Randomness Requirements for Security
draft-eastlake-randomness2-10

Network Working Group Donald E. Eastlake, 3rd
OBSOLETES RFC 1750 Jeffrey I. Schiller
 Steve Crocker
Expires July 2005 January 2005
 Randomness Requirements for Security
 ---------- ------------ --- --------
 <draft-eastlake-randomness2-10.txt>
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 patent or other IPR claims of which I am aware have been disclosed,
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Abstract
 Security systems are built on strong cryptographic algorithms that
 foil pattern analysis attempts. However, the security of these
 systems is dependent on generating secret quantities for passwords,
 cryptographic keys, and similar quantities. The use of pseudo-random
 processes to generate secret quantities can result in pseudo-
 security. The sophisticated attacker of these security systems may
 find it easier to reproduce the environment that produced the secret
 quantities, searching the resulting small set of possibilities, than
 to locate the quantities in the whole of the potential number space.
 Choosing random quantities to foil a resourceful and motivated
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 adversary is surprisingly difficult. This document points out many
 pitfalls in using poor entropy sources or traditional pseudo-random
 number generation techniques for generating such quantities. It
 recommends the use of truly random hardware techniques and shows that
 the existing hardware on many systems can be used for this purpose.
 It provides suggestions to ameliorate the problem when a hardware
 solution is not available. And it gives examples of how large such
 quantities need to be for some applications.
Acknowledgements
 Special thanks to Paul Hoffman and John Kelsey for their extensive
 comments and to Peter Gutmann, who has permitted the incorporation of
 material from his paper "Software Generation of Practically Strong
 Random Numbers".
 The following other persons (in alphabetic order) have also
 contributed substantially to this document:
 Steve Bellovin, Daniel Brown, Don Davis, Peter Gutmann, Tony
 Hansen, Sandy Harris, Paul Hoffman, Scott Hollenback, Russ
 Housley, Christian Huitema, John Kelsey, Mats Naslund, and Damir
 Rajnovic.
 The following persons (in alphabetic order) contributed to RFC 1750,
 the predecessor of this document:
 David M. Balenson, Don T. Davis, Carl Ellison, Marc Horowitz,
 Christian Huitema, Charlie Kaufman, Steve Kent, Hal Murray, Neil
 Haller, Richard Pitkin, Tim Redmond, and Doug Tygar.
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Table of Contents
 Status of This Document....................................1
 Abstract...................................................1
 Acknowledgements...........................................2
 Table of Contents..........................................3
 1. Introduction and Overview...............................5
 2. General Requirements....................................6
 3. Entropy Sources.........................................9
 3.1 Volume Required........................................9
 3.2 Existing Hardware Can Be Used For Randomness..........10
 3.2.1 Using Existing Sound/Video Input....................10
 3.2.2 Using Existing Disk Drives..........................10
 3.3 Ring Oscillator Sources...............................11
 3.4 Problems with Clocks and Serial Numbers...............12
 3.5 Timing and Value of External Events...................13
 3.6 Non-Hardware Sources of Randomness....................14
 4. De-skewing.............................................15
 4.1 Using Stream Parity to De-Skew........................15
 4.2 Using Transition Mappings to De-Skew..................16
 4.3 Using FFT to De-Skew..................................17
 4.4 Using Compression to De-Skew..........................18
 5. Mixing.................................................19
 5.1 A Trivial Mixing Function.............................19
 5.2 Stronger Mixing Functions.............................20
 5.3 Using S-Boxes for Mixing..............................22
 5.4 Diffie-Hellman as a Mixing Function...................22
 5.5 Using a Mixing Function to Stretch Random Bits........23
 5.6 Other Factors in Choosing a Mixing Function...........23
 6. Pseudo Random Number Generators........................25
 6.1 Some Bad Ideas........................................25
 6.1.1 The Fallacy of Complex Manipulation.................25
 6.1.2 The Fallacy of Selection from a Large Database......26
 6.1.3. Traditional Pseudo-Random Sequences................26
 6.2 Cryptographically Strong Sequences....................28
 6.2.1 OFB and CTR Sequences...............................29
 6.2.2 The Blum Blum Shub Sequence Generator...............30
 6.3 Entropy Pool Techniques...............................31
 7. Randomness Generation Examples and Standards...........33
 7.1 Complete Randomness Generators........................33
 7.1.1 US DoD Recommendations for Password Generation......33
 7.1.2 The /dev/random Device..............................34
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 7.1.3 Windows CryptGenRandom..............................35
 7.2 Generators Assuming a Source of Entropy...............36
 7.2.1 X9.82 Pseudo-Random Number Generation...............36
 7,2.1.1 Notation..........................................36
 7.1.2.2 Initializing the Generator........................37
 7.1.2.5 Generating Random Bits............................37
 7.2.2 X9.17 Key Generation................................37
 7.2.3 DSS Pseudo-Random Number Generation.................38
 8. Examples of Randomness Required........................40
 8.1 Password Generation..................................40
 8.2 A Very High Security Cryptographic Key................41
 8.2.1 Effort per Key Trial................................41
 8.2.2 Meet in the Middle Attacks..........................42
 8.2.3 Other Considerations................................43
 9. Conclusion.............................................44
 10. Security Considerations...............................45
 11. Copyright and Disclaimer..............................45
 12. Appendix A: Changes from RFC 1750.....................46
 13. Informative References................................47
 Author's Addresses........................................52
 File Name and Expiration..................................52
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1. Introduction and Overview
 Software cryptography is coming into wider use and is continuing to
 spread, although there is a long way to go until it becomes
 pervasive.
 Systems like SSH, IPSEC, TLS, S/MIME, PGP, DNSSEC, Kerberos, etc. are
 maturing and becoming a part of the network landscape [SSH, IPSEC,
 MAIL*, TLS, DNSSEC]. By comparison, when the previous version of this
 document [RFC 1750] was issued in 1994, about the only Internet
 cryptographic security specification in the IETF was the Privacy
 Enhanced Mail protocol [MAIL PEM *].
 These systems provide substantial protection against snooping and
 spoofing. However, there is a potential flaw. At the heart of all
 cryptographic systems is the generation of secret, unguessable (i.e.,
 random) numbers.
 The lack of generally available facilities for generating such random
 numbers, that is the lack of general availability of truly
 unpredictable sources, forms an open wound in the design of
 cryptographic software. For the software developer who wants to build
 a key or password generation procedure that runs on a wide range of
 hardware, this is a very real problem.
 It is important to keep in mind that the requirement is for data that
 an adversary has a very low probability of guessing or determining.
 This can easily fail if pseudo-random data is used which only meets
 traditional statistical tests for randomness or which is based on
 limited range sources, such as clocks. Sometimes such pseudo-random
 quantities are determinable by an adversary searching through an
 embarrassingly small space of possibilities.
 This Best Current Practice describes techniques for producing random
 quantities that will be resistant to such attack. It recommends that
 future systems include hardware random number generation or provide
 access to existing hardware that can be used for this purpose. It
 suggests methods for use if such hardware is not available. And it
 gives some estimates of the number of random bits required for sample
 applications.
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2. General Requirements
 A commonly encountered randomness requirement today is the user
 password. This is usually a simple character string. Obviously, if a
 password can be guessed, it does not provide security. (For re-usable
 passwords, it is desirable that users be able to remember the
 password. This may make it advisable to use pronounceable character
 strings or phrases composed on ordinary words. But this only affects
 the format of the password information, not the requirement that the
 password be very hard to guess.)
 Many other requirements come from the cryptographic arena.
 Cryptographic techniques can be used to provide a variety of services
 including confidentiality and authentication. Such services are based
 on quantities, traditionally called "keys", that are unknown to and
 unguessable by an adversary.
 There are even TCP/IP protocol uses for randomness in picking initial
 sequence numbers [RFC 1948].
 Generally speaking, the above examples also illustrate two different
 types of random quantities that may be wanted. In the case of human
 usable passwords, the only important characteristic is that it be
 unguessable; it is not important that they may be composed of ASCII
 characters, for example, so the top bit of every byte is zero. On the
 other hand, for fixed length keys and the like, you normally want
 quantities that are indistinguishable from truly random, that is, all
 bits will pass statistical randomness tests.
 In some cases, such as the use of symmetric encryption with the one
 time pads or an algorithm like the US Advanced Encryption Standard
 [AES], the parties who wish to communicate confidentially and/or with
 authentication must all know the same secret key. In other cases,
 using what are called asymmetric or "public key" cryptographic
 techniques, keys come in pairs. One key of the pair is private and
 must be kept secret by one party, the other is public and can be
 published to the world. It is computationally infeasible to determine
 the private key from the public key and knowledge of the public is of
 no help to an adversary [ASYMMETRIC]. [SCHNEIER, FERGUSON, KAUFMAN]
 The frequency and volume of the requirement for random quantities
 differs greatly for different cryptographic systems. Using pure RSA,
 random quantities are required only when a new key pair is generated;
 thereafter any number of messages can be signed without a further
 need for randomness. The public key Digital Signature Algorithm
 devised by the US National Institute of Standards and Technology
 (NIST) requires good random numbers for each signature [DSS]. And
 encrypting with a one time pad, in principle the strongest possible
 encryption technique, requires a volume of randomness equal to all
 the messages to be processed. [SCHNEIER, FERGUSON, KAUFMAN]
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 In most of these cases, an adversary can try to determine the
 "secret" key by trial and error. (This is possible as long as the key
 is enough smaller than the message that the correct key can be
 uniquely identified.) The probability of an adversary succeeding at
 this must be made acceptably low, depending on the particular
 application. The size of the space the adversary must search is
 related to the amount of key "information" present in an information
 theoretic sense [SHANNON]. This depends on the number of different
 secret values possible and the probability of each value as follows:
 -----
 \
 Bits-of-information = \ - p * log ( p )
 / i 2 i
 /
 -----
 where i counts from 1 to the number of possible secret values and p
 sub i is the probability of the value numbered i. (Since p sub i is
 less than one, the log will be negative so each term in the sum will
 be non-negative.)
 If there are 2^n different values of equal probability, then n bits
 of information are present and an adversary would, on the average,
 have to try half of the values, or 2^(n-1) , before guessing the
 secret quantity. If the probability of different values is unequal,
 then there is less information present and fewer guesses will, on
 average, be required by an adversary. In particular, any values that
 the adversary can know are impossible, or are of low probability, can
 be initially ignored by an adversary, who will search through the
 more probable values first.
 For example, consider a cryptographic system that uses 128 bit keys.
 If these 128 bit keys are derived by using a fixed pseudo-random
 number generator that is seeded with an 8 bit seed, then an adversary
 needs to search through only 256 keys (by running the pseudo-random
 number generator with every possible seed), not the 2^128 keys that
 may at first appear to be the case. Only 8 bits of "information" are
 in these 128 bit keys.
 While the above analysis is correct on average, it can be misleading
 in some cases for cryptographic analysis where what is really
 important is the work factor for an adversary. For example, assume
 that there was a pseudo-random number generator generating 128 bit
 keys, as in the previous paragraph, but that it generated 0 half of
 the time and a random selection from the remaining 2**128 - 1 values
 the rest of the time. The Shannon equation above says that there are
 64 bits of information in one of these key values but an adversary,
 by simply trying the values 0, can break the security of half of the
 uses, albeit a random half. Thus for cryptographic purposes, it is
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 also useful to look at other measures, such as min-entropy, defined
 as
 Min-entropy = - log ( maximum ( p ) )
 i
 where i is as above. Using this equation, we get 1 bit of min-
 entropy for our new hypothetical distribution as opposed to 64 bits
 of classical Shannon entropy.
 A continuous spectrum of entropies, sometimes called Renyi entropy,
 have been defined, specified by a parameter r. When r = 1, it is
 Shannon entropy, and with r = infinity, it is min-entropy. When r =
 0, it is just log (n) where n is the number of non-zero
 probabilities. Renyi entropy is a non-increasing function of r, so
 min-entropy is always the most conservative measure of entropy and
 usually the best to use for cryptographic evaluation. [LUBY]
 Statistically tested randomness in the traditional sense is NOT the
 same as the unpredictability required for security use.
 For example, use of a widely available constant sequence, such as
 that from the CRC tables, is very weak against an adversary. Once
 they learn of or guess it, they can easily break all security, future
 and past, based on the sequence. [CRC] As another example, using AES
 to encrypt successive integers such as 1, 2, 3 ... will also produce
 output that has excellent statistical randomness properties but is
 also predictable. On the other hand, taking successive rolls of a
 six-sided die and encoding the resulting values in ASCII would
 produce statistically poor output with a substantial unpredictable
 component. So you should keep in mind that passing or failing
 statistical tests doesn't tell you that something is unpredictable
 or predictable.
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3. Entropy Sources
 Entropy sources tend to be very implementation dependent. Once one
 has gathered sufficient entropy it can be used as the seed to produce
 the required amount of cryptographically strong pseudo-randomness, as
 described in Sections 6 and 7, after being de-skewed and/or mixed if
 necessary as described in Sections 4 and 5.
 Is there any hope for true strong portable randomness in the future?
 There might be. All that's needed is a physical source of
 unpredictable numbers.
 A thermal noise (sometimes called Johnson noise in integrated
 circuits) or radioactive decay source and a fast, free-running
 oscillator would do the trick directly [GIFFORD]. This is a trivial
 amount of hardware, and could easily be included as a standard part
 of a computer system's architecture. Most audio (or video) input
 devices are useable [TURBID]. Furthermore, any system with a
 spinning disk or ring oscillator and a stable (crystal) time source
 or the like has an adequate source of randomness ([DAVIS] and Section
 3.3). All that's needed is the common perception among computer
 vendors that this small additional hardware and the software to
 access it is necessary and useful.
 ANSI X9 is currently developing a standard which includes a part
 devoted to entropy sources. See [X9.82 - Part 2].
3.1 Volume Required
 How much unpredictability is needed? Is it possible to quantify the
 requirement in, say, number of random bits per second?
 The answer is not very much is needed. For AES, the key can be 128
 bits and, as we show in an example in Section 8, even the highest
 security system is unlikely to require strong keying material of much
 over 200 bits. If a series of keys are needed, they can be generated
 from a strong random seed (starting value) using a cryptographically
 strong sequence as explained in Section 6.2. A few hundred random
 bits generated at start up or once a day would be enough using such
 techniques. Even if the random bits are generated as slowly as one
 per second and it is not possible to overlap the generation process,
 it should be tolerable in most high security applications to wait 200
 seconds occasionally.
 These numbers are trivial to achieve. It could be done by a person
 repeatedly tossing a coin. Almost any hardware based process is
 likely to be much faster.
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3.2 Existing Hardware Can Be Used For Randomness
 As described below, many computers come with hardware that can, with
 care, be used to generate truly random quantities.
3.2.1 Using Existing Sound/Video Input
 Many computers are built with inputs that digitize some real world
 analog source, such as sound from a microphone or video input from a
 camera. Under appropriate circumstances, such input can provide
 reasonably high quality random bits. The "input" from a sound
 digitizer with no source plugged in or a camera with the lens cap on,
 if the system has enough gain to detect anything, is essentially
 thermal noise. This method is extremely hardware and implementation
 dependent.
 For example, on some UNIX based systems, one can read from the
 /dev/audio device with nothing plugged into the microphone jack or
 the microphone receiving only low level background noise. Such data
 is essentially random noise although it should not be trusted without
 some checking in case of hardware failure. It will, in any case,
 need to be de-skewed as described elsewhere.
 Combining this with compression to de-skew (see Section 4) one can,
 in UNIXese, generate a huge amount of medium quality random data by
 doing
 cat /dev/audio | compress - >random-bits-file
 A detailed examination of this type of randomness source appears in
 [TURBID].
3.2.2 Using Existing Disk Drives
 Disk drives have small random fluctuations in their rotational speed
 due to chaotic air turbulence [DAVIS, Jakobsson]. By adding low
 level disk seek time instrumentation to a system, a series of
 measurements can be obtained that include this randomness. Such data
 is usually highly correlated so that significant processing is
 needed, such as described in 5.2 below. Nevertheless experimentation
 a decade ago showed that, with such processing, even slow disk drives
 on the slower computers of that day could easily produce 100 bits a
 minute or more of excellent random data.
 Every increase in processor speed, which increases the resolution
 with which disk motion can be timed, or increase in the rate of disk
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 seeks, increases the rate of random bit generation possible with this
 technique. At the time of this paper and using modern hardware, a
 more typical rate of random bit production would be in excess of
 10,000 bits a second. This technique is used in many operating system
 library random number generators.
 Note: the inclusion of cache memories in disk controllers has little
 effect on this technique if very short seek times, which represent
 cache hits, are simply ignored.
3.3 Ring Oscillator Sources
 If an integrated circuit is being designed or field programmed, an
 odd number of gates can be connected in series to produce a free-
 running ring oscillator. By sampling a point in the ring at a fixed
 frequency, say one determined by a stable crystal oscillator, some
 amount of entropy can be extracted due to variations in the free-
 running oscillator timing. It is possible to increase the rate of
 entropy by xor'ing sampled values from a few ring oscillators with
 relatively prime lengths. It is sometimes recommended that an odd
 number of rings be used so that, even if the rings somehow become
 synchronously locked to each other, there will still be sampled bit
 transitions. Another possibility source to sample is the output of a
 noisy diode.
 Sampled bits from such sources will have to be heavily de-skewed, as
 disk rotation timings must be (see Section 4). An engineering study
 would be needed to determine the amount of entropy being produced
 depending on the particular design. In any case, these can be good
 sources whose cost is a trivial amount of hardware by modern
 standards.
 As an example, IEEE 802.11i suggests that the circuit below be
 considered, with due attention in the design to isolation of the
 rings from each other and from clocked circuits to avoid undesired
 synchronization, etc., and extensive post processing. [IEEE 802.11i]
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 |\ |\ |\
 +-->| >0-->| >0-- 19 total --| >0--+-------+
 | |/ |/ |/ | |
 | | |
 +----------------------------------+ V
 +-----+
 |\ |\ |\ | | output
 +-->| >0-->| >0-- 23 total --| >0--+--->| XOR |------>
 | |/ |/ |/ | | |
 | | +-----+
 +----------------------------------+ ^ ^
 | |
 |\ |\ |\ | |
 +-->| >0-->| >0-- 29 total --| >0--+------+ |
 | |/ |/ |/ | |
 | | |
 +----------------------------------+ |
 |
 other randomness if available--------------+
3.4 Problems with Clocks and Serial Numbers
 Computer clocks, or similar operating system or hardware values,
 provide significantly fewer real bits of unpredictability than might
 appear from their specifications.
 Tests have been done on clocks on numerous systems and it was found
 that their behavior can vary widely and in unexpected ways. One
 version of an operating system running on one set of hardware may
 actually provide, say, microsecond resolution in a clock while a
 different configuration of the "same" system may always provide the
 same lower bits and only count in the upper bits at much lower
 resolution. This means that successive reads on the clock may produce
 identical values even if enough time has passed that the value
 "should" change based on the nominal clock resolution. There are also
 cases where frequently reading a clock can produce artificial
 sequential values because of extra code that checks for the clock
 being unchanged between two reads and increases it by one! Designing
 portable application code to generate unpredictable numbers based on
 such system clocks is particularly challenging because the system
 designer does not always know the properties of the system clocks
 that the code will execute on.
 Use of hardware serial numbers such as an Ethernet MAC addresses may
 also provide fewer bits of uniqueness than one would guess. Such
 quantities are usually heavily structured and subfields may have only
 a limited range of possible values or values easily guessable based
 on approximate date of manufacture or other data. For example, it is
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 likely that a company that manufactures both computers and Ethernet
 adapters will, at least internally, use its own adapters, which
 significantly limits the range of built-in addresses.
 Problems such as those described above related to clocks and serial
 numbers make code to produce unpredictable quantities difficult if
 the code is to be ported across a variety of computer platforms and
 systems.
3.5 Timing and Value of External Events
 It is possible to measure the timing and content of mouse movement,
 key strokes, and similar user events. This is a reasonable source of
 unguessable data with some qualifications. On some machines, inputs
 such as key strokes are buffered. Even though the user's inter-
 keystroke timing may have sufficient variation and unpredictability,
 there might not be an easy way to access that variation. Another
 problem is that no standard method exists to sample timing details.
 This makes it hard to build standard software intended for
 distribution to a large range of machines based on this technique.
 The amount of mouse movement or the keys actually hit are usually
 easier to access than timings but may yield less unpredictability as
 the user may provide highly repetitive input.
 Other external events, such as network packet arrival times and
 lengths, can also be used, but only with great care. In particular,
 the possibility of manipulation of such network traffic measurements
 by an adversary and the lack of history at system start up must be
 carefully considered. If this input is subject to manipulation, it
 must not be trusted as a source of entropy.
 Indeed, almost any external sensor, such as raw radio reception or
 temperature sensing in appropriately equipped computers, can be used
 in principle. But in each case careful consideration must be given to
 how much such data is subject to adversarial manipulation and to how
 much entropy it can actually provide.
 The above techniques are quite powerful against any attackers having
 no access to the quantities being measured. For example, they would
 be powerful against offline attackers who had no access to your
 environment and were trying to crack your random seed after the fact.
 In all cases, the more accurately you can measure the timing or value
 of an external sensor, the more rapidly you can generate bits.
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3.6 Non-Hardware Sources of Randomness
 The best source of input entropy would be a hardware randomness such
 as ring oscillators, disk drive timing, thermal noise, or radioactive
 decay. However, if that is not available, there are other
 possibilities. These include system clocks, system or input/output
 buffers, user/system/hardware/network serial numbers and/or addresses
 and timing, and user input. Unfortunately, each of these sources can
 produce very limited or predictable values under some circumstances.
 Some of the sources listed above would be quite strong on multi-user
 systems where, in essence, each user of the system is a source of
 randomness. However, on a small single user or embedded system,
 especially at start up, it might be possible for an adversary to
 assemble a similar configuration. This could give the adversary
 inputs to the mixing process that were sufficiently correlated to
 those used originally as to make exhaustive search practical.
 The use of multiple random inputs with a strong mixing function is
 recommended and can overcome weakness in any particular input. The
 timing and content of requested "random" user keystrokes can yield
 hundreds of random bits but conservative assumptions need to be made.
 For example, assuming at most a few bits of randomness if the inter-
 keystroke interval is unique in the sequence up to that point and a
 similar assumption if the key hit is unique but assuming that no bits
 of randomness are present in the initial key value or if the timing
 or key value duplicate previous values. The results of mixing these
 timings and characters typed could be further combined with clock
 values and other inputs.
 This strategy may make practical portable code to produce good random
 numbers for security even if some of the inputs are very weak on some
 of the target systems. However, it may still fail against a high
 grade attack on small, single user or embedded systems, especially if
 the adversary has ever been able to observe the generation process in
 the past. A hardware based random source is still preferable.
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4. De-skewing
 Is there any specific requirement on the shape of the distribution of
 quantities gathered for the entropy to produce the random numbers?
 The good news is the distribution need not be uniform. All that is
 needed is a conservative estimate of how non-uniform it is to bound
 performance. Simple techniques to de-skew a bit stream are given
 below and stronger cryptographic techniques are described in Section
 5.2 below.
4.1 Using Stream Parity to De-Skew
 As a simple but not particularly practical example, consider taking a
 sufficiently long string of bits and map the string to "zero" or
 "one". The mapping will not yield a perfectly uniform distribution,
 but it can be as close as desired. One mapping that serves the
 purpose is to take the parity of the string. This has the advantages
 that it is robust across all degrees of skew up to the estimated
 maximum skew and is absolutely trivial to implement in hardware.
 The following analysis gives the number of bits that must be sampled:
 Suppose the ratio of ones to zeros is ( 0.5 + E ) to ( 0.5 - E ),
 where E is between 0 and 0.5 and is a measure of the "eccentricity"
 of the distribution. Consider the distribution of the parity function
 of N bit samples. The probabilities that the parity will be one or
 zero will be the sum of the odd or even terms in the binomial
 expansion of (p + q)^N, where p = 0.5 + E, the probability of a one,
 and q = 0.5 - E, the probability of a zero.
 These sums can be computed easily as
 N N
 1/2 * ( ( p + q ) + ( p - q ) )
 and
 N N
 1/2 * ( ( p + q ) - ( p - q ) ).
 (Which one corresponds to the probability the parity will be 1
 depends on whether N is odd or even.)
 Since p + q = 1 and p - q = 2e, these expressions reduce to
 N
 1/2 * [1 + (2E) ]
 and
 N
 1/2 * [1 - (2E) ].
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 Neither of these will ever be exactly 0.5 unless E is zero, but we
 can bring them arbitrarily close to 0.5. If we want the probabilities
 to be within some delta d of 0.5, i.e. then
 N
 ( 0.5 + ( 0.5 * (2E) ) ) < 0.5 + d.
 Solving for N yields N > log(2d)/log(2E). (Note that 2E is less than
 1, so its log is negative. Division by a negative number reverses the
 sense of an inequality.)
 The following table gives the length of the string which must be
 sampled for various degrees of skew in order to come within 0.001 of
 a 50/50 distribution.
 +---------+--------+-------+
 | Prob(1) | E | N |
 +---------+--------+-------+
 | 0.5 | 0.00 | 1 |
 | 0.6 | 0.10 | 4 |
 | 0.7 | 0.20 | 7 |
 | 0.8 | 0.30 | 13 |
 | 0.9 | 0.40 | 28 |
 | 0.95 | 0.45 | 59 |
 | 0.99 | 0.49 | 308 |
 +---------+--------+-------+
 The last entry shows that even if the distribution is skewed 99% in
 favor of ones, the parity of a string of 308 samples will be within
 0.001 of a 50/50 distribution. But, as we shall see in section 6.1.2,
 there are much stronger techniques that extract more of the available
 entropy.
4.2 Using Transition Mappings to De-Skew
 Another technique, originally due to von Neumann [VON NEUMANN], is to
 examine a bit stream as a sequence of non-overlapping pairs. You
 could then discard any 00 or 11 pairs found, interpret 01 as a 0 and
 10 as a 1. Assume the probability of a 1 is 0.5+E and the probability
 of a 0 is 0.5-E where E is the eccentricity of the source and
 described in the previous section. Then the probability of each pair
 is as follows:
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 +------+-----------------------------------------+
 | pair | probability |
 +------+-----------------------------------------+
 | 00 | (0.5 - E)^2 = 0.25 - E + E^2 |
 | 01 | (0.5 - E)*(0.5 + E) = 0.25 - E^2 |
 | 10 | (0.5 + E)*(0.5 - E) = 0.25 - E^2 |
 | 11 | (0.5 + E)^2 = 0.25 + E + E^2 |
 +------+-----------------------------------------+
 This technique will completely eliminate any bias but at the expense
 of taking an indeterminate number of input bits for any particular
 desired number of output bits. The probability of any particular pair
 being discarded is 0.5 + 2E^2 so the expected number of input bits to
 produce X output bits is X/(0.25 - E^2).
 This technique assumes that the bits are from a stream where each bit
 has the same probability of being a 0 or 1 as any other bit in the
 stream and that bits are not correlated, i.e., that the bits are
 identical independent distributions. If alternate bits were from two
 correlated sources, for example, the above analysis breaks down.
 The above technique also provides another illustration of how a
 simple statistical analysis can mislead if one is not always on the
 lookout for patterns that could be exploited by an adversary. If the
 algorithm were mis-read slightly so that overlapping successive bits
 pairs were used instead of non-overlapping pairs, the statistical
 analysis given is the same; however, instead of providing an unbiased
 uncorrelated series of random 1s and 0s, it instead produces a
 totally predictable sequence of exactly alternating 1s and 0s.
4.3 Using FFT to De-Skew
 When real world data consists of strongly correlated bits, it may
 still contain useful amounts of entropy. This entropy can be
 extracted through use of various transforms, the most powerful of
 which are described in section 5.2 below.
 Using the Fourier transform of the data or its optimized variant, the
 FFT, is an technique interesting primarily for theoretical reasons.
 It can be show that this will discard strong correlations. If
 adequate data is processed and remaining correlations decay, spectral
 lines approaching statistical independence and normally distributed
 randomness can be produced [BRILLINGER].
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4.4 Using Compression to De-Skew
 Reversible compression techniques also provide a crude method of de-
 skewing a skewed bit stream. This follows directly from the
 definition of reversible compression and the formula in Section 2
 above for the amount of information in a sequence. Since the
 compression is reversible, the same amount of information must be
 present in the shorter output than was present in the longer input.
 By the Shannon information equation, this is only possible if, on
 average, the probabilities of the different shorter sequences are
 more uniformly distributed than were the probabilities of the longer
 sequences. Therefore the shorter sequences must be de-skewed relative
 to the input.
 However, many compression techniques add a somewhat predictable
 preface to their output stream and may insert such a sequence again
 periodically in their output or otherwise introduce subtle patterns
 of their own. They should be considered only a rough technique
 compared with those described in Section 5.2. At a minimum, the
 beginning of the compressed sequence should be skipped and only later
 bits used for applications requiring roughly random bits.
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5. Mixing
 What is the best overall strategy for meeting the requirement for
 unguessable random numbers in the absence of a strong reliable
 hardware entropy source? It is to obtain input from a number of
 uncorrelated sources and to mix them with a strong mixing function.
 Such a function will preserve the entropy present in any of the
 sources even if other quantities being combined happen to be fixed or
 easily guessable (low entropy). This may be advisable even with a
 good hardware source, as hardware can also fail, though this should
 be weighed against any increase in the chance of overall failure due
 to added software complexity.
 Once you have used good sources, such as some of those listed in
 Section 3, and mixed them as described in this section, you have a
 strong seed. This can then be used to produce large quantities of
 cryptographically strong material as described in Sections 6 and 7.
 A strong mixing function is one which combines inputs and produces an
 output where each output bit is a different complex non-linear
 function of all the input bits. On average, changing any input bit
 will change about half the output bits. But because the relationship
 is complex and non-linear, no particular output bit is guaranteed to
 change when any particular input bit is changed.
 Consider the problem of converting a stream of bits that is skewed
 towards 0 or 1 or which has a somewhat predictable pattern to a
 shorter stream which is more random, as discussed in Section 4 above.
 This is simply another case where a strong mixing function is
 desired, mixing the input bits to produce a smaller number of output
 bits. The technique given in Section 4.1 of using the parity of a
 number of bits is simply the result of successively Exclusive Or'ing
 them which is examined as a trivial mixing function immediately
 below. Use of stronger mixing functions to extract more of the
 randomness in a stream of skewed bits is examined in Section 5.2. See
 also [NASLUND].
5.1 A Trivial Mixing Function
 A trivial example for single bit inputs described only for expository
 purposes is the Exclusive Or function, which is equivalent to
 addition without carry, as show in the table below. This is a
 degenerate case in which the one output bit always changes for a
 change in either input bit. But, despite its simplicity, it provides
 a useful illustration.
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 +-----------+-----------+----------+
 | input 1 | input 2 | output |
 +-----------+-----------+----------+
 | 0 | 0 | 0 |
 | 0 | 1 | 1 |
 | 1 | 0 | 1 |
 | 1 | 1 | 0 |
 +-----------+-----------+----------+
 If inputs 1 and 2 are uncorrelated and combined in this fashion then
 the output will be an even better (less skewed) random bit than the
 inputs. If we assume an "eccentricity" E as defined in Section 5.2
 above, then the output eccentricity relates to the input eccentricity
 as follows:
 E = 2 * E * E
 output input 1 input 2
 Since E is never greater than 1/2, the eccentricity is always
 improved except in the case where at least one input is a totally
 skewed constant. This is illustrated in the following table where the
 top and left side values are the two input eccentricities and the
 entries are the output eccentricity:
 +--------+--------+--------+--------+--------+--------+--------+
 | E | 0.00 | 0.10 | 0.20 | 0.30 | 0.40 | 0.50 |
 +--------+--------+--------+--------+--------+--------+--------+
 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
 | 0.10 | 0.00 | 0.02 | 0.04 | 0.06 | 0.08 | 0.10 |
 | 0.20 | 0.00 | 0.04 | 0.08 | 0.12 | 0.16 | 0.20 |
 | 0.30 | 0.00 | 0.06 | 0.12 | 0.18 | 0.24 | 0.30 |
 | 0.40 | 0.00 | 0.08 | 0.16 | 0.24 | 0.32 | 0.40 |
 | 0.50 | 0.00 | 0.10 | 0.20 | 0.30 | 0.40 | 0.50 |
 +--------+--------+--------+--------+--------+--------+--------+
 However, keep in mind that the above calculations assume that the
 inputs are not correlated. If the inputs were, say, the parity of the
 number of minutes from midnight on two clocks accurate to a few
 seconds, then each might appear random if sampled at random intervals
 much longer than a minute. Yet if they were both sampled and combined
 with xor, the result would be zero most of the time.
5.2 Stronger Mixing Functions
 The US Government Advanced Encryption Standard [AES] is an example of
 a strong mixing function for multiple bit quantities. It takes up to
 384 bits of input (128 bits of "data" and 256 bits of "key") and
 produces 128 bits of output each of which is dependent on a complex
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 non-linear function of all input bits. Other encryption functions
 with this characteristic, such as [DES], can also be used by
 considering them to mix all of their key and data input bits.
 Another good family of mixing functions are the "message digest" or
 hashing functions such as The US Government Secure Hash Standards
 [SHA*] and the MD4, MD5 [MD4, MD5] series. These functions all take a
 practically unlimited amount of input and produce a relatively short
 fixed length output mixing all the input bits. The MD* series produce
 128 bits of output, SHA-1 produces 160 bits, and other SHA functions
 produce up to 512 bits.
 Although the message digest functions are designed for variable
 amounts of input, AES and other encryption functions can also be used
 to combine any number of inputs. If 128 bits of output is adequate,
 the inputs can be packed into a 128-bit data quantity and successive
 AES keys, padding with zeros if needed, which are then used to
 successively encrypt using AES in Electronic Codebook Mode. Or the
 input could be packed into one 128-bit key and multiple data blocks
 and a CBC-MAC calculated [MODES].
 If more than 128 bits of output are needed and you want to employ
 AES, use more complex mixing. But keep in mind that it is absolutely
 impossible to get more bits of "randomness" out than are put in. For
 example, if inputs are packed into three quantities, A, B, and C, use
 AES to encrypt A with B as a key and then with C as a key to produce
 the 1st part of the output, then encrypt B with C and then A for more
 output and, if necessary, encrypt C with A and then B for yet more
 output. Still more output can be produced by reversing the order of
 the keys given above to stretch things. The same can be done with the
 hash functions by hashing various subsets of the input data or
 different copies of the input data with different prefixes and/or
 suffixes to produce multiple outputs.
 An example of using a strong mixing function would be to reconsider
 the case of a string of 308 bits each of which is biased 99% towards
 zero. The parity technique given in Section 4.1 above reduced this to
 one bit with only a 1/1000 deviance from being equally likely a zero
 or one. But, applying the equation for information given in Section
 2, this 308 bit skewed sequence has over 5 bits of information in it.
 Thus hashing it with SHA-1 and taking the bottom 5 bits of the result
 would yield 5 unbiased random bits as opposed to the single bit given
 by calculating the parity of the string. Alternatively, for some
 applications, you could use the entire hash output to retain almost
 all of the 5+ bits of entropy in a 160 bit quantity.
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5.3 Using S-Boxes for Mixing
 Many modern block encryption functions, including DES and AES,
 incorporate modules known as S-Boxes (substitution boxes). These
 produce a smaller number of outputs from a larger number of inputs
 through a complex non-linear mixing function which would have the
 effect of concentrating limited entropy in the inputs into the
 output.
 S-Boxes sometimes incorporate bent Boolean functions (functions of an
 even number of bits producing one output bit with maximum non-
 linearity). Looking at the output for all input pairs differing in
 any particular bit position, exactly half the outputs are different.
 An S-Box in which each output bit is produced by a bent function such
 that any linear combination of these functions is also a bent
 function is called a "perfect S-Box".
 S-boxes and various repeated application or cascades of such boxes
 can be used for mixing. [SBOX*]
5.4 Diffie-Hellman as a Mixing Function
 Diffie-Hellman exponential key exchange is a technique that yields a
 shared secret between two parties that can be made computationally
 infeasible for a third party to determine even if they can observe
 all the messages between the two communicating parties. This shared
 secret is a mixture of initial quantities generated by each of the
 parties [D-H].
 If these initial quantities are random and uncorrelated, then the
 shared secret combines their entropy, but, of course, can not produce
 more randomness than the size of the shared secret generated.
 While this is true if the Diffie-Hellman computation is performed
 privately, an adversary that can observe either of the public keys
 and knows the modulus being used need only search through the space
 of the other secret key in order to be able to calculate the shared
 secret [D-H]. So, conservatively, it would be best to consider public
 Diffie-Hellman to produce a quantity whose guessability corresponds
 to the worst of the two inputs. Because of this and the fact that
 Diffie-Hellman is computationally intensive, its use as a mixing
 function is not recommended.
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5.5 Using a Mixing Function to Stretch Random Bits
 While it is not necessary for a mixing function to produce the same
 or fewer bits than its inputs, mixing bits cannot "stretch" the
 amount of random unpredictability present in the inputs. Thus four
 inputs of 32 bits each where there is 12 bits worth of
 unpredictability (such as 4,096 equally probable values) in each
 input cannot produce more than 48 bits worth of unpredictable output.
 The output can be expanded to hundreds or thousands of bits by, for
 example, mixing with successive integers, but the clever adversary's
 search space is still 2^48 possibilities. Furthermore, mixing to
 fewer bits than are input will tend to strengthen the randomness of
 the output.
 The last table in Section 5.1 shows that mixing a random bit with a
 constant bit with Exclusive Or will produce a random bit. While this
 is true, it does not provide a way to "stretch" one random bit into
 more than one. If, for example, a random bit is mixed with a 0 and
 then with a 1, this produces a two bit sequence but it will always be
 either 01 or 10. Since there are only two possible values, there is
 still only the one bit of original randomness.
5.6 Other Factors in Choosing a Mixing Function
 For local use, AES has the advantages that it has been widely tested
 for flaws, is reasonably efficient in software, and is widely
 documented and implemented with hardware and software implementations
 available all over the world including open source code. The SHA*
 family have had a little less study and tend to require more CPU
 cycles than AES but there is no reason to believe they are flawed.
 Both SHA* and MD5 were derived from the earlier MD4 algorithm. They
 all have source code available [SHA*, MD*]. Some signs of weakness
 have been found in MD4 and MD5. In particular, MD4 has only three
 rounds and there are several independent breaks of the first two or
 last two rounds. And some collisions have been found in MD5 output.
 AES was selected by a robust, public, and international process. It
 and SHA* have been vouched for by the US National Security Agency
 (NSA) on the basis of criteria that mostly remain secret, as was DES.
 While this has been the cause of much speculation and doubt,
 investigation of DES over the years has indicated that NSA
 involvement in modifications to its design, which originated with
 IBM, was primarily to strengthen it. There has been no announcement
 of a concealed or special weakness being found in DES. It is likely
 that the NSA modifications to MD4 to produce the SHA algorithms
 similarly strengthened these algorithms, possibly against threats not
 yet known in the public cryptographic community.
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 Where input lengths are unpredictable, hash algorithms are more
 convenient to use than block encryption algorithms since they are
 generally designed to accept variable length inputs. Block encryption
 algorithms generally require an additional padding algorithm to
 accommodate inputs that are not an even multiple of the block size.
 As of the time of this document, the authors know of no patent claims
 to the basic AES, DES, SHA*, MD4, and MD5 algorithms other than
 patents for which an irrevocable royalty free license has been
 granted to the world. There may, of course, be essential patents of
 which the authors are unaware or patents on implementations or uses
 or other relevant patents issued or to be issued.
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6. Pseudo Random Number Generators
 When you have a seed with sufficient entropy, from input as described
 in Section 3 possibly de-skewed and mixed as described in Sections 4
 and 5, you can algorithmically extend that seed to produce a large
 quantity of cryptographically strong random quantities. Such
 algorithms are platform independent and can operate in the same
 fashion on any computer. To be secure their input and internal
 workings must be protected from adversarial observation.
 The design of such pseudo random number generation algorithms, like
 the design of symmetric encryption algorithms, is not a task for
 amateurs. Section 6.1 below lists a number of bad ideas which failed
 algorithms have used. If you are interested in what works, you can
 skip 6.1 and just read the remainder of this section and Section 7
 below which describes and gives references for some standard pseudo
 random number generation algorithms. See Section 7 and [X9.82 - Part
 3].
6.1 Some Bad Ideas
 The subsections below describe a number of idea which might seem
 reasonable but which lead to insecure pseudo random number
 generation.
6.1.1 The Fallacy of Complex Manipulation
 One strategy which may give a misleading appearance of
 unpredictability is to take a very complex algorithm (or an excellent
 traditional pseudo-random number generator with good statistical
 properties) and calculate a cryptographic key by starting with
 limited data such as the computer system clock value as the seed. An
 adversary who knew roughly when the generator was started would have
 a relatively small number of seed values to test as they would know
 likely values of the system clock. Large numbers of pseudo-random
 bits could be generated but the search space an adversary would need
 to check could be quite small.
 Thus very strong and/or complex manipulation of data will not help if
 the adversary can learn what the manipulation is and there is not
 enough entropy in the starting seed value. They can usually use the
 limited number of results stemming from a limited number of seed
 values to defeat security.
 Another serious strategy error is to assume that a very complex
 pseudo-random number generation algorithm will produce strong random
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 numbers when there has been no theory behind or analysis of the
 algorithm. There is a excellent example of this fallacy right near
 the beginning of Chapter 3 in [KNUTH] where the author describes a
 complex algorithm. It was intended that the machine language program
 corresponding to the algorithm would be so complicated that a person
 trying to read the code without comments wouldn't know what the
 program was doing. Unfortunately, actual use of this algorithm showed
 that it almost immediately converged to a single repeated value in
 one case and a small cycle of values in another case.
 Not only does complex manipulation not help you if you have a limited
 range of seeds but blindly chosen complex manipulation can destroy
 the entropy in a good seed!
6.1.2 The Fallacy of Selection from a Large Database
 Another strategy that can give a misleading appearance of
 unpredictability is selection of a quantity randomly from a database
 and assume that its strength is related to the total number of bits
 in the database. For example, typical USENET servers process many
 megabytes of information per day [USENET]. Assume a random quantity
 was selected by fetching 32 bytes of data from a random starting
 point in this data. This does not yield 32*8 = 256 bits worth of
 unguessability. Even after allowing that much of the data is human
 language and probably has no more than 2 or 3 bits of information per
 byte, it doesn't yield 32*2 = 64 bits of unguessability. For an
 adversary with access to the same Usenet database the unguessability
 rests only on the starting point of the selection. That is perhaps a
 little over a couple of dozen bits of unguessability.
 The same argument applies to selecting sequences from the data on a
 publicly available CD/DVD recording or any other large public
 database. If the adversary has access to the same database, this
 "selection from a large volume of data" step buys little. However,
 if a selection can be made from data to which the adversary has no
 access, such as system buffers on an active multi-user system, it may
 be of help.
6.1.3. Traditional Pseudo-Random Sequences
 This section talks about traditional sources of deterministic of
 "pseudo-random" numbers. These typically start with a "seed" quantity
 and use simple numeric or logical operations to produce a sequence of
 values. Note that none of the techniques discussed in this section is
 suitable for cryptographic use. They are presented for general
 information.
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 [KNUTH] has a classic exposition on pseudo-random numbers.
 Applications he mentions are simulation of natural phenomena,
 sampling, numerical analysis, testing computer programs, decision
 making, and games. None of these have the same characteristics as the
 sort of security uses we are talking about. Only in the last two
 could there be an adversary trying to find the random quantity.
 However, in these cases, the adversary normally has only a single
 chance to use a guessed value. In guessing passwords or attempting to
 break an encryption scheme, the adversary normally has many, perhaps
 unlimited, chances at guessing the correct value. Sometimes they can
 store the message they are trying to break and repeatedly attack it.
 They are also be assumed to be aided by a computer.
 For testing the "randomness" of numbers, Knuth suggests a variety of
 measures including statistical and spectral. These tests check things
 like autocorrelation between different parts of a "random" sequence
 or distribution of its values. But they could be met by a constant
 stored random sequence, such as the "random" sequence printed in the
 CRC Standard Mathematical Tables [CRC]. Despite meeting all the tests
 suggested by Knuth, that sequence is unsuitable for cryptographic use
 as adversaries must be assumed to have copies of all common published
 "random" sequences and will able to spot the source and predict
 future values.
 A typical pseudo-random number generation technique, known as a
 linear congruence pseudo-random number generator, is modular
 arithmetic where the value numbered N+1 is calculated from the value
 numbered N by
 V = ( V * a + b )(Mod c)
 N+1 N
 The above technique has a strong relationship to linear shift
 register pseudo-random number generators, which are well understood
 cryptographically [SHIFT*]. In such generators bits are introduced at
 one end of a shift register as the Exclusive Or (binary sum without
 carry) of bits from selected fixed taps into the register. For
 example:
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 +----+ +----+ +----+ +----+
 | B | <-- | B | <-- | B | <-- . . . . . . <-- | B | <-+
 | 0 | | 1 | | 2 | | n | |
 +----+ +----+ +----+ +----+ |
 | | | |
 | | V +-----+
 | V +----------------> | |
 V +-----------------------------> | XOR |
 +---------------------------------------------------> | |
 +-----+
 V = ( ( V * 2 ) + B .xor. B ... )(Mod 2^n)
 N+1 N 0 2
 The goodness of traditional pseudo-random number generator algorithms
 is measured by statistical tests on such sequences. Carefully chosen
 values a, b, c, and initial V or the placement of shift register tap
 in the above simple processes can produce excellent statistics.
 These sequences may be adequate in simulations (Monte Carlo
 experiments) as long as the sequence is orthogonal to the structure
 of the space being explored. Even there, subtle patterns may cause
 problems. However, such sequences are clearly bad for use in security
 applications. They are fully predictable if the initial state is
 known. Depending on the form of the pseudo-random number generator,
 the sequence may be determinable from observation of a short portion
 of the sequence [SCHNEIER, STERN]. For example, with the generators
 above, one can determine V(n+1) given knowledge of V(n). In fact, it
 has been shown that with these techniques, even if only one bit of
 the pseudo-random values are released, the seed can be determined
 from short sequences.
 Not only have linear congruent generators been broken, but techniques
 are now known for breaking all polynomial congruent generators.
 [KRAWCZYK]
6.2 Cryptographically Strong Sequences
 In cases where a series of random quantities must be generated, an
 adversary may learn some values in the sequence. In general, they
 should not be able to predict other values from the ones that they
 know.
 The correct technique is to start with a strong random seed, take
 cryptographically strong steps from that seed [FERGUSON, SCHNEIER],
 and do not reveal the complete state of the generator in the sequence
 elements. If each value in the sequence can be calculated in a fixed
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 way from the previous value, then when any value is compromised, all
 future values can be determined. This would be the case, for example,
 if each value were a constant function of the previously used values,
 even if the function were a very strong, non-invertible message
 digest function.
 (It should be noted that if your technique for generating a sequence
 of key values is fast enough, it can trivially be used as the basis
 for a confidentiality system. If two parties use the same sequence
 generating technique and start with the same seed material, they will
 generate identical sequences. These could, for example, be xor'ed at
 one end with data being send, encrypting it, and xor'ed with this
 data as received, decrypting it due to the reversible properties of
 the xor operation. This is commonly referred to as a simple stream
 cipher.)
6.2.1 OFB and CTR Sequences
 One way to achieve a strong sequence is to have the values be
 produced by taking a seed value and hashing the quantities produced
 by concatenating the seed with successive integers or the like and
 then mask the values obtained so as to limit the amount of generator
 state available to the adversary.
 It may also be possible to use an "encryption" algorithm with a
 random key and seed value to encrypt successive integers as in
 counter (CTR) mode encryption. Alternatively, you can feedback all of
 the output value from encryption into the value to be encrypted for
 the next iteration. This is a particular example of output feedback
 mode (OFB). [MODES]
 An example is shown below where shifting and masking are used to
 combine part of the output feedback with part of the old input. This
 type of partial feedback should be avoided for reasons described
 below.
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 +---------------+
 | V |
 | | n |--+
 +--+------------+ |
 | | +---------+
 shift| +---> | | +-----+
 +--+ | Encrypt | <--- | Key |
 | +-------- | | +-----+
 | | +---------+
 V V
 +------------+--+
 | V | |
 | n+1 |
 +---------------+
 Note that if a shift of one is used, this is the same as the shift
 register technique described in Section 3 above but with the all
 important difference that the feedback is determined by a complex
 non-linear function of all bits rather than a simple linear or
 polynomial combination of output from a few bit position taps.
 It has been shown by Donald W. Davies that this sort of shifted
 partial output feedback significantly weakens an algorithm compared
 with feeding all of the output bits back as input. In particular, for
 DES, repeated encrypting a full 64 bit quantity will give an expected
 repeat in about 2^63 iterations. Feeding back anything less than 64
 (and more than 0) bits will give an expected repeat in between 2^31
 and 2^32 iterations!
 To predict values of a sequence from others when the sequence was
 generated by these techniques is equivalent to breaking the
 cryptosystem or inverting the "non-invertible" hashing involved with
 only partial information available. The less information revealed
 each iteration, the harder it will be for an adversary to predict the
 sequence. Thus it is best to use only one bit from each value. It has
 been shown that in some cases this makes it impossible to break a
 system even when the cryptographic system is invertible and can be
 broken if all of each generated value was revealed.
6.2.2 The Blum Blum Shub Sequence Generator
 Currently the generator which has the strongest public proof of
 strength is called the Blum Blum Shub generator after its inventors
 [BBS]. It is also very simple and is based on quadratic residues.
 Its only disadvantage is that it is computationally intensive
 compared with the traditional techniques give in 6.1.3 above. This is
 not a major draw back if it is used for moderately infrequent
 purposes, such as generating session keys.
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 Simply choose two large prime numbers, say p and q, which both have
 the property that you get a remainder of 3 if you divide them by 4.
 Let n = p * q. Then you choose a random number x relatively prime to
 n. The initial seed for the generator and the method for calculating
 subsequent values are then
 2
 s = ( x )(Mod n)
 0
 2
 s = ( s )(Mod n)
 i+1 i
 You must be careful to use only a few bits from the bottom of each s.
 It is always safe to use only the lowest order bit. If you use no
 more than the
 log ( log ( s ) )
 2 2 i
 low order bits, then predicting any additional bits from a sequence
 generated in this manner is provable as hard as factoring n. As long
 as the initial x is secret, you can even make n public if you want.
 An interesting characteristic of this generator is that you can
 directly calculate any of the s values. In particular
 i
 ( ( 2 )(Mod (( p - 1 ) * ( q - 1 )) ) )
 s = ( s )(Mod n)
 i 0
 This means that in applications where many keys are generated in this
 fashion, it is not necessary to save them all. Each key can be
 effectively indexed and recovered from that small index and the
 initial s and n.
6.3 Entropy Pool Techniques
 Many modern pseudo-random number sources, such as those describe in
 Sections 7.1.2 and 7.1.3, utilize the technique of maintaining a
 "pool" of bits and providing operations for strongly mixing input
 with some randomness into the pool and extracting pseudo random bits
 from the pool. This is illustrated in the figure below.
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 +--------+ +------+ +---------+
 --->| Mix In |--->| POOL |--->| Extract |--->
 | Bits | | | | Bits |
 +--------+ +------+ +---------+
 ^ V
 | |
 +-----------+
 Bits to be feed into the pool can be any of the various hardware,
 environmental, or user input sources discussed above. It is also
 common to save the state of the pool on system shut down and restore
 it on re-starting, if stable storage is available.
 Care must be taken that enough entropy has been added to the pool to
 support particular output uses desired. See [RSA BULL1] for similar
 suggestions.
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7. Randomness Generation Examples and Standards
 Several public standards and widely deployed examples are now in
 place for the generation of keys or other cryptographically random
 quantities. Some, in section 7.1 below, include an entropy source.
 Others, described in section 7.2, provide the pseudo-random number
 strong sequence generator but assume the input of a random seed or
 input from a source of entropy.
7.1 Complete Randomness Generators
 Three standards are described below. The two older standards use
 DES, with its 64-bit block and key size limit, but any equally strong
 or stronger mixing function could be substituted [DES]. The third is
 a more modern and stronger standard based on SHA-1 [SHA*]. Lastly
 the widely deployed modern UNIX and Windows random number generators
 are described.
7.1.1 US DoD Recommendations for Password Generation
 The United States Department of Defense has specific recommendations
 for password generation [DoD]. They suggest using the US Data
 Encryption Standard [DES] in Output Feedback Mode [MODES] as follows:
 use an initialization vector determined from
 the system clock,
 system ID,
 user ID, and
 date and time;
 use a key determined from
 system interrupt registers,
 system status registers, and
 system counters; and,
 as plain text, use an external randomly generated 64 bit
 quantity such as the ASCII bytes for 8 characters typed in by a
 system administrator.
 The password can then be calculated from the 64 bit "cipher text"
 generated by DES in 64-bit Output Feedback Mode. As many bits as are
 needed can be taken from these 64 bits and expanded into a
 pronounceable word, phrase, or other format if a human being needs to
 remember the password.
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7.1.2 The /dev/random Device
 Several versions of the UNIX operating system provide a kernel-
 resident random number generator. In some cases, these generators
 make use of events captured by the Kernel during normal system
 operation.
 For example, on some versions of Linux, the generator consists of a
 random pool of 512 bytes represented as 128 words of 4-bytes each.
 When an event occurs, such as a disk drive interrupt, the time of the
 event is XORed into the pool and the pool is stirred via a primitive
 polynomial of degree 128. The pool itself is treated as a ring
 buffer, with new data being XORed (after stirring with the
 polynomial) across the entire pool.
 Each call that adds entropy to the pool estimates the amount of
 likely true entropy the input contains. The pool itself contains a
 accumulator that estimates the total over all entropy of the pool.
 Input events come from several sources as listed below.
 Unfortunately, for server machines without human operators, the first
 and third are not available and entropy may be added slowly in that
 case.
 1. Keyboard interrupts. The time of the interrupt as well as the scan
 code are added to the pool. This in effect adds entropy from the
 human operator by measuring inter-keystroke arrival times.
 2. Disk completion and other interrupts. A system being used by a
 person will likely have a hard to predict pattern of disk
 accesses. (But not all disk drivers support capturing this timing
 information with sufficient accuracy to be useful.)
 3. Mouse motion. The timing as well as mouse position is added in.
 When random bytes are required, the pool is hashed with SHA-1 [SHA*]
 to yield the returned bytes of randomness. If more bytes are required
 than the output of SHA-1 (20 bytes), then the hashed output is
 stirred back into the pool and a new hash performed to obtain the
 next 20 bytes. As bytes are removed from the pool, the estimate of
 entropy is similarly decremented.
 To ensure a reasonable random pool upon system startup, the standard
 startup and shutdown scripts save the pool to a disk file at shutdown
 and read this file at system startup.
 There are two user exported interfaces. /dev/random returns bytes
 from the pool, but blocks when the estimated entropy drops to zero.
 As entropy is added to the pool from events, more data becomes
 available via /dev/random. Random data obtained from such a
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 /dev/random device is suitable for key generation for long term keys,
 if enough random bits are in the pool or are added in a reasonable
 amount of time.
 /dev/urandom works like /dev/random, however it provides data even
 when the entropy estimate for the random pool drops to zero. This may
 be adequate for session keys or for other key generation tasks where
 blocking while waiting for more random bits is not acceptable. The
 risk of continuing to take data even when the pool's entropy
 estimate is small in that past output may be computable from current
 output provided an attacker can reverse SHA-1. Given that SHA-1 is
 designed to be non-invertible, this is a reasonable risk.
 To obtain random numbers under Linux, Solaris, or other UNIX systems
 equipped with code as described above, all an application needs to do
 is open either /dev/random or /dev/urandom and read the desired
 number of bytes.
 (The Linux Random device was written by Theodore Ts'o. It was based
 loosely on the random number generator in PGP 2.X and PGP 3.0 (aka
 PGP 5.0).)
7.1.3 Windows CryptGenRandom
 Microsoft's recommendation to users of the widely deployed Windows
 operating system is generally to use the CryptGenRandom pseudo-random
 number generation call with the CryptAPI cryptographic service
 provider. This takes a handle to a cryptographic service provider
 library, a pointer to a buffer by which the caller can provide
 entropy and into which the generated pseudo-randomness is returned,
 and an indication of how many octets of randomness are desired.
 The Windows CryptAPI cryptographic service provider stores a seed
 state variable with every user. When CryptGenRandom is called, this
 is combined with any randomness provided in the call and various
 system and user data such as the process ID, thread ID, system clock,
 system time, system counter, memory status, free disk clusters, and
 hashed user environment block. This data is all feed to SHA-1 and the
 output used to seed an RC4 key stream. That key stream is used to
 produce the pseudo-random data requested and to update the user's
 seed state variable.
 Users of Windows ".NET" will probably find it easier to use the
 RNGCryptoServiceProvider.GetBytes method interface.
 For further information, see [WSC].
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7.2 Generators Assuming a Source of Entropy
 The pseudo-random number generators described in the following three
 sections all assume that a seed value with sufficient entropy is
 provided to them. They then generate a strong sequence (see Section
 6.2) from that seed.
7.2.1 X9.82 Pseudo-Random Number Generation
 The ANSI X9F1 committee is in the final stages of creating a standard
 for random number generation covering both true randomness generators
 and pseudo-random number generators. It includes a number of pseudo-
 random number generators based on hash functions one of which will
 probably be based on HMAC SHA hash constructs [HMAC]. The draft
 version of this generated is as described below omitting a number of
 optional features [X9.82].
 In the description in the subsections below, the HMAC hash construct
 is simply referred to as HMAC but, of course, in an particular use, a
 particular standard SHA function must be selected. Generally
 speaking, if the strength of the pseudo-random values to be generated
 is to be N bits, the SHA function chosen must be one generating N or
 more bits of output and a source of at least N bits of input entropy
 will be required. The same hash function must be used throughout an
 instantiation of this generator.
7,2.1.1 Notation
 In the following sections the notation give below is used:
 hash_length is the output size of the underlying hash function in
 use.
 input_entropy is the input bit string that provides entropy to the
 generator.
 K is a bit string of size hash_length that is part of the state of
 the generator and is updated at least once each time random
 bits are generated.
 V is a bit string of size hash_length and is part of the state of
 the generator which is updated each time hash_length bits of
 output are generated.
 | represents concatenation
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7.1.2.2 Initializing the Generator
 Set V to all zero bytes except that the low order bit of each byte is
 set to one.
 Set K to all zero bytes.
 K = HMAC ( K, V | 0x00 | input_entropy )
 V = HMAC ( K, V )
 K = HMAC ( K, V | 0x01 | input_entropy )
 V = HMAC ( K, V )
 Note: all SHA algorithms produce an integral number of bytes of the
 length of K and V will be an integral number of bytes.
7.1.2.5 Generating Random Bits
 When output is called for simply set
 V = HMAC ( K, V )
 and use leading bits from V. If more bits are needed than the length
 of V, set "temp" to a null bit string and then repeatedly perform
 V = HMAC ( K, V )
 temp = temp | V
 stopping as soon a temp is equal to or longer than the number of
 random bits called for and use the called for number of leading bits
 from temp. The definition of the algorithm prohibits calling from
 more than 2**35 bits.
7.2.2 X9.17 Key Generation
 The American National Standards Institute has specified a method for
 generating a sequence of keys as follows [X9.17]:
 s is the initial 64 bit seed
 0
 g is the sequence of generated 64 bit key quantities
 n
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 k is a random key reserved for generating this key sequence
 t is the time at which a key is generated to as fine a resolution
 as is available (up to 64 bits).
 DES ( K, Q ) is the DES encryption of quantity Q with key K
 g = DES ( k, DES ( k, t ) .xor. s )
 n n
 s = DES ( k, DES ( k, t ) .xor. g )
 n+1 n
 If g sub n is to be used as a DES key, then every eighth bit should
 be adjusted for parity for that use but the entire 64 bit unmodified
 g should be used in calculating the next s.
7.2.3 DSS Pseudo-Random Number Generation
 Appendix 3 of the NIST Digital Signature Standard [DSS] provides a
 method of producing a sequence of pseudo-random 160 bit quantities
 for use as private keys or the like. This has been modified by Change
 Notice 1 [DSS CN1] to produce the following algorithm for generating
 general purpose pseudorandom numbers:
 t = 0x 67452301 EFCDAB89 98BADCFE 10325476 C3D2E1F0
 XKEY = initial seed
 0
 For j = 0 to ...
 XVAL = ( XKEY + optional user input ) (Mod 2^512)
 j
 X = G( t, XVAL )
 j
 XKEY = ( 1 + XKEY + X ) (Mod 2^512)
 j+1 j j
 The quantities X thus produced are the pseudo-random sequence of 160
 bit values. Two functions can be used for "G" above. Each produces
 a 160-bit value and takes two arguments, the first argument a 160-bit
 value and the second a 512 bit value.
 The first is based on SHA-1 and works by setting the 5 linking
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 variables, denoted H with subscripts in the SHA-1 specification, to
 the first argument divided into fifths. Then steps (a) through (e) of
 section 7 of the NIST SHA-1 specification are run over the second
 argument as if it were a 512-bit data block. The values of the
 linking variable after those steps are then concatenated to produce
 the output of G. [SHA*]
 As an alternative second method, NIST also defined an alternate G
 function based on multiple applications of the DES encryption
 function [DSS].
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8. Examples of Randomness Required
 Below are two examples showing rough calculations of needed
 randomness for security. The first is for moderate security passwords
 while the second assumes a need for a very high security
 cryptographic key.
 In addition [ORMAN] and [RSA BULL13] provide information on the
 public key lengths that should be used for exchanging symmetric keys.
8.1 Password Generation
 Assume that user passwords change once a year and it is desired that
 the probability that an adversary could guess the password for a
 particular account be less than one in a thousand. Further assume
 that sending a password to the system is the only way to try a
 password. Then the crucial question is how often an adversary can try
 possibilities. Assume that delays have been introduced into a system
 so that, at most, an adversary can make one password try every six
 seconds. That's 600 per hour or about 15,000 per day or about
 5,000,000 tries in a year. Assuming any sort of monitoring, it is
 unlikely someone could actually try continuously for a year. In fact,
 even if log files are only checked monthly, 500,000 tries is more
 plausible before the attack is noticed and steps taken to change
 passwords and make it harder to try more passwords.
 To have a one in a thousand chance of guessing the password in
 500,000 tries implies a universe of at least 500,000,000 passwords or
 about 2^29. Thus 29 bits of randomness are needed. This can probably
 be achieved using the US DoD recommended inputs for password
 generation as it has 8 inputs which probably average over 5 bits of
 randomness each (see section 7.1). Using a list of 1000 words, the
 password could be expressed as a three word phrase (1,000,000,000
 possibilities) or, using case insensitive letters and digits, six
 would suffice ((26+10)^6 = 2,176,782,336 possibilities).
 For a higher security password, the number of bits required goes up.
 To decrease the probability by 1,000 requires increasing the universe
 of passwords by the same factor which adds about 10 bits. Thus to
 have only a one in a million chance of a password being guessed under
 the above scenario would require 39 bits of randomness and a password
 that was a four word phrase from a 1000 word list or eight
 letters/digits. To go to a one in 10^9 chance, 49 bits of randomness
 are needed implying a five word phrase or ten letter/digit password.
 In a real system, of course, there are also other factors. For
 example, the larger and harder to remember passwords are, the more
 likely users are to write them down resulting in an additional risk
D. Eastlake, J. Schiller, S. Crocker [Page 40]
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 of compromise.
8.2 A Very High Security Cryptographic Key
 Assume that a very high security key is needed for symmetric
 encryption / decryption between two parties. Assume an adversary can
 observe communications and knows the algorithm being used. Within the
 field of random possibilities, the adversary can try key values in
 hopes of finding the one in use. Assume further that brute force
 trial of keys is the best the adversary can do.
8.2.1 Effort per Key Trial
 How much effort will it take to try each key? For very high security
 applications it is best to assume a low value of effort. Even if it
 would clearly take tens of thousands of computer cycles or more to
 try a single key, there may be some pattern that enables huge blocks
 of key values to be tested with much less effort per key. Thus it is
 probably best to assume no more than a couple hundred cycles per key.
 (There is no clear lower bound on this as computers operate in
 parallel on a number of bits and a poor encryption algorithm could
 allow many keys or even groups of keys to be tested in parallel.
 However, we need to assume some value and can hope that a reasonably
 strong algorithm has been chosen for our hypothetical high security
 task.)
 If the adversary can command a highly parallel processor or a large
 network of work stations, 10^11 cycles per second is probably a
 minimum assumption for availability today. Looking forward a few
 years, there should be at least an order of magnitude improvement.
 Thus assuming 10^10 keys could be checked per second or 3.6*10^12 per
 hour or 6*10^14 per week or 2.4*10^15 per month is reasonable. This
 implies a need for a minimum of 63 bits of randomness in keys to be
 sure they cannot be found in a month. Even then it is possible that,
 a few years from now, a highly determined and resourceful adversary
 could break the key in 2 weeks (on average they need try only half
 the keys).
 These questions are considered in detail in "Minimal Key Lengths for
 Symmetric Ciphers to Provide Adequate Commercial Security: A Report
 by an Ad Hoc Group of Cryptographers and Computer Scientists"
 [KeyStudy] which was sponsored by the Business Software Alliance. It
 concluded that a reasonable key length in 1995 for very high security
 is in the range of 75 to 90 bits and, since the cost of cryptography
 does not vary much with they key size, recommends 90 bits. To update
 these recommendations, just add 2/3 of a bit per year for Moore's
D. Eastlake, J. Schiller, S. Crocker [Page 41]
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 law [MOORE]. Thus, in the year 2004, this translates to a
 determination that a reasonable key length is in the 81 to 96 bit
 range. In fact, today, it is increasingly common to use keys longer
 than 96 bits, such as 128-bit (or longer) keys with AES and keys with
 effective lengths of 112-bits using triple-DES.
8.2.2 Meet in the Middle Attacks
 If chosen or known plain text and the resulting encrypted text are
 available, a "meet in the middle" attack is possible if the structure
 of the encryption algorithm allows it. (In a known plain text attack,
 the adversary knows all or part of the messages being encrypted,
 possibly some standard header or trailer fields. In a chosen plain
 text attack, the adversary can force some chosen plain text to be
 encrypted, possibly by "leaking" an exciting text that would then be
 sent by the adversary over an encrypted channel.)
 An oversimplified explanation of the meet in the middle attack is as
 follows: the adversary can half-encrypt the known or chosen plain
 text with all possible first half-keys, sort the output, then half-
 decrypt the encoded text with all the second half-keys. If a match is
 found, the full key can be assembled from the halves and used to
 decrypt other parts of the message or other messages. At its best,
 this type of attack can halve the exponent of the work required by
 the adversary while adding a very large but roughly constant factor
 of effort. Thus, if this attack can be mounted, a doubling of the
 amount of randomness in the very strong key to a minimum of 192 bits
 (96*2) is required for the year 2004 based on the [KeyStudy]
 analysis.
 This amount of randomness is well beyond the limit of that in the
 inputs recommended by the US DoD for password generation and could
 require user typing timing, hardware random number generation, or
 other sources.
 The meet in the middle attack assumes that the cryptographic
 algorithm can be decomposed in this way. Hopefully no modern
 algorithm has this weakness but there may be cases where we are not
 sure of that or even of what algorithm a key will be used with. Even
 if a basic algorithm is not subject to a meet in the middle attack,
 an attempt to produce a stronger algorithm by applying the basic
 algorithm twice (or two different algorithms sequentially) with
 different keys will gain less added security than would be expected.
 Such a composite algorithm would be subject to a meet in the middle
 attack.
 Enormous resources may be required to mount a meet in the middle
 attack but they are probably within the range of the national
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 security services of a major nation. Essentially all nations spy on
 other nations traffic.
8.2.3 Other Considerations
 [KeyStudy] also considers the possibilities of special purpose code
 breaking hardware and having an adequate safety margin.
 It should be noted that key length calculations such at those above
 are controversial and depend on various assumptions about the
 cryptographic algorithms in use. In some cases, a professional with a
 deep knowledge of code breaking techniques and of the strength of the
 algorithm in use could be satisfied with less than half of the 192
 bit key size derived above.
 For further examples of conservative design principles see
 [FERGUSON].
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9. Conclusion
 Generation of unguessable "random" secret quantities for security use
 is an essential but difficult task.
 Hardware techniques to produce the needed entropy would be relatively
 simple. In particular, the volume and quality would not need to be
 high and existing computer hardware, such as audio input or disk
 drives, can be used.
 Widely available computational techniques are available to process
 low quality random quantities from multiple sources or a larger
 quantity of such low quality input from one source and produce a
 smaller quantity of higher quality keying material. In the absence of
 hardware sources of randomness, a variety of user and software
 sources can frequently, with care, be used instead; however, most
 modern systems already have hardware, such as disk drives or audio
 input, that could be used to produce high quality randomness.
 Once a sufficient quantity of high quality seed key material (a
 couple of hundred bits) is available, computational techniques are
 available to produce cryptographically strong sequences of
 computationally unpredictable quantities from this seed material.
D. Eastlake, J. Schiller, S. Crocker [Page 44]
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10. Security Considerations
 The entirety of this document concerns techniques and recommendations
 for generating unguessable "random" quantities for use as passwords,
 cryptographic keys, initialization vectors, sequence numbers, and
 similar security uses.
11. Copyright and Disclaimer
 Copyright (C) The Internet Society 2005. This document is subject to
 the rights, licenses and restrictions contained in BCP 78 and except
 as set forth therein, the authors retain all their rights.
 This document and the information contained herein are provided on an
 "AS IS" basis and THE CONTRIBUTOR, THE ORGANIZATION HE/SHE REPRESENTS
 OR IS SPONSORED BY (IF ANY), THE INTERNET SOCIETY AND THE INTERNET
 ENGINEERING TASK FORCE DISCLAIM ALL WARRANTIES, EXPRESS OR IMPLIED,
 INCLUDING BUT NOT LIMITED TO ANY WARRANTY THAT THE USE OF THE
 INFORMATION HEREIN WILL NOT INFRINGE ANY RIGHTS OR ANY IMPLIED
 WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE.
D. Eastlake, J. Schiller, S. Crocker [Page 45]
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12. Appendix A: Changes from RFC 1750
 1. Additional acknowledgements have been added.
 2. Insertion of section 5.3 on mixing with S-boxes.
 3. Addition of section 3.3 on Ring Oscillator randomness sources.
 4. AES and the members of the SHA series producing more than 160
 bits have been added. Use of AES has been emphasized and the use
 of DES de-emphasized.
 5. Addition of section 6.3 on entropy pool techniques.
 6. Addition of section 7.2.3 on the pseudo-random number generation
 techniques given in FIPS 186-2 (with Change Notice 1), 7.2.1 on
 those given in X9.82, section 7.1.2 on the random number
 generation techniques of the /dev/random device in Linux and
 other UNIX systems, and section 7.1.3 on random number generation
 techniques in the Windows operating system.
 7. Addition of references to the "Minimal Key Lengths for Symmetric
 Ciphers to Provide Adequate Commercial Security" study published
 in January 1996 [KeyStudy] and to [RFC 1948].
 8. Added caveats to using Diffie-Hellman as a mixing function and,
 because of those caveats and its computationally intensive
 nature, recommend against its use.
 9. Addition of references to the X9.82 effort and the [TURBID] and
 [NASLUND] papers.
 10. Addition of discussion of min-entropy and Renyi entropy and
 references to the [LUBY] book.
 11. Major restructuring, minor wording changes, and a variety of
 reference updates.
D. Eastlake, J. Schiller, S. Crocker [Page 46]
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13. Informative References
 [AES] - "Specification of the Advanced Encryption Standard (AES)",
 United States of America, US National Institute of Standards and
 Technology, FIPS 197, November 2001.
 [ASYMMETRIC] - "Secure Communications and Asymmetric Cryptosystems",
 edited by Gustavus J. Simmons, AAAS Selected Symposium 69, Westview
 Press, Inc.
 [BBS] - "A Simple Unpredictable Pseudo-Random Number Generator", SIAM
 Journal on Computing, v. 15, n. 2, 1986, L. Blum, M. Blum, & M. Shub.
 [BRILLINGER] - "Time Series: Data Analysis and Theory", Holden-Day,
 1981, David Brillinger.
 [CRC] - "C.R.C. Standard Mathematical Tables", Chemical Rubber
 Publishing Company.
 [DAVIS] - "Cryptographic Randomness from Air Turbulence in Disk
 Drives", Advances in Cryptology - Crypto '94, Springer-Verlag
 Lecture Notes in Computer Science #839, 1984, Don Davis, Ross Ihaka,
 and Philip Fenstermacher.
 [DES] - "Data Encryption Standard", US National Institute of
 Standards and Technology, FIPS 46-3, October 1999.
 - "Data Encryption Algorithm", American National Standards
 Institute, ANSI X3.92-1981.
 (See also FIPS 112, Password Usage, which includes FORTRAN
 code for performing DES.)
 [D-H] - RFC 2631, "Diffie-Hellman Key Agreement Method", Eric
 Rescrola, June 1999.
 [DNSSEC] - RFC 2535, "Domain Name System Security Extensions", D.
 Eastlake, March 1999.
 [DoD] - "Password Management Guideline", United States of America,
 Department of Defense, Computer Security Center, CSC-STD-002-85.
 (See also FIPS 112, Password Usage, which incorporates CSC-STD-002-85
 as one of its appendices.)
 [DSS] - "Digital Signature Standard (DSS)", US National Institute of
 Standards and Technology, FIPS 186-2, January 2000.
 [DSS CN1] - "Digital Signature Standard Change Notice 1", US National
 Institute of Standards and Technology, FIPS 186-2 Change Notice 1, 5
 October 2001.
 [FERGUSON] - "Practical Cryptography", Niels Ferguson and Bruce
D. Eastlake, J. Schiller, S. Crocker [Page 47]
INTERNET DRAFT Randomness Requirements for Security January 2005
 Schneier, Wiley Publishing Inc., ISBN 047122894X, April 2003.
 [GIFFORD] - "Natural Random Number", MIT/LCS/TM-371, David K.
 Gifford, September 1988.
 [IEEE 802.11i] - "Amendment to Standard for Telecommunications and
 Information Exchange Between Systems - LAN/MAN Specific Requirements
 - Part 11: Wireless Medium Access Control (MAC) and physical layer
 (PHY) specifications: Medium Access Control (MAC) Security
 Enhancements", The Institute for Electrical and Electronics
 Engineers, January 2004.
 [IPSEC] - RFC 2401, "Security Architecture for the Internet
 Protocol", S. Kent, R. Atkinson, November 1998.
 [Jakobsson] - M. Jakobsson, E. Shriver, B. K. Hillyer, and A. Juels,
 "A practical secure random bit generator", Proceedings of the Fifth
 ACM Conference on Computer and Communications Security, 1998. See
 also http://citeseer.ist.psu.edu/article/jakobsson98practical.html.
 [KAUFMAN] - "Network Security: Private Communication in a Public
 World", Charlie Kaufman, Radia Perlman, and Mike Speciner, Prentis
 Hall PTR, ISBN 0-13-046019-2, 2nd Edition 2002.
 [KeyStudy] - "Minimal Key Lengths for Symmetric Ciphers to Provide
 Adequate Commercial Security: A Report by an Ad Hoc Group of
 Cryptographers and Computer Scientists", M. Blaze, W. Diffie, R.
 Rivest, B. Schneier, T. Shimomura, E. Thompson, and M. Weiner,
 January 1996, <www.counterpane.com/keylength.html>.
 [KNUTH] - "The Art of Computer Programming", Volume 2: Seminumerical
 Algorithms, Chapter 3: Random Numbers, Donald E. Knuth, Addison
 Wesley Publishing Company, 3rd Edition November 1997.
 [KRAWCZYK] - "How to Predict Congruential Generators", H. Krawczyk,
 Journal of Algorithms, V. 13, N. 4, December 1992.
 [LUBY] - "Pseudorandomness and Cryptographic Applications", Michael
 Luby, Princeton University Press, ISBN 0691025460, 8 January 1996.
 [MAIL PEM 1] - RFC 1421, "Privacy Enhancement for Internet Electronic
 Mail: Part I: Message Encryption and Authentication Procedures", J.
 Linn, 02/10/1993.
 [MAIL PEM 2] - RFC 1422, "Privacy Enhancement for Internet
 Electronic Mail: Part II: Certificate-Based Key Management", S. Kent,
 02/10/1993.
 [MAIL PEM 3] - RFC 1423, "Privacy Enhancement for Internet
 Electronic Mail: Part III: Algorithms, Modes, and Identifiers", D.
 Balenson, 02/10/1993.
 [MAIL PEM 4] - RFC 1424, "Privacy Enhancement for Internet
D. Eastlake, J. Schiller, S. Crocker [Page 48]
INTERNET DRAFT Randomness Requirements for Security January 2005
 Electronic Mail: Part IV: Key Certification and Related Services", B.
 Kaliski, 02/10/1993.
 [MAIL PGP]
 - RFC 2440, "OpenPGP Message Format", J. Callas, L.
 Donnerhacke, H. Finney, R. Thayer, November 1998.
 - RFC 3156, "MIME Security with OpenPGP" M. Elkins, D. Del
 Torto, R. Levien, T. Roessler, August 2001.
 [MAIL S/MIME] - RFCs 2632 through 2634:
 - RFC 2632, "S/MIME Version 3 Certificate Handling", B.
 Ramsdell, Ed., June 1999.
 - RFC 2633, "S/MIME Version 3 Message Specification", B.
 Ramsdell, Ed., June 1999.
 - RFC 2634, "Enhanced Security Services for S/MIME" P.
 Hoffman, Ed., June 1999.
 [MD4] - "The MD4 Message-Digest Algorithm", RFC1320, April 1992, R.
 Rivest
 [MD5] - "The MD5 Message-Digest Algorithm", RFC1321, April 1992, R.
 Rivest
 [MODES] - "DES Modes of Operation", US National Institute of
 Standards and Technology, FIPS 81, December 1980.
 - "Data Encryption Algorithm - Modes of Operation", American
 National Standards Institute, ANSI X3.106-1983.
 [MOORE] - Moore's Law: the exponential increase in the logic density
 of silicon circuits. Originally formulated by Gordon Moore in 1964 as
 a doubling every year starting in 1962, in the late 1970s the rate
 fell to a doubling every 18 months and has remained there through the
 date of this document. See "The New Hacker's Dictionary", Third
 Edition, MIT Press, ISBN 0-262-18178-9, Eric S. Raymond, 1996.
 [NASLUND] - "Extraction of Optimally Unbiased Bits from a Biased
 Source", M. Naslund and A. Russell, IEEE Transactions on Information
 Theory. 46(3), May 2000.
 <http://www.engr.uconn.edu/~acr/Papers/biasIEEEjour.ps>
 [ORMAN] - "Determining Strengths For Public Keys Used For Exchanging
 Symmetric Keys", RFC 3766, Hilarie Orman, Paul Hoffman, April 2004.
 [RFC 1750] - "Randomness Requirements for Security", D. Eastlake, S.
 Crocker, J. Schiller, December 1994.
 [RFC 1948] - "Defending Against Sequence Number Attacks", S.
 Bellovin, May 1986.
 [RSA BULL1] - "Suggestions for Random Number Generation in Software",
D. Eastlake, J. Schiller, S. Crocker [Page 49]
INTERNET DRAFT Randomness Requirements for Security January 2005
 RSA Laboratories Bulletin #1, January 1996.
 [RSA BULL13] - "A Cost-Based Security Analysis of Symmetric and
 Asymmetric Key Lengths", RSA Laboratories Bulletin #13, Robert
 Silverman, April 2000 (revised November 2001).
 [SBOX1] - "Practical s-box design", S. Mister, C. Adams, Selected
 Areas in Cryptography, 1996.
 [SBOX2] - "Perfect Non-linear S-boxes", K. Nyberg, Advances in
 Cryptography - Eurocrypt '91 Proceedings, Springer-Verland, 1991.
 [SCHNEIER] - "Applied Cryptography: Protocols, Algorithms, and Source
 Code in C", Bruce Schneier, 2nd Edition, John Wiley & Sons, 1996.
 [SHANNON] - "The Mathematical Theory of Communication", University of
 Illinois Press, 1963, Claude E. Shannon. (originally from: Bell
 System Technical Journal, July and October 1948)
 [SHIFT1] - "Shift Register Sequences", Solomon W. Golomb, Aegean Park
 Press, Revised Edition 1982.
 [SHIFT2] - "Cryptanalysis of Shift-Register Generated Stream Cypher
 Systems", Wayne G. Barker, Aegean Park Press, 1984.
 [SHA] - "Secure Hash Standard", US National Institute of Science and
 Technology, FIPS 180-2, 1 August 2002.
 [SHA RFC] - RFC 3174, "US Secure Hash Algorithm 1 (SHA1)", D.
 Eastlake, P. Jones, September 2001.
 [SSH] - draft-ietf-secsh-*, work in progress.
 [STERN] - "Secret Linear Congruential Generators are not
 Cryptographically Secure", J. Stern, Proceedings of IEEE STOC, 1987.
 [TLS] - RFC 2246, "The TLS Protocol Version 1.0", T. Dierks, C.
 Allen, January 1999.
 [TURBID] - "High Entropy Symbol Generator", John S. Denker,
 <http://www.av8n.com/turbid/paper/turbid.htm>, 2003.
 [USENET] - RFC 977, "Network News Transfer Protocol", B. Kantor, P.
 Lapsley, February 1986.
 - RFC 2980, "Common NNTP Extensions", S. Barber, October
 2000.
 [VON NEUMANN] - "Various techniques used in connection with random
 digits", von Neumann's Collected Works, Vol. 5, Pergamon Press,
 1963, J. von Neumann.
D. Eastlake, J. Schiller, S. Crocker [Page 50]
INTERNET DRAFT Randomness Requirements for Security January 2005
 [WSC] - "Writing Secure Code, Second Edition", Michael Howard, David.
 C. LeBlanc, Microsoft Press, ISBN 0735617228, December 2002.
 [X9.17] - "American National Standard for Financial Institution Key
 Management (Wholesale)", American Bankers Association, 1985.
 [X9.82] - "Random Number Generation", American National Standards
 Institute, ANSI X9F1, work in progress.
 Part 1 - Overview and General Principles.
 Part 2 - Non-Deterministic Random Bit Generators
 Part 3 - Deterministic Random Bit Generators
D. Eastlake, J. Schiller, S. Crocker [Page 51]
INTERNET DRAFT Randomness Requirements for Security January 2005
Author's Addresses
 Donald E. Eastlake 3rd
 Motorola Laboratories
 155 Beaver Street
 Milford, MA 01757 USA
 Telephone: +1 508-786-7554 (w)
 +1 508-634-2066 (h)
 EMail: Donald.Eastlake@motorola.com
 Jeffrey I. Schiller
 MIT, Room E40-311
 77 Massachusetts Avenue
 Cambridge, MA 02139-4307 USA
 Telephone: +1 617-253-0161
 E-mail: jis@mit.edu
 Steve Crocker
 EMail: steve@stevecrocker.com
File Name and Expiration
 This is file draft-eastlake-randomness2-10.txt.
 It expires July 2005.
D. Eastlake, J. Schiller, S. Crocker [Page 52]