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Versions: 00 01 02 03 04 05 06 07 08 09 10 RFC 4086

Network Working Group                            Donald E. Eastlake, 3rd
OBSOLETES RFC 1750                                   Jeffrey I. Schiller
                                                           Steve Crocker
Expires April 2005                                          October 2004

                  Randomness Requirements for Security
                  ---------- ------------ --- --------

Status of This Document

   By submitting this Internet-Draft, I certify that any applicable
   patent or other IPR claims of which I am aware have been disclosed,
   or will be disclosed, and any of which I become aware will be
   disclosed, in accordance with RFC 3668.

   This document is intended to become a Best Current Practice.
   Comments should be sent to the authors. Distribution is unlimited.

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   Copyright (C) The Internet Society 2004. All Rights Reserved.


   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

D. Eastlake, J. Schiller, S. Crocker                            [Page 1]

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   adversary is surprisingly difficult. This document points out many
   pitfalls in using traditional pseudo-random number generation
   techniques for choosing 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.


   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:

        Daniel Brown, Don Davis, Peter Gutmann, Tony Hansen, Sandy
        Harris, Paul Hoffman, Scott Hollenback, Russ Housley, Christian
        Huitema, John Kelsey, 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.

D. Eastlake, J. Schiller, S. Crocker                            [Page 2]

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Table of Contents

      Status of This Document....................................1

      Table of Contents..........................................3

      1. Introduction............................................5

      2. General Requirements....................................6

      3. Traditional Pseudo-Random Sequences.....................9

      4. Unpredictability.......................................11
      4.1 Problems with Clocks and Serial Numbers...............11
      4.2 Timing and Value of External Events...................12
      4.3 The Fallacy of Complex Manipulation...................12
      4.4 The Fallacy of Selection from a Large Database........13

      5. Hardware for Randomness................................15
      5.1 Volume Required.......................................15
      5.2 Sensitivity to Skew...................................15
      5.2.1 Using Stream Parity to De-Skew......................16
      5.2.2 Using Transition Mappings to De-Skew................17
      5.2.3 Using FFT to De-Skew................................18
      5.2.4 Using Compression to De-Skew........................18
      5.3 Existing Hardware Can Be Used For Randomness..........19
      5.3.1 Using Existing Sound/Video Input....................19
      5.3.2 Using Existing Disk Drives..........................19
      5.4 Ring Oscillator Sources...............................20

      6. Recommended Software Strategy..........................22
      6.1 Mixing Functions......................................22
      6.1.1 A Trivial Mixing Function...........................22
      6.1.2 Stronger Mixing Functions...........................23
      6.1.3 Using S-Boxes for Mixing............................25
      6.1.4 Diffie-Hellman as a Mixing Function.................25
      6.1.5 Using a Mixing Function to Stretch Random Bits......25
      6.1.6 Other Factors in Choosing a Mixing Function.........26
      6.2 Non-Hardware Sources of Randomness....................27
      6.3 Cryptographically Strong Sequences....................28
      6.3.1 OFB and CTR Sequences...............................28
      6.3.2 The Blum Blum Shub Sequence Generator...............29
      6.3.3 Entropy Pool Techniques.............................30

      7. Key Generation Examples and Standards..................32
      7.1 US DoD Recommendations for Password Generation........32
      7.2 X9.17 Key Generation..................................32
      7.3 DSS Pseudo-Random Number Generation...................33

D. Eastlake, J. Schiller, S. Crocker                            [Page 3]

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      7.4 X9.82 Pseudo-Random Number Generation.................34
      7.5 The /dev/random Device................................34
      7.6 Windows CryptGenRandom................................36

      8. Examples of Randomness Required........................37
      8.1  Password Generation..................................37
      8.2 A Very High Security Cryptographic Key................38
      8.2.1 Effort per Key Trial................................38
      8.2.2 Meet in the Middle Attacks..........................39
      8.2.3 Other Considerations................................40

      9. Conclusion.............................................41

      10. Security Considerations...............................42
      11. Copyright and Disclaimer..............................42

      12. Appendix A: Changes from RFC 1750.....................43

      14. Informative References................................44

      Author's Addresses........................................48
      File Name and Expiration..................................48

D. Eastlake, J. Schiller, S. Crocker                            [Page 4]

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1. Introduction

   Software cryptography is coming into wider use and is continuing to
   spread, although there is a long way to go until it becomes

   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

D. Eastlake, J. Schiller, S. Crocker                            [Page 5]

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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.

   Generally speaking, the above two 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
   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]

   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

D. Eastlake, J. Schiller, S. Crocker                            [Page 6]

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   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 the 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
   also useful to look at other measures, such as min-entropy, defined

D. Eastlake, J. Schiller, S. Crocker                            [Page 7]

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        Min-entropy =  - log  ( maximum ( p  ) )

   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]

D. Eastlake, J. Schiller, S. Crocker                            [Page 8]

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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 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

   [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. 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

D. Eastlake, J. Schiller, S. Crocker                            [Page 9]

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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.

D. Eastlake, J. Schiller, S. Crocker                           [Page 10]

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4. Unpredictability

   Statistically tested randomness in the traditional sense described in
   section 3 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] Yet the statistical properties
   of these tables are good. So you should keep in mind that passing
   statistical tests doesn't tell you that something is unpredictable.

   The following sections describe the limitations of some randomness
   generation techniques and sources. Much better sources are described
   in Section 5.

4.1 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 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
   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.

D. Eastlake, J. Schiller, S. Crocker                           [Page 11]

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   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

4.2 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.

4.3 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

D. Eastlake, J. Schiller, S. Crocker                           [Page 12]

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   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 unpredictability 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
   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 randomness in a good seed!

4.4 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

D. Eastlake, J. Schiller, S. Crocker                           [Page 13]

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   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.

D. Eastlake, J. Schiller, S. Crocker                           [Page 14]

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5. Hardware for Randomness

   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
   5.4). 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.

5.1 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 6.3. 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.

5.2 Sensitivity to Skew

   Is there any specific requirement on the shape of the distribution of
   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 the
   bit stream are given below and stronger cryptographic techniques are

D. Eastlake, J. Schiller, S. Crocker                           [Page 15]

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   described in Section 6.1.2 below.

5.2.1 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 )  )
                         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

        1/2 * [1 + (2e) ]
        1/2 * [1 - (2e) ].

   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

        ( 0.5 + ( 0.5 * (2e)  ) )  <  0.5 + d.

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   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

5.2.2 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:

            | 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

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   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.

5.2.3 Using FFT to De-Skew

   When real world data consists of strongly biased or correlated bits,
   it may still contain useful amounts of randomness. This randomness
   can be extracted through use of various transforms, the most powerful
   of which are described in section 6.1.2 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].

5.2.4 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.

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   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 6.1.2. At a minimum, the
   beginning of the compressed sequence should be skipped and only later
   bits used for applications requiring roughly random bits.

5.3 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.

5.3.1 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

   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 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

5.3.2 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

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   measurements can be obtained that include this randomness. Such data
   is usually highly correlated so that significant processing is
   needed, such as described in 6.1.2 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
   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.

5.4 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 (Section 5.3.2). 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

   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--------------+

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6. Recommended Software Strategy

   What is the best overall strategy for meeting the requirement for
   unguessable random numbers in the absence of a reliable hardware
   source?  It is to obtain random input from a number of uncorrelated
   sources and to mix them with a strong mixing function. Such a
   function will preserve the randomness present in any of the sources
   even if other quantities being combined happen to be fixed or easily
   guessable. 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

6.1 Mixing Functions

   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 5.2
   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 5.2.1 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 6.1.2.

6.1.1 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.

6.1.2 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 5.2.1 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 entropy.

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6.1.3 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

   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*]

6.1.4 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 that randomness, 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, if an adversary can observe either of the public keys and
   knows the modulus being used, they 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.

6.1.5 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

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   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 6.1.1 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.

6.1.6 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. No concealed or special weakness
   has been 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.

   Where input lengths are unpredictable, hash algorithms are a little
   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

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   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 basic patents of which
   the authors are unaware or patents on implementations or uses or
   other relevant patents issued or to be issued.

6.2 Non-Hardware Sources of Randomness

   The best source of input for mixing 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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6.3 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

   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
   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

6.3.1 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 encrypted value 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

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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.

6.3.2 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 6.3.1 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

             s    =  ( x  )(Mod n)

             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

               ( ( 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.

6.3.3 Entropy Pool Techniques

   Many modern pseudo-random number sources 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 Section 7.5 for more
   details on an example implementation and [RSA BULL1] for similar

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7. Key Generation Examples and Standards

   Several public standards and widely deployed examples are now in
   place for the generation of keys without special hardware.  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 random number generators are described.

7.1 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 8 characters typed in by a system

   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.

7.2 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

        g  is the sequence of generated 64 bit key quantities

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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.

7.3 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

        For j = 0 to ...

             XVAL = ( XKEY  + optional user input ) (Mod 2^512)

             X  = G( t, XVAL )

             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].

7.4 X9.82 Pseudo-Random Number Generation

   The National Institute for Standards and Technology (NIST) and the
   American National Standards Institutes (ANSI) X9F1 committee are 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
   for use with AES and other block ciphers.  It also includes random
   number generators based on hash functions and the arithmetic of
   elliptic curves [X9.82].

7.5 The /dev/random Device

   Several versions of the UNIX operating system provides a kernel-
   resident random number generator. In some cases, these generators
   makes use of events captured by the Kernel during normal system

   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 xor'ed 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 very slowly in
   that case.

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   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 scripts (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
   /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).)

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7.6 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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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

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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

   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 law

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   [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]

   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

   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

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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 such randomness 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.

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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 2004.  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

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12. Appendix A: Changes from RFC 1750

    1. Additional acknowledgements have been added.

    2. Insertion of section 5.2.4 on de-skewing with S-boxes.

    3. Addition of section 5.4 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 6.3.3 on entropy pool techniques.

    6. Addition of section 7.3 on the pseudo-random number generation
       techniques given in FIPS 186-2 (with Change Notice 1), 7.4 on
       those given in X9.82, section 7.5 on the random number generation
       techniques of the /dev/random device in Linux and other UNIX
       systems, and section 7.6 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].

    8. Added caveats to using Diffie-Hellman as a mixing function.

    9. Addition of references to the [TURBID] paper and system.

   10. Addition of discussion of min-entropy and Renyi entropy and
       references to the [LUBY] book.

   11. Minor wording changes and reference updates.

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14. 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

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   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,
           [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

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   Electronic Mail: Part IV: Key Certification and Related Services", B.
   Kaliski, 02/10/1993.

           - 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.

   [MD5] - "The MD5 Message-Digest Algorithm", RFC1321, April 1992, R.

   [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.

   [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.

   [RSA BULL1] - "Suggestions for Random Number Generation in Software",
   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.

D. Eastlake, J. Schiller, S. Crocker                           [Page 46]

INTERNET DRAFT    Randomness Requirements for Security      October 2004

   [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

   [VON NEUMANN] - "Various techniques used in connection with random
   digits", von Neumann's Collected Works, Vol. 5, Pergamon Press, 1963,
   J. von Neumann.

   [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.

D. Eastlake, J. Schiller, S. Crocker                           [Page 47]

INTERNET DRAFT    Randomness Requirements for Security      October 2004

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-randomness2-09.txt.

   It expires April 2005.

D. Eastlake, J. Schiller, S. Crocker                           [Page 48]

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