RandomSources: rfc1750.txt

File rfc1750.txt, 72.1 KB (added by pchapin, 8 years ago)

This RFC discusses issues in generating random numbers and includes comments about hardware sources.

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7Network Working Group                                   D. Eastlake, 3rd
8Request for Comments: 1750                                           DEC
9Category: Informational                                       S. Crocker
10                                                               Cybercash
11                                                             J. Schiller
12                                                                     MIT
13                                                           December 1994
14
15
16                Randomness Recommendations for Security
17
18Status of this Memo
19
20   This memo provides information for the Internet community.  This memo
21   does not specify an Internet standard of any kind.  Distribution of
22   this memo is unlimited.
23
24Abstract
25
26   Security systems today are built on increasingly strong cryptographic
27   algorithms that foil pattern analysis attempts. However, the security
28   of these systems is dependent on generating secret quantities for
29   passwords, cryptographic keys, and similar quantities.  The use of
30   pseudo-random processes to generate secret quantities can result in
31   pseudo-security.  The sophisticated attacker of these security
32   systems may find it easier to reproduce the environment that produced
33   the secret quantities, searching the resulting small set of
34   possibilities, than to locate the quantities in the whole of the
35   number space.
36
37   Choosing random quantities to foil a resourceful and motivated
38   adversary is surprisingly difficult.  This paper points out many
39   pitfalls in using traditional pseudo-random number generation
40   techniques for choosing such quantities.  It recommends the use of
41   truly random hardware techniques and shows that the existing hardware
42   on many systems can be used for this purpose.  It provides
43   suggestions to ameliorate the problem when a hardware solution is not
44   available.  And it gives examples of how large such quantities need
45   to be for some particular applications.
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58Eastlake, Crocker & Schiller                                    [Page 1]
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60RFC 1750        Randomness Recommendations for Security    December 1994
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62
63Acknowledgements
64
65   Comments on this document that have been incorporated were received
66   from (in alphabetic order) the following:
67
68        David M. Balenson (TIS)
69        Don Coppersmith (IBM)
70        Don T. Davis (consultant)
71        Carl Ellison (Stratus)
72        Marc Horowitz (MIT)
73        Christian Huitema (INRIA)
74        Charlie Kaufman (IRIS)
75        Steve Kent (BBN)
76        Hal Murray (DEC)
77        Neil Haller (Bellcore)
78        Richard Pitkin (DEC)
79        Tim Redmond (TIS)
80        Doug Tygar (CMU)
81
82Table of Contents
83
84   1. Introduction........................................... 3
85   2. Requirements........................................... 4
86   3. Traditional Pseudo-Random Sequences.................... 5
87   4. Unpredictability....................................... 7
88   4.1 Problems with Clocks and Serial Numbers............... 7
89   4.2 Timing and Content of External Events................  8
90   4.3 The Fallacy of Complex Manipulation..................  8
91   4.4 The Fallacy of Selection from a Large Database.......  9
92   5. Hardware for Randomness............................... 10
93   5.1 Volume Required...................................... 10
94   5.2 Sensitivity to Skew.................................. 10
95   5.2.1 Using Stream Parity to De-Skew..................... 11
96   5.2.2 Using Transition Mappings to De-Skew............... 12
97   5.2.3 Using FFT to De-Skew............................... 13
98   5.2.4 Using Compression to De-Skew....................... 13
99   5.3 Existing Hardware Can Be Used For Randomness......... 14
100   5.3.1 Using Existing Sound/Video Input................... 14
101   5.3.2 Using Existing Disk Drives......................... 14
102   6. Recommended Non-Hardware Strategy..................... 14
103   6.1 Mixing Functions..................................... 15
104   6.1.1 A Trivial Mixing Function.......................... 15
105   6.1.2 Stronger Mixing Functions.......................... 16
106   6.1.3 Diff-Hellman as a Mixing Function.................. 17
107   6.1.4 Using a Mixing Function to Stretch Random Bits..... 17
108   6.1.5 Other Factors in Choosing a Mixing Function........ 18
109   6.2 Non-Hardware Sources of Randomness................... 19
110   6.3 Cryptographically Strong Sequences................... 19
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118
119   6.3.1 Traditional Strong Sequences....................... 20
120   6.3.2 The Blum Blum Shub Sequence Generator.............. 21
121   7. Key Generation Standards.............................. 22
122   7.1 US DoD Recommendations for Password Generation....... 23
123   7.2 X9.17 Key Generation................................. 23
124   8. Examples of Randomness Required....................... 24
125   8.1  Password Generation................................. 24
126   8.2 A Very High Security Cryptographic Key............... 25
127   8.2.1 Effort per Key Trial............................... 25
128   8.2.2 Meet in the Middle Attacks......................... 26
129   8.2.3 Other Considerations............................... 26
130   9. Conclusion............................................ 27
131   10. Security Considerations.............................. 27
132   References............................................... 28
133   Authors' Addresses....................................... 30
134
1351. Introduction
136
137   Software cryptography is coming into wider use.  Systems like
138   Kerberos, PEM, PGP, etc. are maturing and becoming a part of the
139   network landscape [PEM].  These systems provide substantial
140   protection against snooping and spoofing.  However, there is a
141   potential flaw.  At the heart of all cryptographic systems is the
142   generation of secret, unguessable (i.e., random) numbers.
143
144   For the present, the lack of generally available facilities for
145   generating such unpredictable numbers is an open wound in the design
146   of cryptographic software.  For the software developer who wants to
147   build a key or password generation procedure that runs on a wide
148   range of hardware, the only safe strategy so far has been to force
149   the local installation to supply a suitable routine to generate
150   random numbers.  To say the least, this is an awkward, error-prone
151   and unpalatable solution.
152
153   It is important to keep in mind that the requirement is for data that
154   an adversary has a very low probability of guessing or determining.
155   This will fail if pseudo-random data is used which only meets
156   traditional statistical tests for randomness or which is based on
157   limited range sources, such as clocks.  Frequently such random
158   quantities are determinable by an adversary searching through an
159   embarrassingly small space of possibilities.
160
161   This informational document suggests techniques for producing random
162   quantities that will be resistant to such attack.  It recommends that
163   future systems include hardware random number generation or provide
164   access to existing hardware that can be used for this purpose.  It
165   suggests methods for use if such hardware is not available.  And it
166   gives some estimates of the number of random bits required for sample
167
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175   applications.
176
1772. Requirements
178
179   Probably the most commonly encountered randomness requirement today
180   is the user password. This is usually a simple character string.
181   Obviously, if a password can be guessed, it does not provide
182   security.  (For re-usable passwords, it is desirable that users be
183   able to remember the password.  This may make it advisable to use
184   pronounceable character strings or phrases composed on ordinary
185   words.  But this only affects the format of the password information,
186   not the requirement that the password be very hard to guess.)
187
188   Many other requirements come from the cryptographic arena.
189   Cryptographic techniques can be used to provide a variety of services
190   including confidentiality and authentication.  Such services are
191   based on quantities, traditionally called "keys", that are unknown to
192   and unguessable by an adversary.
193
194   In some cases, such as the use of symmetric encryption with the one
195   time pads [CRYPTO*] or the US Data Encryption Standard [DES], the
196   parties who wish to communicate confidentially and/or with
197   authentication must all know the same secret key.  In other cases,
198   using what are called asymmetric or "public key" cryptographic
199   techniques, keys come in pairs.  One key of the pair is private and
200   must be kept secret by one party, the other is public and can be
201   published to the world.  It is computationally infeasible to
202   determine the private key from the public key [ASYMMETRIC, CRYPTO*].
203
204   The frequency and volume of the requirement for random quantities
205   differs greatly for different cryptographic systems.  Using pure RSA
206   [CRYPTO*], random quantities are required when the key pair is
207   generated, but thereafter any number of messages can be signed
208   without any further need for randomness.  The public key Digital
209   Signature Algorithm that has been proposed by the US National
210   Institute of Standards and Technology (NIST) requires good random
211   numbers for each signature.  And encrypting with a one time pad, in
212   principle the strongest possible encryption technique, requires a
213   volume of randomness equal to all the messages to be processed.
214
215   In most of these cases, an adversary can try to determine the
216   "secret" key by trial and error.  (This is possible as long as the
217   key is enough smaller than the message that the correct key can be
218   uniquely identified.)  The probability of an adversary succeeding at
219   this must be made acceptably low, depending on the particular
220   application.  The size of the space the adversary must search is
221   related to the amount of key "information" present in the information
222   theoretic sense [SHANNON].  This depends on the number of different
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231   secret values possible and the probability of each value as follows:
232
233                      -----
234                       \
235        Bits-of-info =  \  - p   * log  ( p  )
236                        /     i       2    i
237                       /
238                      -----
239
240   where i varies from 1 to the number of possible secret values and p
241   sub i is the probability of the value numbered i.  (Since p sub i is
242   less than one, the log will be negative so each term in the sum will
243   be non-negative.)
244
245   If there are 2^n different values of equal probability, then n bits
246   of information are present and an adversary would, on the average,
247   have to try half of the values, or 2^(n-1) , before guessing the
248   secret quantity.  If the probability of different values is unequal,
249   then there is less information present and fewer guesses will, on
250   average, be required by an adversary.  In particular, any values that
251   the adversary can know are impossible, or are of low probability, can
252   be initially ignored by an adversary, who will search through the
253   more probable values first.
254
255   For example, consider a cryptographic system that uses 56 bit keys.
256   If these 56 bit keys are derived by using a fixed pseudo-random
257   number generator that is seeded with an 8 bit seed, then an adversary
258   needs to search through only 256 keys (by running the pseudo-random
259   number generator with every possible seed), not the 2^56 keys that
260   may at first appear to be the case. Only 8 bits of "information" are
261   in these 56 bit keys.
262
2633. Traditional Pseudo-Random Sequences
264
265   Most traditional sources of random numbers use deterministic sources
266   of "pseudo-random" numbers.  These typically start with a "seed"
267   quantity and use numeric or logical operations to produce a sequence
268   of values.
269
270   [KNUTH] has a classic exposition on pseudo-random numbers.
271   Applications he mentions are simulation of natural phenomena,
272   sampling, numerical analysis, testing computer programs, decision
273   making, and games.  None of these have the same characteristics as
274   the sort of security uses we are talking about.  Only in the last two
275   could there be an adversary trying to find the random quantity.
276   However, in these cases, the adversary normally has only a single
277   chance to use a guessed value.  In guessing passwords or attempting
278   to break an encryption scheme, the adversary normally has many,
279
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287   perhaps unlimited, chances at guessing the correct value and should
288   be assumed to be aided by a computer.
289
290   For testing the "randomness" of numbers, Knuth suggests a variety of
291   measures including statistical and spectral.  These tests check
292   things like autocorrelation between different parts of a "random"
293   sequence or distribution of its values.  They could be met by a
294   constant stored random sequence, such as the "random" sequence
295   printed in the CRC Standard Mathematical Tables [CRC].
296
297   A typical pseudo-random number generation technique, known as a
298   linear congruence pseudo-random number generator, is modular
299   arithmetic where the N+1th value is calculated from the Nth value by
300
301        V    = ( V  * a + b )(Mod c)
302         N+1      N
303
304   The above technique has a strong relationship to linear shift
305   register pseudo-random number generators, which are well understood
306   cryptographically [SHIFT*].  In such generators bits are introduced
307   at one end of a shift register as the Exclusive Or (binary sum
308   without carry) of bits from selected fixed taps into the register.
309
310   For example:
311
312      +----+     +----+     +----+                      +----+
313      | B  | <-- | B  | <-- | B  | <--  . . . . . . <-- | B  | <-+
314      |  0 |     |  1 |     |  2 |                      |  n |   |
315      +----+     +----+     +----+                      +----+   |
316        |                     |            |                     |
317        |                     |            V                  +-----+
318        |                     V            +----------------> |     |
319        V                     +-----------------------------> | XOR |
320        +---------------------------------------------------> |     |
321                                                              +-----+
322
323
324       V    = ( ( V  * 2 ) + B .xor. B ... )(Mod 2^n)
325        N+1         N         0       2
326
327   The goodness of traditional pseudo-random number generator algorithms
328   is measured by statistical tests on such sequences.  Carefully chosen
329   values of the initial V and a, b, and c or the placement of shift
330   register tap in the above simple processes can produce excellent
331   statistics.
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343   These sequences may be adequate in simulations (Monte Carlo
344   experiments) as long as the sequence is orthogonal to the structure
345   of the space being explored.  Even there, subtle patterns may cause
346   problems.  However, such sequences are clearly bad for use in
347   security applications.  They are fully predictable if the initial
348   state is known.  Depending on the form of the pseudo-random number
349   generator, the sequence may be determinable from observation of a
350   short portion of the sequence [CRYPTO*, STERN].  For example, with
351   the generators above, one can determine V(n+1) given knowledge of
352   V(n).  In fact, it has been shown that with these techniques, even if
353   only one bit of the pseudo-random values is released, the seed can be
354   determined from short sequences.
355
356   Not only have linear congruent generators been broken, but techniques
357   are now known for breaking all polynomial congruent generators
358   [KRAWCZYK].
359
3604. Unpredictability
361
362   Randomness in the traditional sense described in section 3 is NOT the
363   same as the unpredictability required for security use.
364
365   For example, use of a widely available constant sequence, such as
366   that from the CRC tables, is very weak against an adversary. Once
367   they learn of or guess it, they can easily break all security, future
368   and past, based on the sequence [CRC].  Yet the statistical
369   properties of these tables are good.
370
371   The following sections describe the limitations of some randomness
372   generation techniques and sources.
373
3744.1 Problems with Clocks and Serial Numbers
375
376   Computer clocks, or similar operating system or hardware values,
377   provide significantly fewer real bits of unpredictability than might
378   appear from their specifications.
379
380   Tests have been done on clocks on numerous systems and it was found
381   that their behavior can vary widely and in unexpected ways.  One
382   version of an operating system running on one set of hardware may
383   actually provide, say, microsecond resolution in a clock while a
384   different configuration of the "same" system may always provide the
385   same lower bits and only count in the upper bits at much lower
386   resolution.  This means that successive reads on the clock may
387   produce identical values even if enough time has passed that the
388   value "should" change based on the nominal clock resolution. There
389   are also cases where frequently reading a clock can produce
390   artificial sequential values because of extra code that checks for
391
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399   the clock being unchanged between two reads and increases it by one!
400   Designing portable application code to generate unpredictable numbers
401   based on such system clocks is particularly challenging because the
402   system designer does not always know the properties of the system
403   clocks that the code will execute on.
404
405   Use of a hardware serial number such as an Ethernet address may also
406   provide fewer bits of uniqueness than one would guess.  Such
407   quantities are usually heavily structured and subfields may have only
408   a limited range of possible values or values easily guessable based
409   on approximate date of manufacture or other data.  For example, it is
410   likely that most of the Ethernet cards installed on Digital Equipment
411   Corporation (DEC) hardware within DEC were manufactured by DEC
412   itself, which significantly limits the range of built in addresses.
413
414   Problems such as those described above related to clocks and serial
415   numbers make code to produce unpredictable quantities difficult if
416   the code is to be ported across a variety of computer platforms and
417   systems.
418
4194.2 Timing and Content of External Events
420
421   It is possible to measure the timing and content of mouse movement,
422   key strokes, and similar user events.  This is a reasonable source of
423   unguessable data with some qualifications.  On some machines, inputs
424   such as key strokes are buffered.  Even though the user's inter-
425   keystroke timing may have sufficient variation and unpredictability,
426   there might not be an easy way to access that variation.  Another
427   problem is that no standard method exists to sample timing details.
428   This makes it hard to build standard software intended for
429   distribution to a large range of machines based on this technique.
430
431   The amount of mouse movement or the keys actually hit are usually
432   easier to access than timings but may yield less unpredictability as
433   the user may provide highly repetitive input.
434
435   Other external events, such as network packet arrival times, can also
436   be used with care.  In particular, the possibility of manipulation of
437   such times by an adversary must be considered.
438
4394.3 The Fallacy of Complex Manipulation
440
441   One strategy which may give a misleading appearance of
442   unpredictability is to take a very complex algorithm (or an excellent
443   traditional pseudo-random number generator with good statistical
444   properties) and calculate a cryptographic key by starting with the
445   current value of a computer system clock as the seed.  An adversary
446   who knew roughly when the generator was started would have a
447
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455   relatively small number of seed values to test as they would know
456   likely values of the system clock.  Large numbers of pseudo-random
457   bits could be generated but the search space an adversary would need
458   to check could be quite small.
459
460   Thus very strong and/or complex manipulation of data will not help if
461   the adversary can learn what the manipulation is and there is not
462   enough unpredictability in the starting seed value.  Even if they can
463   not learn what the manipulation is, they may be able to use the
464   limited number of results stemming from a limited number of seed
465   values to defeat security.
466
467   Another serious strategy error is to assume that a very complex
468   pseudo-random number generation algorithm will produce strong random
469   numbers when there has been no theory behind or analysis of the
470   algorithm.  There is a excellent example of this fallacy right near
471   the beginning of chapter 3 in [KNUTH] where the author describes a
472   complex algorithm.  It was intended that the machine language program
473   corresponding to the algorithm would be so complicated that a person
474   trying to read the code without comments wouldn't know what the
475   program was doing.  Unfortunately, actual use of this algorithm
476   showed that it almost immediately converged to a single repeated
477   value in one case and a small cycle of values in another case.
478
479   Not only does complex manipulation not help you if you have a limited
480   range of seeds but blindly chosen complex manipulation can destroy
481   the randomness in a good seed!
482
4834.4 The Fallacy of Selection from a Large Database
484
485   Another strategy that can give a misleading appearance of
486   unpredictability is selection of a quantity randomly from a database
487   and assume that its strength is related to the total number of bits
488   in the database.  For example, typical USENET servers as of this date
489   process over 35 megabytes of information per day.  Assume a random
490   quantity was selected by fetching 32 bytes of data from a random
491   starting point in this data.  This does not yield 32*8 = 256 bits
492   worth of unguessability.  Even after allowing that much of the data
493   is human language and probably has more like 2 or 3 bits of
494   information per byte, it doesn't yield 32*2.5 = 80 bits of
495   unguessability.  For an adversary with access to the same 35
496   megabytes the unguessability rests only on the starting point of the
497   selection.  That is, at best, about 25 bits of unguessability in this
498   case.
499
500   The same argument applies to selecting sequences from the data on a
501   CD ROM or Audio CD recording or any other large public database.  If
502   the adversary has access to the same database, this "selection from a
503
504
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510
511   large volume of data" step buys very little.  However, if a selection
512   can be made from data to which the adversary has no access, such as
513   system buffers on an active multi-user system, it may be of some
514   help.
515
5165. Hardware for Randomness
517
518   Is there any hope for strong portable randomness in the future?
519   There might be.  All that's needed is a physical source of
520   unpredictable numbers.
521
522   A thermal noise or radioactive decay source and a fast, free-running
523   oscillator would do the trick directly [GIFFORD].  This is a trivial
524   amount of hardware, and could easily be included as a standard part
525   of a computer system's architecture.  Furthermore, any system with a
526   spinning disk or the like has an adequate source of randomness
527   [DAVIS].  All that's needed is the common perception among computer
528   vendors that this small additional hardware and the software to
529   access it is necessary and useful.
530
5315.1 Volume Required
532
533   How much unpredictability is needed?  Is it possible to quantify the
534   requirement in, say, number of random bits per second?
535
536   The answer is not very much is needed.  For DES, the key is 56 bits
537   and, as we show in an example in Section 8, even the highest security
538   system is unlikely to require a keying material of over 200 bits.  If
539   a series of keys are needed, it can be generated from a strong random
540   seed using a cryptographically strong sequence as explained in
541   Section 6.3.  A few hundred random bits generated once a day would be
542   enough using such techniques.  Even if the random bits are generated
543   as slowly as one per second and it is not possible to overlap the
544   generation process, it should be tolerable in high security
545   applications to wait 200 seconds occasionally.
546
547   These numbers are trivial to achieve.  It could be done by a person
548   repeatedly tossing a coin.  Almost any hardware process is likely to
549   be much faster.
550
5515.2 Sensitivity to Skew
552
553   Is there any specific requirement on the shape of the distribution of
554   the random numbers?  The good news is the distribution need not be
555   uniform.  All that is needed is a conservative estimate of how non-
556   uniform it is to bound performance.  Two simple techniques to de-skew
557   the bit stream are given below and stronger techniques are mentioned
558   in Section 6.1.2 below.
559
560
561
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565
566
5675.2.1 Using Stream Parity to De-Skew
568
569   Consider taking a sufficiently long string of bits and map the string
570   to "zero" or "one".  The mapping will not yield a perfectly uniform
571   distribution, but it can be as close as desired.  One mapping that
572   serves the purpose is to take the parity of the string.  This has the
573   advantages that it is robust across all degrees of skew up to the
574   estimated maximum skew and is absolutely trivial to implement in
575   hardware.
576
577   The following analysis gives the number of bits that must be sampled:
578
579   Suppose the ratio of ones to zeros is 0.5 + e : 0.5 - e, where e is
580   between 0 and 0.5 and is a measure of the "eccentricity" of the
581   distribution.  Consider the distribution of the parity function of N
582   bit samples.  The probabilities that the parity will be one or zero
583   will be the sum of the odd or even terms in the binomial expansion of
584   (p + q)^N, where p = 0.5 + e, the probability of a one, and q = 0.5 -
585   e, the probability of a zero.
586
587   These sums can be computed easily as
588
589                         N            N
590        1/2 * ( ( p + q )  + ( p - q )  )
591   and
592                         N            N
593        1/2 * ( ( p + q )  - ( p - q )  ).
594
595   (Which one corresponds to the probability the parity will be 1
596   depends on whether N is odd or even.)
597
598   Since p + q = 1 and p - q = 2e, these expressions reduce to
599
600                       N
601        1/2 * [1 + (2e) ]
602   and
603                       N
604        1/2 * [1 - (2e) ].
605
606   Neither of these will ever be exactly 0.5 unless e is zero, but we
607   can bring them arbitrarily close to 0.5.  If we want the
608   probabilities to be within some delta d of 0.5, i.e. then
609
610                            N
611        ( 0.5 + ( 0.5 * (2e)  ) )  <  0.5 + d.
612
613
614
615
616
617
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620RFC 1750        Randomness Recommendations for Security    December 1994
621
622
623   Solving for N yields N > log(2d)/log(2e).  (Note that 2e is less than
624   1, so its log is negative.  Division by a negative number reverses
625   the sense of an inequality.)
626
627   The following table gives the length of the string which must be
628   sampled for various degrees of skew in order to come within 0.001 of
629   a 50/50 distribution.
630
631                       +---------+--------+-------+
632                       | Prob(1) |    e   |    N  |
633                       +---------+--------+-------+
634                       |   0.5   |  0.00  |    1  |
635                       |   0.6   |  0.10  |    4  |
636                       |   0.7   |  0.20  |    7  |
637                       |   0.8   |  0.30  |   13  |
638                       |   0.9   |  0.40  |   28  |
639                       |   0.95  |  0.45  |   59  |
640                       |   0.99  |  0.49  |  308  |
641                       +---------+--------+-------+
642
643   The last entry shows that even if the distribution is skewed 99% in
644   favor of ones, the parity of a string of 308 samples will be within
645   0.001 of a 50/50 distribution.
646
6475.2.2 Using Transition Mappings to De-Skew
648
649   Another technique, originally due to von Neumann [VON NEUMANN], is to
650   examine a bit stream as a sequence of non-overlapping pairs. You
651   could then discard any 00 or 11 pairs found, interpret 01 as a 0 and
652   10 as a 1.  Assume the probability of a 1 is 0.5+e and the
653   probability of a 0 is 0.5-e where e is the eccentricity of the source
654   and described in the previous section.  Then the probability of each
655   pair is as follows:
656
657            +------+-----------------------------------------+
658            | pair |            probability                  |
659            +------+-----------------------------------------+
660            |  00  | (0.5 - e)^2          =  0.25 - e + e^2  |
661            |  01  | (0.5 - e)*(0.5 + e)  =  0.25     - e^2  |
662            |  10  | (0.5 + e)*(0.5 - e)  =  0.25     - e^2  |
663            |  11  | (0.5 + e)^2          =  0.25 + e + e^2  |
664            +------+-----------------------------------------+
665
666   This technique will completely eliminate any bias but at the expense
667   of taking an indeterminate number of input bits for any particular
668   desired number of output bits.  The probability of any particular
669   pair being discarded is 0.5 + 2e^2 so the expected number of input
670   bits to produce X output bits is X/(0.25 - e^2).
671
672
673
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676RFC 1750        Randomness Recommendations for Security    December 1994
677
678
679   This technique assumes that the bits are from a stream where each bit
680   has the same probability of being a 0 or 1 as any other bit in the
681   stream and that bits are not correlated, i.e., that the bits are
682   identical independent distributions.  If alternate bits were from two
683   correlated sources, for example, the above analysis breaks down.
684
685   The above technique also provides another illustration of how a
686   simple statistical analysis can mislead if one is not always on the
687   lookout for patterns that could be exploited by an adversary.  If the
688   algorithm were mis-read slightly so that overlapping successive bits
689   pairs were used instead of non-overlapping pairs, the statistical
690   analysis given is the same; however, instead of provided an unbiased
691   uncorrelated series of random 1's and 0's, it instead produces a
692   totally predictable sequence of exactly alternating 1's and 0's.
693
6945.2.3 Using FFT to De-Skew
695
696   When real world data consists of strongly biased or correlated bits,
697   it may still contain useful amounts of randomness.  This randomness
698   can be extracted through use of the discrete Fourier transform or its
699   optimized variant, the FFT.
700
701   Using the Fourier transform of the data, strong correlations can be
702   discarded.  If adequate data is processed and remaining correlations
703   decay, spectral lines approaching statistical independence and
704   normally distributed randomness can be produced [BRILLINGER].
705
7065.2.4 Using Compression to De-Skew
707
708   Reversible compression techniques also provide a crude method of de-
709   skewing a skewed bit stream.  This follows directly from the
710   definition of reversible compression and the formula in Section 2
711   above for the amount of information in a sequence.  Since the
712   compression is reversible, the same amount of information must be
713   present in the shorter output than was present in the longer input.
714   By the Shannon information equation, this is only possible if, on
715   average, the probabilities of the different shorter sequences are
716   more uniformly distributed than were the probabilities of the longer
717   sequences.  Thus the shorter sequences are de-skewed relative to the
718   input.
719
720   However, many compression techniques add a somewhat predicatable
721   preface to their output stream and may insert such a sequence again
722   periodically in their output or otherwise introduce subtle patterns
723   of their own.  They should be considered only a rough technique
724   compared with those described above or in Section 6.1.2.  At a
725   minimum, the beginning of the compressed sequence should be skipped
726   and only later bits used for applications requiring random bits.
727
728
729
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733
734
7355.3 Existing Hardware Can Be Used For Randomness
736
737   As described below, many computers come with hardware that can, with
738   care, be used to generate truly random quantities.
739
7405.3.1 Using Existing Sound/Video Input
741
742   Increasingly computers are being built with inputs that digitize some
743   real world analog source, such as sound from a microphone or video
744   input from a camera.  Under appropriate circumstances, such input can
745   provide reasonably high quality random bits.  The "input" from a
746   sound digitizer with no source plugged in or a camera with the lens
747   cap on, if the system has enough gain to detect anything, is
748   essentially thermal noise.
749
750   For example, on a SPARCstation, one can read from the /dev/audio
751   device with nothing plugged into the microphone jack.  Such data is
752   essentially random noise although it should not be trusted without
753   some checking in case of hardware failure.  It will, in any case,
754   need to be de-skewed as described elsewhere.
755
756   Combining this with compression to de-skew one can, in UNIXese,
757   generate a huge amount of medium quality random data by doing
758
759        cat /dev/audio | compress - >random-bits-file
760
7615.3.2 Using Existing Disk Drives
762
763   Disk drives have small random fluctuations in their rotational speed
764   due to chaotic air turbulence [DAVIS].  By adding low level disk seek
765   time instrumentation to a system, a series of measurements can be
766   obtained that include this randomness. Such data is usually highly
767   correlated so that significant processing is needed, including FFT
768   (see section 5.2.3).  Nevertheless experimentation has shown that,
769   with such processing, disk drives easily produce 100 bits a minute or
770   more of excellent random data.
771
772   Partly offsetting this need for processing is the fact that disk
773   drive failure will normally be rapidly noticed.  Thus, problems with
774   this method of random number generation due to hardware failure are
775   very unlikely.
776
7776. Recommended Non-Hardware Strategy
778
779   What is the best overall strategy for meeting the requirement for
780   unguessable random numbers in the absence of a reliable hardware
781   source?  It is to obtain random input from a large number of
782   uncorrelated sources and to mix them with a strong mixing function.
783
784
785
786Eastlake, Crocker & Schiller                                   [Page 14]
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788RFC 1750        Randomness Recommendations for Security    December 1994
789
790
791   Such a function will preserve the randomness present in any of the
792   sources even if other quantities being combined are fixed or easily
793   guessable.  This may be advisable even with a good hardware source as
794   hardware can also fail, though this should be weighed against any
795   increase in the chance of overall failure due to added software
796   complexity.
797
7986.1 Mixing Functions
799
800   A strong mixing function is one which combines two or more inputs and
801   produces an output where each output bit is a different complex non-
802   linear function of all the input bits.  On average, changing any
803   input bit will change about half the output bits.  But because the
804   relationship is complex and non-linear, no particular output bit is
805   guaranteed to change when any particular input bit is changed.
806
807   Consider the problem of converting a stream of bits that is skewed
808   towards 0 or 1 to a shorter stream which is more random, as discussed
809   in Section 5.2 above.  This is simply another case where a strong
810   mixing function is desired, mixing the input bits to produce a
811   smaller number of output bits.  The technique given in Section 5.2.1
812   of using the parity of a number of bits is simply the result of
813   successively Exclusive Or'ing them which is examined as a trivial
814   mixing function immediately below.  Use of stronger mixing functions
815   to extract more of the randomness in a stream of skewed bits is
816   examined in Section 6.1.2.
817
8186.1.1 A Trivial Mixing Function
819
820   A trivial example for single bit inputs is the Exclusive Or function,
821   which is equivalent to addition without carry, as show in the table
822   below.  This is a degenerate case in which the one output bit always
823   changes for a change in either input bit.  But, despite its
824   simplicity, it will still provide a useful illustration.
825
826                   +-----------+-----------+----------+
827                   |  input 1  |  input 2  |  output  |
828                   +-----------+-----------+----------+
829                   |     0     |     0     |     0    |
830                   |     0     |     1     |     1    |
831                   |     1     |     0     |     1    |
832                   |     1     |     1     |     0    |
833                   +-----------+-----------+----------+
834
835   If inputs 1 and 2 are uncorrelated and combined in this fashion then
836   the output will be an even better (less skewed) random bit than the
837   inputs.  If we assume an "eccentricity" e as defined in Section 5.2
838   above, then the output eccentricity relates to the input eccentricity
839
840
841
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844RFC 1750        Randomness Recommendations for Security    December 1994
845
846
847   as follows:
848
849        e       = 2 * e        * e
850         output        input 1    input 2
851
852   Since e is never greater than 1/2, the eccentricity is always
853   improved except in the case where at least one input is a totally
854   skewed constant.  This is illustrated in the following table where
855   the top and left side values are the two input eccentricities and the
856   entries are the output eccentricity:
857
858     +--------+--------+--------+--------+--------+--------+--------+
859     |    e   |  0.00  |  0.10  |  0.20  |  0.30  |  0.40  |  0.50  |
860     +--------+--------+--------+--------+--------+--------+--------+
861     |  0.00  |  0.00  |  0.00  |  0.00  |  0.00  |  0.00  |  0.00  |
862     |  0.10  |  0.00  |  0.02  |  0.04  |  0.06  |  0.08  |  0.10  |
863     |  0.20  |  0.00  |  0.04  |  0.08  |  0.12  |  0.16  |  0.20  |
864     |  0.30  |  0.00  |  0.06  |  0.12  |  0.18  |  0.24  |  0.30  |
865     |  0.40  |  0.00  |  0.08  |  0.16  |  0.24  |  0.32  |  0.40  |
866     |  0.50  |  0.00  |  0.10  |  0.20  |  0.30  |  0.40  |  0.50  |
867     +--------+--------+--------+--------+--------+--------+--------+
868
869   However, keep in mind that the above calculations assume that the
870   inputs are not correlated.  If the inputs were, say, the parity of
871   the number of minutes from midnight on two clocks accurate to a few
872   seconds, then each might appear random if sampled at random intervals
873   much longer than a minute.  Yet if they were both sampled and
874   combined with xor, the result would be zero most of the time.
875
8766.1.2 Stronger Mixing Functions
877
878   The US Government Data Encryption Standard [DES] is an example of a
879   strong mixing function for multiple bit quantities.  It takes up to
880   120 bits of input (64 bits of "data" and 56 bits of "key") and
881   produces 64 bits of output each of which is dependent on a complex
882   non-linear function of all input bits.  Other strong encryption
883   functions with this characteristic can also be used by considering
884   them to mix all of their key and data input bits.
885
886   Another good family of mixing functions are the "message digest" or
887   hashing functions such as The US Government Secure Hash Standard
888   [SHS] and the MD2, MD4, MD5 [MD2, MD4, MD5] series.  These functions
889   all take an arbitrary amount of input and produce an output mixing
890   all the input bits. The MD* series produce 128 bits of output and SHS
891   produces 160 bits.
892
893
894
895
896
897
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900RFC 1750        Randomness Recommendations for Security    December 1994
901
902
903   Although the message digest functions are designed for variable
904   amounts of input, DES and other encryption functions can also be used
905   to combine any number of inputs.  If 64 bits of output is adequate,
906   the inputs can be packed into a 64 bit data quantity and successive
907   56 bit keys, padding with zeros if needed, which are then used to
908   successively encrypt using DES in Electronic Codebook Mode [DES
909   MODES].  If more than 64 bits of output are needed, use more complex
910   mixing.  For example, if inputs are packed into three quantities, A,
911   B, and C, use DES to encrypt A with B as a key and then with C as a
912   key to produce the 1st part of the output, then encrypt B with C and
913   then A for more output and, if necessary, encrypt C with A and then B
914   for yet more output.  Still more output can be produced by reversing
915   the order of the keys given above to stretch things. The same can be
916   done with the hash functions by hashing various subsets of the input
917   data to produce multiple outputs.  But keep in mind that it is
918   impossible to get more bits of "randomness" out than are put in.
919
920   An example of using a strong mixing function would be to reconsider
921   the case of a string of 308 bits each of which is biased 99% towards
922   zero.  The parity technique given in Section 5.2.1 above reduced this
923   to one bit with only a 1/1000 deviance from being equally likely a
924   zero or one.  But, applying the equation for information given in
925   Section 2, this 308 bit sequence has 5 bits of information in it.
926   Thus hashing it with SHS or MD5 and taking the bottom 5 bits of the
927   result would yield 5 unbiased random bits as opposed to the single
928   bit given by calculating the parity of the string.
929
9306.1.3 Diffie-Hellman as a Mixing Function
931
932   Diffie-Hellman exponential key exchange is a technique that yields a
933   shared secret between two parties that can be made computationally
934   infeasible for a third party to determine even if they can observe
935   all the messages between the two communicating parties.  This shared
936   secret is a mixture of initial quantities generated by each of them
937   [D-H].  If these initial quantities are random, then the shared
938   secret contains the combined randomness of them both, assuming they
939   are uncorrelated.
940
9416.1.4 Using a Mixing Function to Stretch Random Bits
942
943   While it is not necessary for a mixing function to produce the same
944   or fewer bits than its inputs, mixing bits cannot "stretch" the
945   amount of random unpredictability present in the inputs.  Thus four
946   inputs of 32 bits each where there is 12 bits worth of
947   unpredicatability (such as 4,096 equally probable values) in each
948   input cannot produce more than 48 bits worth of unpredictable output.
949   The output can be expanded to hundreds or thousands of bits by, for
950   example, mixing with successive integers, but the clever adversary's
951
952
953
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957
958
959   search space is still 2^48 possibilities.  Furthermore, mixing to
960   fewer bits than are input will tend to strengthen the randomness of
961   the output the way using Exclusive Or to produce one bit from two did
962   above.
963
964   The last table in Section 6.1.1 shows that mixing a random bit with a
965   constant bit with Exclusive Or will produce a random bit.  While this
966   is true, it does not provide a way to "stretch" one random bit into
967   more than one.  If, for example, a random bit is mixed with a 0 and
968   then with a 1, this produces a two bit sequence but it will always be
969   either 01 or 10.  Since there are only two possible values, there is
970   still only the one bit of original randomness.
971
9726.1.5 Other Factors in Choosing a Mixing Function
973
974   For local use, DES has the advantages that it has been widely tested
975   for flaws, is widely documented, and is widely implemented with
976   hardware and software implementations available all over the world
977   including source code available by anonymous FTP.  The SHS and MD*
978   family are younger algorithms which have been less tested but there
979   is no particular reason to believe they are flawed.  Both MD5 and SHS
980   were derived from the earlier MD4 algorithm.  They all have source
981   code available by anonymous FTP [SHS, MD2, MD4, MD5].
982
983   DES and SHS have been vouched for the the US National Security Agency
984   (NSA) on the basis of criteria that primarily remain secret.  While
985   this is the cause of much speculation and doubt, investigation of DES
986   over the years has indicated that NSA involvement in modifications to
987   its design, which originated with IBM, was primarily to strengthen
988   it.  No concealed or special weakness has been found in DES.  It is
989   almost certain that the NSA modification to MD4 to produce the SHS
990   similarly strengthened the algorithm, possibly against threats not
991   yet known in the public cryptographic community.
992
993   DES, SHS, MD4, and MD5 are royalty free for all purposes.  MD2 has
994   been freely licensed only for non-profit use in connection with
995   Privacy Enhanced Mail [PEM].  Between the MD* algorithms, some people
996   believe that, as with "Goldilocks and the Three Bears", MD2 is strong
997   but too slow, MD4 is fast but too weak, and MD5 is just right.
998
999   Another advantage of the MD* or similar hashing algorithms over
1000   encryption algorithms is that they are not subject to the same
1001   regulations imposed by the US Government prohibiting the unlicensed
1002   export or import of encryption/decryption software and hardware.  The
1003   same should be true of DES rigged to produce an irreversible hash
1004   code but most DES packages are oriented to reversible encryption.
1005
1006
1007
1008
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1013
1014
10156.2 Non-Hardware Sources of Randomness
1016
1017   The best source of input for mixing would be a hardware randomness
1018   such as disk drive timing affected by air turbulence, audio input
1019   with thermal noise, or radioactive decay.  However, if that is not
1020   available there are other possibilities.  These include system
1021   clocks, system or input/output buffers, user/system/hardware/network
1022   serial numbers and/or addresses and timing, and user input.
1023   Unfortunately, any of these sources can produce limited or
1024   predicatable values under some circumstances.
1025
1026   Some of the sources listed above would be quite strong on multi-user
1027   systems where, in essence, each user of the system is a source of
1028   randomness.  However, on a small single user system, such as a
1029   typical IBM PC or Apple Macintosh, it might be possible for an
1030   adversary to assemble a similar configuration.  This could give the
1031   adversary inputs to the mixing process that were sufficiently
1032   correlated to those used originally as to make exhaustive search
1033   practical.
1034
1035   The use of multiple random inputs with a strong mixing function is
1036   recommended and can overcome weakness in any particular input.  For
1037   example, the timing and content of requested "random" user keystrokes
1038   can yield hundreds of random bits but conservative assumptions need
1039   to be made.  For example, assuming a few bits of randomness if the
1040   inter-keystroke interval is unique in the sequence up to that point
1041   and a similar assumption if the key hit is unique but assuming that
1042   no bits of randomness are present in the initial key value or if the
1043   timing or key value duplicate previous values.  The results of mixing
1044   these timings and characters typed could be further combined with
1045   clock values and other inputs.
1046
1047   This strategy may make practical portable code to produce good random
1048   numbers for security even if some of the inputs are very weak on some
1049   of the target systems.  However, it may still fail against a high
1050   grade attack on small single user systems, especially if the
1051   adversary has ever been able to observe the generation process in the
1052   past.  A hardware based random source is still preferable.
1053
10546.3 Cryptographically Strong Sequences
1055
1056   In cases where a series of random quantities must be generated, an
1057   adversary may learn some values in the sequence.  In general, they
1058   should not be able to predict other values from the ones that they
1059   know.
1060
1061
1062
1063
1064
1065
1066Eastlake, Crocker & Schiller                                   [Page 19]
1067
1068RFC 1750        Randomness Recommendations for Security    December 1994
1069
1070
1071   The correct technique is to start with a strong random seed, take
1072   cryptographically strong steps from that seed [CRYPTO2, CRYPTO3], and
1073   do not reveal the complete state of the generator in the sequence
1074   elements.  If each value in the sequence can be calculated in a fixed
1075   way from the previous value, then when any value is compromised, all
1076   future values can be determined.  This would be the case, for
1077   example, if each value were a constant function of the previously
1078   used values, even if the function were a very strong, non-invertible
1079   message digest function.
1080
1081   It should be noted that if your technique for generating a sequence
1082   of key values is fast enough, it can trivially be used as the basis
1083   for a confidentiality system.  If two parties use the same sequence
1084   generating technique and start with the same seed material, they will
1085   generate identical sequences.  These could, for example, be xor'ed at
1086   one end with data being send, encrypting it, and xor'ed with this
1087   data as received, decrypting it due to the reversible properties of
1088   the xor operation.
1089
10906.3.1 Traditional Strong Sequences
1091
1092   A traditional way to achieve a strong sequence has been to have the
1093   values be produced by hashing the quantities produced by
1094   concatenating the seed with successive integers or the like and then
1095   mask the values obtained so as to limit the amount of generator state
1096   available to the adversary.
1097
1098   It may also be possible to use an "encryption" algorithm with a
1099   random key and seed value to encrypt and feedback some or all of the
1100   output encrypted value into the value to be encrypted for the next
1101   iteration.  Appropriate feedback techniques will usually be
1102   recommended with the encryption algorithm.  An example is shown below
1103   where shifting and masking are used to combine the cypher output
1104   feedback.  This type of feedback is recommended by the US Government
1105   in connection with DES [DES MODES].
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122Eastlake, Crocker & Schiller                                   [Page 20]
1123
1124RFC 1750        Randomness Recommendations for Security    December 1994
1125
1126
1127      +---------------+
1128      |       V       |
1129      |  |     n      |
1130      +--+------------+
1131            |      |           +---------+
1132            |      +---------> |         |      +-----+
1133         +--+                  | Encrypt | <--- | Key |
1134         |           +-------- |         |      +-----+
1135         |           |         +---------+
1136         V           V
1137      +------------+--+
1138      |      V     |  |
1139      |       n+1     |
1140      +---------------+
1141
1142   Note that if a shift of one is used, this is the same as the shift
1143   register technique described in Section 3 above but with the all
1144   important difference that the feedback is determined by a complex
1145   non-linear function of all bits rather than a simple linear or
1146   polynomial combination of output from a few bit position taps.
1147
1148   It has been shown by Donald W. Davies that this sort of shifted
1149   partial output feedback significantly weakens an algorithm compared
1150   will feeding all of the output bits back as input.  In particular,
1151   for DES, repeated encrypting a full 64 bit quantity will give an
1152   expected repeat in about 2^63 iterations.  Feeding back anything less
1153   than 64 (and more than 0) bits will give an expected repeat in
1154   between 2**31 and 2**32 iterations!
1155
1156   To predict values of a sequence from others when the sequence was
1157   generated by these techniques is equivalent to breaking the
1158   cryptosystem or inverting the "non-invertible" hashing involved with
1159   only partial information available.  The less information revealed
1160   each iteration, the harder it will be for an adversary to predict the
1161   sequence.  Thus it is best to use only one bit from each value.  It
1162   has been shown that in some cases this makes it impossible to break a
1163   system even when the cryptographic system is invertible and can be
1164   broken if all of each generated value was revealed.
1165
11666.3.2 The Blum Blum Shub Sequence Generator
1167
1168   Currently the generator which has the strongest public proof of
1169   strength is called the Blum Blum Shub generator after its inventors
1170   [BBS].  It is also very simple and is based on quadratic residues.
1171   It's only disadvantage is that is is computationally intensive
1172   compared with the traditional techniques give in 6.3.1 above.  This
1173   is not a serious draw back if it is used for moderately infrequent
1174   purposes, such as generating session keys.
1175
1176
1177
1178Eastlake, Crocker & Schiller                                   [Page 21]
1179
1180RFC 1750        Randomness Recommendations for Security    December 1994
1181
1182
1183   Simply choose two large prime numbers, say p and q, which both have
1184   the property that you get a remainder of 3 if you divide them by 4.
1185   Let n = p * q.  Then you choose a random number x relatively prime to
1186   n.  The initial seed for the generator and the method for calculating
1187   subsequent values are then
1188
1189                   2
1190        s    =  ( x  )(Mod n)
1191         0
1192
1193                   2
1194        s    = ( s   )(Mod n)
1195         i+1      i
1196
1197   You must be careful to use only a few bits from the bottom of each s.
1198   It is always safe to use only the lowest order bit.  If you use no
1199   more than the
1200
1201                  log  ( log  ( s  ) )
1202                     2      2    i
1203
1204   low order bits, then predicting any additional bits from a sequence
1205   generated in this manner is provable as hard as factoring n.  As long
1206   as the initial x is secret, you can even make n public if you want.
1207
1208   An intersting characteristic of this generator is that you can
1209   directly calculate any of the s values.  In particular
1210
1211                     i
1212               ( ( 2  )(Mod (( p - 1 ) * ( q - 1 )) ) )
1213      s  = ( s                                          )(Mod n)
1214       i      0
1215
1216   This means that in applications where many keys are generated in this
1217   fashion, it is not necessary to save them all.  Each key can be
1218   effectively indexed and recovered from that small index and the
1219   initial s and n.
1220
12217. Key Generation Standards
1222
1223   Several public standards are now in place for the generation of keys.
1224   Two of these are described below.  Both use DES but any equally
1225   strong or stronger mixing function could be substituted.
1226
1227
1228
1229
1230
1231
1232
1233
1234Eastlake, Crocker & Schiller                                   [Page 22]
1235
1236RFC 1750        Randomness Recommendations for Security    December 1994
1237
1238
12397.1 US DoD Recommendations for Password Generation
1240
1241   The United States Department of Defense has specific recommendations
1242   for password generation [DoD].  They suggest using the US Data
1243   Encryption Standard [DES] in Output Feedback Mode [DES MODES] as
1244   follows:
1245
1246        use an initialization vector determined from
1247             the system clock,
1248             system ID,
1249             user ID, and
1250             date and time;
1251        use a key determined from
1252             system interrupt registers,
1253             system status registers, and
1254             system counters; and,
1255        as plain text, use an external randomly generated 64 bit
1256        quantity such as 8 characters typed in by a system
1257        administrator.
1258
1259   The password can then be calculated from the 64 bit "cipher text"
1260   generated in 64-bit Output Feedback Mode.  As many bits as are needed
1261   can be taken from these 64 bits and expanded into a pronounceable
1262   word, phrase, or other format if a human being needs to remember the
1263   password.
1264
12657.2 X9.17 Key Generation
1266
1267   The American National Standards Institute has specified a method for
1268   generating a sequence of keys as follows:
1269
1270        s  is the initial 64 bit seed
1271         0
1272
1273        g  is the sequence of generated 64 bit key quantities
1274         n
1275
1276        k is a random key reserved for generating this key sequence
1277
1278        t is the time at which a key is generated to as fine a resolution
1279            as is available (up to 64 bits).
1280
1281        DES ( K, Q ) is the DES encryption of quantity Q with key K
1282
1283
1284
1285
1286
1287
1288
1289
1290Eastlake, Crocker & Schiller                                   [Page 23]
1291
1292RFC 1750        Randomness Recommendations for Security    December 1994
1293
1294
1295        g    = DES ( k, DES ( k, t ) .xor. s  )
1296         n                                  n
1297
1298        s    = DES ( k, DES ( k, t ) .xor. g  )
1299         n+1                                n
1300
1301   If g sub n is to be used as a DES key, then every eighth bit should
1302   be adjusted for parity for that use but the entire 64 bit unmodified
1303   g should be used in calculating the next s.
1304
13058. Examples of Randomness Required
1306
1307   Below are two examples showing rough calculations of needed
1308   randomness for security.  The first is for moderate security
1309   passwords while the second assumes a need for a very high security
1310   cryptographic key.
1311
13128.1  Password Generation
1313
1314   Assume that user passwords change once a year and it is desired that
1315   the probability that an adversary could guess the password for a
1316   particular account be less than one in a thousand.  Further assume
1317   that sending a password to the system is the only way to try a
1318   password.  Then the crucial question is how often an adversary can
1319   try possibilities.  Assume that delays have been introduced into a
1320   system so that, at most, an adversary can make one password try every
1321   six seconds.  That's 600 per hour or about 15,000 per day or about
1322   5,000,000 tries in a year.  Assuming any sort of monitoring, it is
1323   unlikely someone could actually try continuously for a year.  In
1324   fact, even if log files are only checked monthly, 500,000 tries is
1325   more plausible before the attack is noticed and steps taken to change
1326   passwords and make it harder to try more passwords.
1327
1328   To have a one in a thousand chance of guessing the password in
1329   500,000 tries implies a universe of at least 500,000,000 passwords or
1330   about 2^29.  Thus 29 bits of randomness are needed. This can probably
1331   be achieved using the US DoD recommended inputs for password
1332   generation as it has 8 inputs which probably average over 5 bits of
1333   randomness each (see section 7.1).  Using a list of 1000 words, the
1334   password could be expressed as a three word phrase (1,000,000,000
1335   possibilities) or, using case insensitive letters and digits, six
1336   would suffice ((26+10)^6 = 2,176,782,336 possibilities).
1337
1338   For a higher security password, the number of bits required goes up.
1339   To decrease the probability by 1,000 requires increasing the universe
1340   of passwords by the same factor which adds about 10 bits.  Thus to
1341   have only a one in a million chance of a password being guessed under
1342   the above scenario would require 39 bits of randomness and a password
1343
1344
1345
1346Eastlake, Crocker & Schiller                                   [Page 24]
1347
1348RFC 1750        Randomness Recommendations for Security    December 1994
1349
1350
1351   that was a four word phrase from a 1000 word list or eight
1352   letters/digits.  To go to a one in 10^9 chance, 49 bits of randomness
1353   are needed implying a five word phrase or ten letter/digit password.
1354
1355   In a real system, of course, there are also other factors.  For
1356   example, the larger and harder to remember passwords are, the more
1357   likely users are to write them down resulting in an additional risk
1358   of compromise.
1359
13608.2 A Very High Security Cryptographic Key
1361
1362   Assume that a very high security key is needed for symmetric
1363   encryption / decryption between two parties.  Assume an adversary can
1364   observe communications and knows the algorithm being used.  Within
1365   the field of random possibilities, the adversary can try key values
1366   in hopes of finding the one in use.  Assume further that brute force
1367   trial of keys is the best the adversary can do.
1368
13698.2.1 Effort per Key Trial
1370
1371   How much effort will it take to try each key?  For very high security
1372   applications it is best to assume a low value of effort.  Even if it
1373   would clearly take tens of thousands of computer cycles or more to
1374   try a single key, there may be some pattern that enables huge blocks
1375   of key values to be tested with much less effort per key.  Thus it is
1376   probably best to assume no more than a couple hundred cycles per key.
1377   (There is no clear lower bound on this as computers operate in
1378   parallel on a number of bits and a poor encryption algorithm could
1379   allow many keys or even groups of keys to be tested in parallel.
1380   However, we need to assume some value and can hope that a reasonably
1381   strong algorithm has been chosen for our hypothetical high security
1382   task.)
1383
1384   If the adversary can command a highly parallel processor or a large
1385   network of work stations, 2*10^10 cycles per second is probably a
1386   minimum assumption for availability today.  Looking forward just a
1387   couple years, there should be at least an order of magnitude
1388   improvement.  Thus assuming 10^9 keys could be checked per second or
1389   3.6*10^11 per hour or 6*10^13 per week or 2.4*10^14 per month is
1390   reasonable.  This implies a need for a minimum of 51 bits of
1391   randomness in keys to be sure they cannot be found in a month.  Even
1392   then it is possible that, a few years from now, a highly determined
1393   and resourceful adversary could break the key in 2 weeks (on average
1394   they need try only half the keys).
1395
1396
1397
1398
1399
1400
1401
1402Eastlake, Crocker & Schiller                                   [Page 25]
1403
1404RFC 1750        Randomness Recommendations for Security    December 1994
1405
1406
14078.2.2 Meet in the Middle Attacks
1408
1409   If chosen or known plain text and the resulting encrypted text are
1410   available, a "meet in the middle" attack is possible if the structure
1411   of the encryption algorithm allows it.  (In a known plain text
1412   attack, the adversary knows all or part of the messages being
1413   encrypted, possibly some standard header or trailer fields.  In a
1414   chosen plain text attack, the adversary can force some chosen plain
1415   text to be encrypted, possibly by "leaking" an exciting text that
1416   would then be sent by the adversary over an encrypted channel.)
1417
1418   An oversimplified explanation of the meet in the middle attack is as
1419   follows: the adversary can half-encrypt the known or chosen plain
1420   text with all possible first half-keys, sort the output, then half-
1421   decrypt the encoded text with all the second half-keys.  If a match
1422   is found, the full key can be assembled from the halves and used to
1423   decrypt other parts of the message or other messages.  At its best,
1424   this type of attack can halve the exponent of the work required by
1425   the adversary while adding a large but roughly constant factor of
1426   effort.  To be assured of safety against this, a doubling of the
1427   amount of randomness in the key to a minimum of 102 bits is required.
1428
1429   The meet in the middle attack assumes that the cryptographic
1430   algorithm can be decomposed in this way but we can not rule that out
1431   without a deep knowledge of the algorithm.  Even if a basic algorithm
1432   is not subject to a meet in the middle attack, an attempt to produce
1433   a stronger algorithm by applying the basic algorithm twice (or two
1434   different algorithms sequentially) with different keys may gain less
1435   added security than would be expected.  Such a composite algorithm
1436   would be subject to a meet in the middle attack.
1437
1438   Enormous resources may be required to mount a meet in the middle
1439   attack but they are probably within the range of the national
1440   security services of a major nation.  Essentially all nations spy on
1441   other nations government traffic and several nations are believed to
1442   spy on commercial traffic for economic advantage.
1443
14448.2.3 Other Considerations
1445
1446   Since we have not even considered the possibilities of special
1447   purpose code breaking hardware or just how much of a safety margin we
1448   want beyond our assumptions above, probably a good minimum for a very
1449   high security cryptographic key is 128 bits of randomness which
1450   implies a minimum key length of 128 bits.  If the two parties agree
1451   on a key by Diffie-Hellman exchange [D-H], then in principle only
1452   half of this randomness would have to be supplied by each party.
1453   However, there is probably some correlation between their random
1454   inputs so it is probably best to assume that each party needs to
1455
1456
1457
1458Eastlake, Crocker & Schiller                                   [Page 26]
1459
1460RFC 1750        Randomness Recommendations for Security    December 1994
1461
1462
1463   provide at least 96 bits worth of randomness for very high security
1464   if Diffie-Hellman is used.
1465
1466   This amount of randomness is beyond the limit of that in the inputs
1467   recommended by the US DoD for password generation and could require
1468   user typing timing, hardware random number generation, or other
1469   sources.
1470
1471   It should be noted that key length calculations such at those above
1472   are controversial and depend on various assumptions about the
1473   cryptographic algorithms in use.  In some cases, a professional with
1474   a deep knowledge of code breaking techniques and of the strength of
1475   the algorithm in use could be satisfied with less than half of the
1476   key size derived above.
1477
14789. Conclusion
1479
1480   Generation of unguessable "random" secret quantities for security use
1481   is an essential but difficult task.
1482
1483   We have shown that hardware techniques to produce such randomness
1484   would be relatively simple.  In particular, the volume and quality
1485   would not need to be high and existing computer hardware, such as
1486   disk drives, can be used.  Computational techniques are available to
1487   process low quality random quantities from multiple sources or a
1488   larger quantity of such low quality input from one source and produce
1489   a smaller quantity of higher quality, less predictable key material.
1490   In the absence of hardware sources of randomness, a variety of user
1491   and software sources can frequently be used instead with care;
1492   however, most modern systems already have hardware, such as disk
1493   drives or audio input, that could be used to produce high quality
1494   randomness.
1495
1496   Once a sufficient quantity of high quality seed key material (a few
1497   hundred bits) is available, strong computational techniques are
1498   available to produce cryptographically strong sequences of
1499   unpredicatable quantities from this seed material.
1500
150110. Security Considerations
1502
1503   The entirety of this document concerns techniques and recommendations
1504   for generating unguessable "random" quantities for use as passwords,
1505   cryptographic keys, and similar security uses.
1506
1507
1508
1509
1510
1511
1512
1513
1514Eastlake, Crocker & Schiller                                   [Page 27]
1515
1516RFC 1750        Randomness Recommendations for Security    December 1994
1517
1518
1519References
1520
1521   [ASYMMETRIC] - Secure Communications and Asymmetric Cryptosystems,
1522   edited by Gustavus J. Simmons, AAAS Selected Symposium 69, Westview
1523   Press, Inc.
1524
1525   [BBS] - A Simple Unpredictable Pseudo-Random Number Generator, SIAM
1526   Journal on Computing, v. 15, n. 2, 1986, L. Blum, M. Blum, & M. Shub.
1527
1528   [BRILLINGER] - Time Series: Data Analysis and Theory, Holden-Day,
1529   1981, David Brillinger.
1530
1531   [CRC] - C.R.C. Standard Mathematical Tables, Chemical Rubber
1532   Publishing Company.
1533
1534   [CRYPTO1] - Cryptography: A Primer, A Wiley-Interscience Publication,
1535   John Wiley & Sons, 1981, Alan G. Konheim.
1536
1537   [CRYPTO2] - Cryptography:  A New Dimension in Computer Data Security,
1538   A Wiley-Interscience Publication, John Wiley & Sons, 1982, Carl H.
1539   Meyer & Stephen M. Matyas.
1540
1541   [CRYPTO3] - Applied Cryptography: Protocols, Algorithms, and Source
1542   Code in C, John Wiley & Sons, 1994, Bruce Schneier.
1543
1544   [DAVIS] - Cryptographic Randomness from Air Turbulence in Disk
1545   Drives, Advances in Cryptology - Crypto '94, Springer-Verlag Lecture
1546   Notes in Computer Science #839, 1984, Don Davis, Ross Ihaka, and
1547   Philip Fenstermacher.
1548
1549   [DES] -  Data Encryption Standard, United States of America,
1550   Department of Commerce, National Institute of Standards and
1551   Technology, Federal Information Processing Standard (FIPS) 46-1.
1552   - Data Encryption Algorithm, American National Standards Institute,
1553   ANSI X3.92-1981.
1554   (See also FIPS 112, Password Usage, which includes FORTRAN code for
1555   performing DES.)
1556
1557   [DES MODES] - DES Modes of Operation, United States of America,
1558   Department of Commerce, National Institute of Standards and
1559   Technology, Federal Information Processing Standard (FIPS) 81.
1560   - Data Encryption Algorithm - Modes of Operation, American National
1561   Standards Institute, ANSI X3.106-1983.
1562
1563   [D-H] - New Directions in Cryptography, IEEE Transactions on
1564   Information Technology, November, 1976, Whitfield Diffie and Martin
1565   E. Hellman.
1566
1567
1568
1569
1570Eastlake, Crocker & Schiller                                   [Page 28]
1571
1572RFC 1750        Randomness Recommendations for Security    December 1994
1573
1574
1575   [DoD] - Password Management Guideline, United States of America,
1576   Department of Defense, Computer Security Center, CSC-STD-002-85.
1577   (See also FIPS 112, Password Usage, which incorporates CSC-STD-002-85
1578   as one of its appendices.)
1579
1580   [GIFFORD] - Natural Random Number, MIT/LCS/TM-371, September 1988,
1581   David K. Gifford
1582
1583   [KNUTH] - The Art of Computer Programming, Volume 2: Seminumerical
1584   Algorithms, Chapter 3: Random Numbers. Addison Wesley Publishing
1585   Company, Second Edition 1982, Donald E. Knuth.
1586
1587   [KRAWCZYK] - How to Predict Congruential Generators, Journal of
1588   Algorithms, V. 13, N. 4, December 1992, H. Krawczyk
1589
1590   [MD2] - The MD2 Message-Digest Algorithm, RFC1319, April 1992, B.
1591   Kaliski
1592   [MD4] - The MD4 Message-Digest Algorithm, RFC1320, April 1992, R.
1593   Rivest
1594   [MD5] - The MD5 Message-Digest Algorithm, RFC1321, April 1992, R.
1595   Rivest
1596
1597   [PEM] - RFCs 1421 through 1424:
1598   - RFC 1424, Privacy Enhancement for Internet Electronic Mail: Part
1599   IV: Key Certification and Related Services, 02/10/1993, B. Kaliski
1600   - RFC 1423, Privacy Enhancement for Internet Electronic Mail: Part
1601   III: Algorithms, Modes, and Identifiers, 02/10/1993, D. Balenson
1602   - RFC 1422, Privacy Enhancement for Internet Electronic Mail: Part
1603   II: Certificate-Based Key Management, 02/10/1993, S. Kent
1604   - RFC 1421, Privacy Enhancement for Internet Electronic Mail: Part I:
1605   Message Encryption and Authentication Procedures, 02/10/1993, J. Linn
1606
1607   [SHANNON] - The Mathematical Theory of Communication, University of
1608   Illinois Press, 1963, Claude E. Shannon.  (originally from:  Bell
1609   System Technical Journal, July and October 1948)
1610
1611   [SHIFT1] - Shift Register Sequences, Aegean Park Press, Revised
1612   Edition 1982, Solomon W. Golomb.
1613
1614   [SHIFT2] - Cryptanalysis of Shift-Register Generated Stream Cypher
1615   Systems, Aegean Park Press, 1984, Wayne G. Barker.
1616
1617   [SHS] - Secure Hash Standard, United States of American, National
1618   Institute of Science and Technology, Federal Information Processing
1619   Standard (FIPS) 180, April 1993.
1620
1621   [STERN] - Secret Linear Congruential Generators are not
1622   Cryptograhically Secure, Proceedings of IEEE STOC, 1987, J. Stern.
1623
1624
1625
1626Eastlake, Crocker & Schiller                                   [Page 29]
1627
1628RFC 1750        Randomness Recommendations for Security    December 1994
1629
1630
1631   [VON NEUMANN] - Various techniques used in connection with random
1632   digits, von Neumann's Collected Works, Vol. 5, Pergamon Press, 1963,
1633   J. von Neumann.
1634
1635Authors' Addresses
1636
1637   Donald E. Eastlake 3rd
1638   Digital Equipment Corporation
1639   550 King Street, LKG2-1/BB3
1640   Littleton, MA 01460
1641
1642   Phone:   +1 508 486 6577(w)  +1 508 287 4877(h)
1643   EMail:   dee@lkg.dec.com
1644
1645
1646   Stephen D. Crocker
1647   CyberCash Inc.
1648   2086 Hunters Crest Way
1649   Vienna, VA 22181
1650
1651   Phone:   +1 703-620-1222(w)  +1 703-391-2651 (fax)
1652   EMail:   crocker@cybercash.com
1653
1654
1655   Jeffrey I. Schiller
1656   Massachusetts Institute of Technology
1657   77 Massachusetts Avenue
1658   Cambridge, MA 02139
1659
1660   Phone:   +1 617 253 0161(w)
1661   EMail:   jis@mit.edu
1662
1663
1664
1665
1666
1667
1668
1669
1670
1671
1672
1673
1674
1675
1676
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1682Eastlake, Crocker & Schiller                                   [Page 30]
1683