7
Network Working Group D. Eastlake, 3rd
8
Request for Comments: 1750 DEC
9
Category: Informational S. Crocker
16
Randomness Recommendations for Security
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.
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
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.
58
Eastlake, Crocker & Schiller [Page 1]
60
RFC 1750 Randomness Recommendations for Security December 1994
65
Comments on this document that have been incorporated were received
66
from (in alphabetic order) the following:
68
David M. Balenson (TIS)
70
Don T. Davis (consultant)
71
Carl Ellison (Stratus)
73
Christian Huitema (INRIA)
74
Charlie Kaufman (IRIS)
77
Neil Haller (Bellcore)
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
114
Eastlake, Crocker & Schiller [Page 2]
116
RFC 1750 Randomness Recommendations for Security December 1994
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
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.
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.
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.
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
170
Eastlake, Crocker & Schiller [Page 3]
172
RFC 1750 Randomness Recommendations for Security December 1994
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.)
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.
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*].
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.
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
226
Eastlake, Crocker & Schiller [Page 4]
228
RFC 1750 Randomness Recommendations for Security December 1994
231
secret values possible and the probability of each value as follows:
235
Bits-of-info = \ - p * log ( p )
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
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.
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.
263
3. Traditional Pseudo-Random Sequences
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
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,
282
Eastlake, Crocker & Schiller [Page 5]
284
RFC 1750 Randomness Recommendations for Security December 1994
287
perhaps unlimited, chances at guessing the correct value and should
288
be assumed to be aided by a computer.
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].
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
301
V = ( V * a + b )(Mod c)
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.
312
+----+ +----+ +----+ +----+
313
| B | <-- | B | <-- | B | <-- . . . . . . <-- | B | <-+
314
| 0 | | 1 | | 2 | | n | |
315
+----+ +----+ +----+ +----+ |
318
| V +----------------> | |
319
V +-----------------------------> | XOR |
320
+---------------------------------------------------> | |
324
V = ( ( V * 2 ) + B .xor. B ... )(Mod 2^n)
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
338
Eastlake, Crocker & Schiller [Page 6]
340
RFC 1750 Randomness Recommendations for Security December 1994
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.
356
Not only have linear congruent generators been broken, but techniques
357
are now known for breaking all polynomial congruent generators
362
Randomness in the traditional sense described in section 3 is NOT the
363
same as the unpredictability required for security use.
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.
371
The following sections describe the limitations of some randomness
372
generation techniques and sources.
374
4.1 Problems with Clocks and Serial Numbers
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.
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
394
Eastlake, Crocker & Schiller [Page 7]
396
RFC 1750 Randomness Recommendations for Security December 1994
399
the clock being unchanged between two reads and increases it by one!
400
Designing portable application code to generate unpredictable numbers
401
based on such system clocks is particularly challenging because the
402
system designer does not always know the properties of the system
403
clocks that the code will execute on.
405
Use of a hardware serial number such as an Ethernet address may also
406
provide fewer bits of uniqueness than one would guess. Such
407
quantities are usually heavily structured and subfields may have only
408
a limited range of possible values or values easily guessable based
409
on approximate date of manufacture or other data. For example, it is
410
likely that 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.
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
419
4.2 Timing and Content of External Events
421
It is possible to measure the timing and content of mouse movement,
422
key strokes, and similar user events. This is a reasonable source of
423
unguessable data with some qualifications. On some machines, inputs
424
such as key strokes are buffered. Even though the user's inter-
425
keystroke timing may have sufficient variation and unpredictability,
426
there might not be an easy way to access that variation. Another
427
problem is that no standard method exists to sample timing details.
428
This makes it hard to build standard software intended for
429
distribution to a large range of machines based on this technique.
431
The amount of mouse movement or the keys actually hit are usually
432
easier to access than timings but may yield less unpredictability as
433
the user may provide highly repetitive input.
435
Other external events, such as network packet arrival times, can also
436
be used with care. In particular, the possibility of manipulation of
437
such times by an adversary must be considered.
439
4.3 The Fallacy of Complex Manipulation
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
450
Eastlake, Crocker & Schiller [Page 8]
452
RFC 1750 Randomness Recommendations for Security December 1994
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.
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.
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.
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!
483
4.4 The Fallacy of Selection from a Large Database
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
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
506
Eastlake, Crocker & Schiller [Page 9]
508
RFC 1750 Randomness Recommendations for Security December 1994
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
516
5. Hardware for Randomness
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.
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.
533
How much unpredictability is needed? Is it possible to quantify the
534
requirement in, say, number of random bits per second?
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.
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
551
5.2 Sensitivity to Skew
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.
562
Eastlake, Crocker & Schiller [Page 10]
564
RFC 1750 Randomness Recommendations for Security December 1994
567
5.2.1 Using Stream Parity to De-Skew
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
577
The following analysis gives the number of bits that must be sampled:
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.
587
These sums can be computed easily as
590
1/2 * ( ( p + q ) + ( p - q ) )
593
1/2 * ( ( p + q ) - ( p - q ) ).
595
(Which one corresponds to the probability the parity will be 1
596
depends on whether N is odd or even.)
598
Since p + q = 1 and p - q = 2e, these expressions reduce to
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
611
( 0.5 + ( 0.5 * (2e) ) ) < 0.5 + d.
618
Eastlake, Crocker & Schiller [Page 11]
620
RFC 1750 Randomness Recommendations for Security December 1994
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.)
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.
631
+---------+--------+-------+
633
+---------+--------+-------+
640
| 0.99 | 0.49 | 308 |
641
+---------+--------+-------+
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.
647
5.2.2 Using Transition Mappings to De-Skew
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
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
+------+-----------------------------------------+
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).
674
Eastlake, Crocker & Schiller [Page 12]
676
RFC 1750 Randomness Recommendations for Security December 1994
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.
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.
694
5.2.3 Using FFT to De-Skew
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.
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].
706
5.2.4 Using Compression to De-Skew
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
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.
730
Eastlake, Crocker & Schiller [Page 13]
732
RFC 1750 Randomness Recommendations for Security December 1994
735
5.3 Existing Hardware Can Be Used For Randomness
737
As described below, many computers come with hardware that can, with
738
care, be used to generate truly random quantities.
740
5.3.1 Using Existing Sound/Video Input
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.
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.
756
Combining this with compression to de-skew one can, in UNIXese,
757
generate a huge amount of medium quality random data by doing
759
cat /dev/audio | compress - >random-bits-file
761
5.3.2 Using Existing Disk Drives
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.
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
777
6. Recommended Non-Hardware Strategy
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.
786
Eastlake, Crocker & Schiller [Page 14]
788
RFC 1750 Randomness Recommendations for Security December 1994
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
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.
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.
818
6.1.1 A Trivial Mixing Function
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.
826
+-----------+-----------+----------+
827
| input 1 | input 2 | output |
828
+-----------+-----------+----------+
833
+-----------+-----------+----------+
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
842
Eastlake, Crocker & Schiller [Page 15]
844
RFC 1750 Randomness Recommendations for Security December 1994
850
output input 1 input 2
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:
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
+--------+--------+--------+--------+--------+--------+--------+
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.
876
6.1.2 Stronger Mixing Functions
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.
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
898
Eastlake, Crocker & Schiller [Page 16]
900
RFC 1750 Randomness Recommendations for Security December 1994
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.
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.
930
6.1.3 Diffie-Hellman as a Mixing Function
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
941
6.1.4 Using a Mixing Function to Stretch Random Bits
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
954
Eastlake, Crocker & Schiller [Page 17]
956
RFC 1750 Randomness Recommendations for Security December 1994
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
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.
972
6.1.5 Other Factors in Choosing a Mixing Function
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].
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.
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.
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.
1010
Eastlake, Crocker & Schiller [Page 18]
1012
RFC 1750 Randomness Recommendations for Security December 1994
1015
6.2 Non-Hardware Sources of Randomness
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.
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
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.
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.
1054
6.3 Cryptographically Strong Sequences
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
1066
Eastlake, Crocker & Schiller [Page 19]
1068
RFC 1750 Randomness Recommendations for Security December 1994
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.
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
1090
6.3.1 Traditional Strong Sequences
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.
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].
1122
Eastlake, Crocker & Schiller [Page 20]
1124
RFC 1750 Randomness Recommendations for Security December 1994
1132
| +---------> | | +-----+
1133
+--+ | Encrypt | <--- | Key |
1134
| +-------- | | +-----+
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.
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!
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.
1166
6.3.2 The Blum Blum Shub Sequence Generator
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.
1178
Eastlake, Crocker & Schiller [Page 21]
1180
RFC 1750 Randomness Recommendations for Security December 1994
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
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
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.
1208
An intersting characteristic of this generator is that you can
1209
directly calculate any of the s values. In particular
1212
( ( 2 )(Mod (( p - 1 ) * ( q - 1 )) ) )
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
1221
7. Key Generation Standards
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.
1234
Eastlake, Crocker & Schiller [Page 22]
1236
RFC 1750 Randomness Recommendations for Security December 1994
1239
7.1 US DoD Recommendations for Password Generation
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
1246
use an initialization vector determined from
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
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
1265
7.2 X9.17 Key Generation
1267
The American National Standards Institute has specified a method for
1268
generating a sequence of keys as follows:
1270
s is the initial 64 bit seed
1273
g is the sequence of generated 64 bit key quantities
1276
k is a random key reserved for generating this key sequence
1278
t is the time at which a key is generated to as fine a resolution
1279
as is available (up to 64 bits).
1281
DES ( K, Q ) is the DES encryption of quantity Q with key K
1290
Eastlake, Crocker & Schiller [Page 23]
1292
RFC 1750 Randomness Recommendations for Security December 1994
1295
g = DES ( k, DES ( k, t ) .xor. s )
1298
s = DES ( k, DES ( k, t ) .xor. g )
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.
1305
8. Examples of Randomness Required
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
1312
8.1 Password Generation
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.
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).
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
1346
Eastlake, Crocker & Schiller [Page 24]
1348
RFC 1750 Randomness Recommendations for Security December 1994
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.
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
1360
8.2 A Very High Security Cryptographic Key
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.
1369
8.2.1 Effort per Key Trial
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
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).
1402
Eastlake, Crocker & Schiller [Page 25]
1404
RFC 1750 Randomness Recommendations for Security December 1994
1407
8.2.2 Meet in the Middle Attacks
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.)
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.
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.
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.
1444
8.2.3 Other Considerations
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
1458
Eastlake, Crocker & Schiller [Page 26]
1460
RFC 1750 Randomness Recommendations for Security December 1994
1463
provide at least 96 bits worth of randomness for very high security
1464
if Diffie-Hellman is used.
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
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.
1480
Generation of unguessable "random" secret quantities for security use
1481
is an essential but difficult task.
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
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.
1501
10. Security Considerations
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.
1514
Eastlake, Crocker & Schiller [Page 27]
1516
RFC 1750 Randomness Recommendations for Security December 1994
1521
[ASYMMETRIC] - Secure Communications and Asymmetric Cryptosystems,
1522
edited by Gustavus J. Simmons, AAAS Selected Symposium 69, Westview
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.
1528
[BRILLINGER] - Time Series: Data Analysis and Theory, Holden-Day,
1529
1981, David Brillinger.
1531
[CRC] - C.R.C. Standard Mathematical Tables, Chemical Rubber
1534
[CRYPTO1] - Cryptography: A Primer, A Wiley-Interscience Publication,
1535
John Wiley & Sons, 1981, Alan G. Konheim.
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.
1541
[CRYPTO3] - Applied Cryptography: Protocols, Algorithms, and Source
1542
Code in C, John Wiley & Sons, 1994, Bruce Schneier.
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.
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,
1554
(See also FIPS 112, Password Usage, which includes FORTRAN code for
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.
1563
[D-H] - New Directions in Cryptography, IEEE Transactions on
1564
Information Technology, November, 1976, Whitfield Diffie and Martin
1570
Eastlake, Crocker & Schiller [Page 28]
1572
RFC 1750 Randomness Recommendations for Security December 1994
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.)
1580
[GIFFORD] - Natural Random Number, MIT/LCS/TM-371, September 1988,
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.
1587
[KRAWCZYK] - How to Predict Congruential Generators, Journal of
1588
Algorithms, V. 13, N. 4, December 1992, H. Krawczyk
1590
[MD2] - The MD2 Message-Digest Algorithm, RFC1319, April 1992, B.
1592
[MD4] - The MD4 Message-Digest Algorithm, RFC1320, April 1992, R.
1594
[MD5] - The MD5 Message-Digest Algorithm, RFC1321, April 1992, R.
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
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)
1611
[SHIFT1] - Shift Register Sequences, Aegean Park Press, Revised
1612
Edition 1982, Solomon W. Golomb.
1614
[SHIFT2] - Cryptanalysis of Shift-Register Generated Stream Cypher
1615
Systems, Aegean Park Press, 1984, Wayne G. Barker.
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.
1621
[STERN] - Secret Linear Congruential Generators are not
1622
Cryptograhically Secure, Proceedings of IEEE STOC, 1987, J. Stern.
1626
Eastlake, Crocker & Schiller [Page 29]
1628
RFC 1750 Randomness Recommendations for Security December 1994
1631
[VON NEUMANN] - Various techniques used in connection with random
1632
digits, von Neumann's Collected Works, Vol. 5, Pergamon Press, 1963,
1637
Donald E. Eastlake 3rd
1638
Digital Equipment Corporation
1639
550 King Street, LKG2-1/BB3
1642
Phone: +1 508 486 6577(w) +1 508 287 4877(h)
1643
EMail: dee@lkg.dec.com
1648
2086 Hunters Crest Way
1651
Phone: +1 703-620-1222(w) +1 703-391-2651 (fax)
1652
EMail: crocker@cybercash.com
1656
Massachusetts Institute of Technology
1657
77 Massachusetts Avenue
1660
Phone: +1 617 253 0161(w)
1682
Eastlake, Crocker & Schiller [Page 30]
added
removed
doc/rfc/rfc1750.txt
