as key strokes are buffered. Even though the user's inter- 419 keystroke timing may have sufficient variation and unpredictability, 420 there might not be an easy way to access that variation. Another 421 problem is that no standard method exists to sample timing details. 422 This makes it hard to build standard software intended for 423 distribution to a large range of machines based on this technique. 425 The amount of mouse movement or the keys actually hit are usually 426 easier to access than timings but may yield less unpredictability as 427 the user may provide highly repetitive input. 429 Other external events, such as network packet arrival times, can also 430 be used with care. In particular, the possibility of manipulation of 431 such times by an adversary and the lack of history on system start up 432 must be considered. 434 4.3 The Fallacy of Complex Manipulation 436 One strategy which may give a misleading appearance of 437 unpredictability is to take a very complex algorithm (or an excellent 438 traditional pseudo-random number generator with good statistical 439 properties) and calculate a cryptographic key by starting with the 440 current value of a computer system clock as the seed. An adversary 441 who knew roughly when the generator was started would have a 442 relatively small number of seed values to test as they would know 443 likely values of the system clock. Large numbers of pseudo-random 444 bits could be generated but the search space an adversary would need 445 to check could be quite small. 447 Thus very strong and/or complex manipulation of data will not help if 448 the adversary can learn what the manipulation is and there is not 449 enough unpredictability in the starting seed value. Even if they can 450 not learn what the manipulation is, they may be able to use the 451 limited number of results stemming from a limited number of seed 452 values to defeat security. 454 Another serious strategy error is to assume that a very complex 455 pseudo-random number generation algorithm will produce strong random 456 numbers when there has been no theory behind or analysis of the 457 algorithm. There is a excellent example of this fallacy right near 458 the beginning of chapter 3 in [KNUTH] where the author describes a 459 complex algorithm. It was intended that the machine language program 460 corresponding to the algorithm would be so complicated that a person 461 trying to read the code without comments wouldn't know what the 462 program was doing. Unfortunately, actual use of this algorithm 463 showed that it almost immediately converged to a single repeated 464 value in one case and a small cycle of values in another case. 466 Not only does complex manipulation not help you if you have a limited 467 range of seeds but blindly chosen complex manipulation can destroy 468 the randomness in a good seed! 470 4.4 The Fallacy of Selection from a Large Database 472 Another strategy that can give a misleading appearance of 473 unpredictability is selection of a quantity randomly from a database 474 and assume that its strength is related to the total number of bits 475 in the database. For example, typical USENET servers process many 476 megabytes of information per day. Assume a random quantity was 477 selected by fetching 32 bytes of data from a random starting point in 478 this data. This does not yield 32*8 = 256 bits worth of 479 unguessability. Even after allowing that much of the data is human 480 language and probably has no more than 2 or 3 bits of information per 481 byte, it doesn't yield 32*2 = 64 bits of unguessability. For an 482 adversary with access to the same usenet database the unguessability 483 rests only on the starting point of the selection. That is perhaps a 484 little over a couple of dozen bits of unguessability. 486 The same argument applies to selecting sequences from the data on a 487 CD/DVD recording or any other large public database. If the 488 adversary has access to the same database, this "selection from a 489 large volume of data" step buys very little. However, if a selection 490 can be made from data to which the adversary has no access, such as 491 system buffers on an active multi-user system, it may be of help. 493 5. Hardware for Randomness 495 Is there any hope for true strong portable randomness in the future? 496 There might be. All that's needed is a physical source of 497 unpredictable numbers. 499 A thermal noise (sometimes called Johnson noise in integrated 500 circuits) or radioactive decay source and a fast, free-running 501 oscillator would do the trick directly [GIFFORD]. This is a trivial 502 amount of hardware, and could easily be included as a standard part 503 of a computer system's architecture. Furthermore, any system with a 504 spinning disk or ring oscillator and a stable (crystal) time source 505 or the like has an adequate source of randomness ([DAVIS] and Section 506 5.4). All that's needed is the common perception among computer 507 vendors that this small additional hardware and the software to 508 access it is necessary and useful. 510 5.1 Volume Required 512 How much unpredictability is needed? Is it possible to quantify the 513 requirement in, say, number of random bits per second? 515 The answer is not very much is needed. For AES, the key can be 128 516 bits and, as we show in an example in Section 8, even the highest 517 security system is unlikely to require a keying material of much over 518 200 bits. If a series of keys are needed, they can be generated from 519 a strong random seed using a cryptographically strong sequence as 520 explained in Section 6.3. A few hundred random bits generated at 521 start up or once a day would be enough using such techniques. Even 522 if the random bits are generated as slowly as one per second and it 523 is not possible to overlap the generation process, it should be 524 tolerable in high security applications to wait 200 seconds 525 occasionally. 527 These numbers are trivial to achieve. It could be done by a person 528 repeatedly tossing a coin. Almost any hardware process is likely to 529 be much faster. 531 5.2 Sensitivity to Skew 533 Is there any specific requirement on the shape of the distribution of 534 the random numbers? The good news is the distribution need not be 535 uniform. All that is needed is a conservative estimate of how non- 536 uniform it is to bound performance. Simple techniques to de-skew the 537 bit stream are given below and stronger techniques are mentioned in 538 Section 6.1.2 below. 540 5.2.1 Using Stream Parity to De-Skew 542 Consider taking a sufficiently long string of bits and map the string 543 to "zero" or "one". The mapping will not yield a perfectly uniform 544 distribution, but it can be as close as desired. One mapping that 545 serves the purpose is to take the parity of the string. This has the 546 advantages that it is robust across all degrees of skew up to the 547 estimated maximum skew and is absolutely trivial to implement in 548 hardware. 550 The following analysis gives the number of bits that must be sampled: 552 Suppose the ratio of ones to zeros is 0.5 + e : 0.5 - e, where e is 553 between 0 and 0.5 and is a measure of the "eccentricity" of the 554 distribution. Consider the distribution of the parity function of N 555 bit samples. The probabilities that the parity will be one or zero 556 will be the sum of the odd or even terms in the binomial expansion of 557 (p + q)^N, where p = 0.5 + e, the probability of a one, and q = 0.5 - 558 e, the probability of a zero. 560 These sums can be computed easily as 562 N N 563 1/2 * ( ( p + q ) + ( p - q ) ) 564 and 565 N N 566 1/2 * ( ( p + q ) - ( p - q ) ). 568 (Which one corresponds to the probability the parity will be 1 569 depends on whether N is odd or even.) 571 Since p + q = 1 and p - q = 2e, these expressions reduce to 573 N 574 1/2 * [1 + (2e) ] 575 and 576 N 577 1/2 * [1 - (2e) ]. 579 Neither of these will ever be exactly 0.5 unless e is zero, but we 580 can bring them arbitrarily close to 0.5. If we want the 581 probabilities to be within some delta d of 0.5, i.e. then 583 N 584 ( 0.5 + ( 0.5 * (2e) ) ) < 0.5 + d. 586 Solving for N yields N > log(2d)/log(2e). (Note that 2e is less than 587 1, so its log is negative. Division by a negative number reverses 588 the sense of an inequality.) 589 The following table gives the length of the string which must be 590 sampled for various degrees of skew in order to come within 0.001 of 591 a 50/50 distribution. 593 +---------+--------+-------+ 594 | Prob(1) | e | N | 595 +---------+--------+-------+ 596 | 0.5 | 0.00 | 1 | 597 | 0.6 | 0.10 | 4 | 598 | 0.7 | 0.20 | 7 | 599 | 0.8 | 0.30 | 13 | 600 | 0.9 | 0.40 | 28 | 601 | 0.95 | 0.45 | 59 | 602 | 0.99 | 0.49 | 308 | 603 +---------+--------+-------+ 605 The last entry shows that even if the distribution is skewed 99% in 606 favor of ones, the parity of a string of 308 samples will be within 607 0.001 of a 50/50 distribution. 609 5.2.2 Using Transition Mappings to De-Skew 611 Another technique, originally due to von Neumann [VON NEUMANN], is to 612 examine a bit stream as a sequence of non-overlapping pairs. You 613 could then discard any 00 or 11 pairs found, interpret 01 as a 0 and 614 10 as a 1. Assume the probability of a 1 is 0.5+e and the 615 probability of a 0 is 0.5-e where e is the eccentricity of the source 616 and described in the previous section. Then the probability of each 617 pair is as follows: 619 +------+-----------------------------------------+ 620 | pair | probability | 621 +------+-----------------------------------------+ 622 | 00 | (0.5 - e)^2 = 0.25 - e + e^2 | 623 | 01 | (0.5 - e)*(0.5 + e) = 0.25 - e^2 | 624 | 10 | (0.5 + e)*(0.5 - e) = 0.25 - e^2 | 625 | 11 | (0.5 + e)^2 = 0.25 + e + e^2 | 626 +------+-----------------------------------------+ 628 This technique will completely eliminate any bias but at the expense 629 of taking an indeterminate number of input bits for any particular 630 desired number of output bits. The probability of any particular 631 pair being discarded is 0.5 + 2e^2 so the expected number of input 632 bits to produce X output bits is X/(0.25 - e^2). 634 This technique assumes that the bits are from a stream where each bit 635 has the same probability of being a 0 or 1 as any other bit in the 636 stream and that bits are not correlated, i.e., that the bits are 637 identical independent distributions. If alternate bits were from two 638 correlated sources, for example, the above analysis breaks down. 640 The above technique also provides another illustration of how a 641 simple statistical analysis can mislead if one is not always on the 642 lookout for patterns that could be exploited by an adversary. If the 643 algorithm were mis-read slightly so that overlapping successive bits 644 pairs were used instead of non-overlapping pairs, the statistical 645 analysis given is the same; however, instead of providing an unbiased 646 uncorrelated series of random 1's and 0's, it instead produces a 647 totally predictable sequence of exactly alternating 1's and 0's. 649 5.2.3 Using FFT to De-Skew 651 When real world data consists of strongly biased or correlated bits, 652 it may still contain useful amounts of randomness. This randomness 653 can be extracted through use of the discrete Fourier transform or its 654 optimized variant, the FFT. 656 Using the Fourier transform of the data, strong correlations can be 657 discarded. If adequate data is processed and remaining correlations 658 decay, spectral lines approaching statistical independence and 659 normally distributed randomness can be produced [BRILLINGER]. 661 5.2.4 Using S-Boxes to De-Skew 663 Many modern block encryption functions, including DES and AES, 664 incorporate modules known as S-Boxes (substitution boxes). These 665 produce a smaller number of outputs from a larger number of inputs 666 through a complex non-linear mixing function which would have the 667 effect of concentrating limited entropy in the inputs into the 668 output. 670 S-Boxes sometimes incorporate bent boolean functions which are 671 functions of an even number of bits producing one output bit with 672 maximum non-linearity. Looking at the output for all input pairs 673 differing in any particular bit position, exactly half the outputs 674 are different. 676 An S-Box in which each output bit is produced by a bent function such 677 that any linear combination of these functions is also a bent 678 function is called a "perfect S-Box". Repeated application or 679 cascades of such boxes can be used to de-skew. [SBOX*] 681 5.2.5 Using Compression to De-Skew 683 Reversible compression techniques also provide a crude method of de- 684 skewing a skewed bit stream. This follows directly from the 685 definition of reversible compression and the formula in Section 2 686 above for the amount of information in a sequence. Since the 687 compression is reversible, the same amount of information must be 688 present in the shorter output than was present in the longer input. 689 By the Shannon information equation, this is only possible if, on 690 average, the probabilities of the different shorter sequences are 691 more uniformly distributed than were the probabilities of the longer 692 sequences. Thus the shorter sequences must be de-skewed relative to 693 the input. 695 However, many compression techniques add a somewhat predictable 696 preface to their output stream and may insert such a sequence again 697 periodically in their output or otherwise introduce subtle patterns 698 of their own. They should be considered only a rough technique 699 compared with those described above or in Section 6.1.2. At a 700 minimum, the beginning of the compressed sequence should be skipped 701 and only later bits used for applications requiring random bits. 703 5.3 Existing Hardware Can Be Used For Randomness 705 As described below, many computers come with hardware that can, with 706 care, be used to generate truly random quantities. 708 5.3.1 Using Existing Sound/Video Input 710 Increasingly computers are being built with inputs that digitize some 711 real world analog source, such as sound from a microphone or video 712 input from a camera. Under appropriate circumstances, such input can 713 provide reasonably high quality random bits. The "input" from a 714 sound digitizer with no source plugged in or a camera with the lens 715 cap on, if the system has enough gain to detect anything, is 716 essentially thermal noise. 718 For example, on a SPARCstation, one can read from the /dev/audio 719 device with nothing plugged into the microphone jack. Such data is 720 essentially random noise although it should not be trusted without 721 some checking in case of hardware failure. It will, in any case, 722 need to be de-skewed as described elsewhere. 724 Combining this with compression to de-skew one can, in UNIXese, 725 generate a huge amount of medium quality random data by doing 726 cat /dev/audio | compress - >random-bits-file 728 5.3.2 Using Existing Disk Drives 730 Disk drives have small random fluctuations in their rotational speed 731 due to chaotic air turbulence [DAVIS]. By adding low level disk seek 732 time instrumentation to a system, a series of measurements can be 733 obtained that include this randomness. Such data is usually highly 734 correlated so that significant processing is needed, such as FFT (see 735 section 5.2.3). Nevertheless experimentation has shown that, with 736 such processing, most disk drives easily produce 100 bits a minute or 737 more of excellent random data. 739 Partly offsetting this need for processing is the fact that disk 740 drive failure will normally be rapidly noticed. Thus, problems with 741 this method of random number generation due to hardware failure are 742 unlikely. 744 5.4 Ring Oscillator Sources 746 If an integrated circuit is being designed or field programmed, an 747 odd number of gates can be connected in series to produce a free- 748 running ring oscillator. By sampling a point in the ring at a 749 precise fixed frequency, say one determined by a stable crystal 750 oscialltor, some amount of entropy can be extracted due to slight 751 variations in the free-running osciallator. 753 Such bits will have to be heavily de-skewed as disk rotation timings 754 must be (Section 5.3.2). An engineering study would be needed to 755 determine the amount of entropy being produced depending on the 756 particular design. It may be possible to increase the rate of entropy 757 by xor'ing sampled values from a few ring osciallators with 758 relatively prime lengths or the like. In any case, this can be a 759 good, medium speed source whose cost is a trivial number of gates by 760 modern standards. 762 6. Recommended Software Strategy 764 What is the best overall strategy for meeting the requirement for 765 unguessable random numbers in the absence of a reliable hardware 766 source? It is to obtain random input from a number of uncorrelated 767 sources and to mix them with a strong mixing function. Such a 768 function will preserve the randomness present in any of the sources 769 even if other quantities being combined happen to be fixed or easily 770 guessable. This may be advisable even with a good hardware source, 771 as hardware can also fail, though this should be weighed against any 772 increase in the chance of overall failure due to added software 773 complexity. 775 6.1 Mixing Functions 777 A strong mixing function is one which combines two or more inputs and 778 produces an output where each output bit is a different complex non- 779 linear function of all the input bits. On average, changing any 780 input bit will change about half the output bits. But because the 781 relationship is complex and non-linear, no particular output bit is 782 guaranteed to change when any particular input bit is changed. 784 Consider the problem of converting a stream of bits that is skewed 785 towards 0 or 1 to a shorter stream which is more random, as discussed 786 in Section 5.2 above. This is simply another case where a strong 787 mixing function is desired, mixing the input bits to produce a 788 smaller number of output bits. The technique given in Section 5.2.1 789 of using the parity of a number of bits is simply the result of 790 successively Exclusive Or'ing them which is examined as a trivial 791 mixing function immediately below. Use of stronger mixing functions 792 to extract more of the randomness in a stream of skewed bits is 793 examined in Section 6.1.2. 795 6.1.1 A Trivial Mixing Function 797 A trivial example for single bit inputs is the Exclusive Or function, 798 which is equivalent to addition without carry, as show in the table 799 below. This is a degenerate case in which the one output bit always 800 changes for a change in either input bit. But, despite its 801 simplicity, it will still provide a useful illustration. 803 +-----------+-----------+----------+ 804 | input 1 | input 2 | output | 805 +-----------+-----------+----------+ 806 | 0 | 0 | 0 | 807 | 0 | 1 | 1 | 808 | 1 | 0 | 1 | 809 | 1 | 1 | 0 | 810 +-----------+-----------+----------+ 812 If inputs 1 and 2 are uncorrelated and combined in this fashion then 813 the output will be an even better (less skewed) random bit than the 814 inputs. If we assume an "eccentricity" e as defined in Section 5.2 815 above, then the output eccentricity relates to the input eccentricity 816 as follows: 818 e = 2 * e * e 819 output input 1 input 2 821 Since e is never greater than 1/2, the eccentricity is always 822 improved except in the case where at least one input is a totally 823 skewed constant. This is illustrated in the following table where 824 the top and left side values are the two input eccentricities and the 825 entries are the output eccentricity: 827 +--------+--------+--------+--------+--------+--------+--------+ 828 | e | 0.00 | 0.10 | 0.20 | 0.30 | 0.40 | 0.50 | 829 +--------+--------+--------+--------+--------+--------+--------+ 830 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 831 | 0.10 | 0.00 | 0.02 | 0.04 | 0.06 | 0.08 | 0.10 | 832 | 0.20 | 0.00 | 0.04 | 0.08 | 0.12 | 0.16 | 0.20 | 833 | 0.30 | 0.00 | 0.06 | 0.12 | 0.18 | 0.24 | 0.30 | 834 | 0.40 | 0.00 | 0.08 | 0.16 | 0.24 | 0.32 | 0.40 | 835 | 0.50 | 0.00 | 0.10 | 0.20 | 0.30 | 0.40 | 0.50 | 836 +--------+--------+--------+--------+--------+--------+--------+ 838 However, keep in mind that the above calculations assume that the 839 inputs are not correlated. If the inputs were, say, the parity of 840 the number of minutes from midnight on two clocks accurate to a few 841 seconds, then each might appear random if sampled at random intervals 842 much longer than a minute. Yet if they were both sampled and 843 combined with xor, the result would be zero most of the time. 845 6.1.2 Stronger Mixing Functions 847 The US Government Advanced Encryption Standard [AES] is an example of 848 a strong mixing function for multiple bit quantities. It takes up to 849 384 bits of input (128 bits of "data" and 256 bits of "key") and 850 produces 128 bits of output each of which is dependent on a complex 851 non-linear function of all input bits. Other encryption functions 852 with this characteristic, such as [DES], can also be used by 853 considering them to mix all of their key and data input bits. 855 Another good family of mixing functions are the "message digest" or 856 hashing functions such as The US Government Secure Hash Standards 857 [SHA*] and the MD4, MD5 [MD4, MD5] series. These functions all take 858 an arbitrary amount of input and produce an output mixing all the 859 input bits. The MD* series produce 128 bits of output, SHA-1 produces 860 160 bits, and other SHA functions produce larger numbers of bits. 862 Although the message digest functions are designed for variable 863 amounts of input, AES and other encryption functions can also be used 864 to combine any number of inputs. If 128 bits of output is adequate, 865 the inputs can be packed into a 128 bit data quantity and successive 866 AES keys, padding with zeros if needed, which are then used to 867 successively encrypt using AES in Electronic Codebook Mode [DES 868 MODES]. If more than 128 bits of output are needed, use more complex 869 mixing. For example, if inputs are packed into three quantities, A, 870 B, and C, use AES to encrypt A with B as a key and then with C as a 871 key to produce the 1st part of the output, then encrypt B with C and 872 then A for more output and, if necessary, encrypt C with A and then B 873 for yet more output. Still more output can be produced by reversing 874 the order of the keys given above to stretch things. The same can be 875 done with the hash functions by hashing various subsets of the input 876 data to produce multiple outputs. But keep in mind that it is 877 impossible to get more bits of "randomness" out than are put in. 879 An example of using a strong mixing function would be to reconsider 880 the case of a string of 308 bits each of which is biased 99% towards 881 zero. The parity technique given in Section 5.2.1 above reduced this 882 to one bit with only a 1/1000 deviance from being equally likely a 883 zero or one. But, applying the equation for information given in 884 Section 2, this 308 bit skewed sequence has over 5 bits of 885 information in it. Thus hashing it with SHA-1 and taking the bottom 886 5 bits of the result would yield 5 unbiased random bits as opposed to 887 the single bit given by calculating the parity of the string. 889 6.1.3 Diff-Hellman as a Mixing Function 891 Diffie-Hellman exponential key exchange is a technique that yields a 892 shared secret between two parties that can be made computationally 893 infeasible for a third party to determine even if they can observe 894 all the messages between the two communicating parties. This shared 895 secret is a mixture of initial quantities generated by each of them 896 [D-H]. If these initial quantities are random, then the shared 897 secret contains the combined randomness of them both, assuming they 898 are uncorrelated. 900 6.1.4 Using a Mixing Function to Stretch Random Bits 902 While it is not necessary for a mixing function to produce the same 903 or fewer bits than its inputs, mixing bits cannot "stretch" the 904 amount of random unpredictability present in the inputs. Thus four 905 inputs of 32 bits each where there is 12 bits worth of 906 unpredicatability (such as 4,096 equally probable values) in each 907 input cannot produce more than 48 bits worth of unpredictable output. 908 The output can be expanded to hundreds or thousands of bits by, for 909 example, mixing with successive integers, but the clever adversary's 910 search space is still 2^48 possibilities. Furthermore, mixing to 911 fewer bits than are input will tend to strengthen the randomness of 912 the output the way using Exclusive Or to produce one bit from two did 913 above. 915 The last table in Section 6.1.1 shows that mixing a random bit with a 916 constant bit with Exclusive Or will produce a random bit. While this 917 is true, it does not provide a way to "stretch" one random bit into 918 more than one. If, for example, a random bit is mixed with a 0 and 919 then with a 1, this produces a two bit sequence but it will always be 920 either 01 or 10. Since there are only two possible values, there is 921 still only the one bit of original randomness. 923 6.1.5 Other Factors in Choosing a Mixing Function 925 For local use, AES has the advantages that it has been widely tested 926 for flaws, is reasonably efficient in software, and will be widely 927 documented and implemented with hardware and software implementations 928 available all over the world including source code available on the 929 Internet. The SHA* family are younger algorithms but there is no 930 particular reason to believe they are flawed. Both SHA* and MD5 were 931 derived from the earlier MD4 algorithm. Some signs of weakness have 932 been found in MD4 and MD5. They all have source code available [SHA*, 933 MD*]. 935 AES and SHA* have been vouched for the the US National Security 936 Agency (NSA) on the basis of criteria that primarily remain secret, 937 as was DES. While this has been the cause of much speculation and 938 doubt, investigation of DES over the years has indicated that NSA 939 involvement in modifications to its design, which originated with 940 IBM, was primarily to strengthen it. No concealed or special 941 weakness has been found in DES. It is almost certain that the NSA 942 modifications to MD4 to produce the SHA* similarly strengthened these 943 algorithms, possibly against threats not yet known in the public 944 cryptographic community. 946 AES, DES, SHA*, MD4, and MD5 are believed to be royalty free for all 947 purposes. Continued advances in crypography and computing power have 948 cast doubts on MD4 and MD5 so their use is generally not recommended. 950 Another advantage of the SHA* or similar hashing algorithms over 951 encryption algorithms in the past was that they are not subject to 952 the same regulations imposed by the US Government prohibiting the 953 unlicensed export or import of encryption/decryption software and 954 hardware. 956 6.2 Non-Hardware Sources of Randomness 958 The best source of input for mixing would be a hardware randomness 959 such as disk drive timing effected by air turbulence, audio input 960 with thermal noise, or radioactive decay. However, if that is not 961 available there are other possibilities. These include system 962 clocks, system or input/output buffers, user/system/hardware/network 963 serial numbers and/or addresses and timing, and user input. 964 Unfortunately, any of these sources can produce limited or 965 predicatable values under some circumstances. 967 Some of the sources listed above would be quite strong on multi-user 968 systems where, in essence, each user of the system is a source of 969 randomness. However, on a small single user or embedded system, 970 especially at start up, it might be possible for an adversary to 971 assemble a similar configuration. This could give the adversary 972 inputs to the mixing process that were sufficiently correlated to 973 those used originally as to make exhaustive search practical. 975 The use of multiple random inputs with a strong mixing function is 976 recommended and can overcome weakness in any particular input. For 977 example, the timing and content of requested "random" user keystrokes 978 can yield hundreds of random bits but conservative assumptions need 979 to be made. For example, assuming at most a few bits of randomness 980 if the inter-keystroke interval is unique in the sequence up to that 981 point and a similar assumption if the key hit is unique but assuming 982 that no bits of randomness are present in the initial key value or if 983 the timing or key value duplicate previous values. The results of 984 mixing these timings and characters typed could be further combined 985 with clock values and other inputs. 987 This strategy may make practical portable code to produce good random 988 numbers for security even if some of the inputs are very weak on some 989 of the target systems. However, it may still fail against a high 990 grade attack on small single user or embedded systems, especially if 991 the adversary has ever been able to observe the generation process in 992 the past. A hardware based random source is still preferable. 994 6.3 Cryptographically Strong Sequences 996 In cases where a series of random quantities must be generated, an 997 adversary may learn some values in the sequence. In general, they 998 should not be able to predict other values from the ones that they 999 know. 1001 The correct technique is to start with a strong random seed, take 1002 cryptographically strong steps from that seed [CRYPTO2, CRYPTO3], and 1003 do not reveal the complete state of the generator in the sequence 1004 elements. If each value in the sequence can be calculated in a fixed 1005 way from the previous value, then when any value is compromised, all 1006 future values can be determined. This would be the case, for 1007 example, if each value were a constant function of the previously 1008 used values, even if the function were a very strong, non-invertible 1009 message digest function. 1011 (It should be noted that if your technique for generating a sequence 1012 of key values is fast enough, it can trivially be used as the basis 1013 for a confidentiality system. If two parties use the same sequence 1014 generating technique and start with the same seed material, they will 1015 generate identical sequences. These could, for example, be xor'ed at 1016 one end with data being send, encrypting it, and xor'ed with this 1017 data as received, decrypting it due to the reversible properties of 1018 the xor operation.) 1020 6.3.1 Traditional Strong Sequences 1022 A traditional way to achieve a strong sequence has been to have the 1023 values be produced by hashing the quantities produced by 1024 concatenating the seed with successive integers or the like and then 1025 mask the values obtained so as to limit the amount of generator state 1026 available to the adversary. 1028 It may also be possible to use an "encryption" algorithm with a 1029 random key and seed value to encrypt and feedback some or all of the 1030 output encrypted value into the value to be encrypted for the next 1031 iteration. Appropriate feedback techniques will usually be 1032 recommended with the encryption algorithm. An example is shown below 1033 where shifting and masking are used to combine the cypher output 1034 feedback. This type of feedback was recommended by the US Government 1035 in connection with DES [DES MODES] but should be avoided for reasons 1036 described below. 1038 +---------------+ 1039 | V | 1040 | | n |--+ 1041 +--+------------+ | 1042 | | +---------+ 1043 | +---> | | +-----+ 1044 +--+ | Encrypt | <--- | Key | 1045 | +-------- | | +-----+ 1046 | | +---------+ 1047 V V 1048 +------------+--+ 1049 | V | | 1050 | n+1 | 1051 +---------------+ 1053 Note that if a shift of one is used, this is the same as the shift 1054 register technique described in Section 3 above but with the all 1055 important difference that the feedback is determined by a complex 1056 non-linear function of all bits rather than a simple linear or 1057 polynomial combination of output from a few bit position taps. 1059 It has been shown by Donald W. Davies that this sort of shifted 1060 partial output feedback significantly weakens an algorithm compared 1061 will feeding all of the output bits back as input. In particular, 1062 for DES, repeated encrypting a full 64 bit quantity will give an 1063 expected repeat in about 2^63 iterations. Feeding back anything less 1064 than 64 (and more than 0) bits will give an expected repeat in 1065 between 2**31 and 2**32 iterations! 1067 To predict values of a sequence from others when the sequence was 1068 generated by these techniques is equivalent to breaking the 1069 cryptosystem or inverting the "non-invertible" hashing involved with 1070 only partial information available. The less information revealed 1071 each iteration, the harder it will be for an adversary to predict the 1072 sequence. Thus it is best to use only one bit from each value. It 1073 has been shown that in some cases this makes it impossible to break a 1074 system even when the cryptographic system is invertible and can be 1075 broken if all of each generated value was revealed. 1077 6.3.2 The Blum Blum Shub Sequence Generator 1079 Currently the generator which has the strongest public proof of 1080 strength is called the Blum Blum Shub generator after its inventors 1081 [BBS]. It is also very simple and is based on quadratic residues. 1082 It's only disadvantage is that is is computationally intensive 1083 compared with the traditional techniques give in 6.3.1 above. This 1084 is not a major draw back if it is used for moderately infrequent 1085 purposes, such as generating session keys. 1087 Simply choose two large prime numbers, say p and q, which both have 1088 the property that you get a remainder of 3 if you divide them by 4. 1089 Let n = p * q. Then you choose a random number x relatively prime to 1090 n. The initial seed for the generator and the method for calculating 1091 subsequent values are then 1093 2 1094 s = ( x )(Mod n) 1095 0 1097 2 1098 s = ( s )(Mod n) 1099 i+1 i 1101 You must be careful to use only a few bits from the bottom of each s. 1102 It is always safe to use only the lowest order bit. If you use no 1103 more than the 1104 log ( log ( s ) ) 1105 2 2 i 1106 low order bits, then predicting any additional bits from a sequence 1107 generated in this manner is provable as hard as factoring n. As long 1108 as the initial x is secret, you can even make n public if you want. 1110 An intersting characteristic of this generator is that you can 1111 directly calculate any of the s values. In particular 1113 i 1114 ( ( 2 )(Mod (( p - 1 ) * ( q - 1 )) ) ) 1115 s = ( s )(Mod n) 1116 i 0 1118 This means that in applications where many keys are generated in this 1119 fashion, it is not necessary to save them all. Each key can be 1120 effectively indexed and recovered from that small index and the 1121 initial s and n. 1123 6.3.3 Entropy Pool Techniques 1125 Many modern pseudo random number sources utilize the technique of 1126 maintaining a "pool" of bits and providing operations for strongly 1127 mixing input with some randomness into the pool and extracting psuedo 1128 random bits from the pool. This is illustred in the figure below. 1130 +--------+ +------+ +---------+ 1131 --->| Mix In |--->| POOL |--->| Extract |---> 1132 | Bits | | | | Bits | 1133 +--------+ +------+ +---------+ 1134 ^ V 1135 | | 1136 +-----------+ 1138 Bits to be feed into the pool can be any of the various hardware, 1139 environmental, or user input sources discussed above. It is also 1140 common to save the state of the pool on shut down and restore it on 1141 re-starting, if stable storage is available. 1143 In fact, all of the [MD*] and [SHA*] message digest functions are 1144 implemented by internally maintaining a pool substantially larger 1145 than their ultimate output into which the bytes of the message are 1146 mixed and from which the ultimate message digest is extracted. Thus 1147 the figure above can be implemented by using parts of the message 1148 digest code to strongly mix in any new bit supplied and to compute 1149 output bits based on the pool. However, additional code is needed so 1150 that any number of bits can be extracted and appropriate feedback 1151 from the output process is mixed into the pool so as to produce a 1152 strong pseudo-random output stream. 1154 Care must be taken that enough entropy has been added to the pool to 1155 support particular output uses desired. See Section 7.3 for for more 1156 details on an example implementation and [RSA BULL1] for similar 1157 suggestions. 1159 7. Key Generation Standards and Examples 1161 Several public standards and widely deplyed examples are now in place 1162 for the generation of keys without special hardware. Two standards 1163 are described below. Both use DES but any equally strong or stronger 1164 mixing function could be substituted. Then a few widely deployed 1165 examples are described. 1167 7.1 US DoD Recommendations for Password Generation 1169 The United States Department of Defense has specific recommendations 1170 for password generation [DoD]. They suggest using the US Data 1171 Encryption Standard [DES] in Output Feedback Mode [DES MODES] as 1172 follows: 1174 use an initialization vector determined from 1175 the system clock, 1176 system ID, 1177 user ID, and 1178 date and time; 1179 use a key determined from 1180 system interrupt registers, 1181 system status registers, and 1182 system counters; and, 1183 as plain text, use an external randomly generated 64 bit 1184 quantity such as 8 characters typed in by a system 1185 administrator. 1187 The password can then be calculated from the 64 bit "cipher text" 1188 generated in 64-bit Output Feedback Mode. As many bits as are needed 1189 can be taken from these 64 bits and expanded into a pronounceable 1190 word, phrase, or other format if a human being needs to remember the 1191 password. 1193 7.2 X9.17 Key Generation 1195 The American National Standards Institute has specified a method for 1196 generating a sequence of keys as follows: 1198 s is the initial 64 bit seed 1199 0 1201 g is the sequence of generated 64 bit key quantities 1202 n 1204 k is a random key reserved for generating this key sequence 1205 t is the time at which a key is generated to as fine a resolution 1206 as is available (up to 64 bits). 1208 DES ( K, Q ) is the DES encryption of quantity Q with key K 1210 g = DES ( k, DES ( k, t ) .xor. s ) 1211 n n 1213 s = DES ( k, DES ( k, t ) .xor. g ) 1214 n+1 n 1216 If g sub n is to be used as a DES key, then every eighth bit should 1217 be adjusted for parity for that use but the entire 64 bit unmodified 1218 g should be used in calculating the next s. 1220 7.3 The /dev/random Device under Linux 1222 The Linux operating system provides a Kernel resident random number 1223 generator. This generator makes use of events captured by the Kernel 1224 during normal system operation. 1226 The generator consists of a random pool of bytes, by default 512 1227 bytes (represented as 128, 4 byte integers). When an event occurs, 1228 such as a disk drive interrupt, the time of the event is xor'ed into 1229 the pool and the pool is stirred via a primitive polynomial of degree 1230 128. The pool itself is treated as a ring buffer, with new data 1231 being xor'ed (after stirring with the polynomial) across the entire 1232 pool. 1234 Each call that adds entropy to the pool estimates the amount of 1235 likely true entropy the input contains. The pool itself contains a 1236 accumulator that estimates the total over all entropy of the pool. 1238 Input events come from several sources: 1240 1. Keyboard interrupts. The time of the interrupt as well as the scan 1241 code are added to the pool. This in effect adds entropy from the 1242 human operator by measuring inter-keystroke arrival times. 1244 2. Disk completion and other interrupts. A system being used by a 1245 person will likely have a hard to predict pattern of disk 1246 accesses. 1248 3. Mouse motion. The timing as well as mouse position is added in. 1250 When random bytes are required, the pool is hashed with SHA-1 [SHA1] 1251 to yield the returned bytes of randomness. If more bytes are required 1252 than the output of SHA-1 (20 bytes), then the hashed output is 1253 stirred back into the pool and a new hash performed to obtain the 1254 next 20 bytes. As bytes are removed from the pool, the estimate of 1255 entropy is similarly decremented. 1257 To ensure a reasonable random pool upon system startup, the standard 1258 Linux startup scripts (and shutdown scripts) save the pool to a disk 1259 file at shutdown and read this file at system startup. 1261 There are two user exported interfaces. /dev/random returns bytes 1262 from the pool, but blocks when the estimated entropy drops to zero. 1263 As entropy is added to the pool from events, more data becomes 1264 available via /dev/random. Random data obtained /dev/random is 1265 suitable for key generation for long term keys. 1267 /dev/urandom works like /dev/random, however it provides data even 1268 when the entropy estimate for the random pool drops to zero. This 1269 should be fine for session keys. The risk of continuing to take data 1270 even when the pools entropy estimate is small is that past output may 1271 be computable from current output provided an attacker can reverse 1272 SHA-1. Given that SHA-1 should not be invertible, this is a 1273 reasonable risk. 1275 To obtain random numbers under Linux, all an application needs to do 1276 is open either /dev/random or /dev/urandom and read the desired 1277 number of bytes. 1279 The Linux Random device was written by Theodore Ts'o. It is based 1280 loosely on the random number generator in PGP 2.X and PGP 3.0 (aka 1281 PGP 5.0). 1283 8. Examples of Randomness Required 1285 Below are two examples showing rough calculations of needed 1286 randomness for security. The first is for moderate security 1287 passwords while the second assumes a need for a very high security 1288 cryptographic key. 1290 In addition [ORMAN] and [RSA BULL13] provide information on the 1291 public key lengths that should be used for exchanging symmetric keys. 1293 8.1 Password Generation 1295 Assume that user passwords change once a year and it is desired that 1296 the probability that an adversary could guess the password for a 1297 particular account be less than one in a thousand. Further assume 1298 that sending a password to the system is the only way to try a 1299 password. Then the crucial question is how often an adversary can 1300 try possibilities. Assume that delays have been introduced into a 1301 system so that, at most, an adversary can make one password try every 1302 six seconds. That's 600 per hour or about 15,000 per day or about 1303 5,000,000 tries in a year. Assuming any sort of monitoring, it is 1304 unlikely someone could actually try continuously for a year. In 1305 fact, even if log files are only checked monthly, 500,000 tries is 1306 more plausible before the attack is noticed and steps taken to change 1307 passwords and make it harder to try more passwords. 1309 To have a one in a thousand chance of guessing the password in 1310 500,000 tries implies a universe of at least 500,000,000 passwords or 1311 about 2^29. Thus 29 bits of randomness are needed. This can probably 1312 be achieved using the US DoD recommended inputs for password 1313 generation as it has 8 inputs which probably average over 5 bits of 1314 randomness each (see section 7.1). Using a list of 1000 words, the 1315 password could be expressed as a three word phrase (1,000,000,000 1316 possibilities) or, using case insensitive letters and digits, six 1317 would suffice ((26+10)^6 = 2,176,782,336 possibilities). 1319 For a higher security password, the number of bits required goes up. 1320 To decrease the probability by 1,000 requires increasing the universe 1321 of passwords by the same factor which adds about 10 bits. Thus to 1322 have only a one in a million chance of a password being guessed under 1323 the above scenario would require 39 bits of randomness and a password 1324 that was a four word phrase from a 1000 word list or eight 1325 letters/digits. To go to a one in 10^9 chance, 49 bits of randomness 1326 are needed implying a five word phrase or ten letter/digit password. 1328 In a real system, of course, there are also other factors. For 1329 example, the larger and harder to remember passwords are, the more 1330 likely users are to write them down resulting in an additional risk 1331 of compromise. 1333 8.2 A Very High Security Cryptographic Key 1335 Assume that a very high security key is needed for symmetric 1336 encryption / decryption between two parties. Assume an adversary can 1337 observe communications and knows the algorithm being used. Within 1338 the field of random possibilities, the adversary can try key values 1339 in hopes of finding the one in use. Assume further that brute force 1340 trial of keys is the best the adversary can do. 1342 8.2.1 Effort per Key Trial 1344 How much effort will it take to try each key? For very high security 1345 applications it is best to assume a low value of effort. This 1346 question is considered in detail in Appendix A. It concludes that a 1347 reasonable key length in 1995 for very high security is in the range 1348 of 75 to 90 bits and, since the cost of cryptography does not vary 1349 much with they key size, recommends 90 bits. To update these 1350 recommendations, just add 2/3 of a bit per year for Moore's law 1351 [MOORE]. Thus, in the year 2002, this translates to a determination 1352 that a reasonable key length is in 79 to 94 bit range. 1354 8.2.2 Meet in the Middle Attacks 1356 If chosen or known plain text and the resulting encrypted text are 1357 available, a "meet in the middle" attack is possible if the structure 1358 of the encryption algorithm allows it. (In a known plain text 1359 attack, the adversary knows all or part of the messages being 1360 encrypted, possibly some standard header or trailer fields. In a 1361 chosen plain text attack, the adversary can force some chosen plain 1362 text to be encrypted, possibly by "leaking" an exciting text that 1363 would then be sent by the adversary over an encrypted channel.) 1365 An oversimplified explanation of the meet in the middle attack is as 1366 follows: the adversary can half-encrypt the known or chosen plain 1367 text with all possible first half-keys, sort the output, then half- 1368 decrypt the encoded text with all the second half-keys. If a match 1369 is found, the full key can be assembled from the halves and used to 1370 decrypt other parts of the message or other messages. At its best, 1371 this type of attack can halve the exponent of the work required by 1372 the adversary while adding a large but roughly constant factor of 1373 effort. To be assured of safety against this, a doubling of the 1374 amount of randomness in the very strong key to a minimum of 178 bits 1375 is required for the year 2002 based on the Appendix A analysis. 1377 This amount of randomness is beyond the limit of that in the inputs 1378 recommended by the US DoD for password generation and could require 1379 user typing timing, hardware random number generation, or other 1380 sources. 1382 The meet in the middle attack assumes that the cryptographic 1383 algorithm can be decomposed in this way but we can not rule that out 1384 without a deep knowledge of the algorithm. Even if a basic algorithm 1385 is not subject to a meet in the middle attack, an attempt to produce 1386 a stronger algorithm by applying the basic algorithm twice (or two 1387 different algorithms sequentially) with different keys may gain less 1388 added security than would be expected. Such a composite algorithm 1389 would be subject to a meet in the middle attack. 1391 Enormous resources may be required to mount a meet in the middle 1392 attack but they are probably within the range of the national 1393 security services of a major nation. Essentially all nations spy on 1394 other nations government traffic and several nations are believed to 1395 spy on commercial traffic for economic advantage. 1397 It should be noted that key length calculations such at those above 1398 are controversial and depend on various assumptions about the 1399 cryptographic algorithms in use. In some cases, a professional with 1400 a deep knowledge of code breaking techniques and of the strength of 1401 the algorithm in use could be satisfied with less than half of the 1402 178 bit key size derived above. 1404 9. Conclusion 1406 Generation of unguessable "random" secret quantities for security use 1407 is an essential but difficult task. 1409 Hardware techniques to produce such randomness would be relatively 1410 simple. In particular, the volume and quality would not need to be 1411 high and existing computer hardware, such as disk drives, can be 1412 used. 1414 Computational techniques are available to process low quality random 1415 quantities from multiple sources or a larger quantity of such low 1416 quality input from one source and produce a smaller quantity of 1417 higher quality keying material. In the absence of hardware sources 1418 of randomness, a variety of user and software sources can frequently, 1419 with care, be used instead; however, most modern systems already have 1420 hardware, such as disk drives or audio input, that could be used to 1421 produce high quality randomness. 1423 Once a sufficient quantity of high quality seed key material (a 1424 couple of hundred bits) is available, computational techniques are 1425 available to produce cryptographically strong sequences of 1426 unpredicatable quantities from this seed material. 1428 10. Security Considerations 1430 The entirety of this document concerns techniques and recommendations 1431 for generating unguessable "random" quantities for use as passwords, 1432 cryptographic keys, initialiazation vectors, sequence numbers, and 1433 similar security uses. 1435 Appendix: Minimal Secure Key Lengths Study 1437 Minimal Key Lengths for Symmetric Ciphers 1438 to Provide Adequate Commercial Security 1440 A Report by an Ad Hoc Group of 1441 Cryptographers and Computer Scientists 1443 Matt Blaze, AT&T Research, mab@research.att.com 1444 Whitfield Diffie, Sun Microsystems, diffie@eng.sun.com 1445 Ronald L. Rivest, MIT LCS, rivest@lcs.mit.edu 1446 Bruce Schneier, Counterpane Systems, schneier@counterpane.com 1447 Tsutomu Shimomura, San Diego Supercomputer Center, tsutomu@sdsc.edu 1448 Eric Thompson Access Data, Inc., eric@accessdata.com 1449 Michael Wiener, Bell Northern Research, wiener@bnr.ca 1451 January 1996 1453 A.0 Abstract 1455 Encryption plays an essential role in protecting the privacy of 1456 electronic information against threats from a variety of potential 1457 attackers. In so doing, modern cryptography employs a combination of 1458 _conventional_ or _symmetric_ cryptographic systems for encrypting 1459 data and _public key_ or _asymmetric_ systems for managing the _keys_ 1460 used by the symmetric systems. Assessing the strength required of 1461 the symmetric cryptographic systems is therefore an essential step in 1462 employing cryptography for computer and communication security. 1464 Technology readily available today (late 1995) makes _brute- 1465 force_ attacks against cryptographic systems considered adequate for 1466 the past several years both fast and cheap. General purpose 1467 computers can be used, but a much more efficient approach is to 1468 employ commercially available _Field Programmable Gate Array (FPGA)_ 1469 technology. For attackers prepared to make a higher initial 1470 investment, custom-made, special-purpose chips make such calculations 1471 much faster and significantly lower the amortized cost per solution. 1473 As a result, cryptosystems with 40-bit keys offer virtually no 1474 protection at this point against brute-force attacks. Even the U.S. 1475 Data Encryption Standard with 56-bit keys is increasingly inadequate. 1476 As cryptosystems often succumb to `smarter' attacks than brute-force 1477 key search, it is also important to remember that the keylengths 1478 discussed here are the minimum needed for security against the 1479 computational threats considered. 1481 Fortunately, the cost of very strong encryption is not 1483 significantly greater than that of weak encryption. Therefore, to 1484 provide adequate protection against the most serious threats --- 1485 well-funded commercial enterprises or government intelligence 1486 agencies --- keys used to protect data today should be at least 75 1487 bits long. To protect information adequately for the next 20 years 1488 in the face of expected advances in computing power, keys in newly- 1489 deployed systems should be at least 90 bits long. 1491 A.1. Encryption Plays an Essential Role in Protecting 1492 the Privacy of Electronic Information" 1494 A.1.1 There is a need for information security 1496 As we write this paper in late 1995, the development of 1497 electronic commerce and the Global Information Infrastructure is at a 1498 critical juncture. The dirt paths of the middle ages only became 1499 highways of business and culture after the security of travelers and 1500 the merchandise they carried could be assured. So too the 1501 information superhighway will be an ill-traveled road unless 1502 information, the goods of the Information Age, can be moved, stored, 1503 bought, and sold securely. Neither corporations nor individuals will 1504 entrust their private business or personal data to computer networks 1505 unless they can assure their information's security. 1507 Today, most forms of information can be stored and processed 1508 electronically. This means a wide variety of information, with 1509 varying economic values and privacy aspects and with a wide variation 1510 in the time over which the information needs to be protected, will be 1511 found on computer networks. Consider the spectrum: 1513 o Electronic Funds Transfers of millions or even billions of 1514 dollars, whose short term security is essential but whose 1515 exposure is brief; 1517 o A company's strategic corporate plans, whose confidentiality 1518 must be preserved for a small number of years; 1520 o A proprietary product (Coke formula, new drug design) that 1521 needs to be protected over its useful life, often decades; 1522 and 1524 o Information private to an individual (medical condition, 1525 employment evaluation) that may need protection for the 1526 lifetime of the individual. 1528 A.1.2 Encryption to protect confidentiality 1530 Encryption Can Provide Strong Confidentiality Protection 1532 Encryption is accomplished by scrambling data using mathematical 1533 procedures that make it extremely difficult and time consuming for 1534 anyone other than authorized recipients --- those with the correct 1535 decryption _keys_ --- to recover the _plain text_. Proper encryption 1536 guarantees that the information will be safe even if it falls into 1537 hostile hands. 1539 Encryption --- and decryption --- can be performed by either 1540 computer software or hardware. Common approaches include writing the 1541 algorithm on a disk for execution by a computer central processor; 1542 placing it in ROM or PROM for execution by a microprocessor; and 1543 isolating storage and execution in a computer accessory device (smart 1544 card or PCMCIA card). 1546 The degree of protection obtained depends on several factors. 1547 These include: the quality of the cryptosystem; the way it is 1548 implemented in software or hardware (especially its reliability and 1549 the manner in which the keys are chosen); and the total number of 1550 possible keys that can be used to encrypt the information. A 1551 cryptographic algorithm is considered strong if: 1553 1. There is no shortcut that allows the opponent to recover the 1554 plain text without using brute force to test keys until the 1555 correct one is found; and 1557 2. The number of possible keys is sufficiently large to make 1558 such an attack infeasible. 1560 The principle here is similar to that of a combination lock on a 1561 safe. If the lock is well designed so that a burglar cannot hear or 1562 feel its inner workings, a person who does not know the combination 1563 can open it only by dialing one set of numbers after another until it 1564 yields. 1566 The sizes of encryption keys are measured in bits and the 1567 difficulty of trying all possible keys grows exponentially with the 1568 number of bits used. Adding one bit to the key doubles the number of 1569 possible keys; adding ten increases it by a factor of more than a 1570 thousand. 1572 There is no definitive way to look at a cipher and determine 1573 whether a shortcut exists. Nonetheless, several encryption 1574 algorithms --- most notably the U.S Data Encryption Standard (DES) 1575 --- have been extensively studied in the public literature and are 1576 widely believed to be of very high quality. An essential element in 1577 cryptographic algorithm design is thus the length of the key, whose 1578 size places an upper bound on the system's strength. 1580 Throughout this paper, we will assume that there are no shortcuts 1581 and treat the length of the key as representative of the 1582 cryptosystem's _workfactor_ --- the minimum amount of effort required 1583 to break the system. It is important to bear in mind, however, that 1584 cryptographers regard this as a rash assumption and many would 1585 recommend keys two or more times as long as needed to resist brute- 1586 force attacks. Prudent cryptographic designs not only employ longer 1587 keys than might appear to be needed, but devote more computation to 1588 encrypting and decrypting. A good example of this is the popular 1589 approach of using _triple-DES_: encrypting the output of DES twice 1590 more, using a total of three distinct keys. 1592 Encryption systems fall into two broad classes. Conventional or 1593 symmetric cryptosystems --- those in which an entity with the ability 1594 to encrypt also has the ability to decrypt and vice versa --- are the 1595 systems under consideration in this paper. The more recent public 1596 key or asymmetric cryptosystems have the property that the ability to 1597 encrypt does not imply the ability to decrypt. In contemporary 1598 cryptography, public-key systems are indispensable for managing the 1599 keys of conventional cryptosystems. All known public key 1600 cryptosystems, however, are subject to shortcut attacks and must 1601 therefore use keys ten or more times the lengths of those discussed 1602 here to achieve the an equivalent level of security. 1604 Although computers permit electronic information to be encrypted 1605 using very large keys, advances in computing power keep pushing up 1606 the size of keys that can be considered large and thus keep making it 1607 easier for individuals and organizations to attack encrypted 1608 information without the expenditure of unreasonable resources. 1610 A.1.3 There are a variety of attackers 1612 There Are Threats from a Variety of Potential Attackers. 1614 Threats to confidentiality of information come from a number of 1615 directions and their forms depend on the resources of the attackers. 1616 `Hackers,' who might be anything from high school students to 1617 commercial programmers, may have access to mainframe computers or 1618 networks of workstations. The same people can readily buy 1619 inexpensive, off-the-shelf, boards, containing _Field Programmable 1620 Gate Array (FPGA)_ chips that function as `programmable hardware' and 1621 vastly increase the effectiveness of a cryptanalytic effort. A 1622 startup company or even a well-heeled individual could afford large 1623 numbers of these chips. A major corporation or organized crime 1624 operation with `serious money' to spend could acquire custom computer 1625 chips specially designed for decryption. An intelligence agency, 1626 engaged in espionage for national economic advantage, could build a 1627 machine employing millions of such chips. 1629 A.1.4 Strong encryption is not expensive 1631 Current Technology Permits Very Strong Encryption for Effectively the 1632 Same Cost As Weaker Encryption. 1634 It is a property of computer encryption that modest increases in 1635 computational cost can produce vast increases in security. 1636 Encrypting information very securely (e.g., with 128-bit keys) 1637 typically requires little more computing than encrypting it weakly 1638 (e.g., with 40-bit keys). In many applications, the cryptography 1639 itself accounts for only a small fraction of the computing costs, 1640 compared to such processes as voice or image compression required to 1641 prepare material for encryption. 1643 One consequence of this uniformity of costs is that there is 1644 rarely any need to tailor the strength of cryptography to the 1645 sensitivity of the information being protected. Even if most of the 1646 information in a system has neither privacy implications nor monetary 1647 value, there is no practical or economic reason to design computer 1648 hardware or software to provide differing levels of encryption for 1649 different messages. It is simplest, most prudent, and thus 1650 fundamentally most economical, to employ a uniformly high level of 1651 encryption: the strongest encryption required for any information 1652 that might be stored or transmitted by a secure system. 1654 A.2. Brute-Forece is becoming easier 1656 Readily Available Technology Makes Brute-Force Decryption Attacks 1657 Faster and Cheaper. 1659 The kind of hardware used to mount a brute-force attack against 1660 an encryption algorithm depends on the scale of the cryptanalytic 1661 operation and the total funds available to the attacking enterprise. 1662 In the analysis that follows, we consider three general classes of 1663 technology that are likely to be employed by attackers with differing 1664 resources available to them. Not surprisingly, the cryptanalytic 1665 technologies that require larger up-front investments yield the 1666 lowest cost per recovered key, amortized over the life of the 1667 hardware. 1669 It is the nature of brute-force attacks that they can be 1670 parallelized indefinitely. It is possible to use as many machines as 1671 are available, assigning each to work on a separate part of the 1672 problem. Thus regardless of the technology employed, the search time 1673 can be reduced by adding more equipment; twice as much hardware can 1674 be expected to find the right key in half the time. The total 1675 investment will have doubled, but if the hardware is kept constantly 1676 busy finding keys, the cost per key recovered is unchanged. 1678 At the low end of the technology spectrum is the use of 1679 conventional personal computers or workstations programmed to test 1680 keys. Many people, by virtue of already owning or having access to 1681 the machines, are in a position use such resources at little or no 1682 cost. However, general purpose computers --- laden with such 1683 ancillary equipment as video controllers, keyboards, interfaces, 1684 memory, and disk storage --- make expensive search engines. They are 1685 therefore likely to be employed only by casual attackers who are 1686 unable or unwilling to invest in more specialized equipment. 1688 A more efficient technological approach is to take advantage of 1689 commercially available Field Programmable Gate Arrays. FPGAs 1690 function as programmable hardware and allow faster implementations of 1691 such tasks as encryption and decryption than conventional processors. 1692 FPGAs are a commonly used tool for simple computations that need to 1693 be done very quickly, particularly simulating integrated circuits 1694 during development. 1696 FPGA technology is fast and cheap. The cost of an AT&T ORCA chip 1697 that can test 30 million DES keys per second is $200. This is 1,000 1698 times faster than a PC at about one-tenth the cost! FPGAs are widely 1699 available and, mounted on cards, can be installed in standard PCs 1700 just like sound cards, modems, or extra memory. 1702 FPGA technology may be optimal when the same tool must be used 1703 for attacking a variety of different cryptosystems. Often, as with 1704 DES, a cryptosystem is sufficiently widely used to justify the 1705 construction of more specialized facilities. In these circumstances, 1706 the most cost-effective technology, but the one requiring the largest 1707 initial investment, is the use of _Application-Specific Integrated 1708 Circuits (ASICs)_. A $10 chip can test 200 million keys per second. 1709 This is seven times faster than an FPGA chip at one-twentieth the 1710 cost. 1712 Because ASICs require a far greater engineering investment than 1713 FPGAs and must be fabricated in quantity before they are economical, 1714 this approach is only available to serious, well-funded operations 1715 such as dedicated commercial (or criminal) enterprises and government 1716 intelligence agencies. 1718 A.3. 40-Bit Key Lengths Offer Virtually No Protection 1720 Current U.S. Government policy generally limits exportable mass 1721 market software that incorporates encryption for confidentiality to 1722 using the RC2 or RC4 algorithms with 40-bit keys. A 40-bit key 1723 length means that there are 2^40 possible keys. On average, half of 1724 these (2^39) must be tried to find the correct one. Export of other 1725 algorithms and key lengths must be approved on a case by case basis. 1726 For example, DES with a 56-bit key has been approved for certain 1727 applications such as financial transactions. 1729 The recent successful brute-force attack by two French graduate 1730 students on Netscape's 40-bit RC4 algorithm demonstrates the dangers 1731 of such short keys. These students at the Ecole Polytechnique in 1732 Paris used `idle time' on the school's computers, incurring no cost 1733 to themselves or their school. Even with these limited resources, 1734 they were able to recover the 40-bit key in a few days. 1736 There is no need to have the resources of an institution of 1737 higher education at hand, however. Anyone with a modicum of computer 1738 expertise and a few hundred dollars would be able to attack 40-bit 1739 encryption much faster. An FPGA chip --- costing approximately $400 1740 mounted on a card --- would on average recover a 40-bit key in five 1741 hours. Assuming the FPGA lasts three years and is used continuously 1742 to find keys, the average cost per key is eight cents. 1744 A more determined commercial predator, prepared to spend $10,000 1745 for a set-up with 25 ORCA chips, can find 40-bit keys in an average 1746 of 12 minutes, at the same average eight cent cost. Spending more 1747 money to buy more chips reduces the time accordingly: $300,000 1748 results in a solution in an average of 24 seconds; $10,000,000 1749 results in an average solution in 0.7 seconds. 1751 As already noted, a corporation with substantial resources can 1752 design and commission custom chips that are much faster. By doing 1753 this, a company spending $300,000 could find the right 40-bit key in 1754 an average of 0.18 seconds at 1/10th of a cent per solution; a larger 1755 company or government agency willing to spend $10,000,000 could find 1756 the right key on average in 0.005 seconds (again at 1/10th of a cent 1757 per solution). (Note that the cost per solution remains constant 1758 because we have conservatively assumed constant costs for chip 1759 acquisition --- in fact increasing the quantities purchased of a 1760 custom chip reduces the average chip cost as the initial design and 1761 set-up costs are spread over a greater number of chips.) 1763 These results are summarized in Table I (below). 1765 A.4. Even DES with 56-Bit Keys Is Increasingly Inadequate 1767 A.4.1 DES is no panacea today 1769 The Data Encryption Standard (DES) was developed in the 1970s by 1770 IBM and NSA and adopted by the U.S. Government as a Federal 1771 Information Processing Standard for data encryption. It was intended 1772 to provide strong encryption for the government's sensitive but 1773 unclassified information. It was recognized by many, even at the 1774 time DES was adopted, that technological developments would make 1775 DES's 56-bit key exceedingly vulnerable to attack before the end of 1776 the century. 1778 Today, DES may be the most widely employed encryption algorithm 1779 and continues to be a commonly cited benchmark. Yet DES-like 1780 encryption strength is no panacea. Calculations show that DES is 1781 inadequate against a corporate or government attacker committing 1782 serious resources. The bottom line is that DES is cheaper and easier 1783 to break than many believe. 1785 As explained above, 40-bit encryption provides inadequate 1786 protection against even the most casual of intruders, content to 1787 scavenge time on idle machines or to spend a few hundred dollars. 1788 Against such opponents, using DES with a 56-bit key will provide a 1789 substantial measure of security. At present, it would take a year 1790 and a half for someone using $10,000 worth of FPGA technology to 1791 search out a DES key. In ten years time an investment of this size 1792 would allow one to find a DES key in less than a week. 1794 The real threat to commercial transactions and to privacy on the 1795 Internet is from individuals and organizations willing to invest 1796 substantial time and money. As more and more business and personal 1797 information becomes electronic, the potential rewards to a dedicated 1798 commercial predator also increase significantly and may justify the 1799 commitment of adequate resources. 1801 A serious effort --- on the order of $300,000 --- by a legitimate 1802 or illegitimate business could find a DES key in an average of 19 1803 days using off-the-shelf technology and in only 3 hours using a 1804 custom developed chip. In the latter case, it would cost $38 to find 1805 each key (again assuming a 3 year life to the chip and continual 1806 use). A business or government willing to spend $10,000,000 on 1807 custom chips, could recover DES keys in an average of 6 minutes, for 1808 the same $38 per key. 1810 At the very high end, an organization --- presumably a government 1811 intelligence agency --- willing to spend $300,000,000 could recover 1812 DES keys in 12 seconds each! The investment required is large but 1813 not unheard of in the intelligence community. It is less than the 1814 cost of the Glomar Explorer, built to salvage a single Russian 1815 submarine, and far less than the cost of many spy satellites. Such 1816 an expense might be hard to justify in attacking a single target, but 1817 seems entirely appropriate against a cryptographic algorithm, like 1818 DES, enjoying extensive popularity around the world. 1820 There is ample evidence of the danger presented by government 1821 intelligence agencies seeking to obtain information not only for 1822 military purposes but for commercial advantage. Congressional 1823 hearings in 1993 highlighted instances in which the French and 1824 Japanese governments spied on behalf of their countries' own 1825 businesses. Thus, having to protect commercial information against 1826 such threats is not a hypothetical proposition. 1828 A.4.2 There are smarter avenues of attack than brute force 1830 It is easier to walk around a tree than climb up and down it. 1831 There is no need to break the window of a house to get in if the 1832 front door is unlocked. 1834 Calculations regarding the strength of encryption against brute- 1835 force attack are _worst case_ scenarios. They assume that the 1836 ciphers are in a sense perfect and that attempts to find shortcuts 1837 have failed. One important point is that the crudest approach --- 1838 searching through the keys --- is entirely feasible against many 1839 widely used systems. Another is that the keylengths we discuss are 1840 always minimal. As discussed earlier, prudent designs might use keys 1841 twice or three times as long to provide a margin of safety. 1843 A.4.3 Other algorithms are similar 1845 The Analysis for Other Algorithms Is Roughly Comparable. 1847 The above analysis has focused on the time and money required to 1848 find a key to decrypt information using the RC4 algorithm with a 1849 40-bit key or the DES algorithm with its 56-bit key, but the results 1850 are not peculiar to these ciphers. Although each algorithm has its 1851 own particular characteristics, the effort required to find the keys 1852 of other ciphers is comparable. There may be some differences as the 1853 result of implementation procedures, but these do not materially 1854 affect the brute-force breakability of algorithms with roughly 1855 comparable key lengths. 1857 Specifically, it has been suggested at times that differences in 1858 set-up procedures, such as the long key-setup process in RC4, result 1859 in some algorithms having effectively longer keys than others. For 1860 the purpose of our analysis, such factors appear to vary the 1861 effective key length by no more than about eight bits. 1863 A.5. Appropriate Key Lengths for the Future --- A Proposal 1865 Table I summarizes the costs of carrying out brute-force attacks 1866 against symmetric cryptosystems with 40-bit and 56-bit keys using 1867 networks of general purpose computers, Field Programmable Gate 1868 Arrays, and special-purpose chips. 1870 It shows that 56 bits provides a level of protection --- about a 1871 year and a half --- that would be adequate for many commercial 1872 purposes against an opponent prepared to invest $10,000. Against an 1873 opponent prepared to invest $300,000, the period of protection has 1874 dropped to the barest minimum of 19 days. Above this, the protection 1875 quickly declines to negligible. A very large, but easily imaginable, 1876 investment by an intelligence agency would clearly allow it to 1877 recover keys in real time. 1879 What workfactor would be required for security today? For an 1880 opponent whose budget lay in the $10 to 300 million range, the time 1881 required to search out keys in a 75-bit keyspace would be between 6 1882 years and 70 days. Although the latter figure may seem comparable to 1883 the `barest minimum' 19 days mentioned earlier, it represents --- 1884 under our amortization assumptions --- a cost of $19 million and a 1885 recovery rate of only five keys a year. The victims of such an 1886 attack would have to be fat targets indeed. 1888 Because many kinds of information must be kept confidential for 1889 long periods of time, assessment cannot be limited to the protection 1890 required today. Equally important, cryptosystems --- especially if 1891 they are standards --- often remain in use for years or even decades. 1892 DES, for example, has been in use for more than 20 years and will 1893 probably continue to be employed for several more. In particular, 1894 the lifetime of a cryptosystem is likely to exceed the lifetime of 1895 any individual product embodying it. 1897 A rough estimate of the minimum strength required as a function 1898 of time can be obtained by applying an empirical rule, popularly 1899 called `Moore's Law,' which holds that the computing power available 1900 for a given cost doubles every 18 months. Taking into account both 1901 the lifetime of cryptographic equipment and the lifetime of the 1902 secrets it protects, we believe it is prudent to require that 1903 encrypted data should still be secure in 20 years. Moore's Law thus 1904 predicts that the keys should be approximately 14 bits longer than 1905 required to protect against an attack today. 1907 *Bearing in mind that the additional computational costs of 1908 stronger encryption are modest, we strongly recommend a minimum key- 1909 length of 90 bits for symmetric cryptosystems.* 1911 It is instructive to compare this recommendation with both 1912 Federal Information Processing Standard 46, The Data Encryption 1913 Standard (DES), and Federal Information Processing Standard 185, The 1914 Escrowed Encryption Standard (EES). DES was proposed 21 years ago 1915 and used a 56-bit key. Applying Moore's Law and adding 14 bits, we 1916 see that the strength of DES when it was proposed in 1975 was 1917 comparable to that of a 70-bit system today. Furthermore, it was 1918 estimated at the time that DES was not strong enough and that keys 1919 could be recovered at a rate of one per day for an investment of 1920 about twenty-million dollars. Our 75-bit estimate today corresponds 1921 to 61 bits in 1975, enough to have moved the cost of key recovery 1922 just out of reach. The Escrowed Encryption Standard, while 1923 unacceptable to many potential users for other reasons, embodies a 1924 notion of appropriate key length that is similar to our own. It uses 1925 80-bit keys, a number that lies between our figures of 75 and 90 1926 bits. 1928 Table I 1930 Time and cost Length Needed 1931 Type of Budget Tool per key recovered for protection 1932 Attacker 40bits 56bits in Late 1995 1934 Pedestrian Hacker 1936 tiny scavenged 1 week infeasible 45 1937 computer 1938 time 1940 $400 FPGA 5 hours 38 years 50 1941 ($0.08) ($5,000) 1943 Small Business 1945 $10,000 FPGA 12 minutes 556 days 55 1946 ($0.08) ($5,000) 1948 Corporate Department 1950 $300K FPGA 24 seconds 19 days 60 1951 or ($0.08) ($5,000) 1952 ASIC .18 seconds 3 hours 1953 ($0.001) ($38) 1955 Big Company 1956 $10M FPGA .7 seconds 13 hours 70 1957 or ($0.08) ($5,000) 1958 ASIC .005 seconds 6 minutes 1959 ($0.001) ($38) 1961 Intellegence Agency 1963 $300M ASIC .0002 seconds 12 seconds 75 1964 ($0.001) ($38) 1966 A.6 About the Authors 1968 *Matt Blaze* is a senior research scientist at AT&T Research in the 1969 area of computer security and cryptography. Recently Blaze 1970 demonstrated weaknesses in the U.S. government's `Clipper Chip' key 1971 escrow encryption system. His current interests include large-scale 1972 trust management and the applications of smartcards. 1974 *Whitfield Diffie* is a distinguished Engineer at Sun Microsystems 1975 specializing in security. In 1976 Diffie and Martin Hellman created 1976 public key cryptography, which solved the problem of sending coded 1977 information between individuals with no prior relationship and is the 1978 basis for widespread encryption in the digital information age. 1980 *Ronald L. Rivest* is a professor of computer science at the 1981 Massachusetts Institute of Technology, and is Associate Director of 1982 MIT's Laboratory for Computer Science. Rivest, together with Leonard 1983 Adleman and Adi Shamir, invented the RSA public-key cryptosystem that 1984 is used widely throughout industry. Ron Rivest is one of the 1985 founders of RSA Data Security Inc. and is the creator of variable key 1986 length symmetric key ciphers (e.g., RC4). 1988 *Bruce Schneier* is president of Counterpane Systems, a consulting 1989 firm specializing in cryptography and computer security. Schneier 1990 writes and speaks frequently on computer security and privacy and is 1991 the author of a leading cryptography textbook, Applied Cryptography, 1992 and is the creator of the symmetric key cipher Blowfish. 1994 *Tsutomu Shimomura* is a computational physicist employed by the San 1995 Diego Supercomputer Center who is an expert in designing software 1996 security tools. Last year, Shimomura was responsible for tracking 1997 down the computer outlaw Kevin Mitnick, who electronically stole and 1998 altered valuable electronic information around the country. 2000 *Eric Thompson* heads AccessData Corporation's cryptanalytic team and 2001 is a frequent lecturer on applied crytography. AccessData 2002 specializes in data recovery and decrypting information utilizing 2003 brute force as well as `smarter' attacks. Regular clients include 2004 the FBI and other law enforcement agencies as well as corporations. 2006 *Michael Wiener* is a cryptographic advisor at Bell-Northern Research 2007 where he focuses on cryptanalysis, security architectures, and 2008 public-key infrastructures. His influential 1993 paper, Efficient 2009 DES Key Search, describes in detail how to construct a machine to 2010 brute force crack DES coded information (and provides cost estimates 2011 as well). 2013 A.7 Acknowledgement 2015 The [Appendix] authors would like to thank the Business Software 2016 Alliance, which provided support for a one-day meeting, held in 2017 Chicago on 20 November 1995. 2019 References 2021 [AES] - "Specification of theAdvanced Encryption Standard (AES)", 2022 United States of America, Department of Commerce, National Institute 2023 of Standards and Technology, Federal Information Processing Standard 2024 197, November 2001. 2026 [ASYMMETRIC] - "Secure Communications and Asymmetric Cryptosystems", 2027 edited by Gustavus J. Simmons, AAAS Selected Symposium 69, Westview 2028 Press, Inc. 2030 [BBS] - "A Simple Unpredictable Pseudo-Random Number Generator", SIAM 2031 Journal on Computing, v. 15, n. 2, 1986, L. Blum, M. Blum, & M. Shub. 2033 [BRILLINGER] - "Time Series: Data Analysis and Theory", Holden-Day, 2034 1981, David Brillinger. 2036 [CRC] - "C.R.C. Standard Mathematical Tables", Chemical Rubber 2037 Publishing Company. 2039 [CRYPTO1] - "Cryptography: A Primer", A Wiley-Interscience 2040 Publication, John Wiley & Sons, 1981, Alan G. Konheim. 2042 [CRYPTO2] - "Cryptography: A New Dimension in Computer Data 2043 Security", A Wiley-Interscience Publication, John Wiley & Sons, 1982, 2044 Carl H. Meyer & Stephen M. Matyas. 2046 [CRYPTO3] - "Applied Cryptography: Protocols, Algorithsm, and Source 2047 Code in C", Second Edition, John Wiley & Sons, 1996, Bruce Schneier. 2049 [DAVIS] - "Cryptographic Randomness from Air Turbulence in Disk 2050 Drives", Advances in Cryptology - Crypto '94, Springer-Verlag Lecture 2051 Notes in Computer Science #839, 1984, Don Davis, Ross Ihaka, and 2052 Philip Fenstermacher. 2054 [DES] - "Data Encryption Standard", United States of America, 2055 Department of Commerce, National Institute of Standards and 2056 Technology, Federal Information Processing Standard (FIPS) 46-3, 2057 October 1999. 2058 - "Data Encryption Algorithm", American National Standards Institute, 2059 ANSI X3.92-1981. 2060 (See also FIPS 112, Password Usage, which includes FORTRAN code for 2061 performing DES.) 2063 [DES MODES] - "DES Modes of Operation", United States of America, 2064 Department of Commerce, National Institute of Standards and 2065 Technology, Federal Information Processing Standard (FIPS) 81, 2066 December 1980. 2067 - Data Encryption Algorithm - Modes of Operation, American National 2068 Standards Institute, ANSI X3.106-1983. 2070 [D-H] - "New Directions in Cryptography", IEEE Transactions on 2071 Information Technology, November, 1976, Whitfield Diffie and Martin 2072 E. Hellman. 2074 [DNSSEC] - RFC 2535, "Domain Name System Security Extensions", D. 2075 Eastlake, March 1999. 2077 [DoD] - "Password Management Guideline", United States of America, 2078 Department of Defense, Computer Security Center, CSC-STD-002-85. 2079 (See also FIPS 112, Password Usage, which incorporates CSC-STD-002-85 2080 as one of its appendices.) 2082 [DSS] - "Digital Signature Standard (DSS)", United States of America, 2083 Department of Commerce, National Institute of Standards and 2084 Technoloy, Federal Information Processing Standard (FIPS) 186-2, 2085 January 2000. 2087 [GIFFORD] - "Natural Random Number", MIT/LCS/TM-371, September 1988, 2088 David K. Gifford 2090 [IPSEC] - RFC 2401, "Security Architecture for the Internet 2091 Protocol", S. Kent, R. Atkinson, November 1998 2093 [KNUTH] - "The Art of Computer Programming", Volume 2: Seminumerical 2094 Algorithms, Chapter 3: Random Numbers. Addison Wesley Publishing 2095 Company, Second Edition 1982, Donald E. Knuth. 2097 [KRAWCZYK] - "How to Predict Congruential Generators", Journal of 2098 Algorithms, V. 13, N. 4, December 1992, H. Krawczyk 2100 [MAIL PEM] - RFCs 1421 through 1424: 2101 - RFC 1424, Privacy Enhancement for Internet Electronic Mail: Part 2102 IV: Key Certification and Related Services, 02/10/1993, B. Kaliski 2103 - RFC 1423, Privacy Enhancement for Internet Electronic Mail: Part 2104 III: Algorithms, Modes, and Identifiers, 02/10/1993, D. Balenson 2105 - RFC 1422, Privacy Enhancement for Internet Electronic Mail: Part 2106 II: Certificate-Based Key Management, 02/10/1993, S. Kent 2107 - RFC 1421, Privacy Enhancement for Internet Electronic Mail: Part I: 2108 Message Encryption and Authentication Procedures, 02/10/1993, J. Linn 2110 [MAIL PGP] - RFC 2440, "OpenPGP Message Format", J. Callas, L. 2111 Donnerhacke, H. Finney, R. Thayer", November 1998 2113 [MAIL S/MIME] - RFC 2633, "S/MIME Version 3 Message Specification", 2114 B. Ramsdell, Ed., June 1999. 2116 [MD4] - "The MD4 Message-Digest Algorithm", RFC1320, April 1992, R. 2117 Rivest 2118 [MD5] - "The MD5 Message-Digest Algorithm", RFC1321, April 1992, R. 2119 Rivest 2121 [MOORE] - Moore's Law: the exponential increase the logic density of 2122 silicon circuts. Originally formulated by Gordon Moore in 1964 as a 2123 doubling every year starting in 1962, in the late 1970s the rate fell 2124 to a doubling every 18 months and has remained there through the date 2125 of this document. See "The New Hacker's Dictionary", Third Edition, 2126 MIT Press, ISBN 0-262-18178-9, Eric S. Raymondm 1996. 2128 [ORMAN] - "Determining Strengths For Public Keys Used For Exchanging 2129 Symmetric Keys", draft-orman-public-key-lengths-*.txt, Hilarie Orman, 2130 Paul Hoffman, work in progress. 2132 [RFC 1750] - "Randomness Requirements for Security", D. Eastlake, S. 2133 Crocker, J. Schiller, December 1994. 2135 [RSA BULL1] - "Suggestions for Random Number Generation in Software", 2136 RSA Laboratories Bulletin #1, January 1996. 2138 [RSA BULL13] - "A Cost-Based Security Analysis of Symmetric and 2139 Asymmetric Key Lengths", RSA Laboratories Bulletin #13, Robert 2140 Silverman, April 2000 (revised November 2001). 2142 [SBOX1] - "Practical s-box design", S. Mister, C. Adams, Selected 2143 Areas in Cryptography, 1996. 2144 [SBOX2] - "Perfect Non-linear S-boxes", K. Nyberg, Advances in 2145 Cryptography - Eurocrypt '91 Proceedings, Springer-Verland, 1991. 2147 [SHANNON] - "The Mathematical Theory of Communication", University of 2148 Illinois Press, 1963, Claude E. Shannon. (originally from: Bell 2149 System Technical Journal, July and October 1948) 2151 [SHIFT1] - "Shift Register Sequences", Aegean Park Press, Revised 2152 Edition 1982, Solomon W. Golomb. 2154 [SHIFT2] - "Cryptanalysis of Shift-Register Generated Stream Cypher 2155 Systems", Aegean Park Press, 1984, Wayne G. Barker. 2157 [SHA-1] - "Secure Hash Standard (SHA-1)", United States of American, 2158 National Institute of Science and Technology, Federal Information 2159 Processing Standard (FIPS) 180-1, April 1993. 2160 - RFC 3174, "US Secure Hash Algorithm 1 (SHA1)", D. Eastlake, P. 2161 Jones, September 2001. 2163 [SHA-2] - "Secure Hash Standard", Draft (SHA-2156/384/512), Federal 2164 Information Processing Standard 180-2, not yet issued. 2166 [SSH] - draft-ietf-secsh-*, work in progress. 2168 [STERN] - "Secret Linear Congruential Generators are not 2169 Cryptograhically Secure", Proceedings of IEEE STOC, 1987, J. Stern. 2171 [TLS] - RFC 2246, "The TLS Protocol Version 1.0", T. Dierks, C. 2172 Allen, January 1999. 2174 [VON NEUMANN] - "Various techniques used in connection with random 2175 digits", von Neumann's Collected Works, Vol. 5, Pergamon Press, 1963, 2176 J. von Neumann. 2178 Authors Addresses 2180 Donald E. Eastlake 3rd 2181 Motorola 2182 155 Beaver Street 2183 Milford, MA 01757 USA 2185 Telephone: +1 508-851-8280 (w) 2186 +1 508-634-2066 (h) 2187 EMail: Donald.Eastlake@motorola.com 2189 Jeffrey I. Schiller 2190 MIT Room E40-311 2191 77 Massachusetts Avenue 2192 Cambridge, MA 02139-4307 USA 2194 Telephone: +1 617-253-0161 2195 E-mail: jis@mit.edu 2197 Steve Crocker 2199 EMail: steve@stevecrocker.com 2201 File Name and Expiration 2203 This is file draft-eastlake-randomness2-03.txt. 2205 It expires January 2003.