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rfc:rfc1750

Network Working Group D. Eastlake, 3rd Request for Comments: 1750 DEC Category: Informational S. Crocker

                                                             Cybercash
                                                           J. Schiller
                                                                   MIT
                                                         December 1994
              Randomness Recommendations for Security

Status of this Memo

 This memo provides information for the Internet community.  This memo
 does not specify an Internet standard of any kind.  Distribution of
 this memo is unlimited.

Abstract

 Security systems today are built on increasingly 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
 number space.
 Choosing random quantities to foil a resourceful and motivated
 adversary is surprisingly difficult.  This paper 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 particular applications.

Eastlake, Crocker & Schiller [Page 1] RFC 1750 Randomness Recommendations for Security December 1994

Acknowledgements

 Comments on this document that have been incorporated were received
 from (in alphabetic order) the following:
      David M. Balenson (TIS)
      Don Coppersmith (IBM)
      Don T. Davis (consultant)
      Carl Ellison (Stratus)
      Marc Horowitz (MIT)
      Christian Huitema (INRIA)
      Charlie Kaufman (IRIS)
      Steve Kent (BBN)
      Hal Murray (DEC)
      Neil Haller (Bellcore)
      Richard Pitkin (DEC)
      Tim Redmond (TIS)
      Doug Tygar (CMU)

Table of Contents

 1. Introduction........................................... 3
 2. Requirements........................................... 4
 3. Traditional Pseudo-Random Sequences.................... 5
 4. Unpredictability....................................... 7
 4.1 Problems with Clocks and Serial Numbers............... 7
 4.2 Timing and Content of External Events................  8
 4.3 The Fallacy of Complex Manipulation..................  8
 4.4 The Fallacy of Selection from a Large Database.......  9
 5. Hardware for Randomness............................... 10
 5.1 Volume Required...................................... 10
 5.2 Sensitivity to Skew.................................. 10
 5.2.1 Using Stream Parity to De-Skew..................... 11
 5.2.2 Using Transition Mappings to De-Skew............... 12
 5.2.3 Using FFT to De-Skew............................... 13
 5.2.4 Using Compression to De-Skew....................... 13
 5.3 Existing Hardware Can Be Used For Randomness......... 14
 5.3.1 Using Existing Sound/Video Input................... 14
 5.3.2 Using Existing Disk Drives......................... 14
 6. Recommended Non-Hardware Strategy..................... 14
 6.1 Mixing Functions..................................... 15
 6.1.1 A Trivial Mixing Function.......................... 15
 6.1.2 Stronger Mixing Functions.......................... 16
 6.1.3 Diff-Hellman as a Mixing Function.................. 17
 6.1.4 Using a Mixing Function to Stretch Random Bits..... 17
 6.1.5 Other Factors in Choosing a Mixing Function........ 18
 6.2 Non-Hardware Sources of Randomness................... 19
 6.3 Cryptographically Strong Sequences................... 19

Eastlake, Crocker & Schiller [Page 2] RFC 1750 Randomness Recommendations for Security December 1994

 6.3.1 Traditional Strong Sequences....................... 20
 6.3.2 The Blum Blum Shub Sequence Generator.............. 21
 7. Key Generation Standards.............................. 22
 7.1 US DoD Recommendations for Password Generation....... 23
 7.2 X9.17 Key Generation................................. 23
 8. Examples of Randomness Required....................... 24
 8.1  Password Generation................................. 24
 8.2 A Very High Security Cryptographic Key............... 25
 8.2.1 Effort per Key Trial............................... 25
 8.2.2 Meet in the Middle Attacks......................... 26
 8.2.3 Other Considerations............................... 26
 9. Conclusion............................................ 27
 10. Security Considerations.............................. 27
 References............................................... 28
 Authors' Addresses....................................... 30

1. Introduction

 Software cryptography is coming into wider use.  Systems like
 Kerberos, PEM, PGP, etc. are maturing and becoming a part of the
 network landscape [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.
 For the present, the lack of generally available facilities for
 generating such unpredictable numbers is 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, the only safe strategy so far has been to force
 the local installation to supply a suitable routine to generate
 random numbers.  To say the least, this is an awkward, error-prone
 and unpalatable solution.
 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 will 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.  Frequently such random
 quantities are determinable by an adversary searching through an
 embarrassingly small space of possibilities.
 This informational document suggests 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

Eastlake, Crocker & Schiller [Page 3] RFC 1750 Randomness Recommendations for Security December 1994

 applications.

2. Requirements

 Probably the most 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.
 In some cases, such as the use of symmetric encryption with the one
 time pads [CRYPTO*] or the US Data Encryption Standard [DES], 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 [ASYMMETRIC, CRYPTO*].
 The frequency and volume of the requirement for random quantities
 differs greatly for different cryptographic systems.  Using pure RSA
 [CRYPTO*], random quantities are required when the key pair is
 generated, but thereafter any number of messages can be signed
 without any further need for randomness.  The public key Digital
 Signature Algorithm that has been proposed by the US National
 Institute of Standards and Technology (NIST) requires good random
 numbers for each signature.  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.
 In most of these cases, an adversary can try to determine the
 "secret" key by trial and error.  (This is possible as long as the
 key is enough smaller than the message that the correct key can be
 uniquely identified.)  The probability of an adversary succeeding at
 this must be made acceptably low, depending on the particular
 application.  The size of the space the adversary must search is
 related to the amount of key "information" present in the information
 theoretic sense [SHANNON].  This depends on the number of different

Eastlake, Crocker & Schiller [Page 4] RFC 1750 Randomness Recommendations for Security December 1994

 secret values possible and the probability of each value as follows:
  1. —-

\

      Bits-of-info =  \  - p   * log  ( p  )
                      /     i       2    i
                     /
                    -----
 where i varies 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 56 bit keys.
 If these 56 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 2^56 keys that
 may at first appear to be the case. Only 8 bits of "information" are
 in these 56 bit keys.

3. Traditional Pseudo-Random Sequences

 Most traditional sources of random numbers use deterministic sources
 of "pseudo-random" numbers.  These typically start with a "seed"
 quantity and use numeric or logical operations to produce a sequence
 of values.
 [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,

Eastlake, Crocker & Schiller [Page 5] RFC 1750 Randomness Recommendations for Security December 1994

 perhaps unlimited, chances at guessing the correct value and should
 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.  They could be met by a
 constant stored random sequence, such as the "random" sequence
 printed in the CRC Standard Mathematical Tables [CRC].
 A typical pseudo-random number generation technique, known as a
 linear congruence pseudo-random number generator, is modular
 arithmetic where the N+1th value is calculated from the Nth value by
      V    = ( V  * a + b )(Mod c)
       N+1      N
 The above technique has a strong relationship to linear shift
 register pseudo-random number generators, which are well understood
 cryptographically [SHIFT*].  In such generators bits are introduced
 at one end of a shift register as the Exclusive Or (binary sum
 without carry) of bits from selected fixed taps into the register.
 For example:
    +----+     +----+     +----+                      +----+
    | 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 of the initial V and a, b, and c or the placement of shift
 register tap in the above simple processes can produce excellent
 statistics.

Eastlake, Crocker & Schiller [Page 6] RFC 1750 Randomness Recommendations for Security December 1994

 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 [CRYPTO*, 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 is released, the seed can be
 determined from short sequences.
 Not only have linear congruent generators been broken, but techniques
 are now known for breaking all polynomial congruent generators
 [KRAWCZYK].

4. Unpredictability

 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.
 The following sections describe the limitations of some randomness
 generation techniques and sources.

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

Eastlake, Crocker & Schiller [Page 7] RFC 1750 Randomness Recommendations for Security December 1994

 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 a hardware serial number such as an Ethernet address 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 most of the Ethernet cards installed on Digital Equipment
 Corporation (DEC) hardware within DEC were manufactured by DEC
 itself, which significantly limits the range of built in addresses.
 Problems such as those described above related to clocks and serial
 numbers make code to produce unpredictable quantities difficult if
 the code is to be ported across a variety of computer platforms and
 systems.

4.2 Timing and Content 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, can also
 be used with care.  In particular, the possibility of manipulation of
 such times by an adversary must be considered.

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
 properties) and calculate a cryptographic key by starting with the
 current value of a computer system clock as the seed.  An adversary
 who knew roughly when the generator was started would have a

Eastlake, Crocker & Schiller [Page 8] RFC 1750 Randomness Recommendations for Security December 1994

 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.  Even if they can
 not learn what the manipulation is, they may be able to 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 as of this date
 process over 35 megabytes of information per day.  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 more like 2 or 3 bits of
 information per byte, it doesn't yield 32*2.5 = 80 bits of
 unguessability.  For an adversary with access to the same 35
 megabytes the unguessability rests only on the starting point of the
 selection.  That is, at best, about 25 bits of unguessability in this
 case.
 The same argument applies to selecting sequences from the data on a
 CD ROM or Audio CD recording or any other large public database.  If
 the adversary has access to the same database, this "selection from a

Eastlake, Crocker & Schiller [Page 9] RFC 1750 Randomness Recommendations for Security December 1994

 large volume of data" step buys very 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 some
 help.

5. Hardware for Randomness

 Is there any hope for strong portable randomness in the future?
 There might be.  All that's needed is a physical source of
 unpredictable numbers.
 A thermal noise 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.  Furthermore, any system with a
 spinning disk or the like has an adequate source of randomness
 [DAVIS].  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 DES, the key is 56 bits
 and, as we show in an example in Section 8, even the highest security
 system is unlikely to require a keying material of over 200 bits.  If
 a series of keys are needed, it can be generated from a strong random
 seed using a cryptographically strong sequence as explained in
 Section 6.3.  A few hundred random bits generated 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 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 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.  Two simple techniques to de-skew
 the bit stream are given below and stronger techniques are mentioned
 in Section 6.1.2 below.

Eastlake, Crocker & Schiller [Page 10] RFC 1750 Randomness Recommendations for Security December 1994

5.2.1 Using Stream Parity to De-Skew

 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 : 0.5 - e, where e is
 between 0 and 0.5 and is a measure of the "eccentricity" of the
 distribution.  Consider the distribution of the parity function of N
 bit samples.  The probabilities that the parity will be one or zero
 will be the sum of the odd or even terms in the binomial expansion of
 (p + q)^N, where p = 0.5 + e, the probability of a one, and q = 0.5 -
 e, the probability of a zero.
 These sums can be computed easily as
                       N            N
      1/2 * ( ( p + q )  + ( p - q )  )
 and
                       N            N
      1/2 * ( ( p + q )  - ( p - q )  ).
 (Which one corresponds to the probability the parity will be 1
 depends on whether N is odd or even.)
 Since p + q = 1 and p - q = 2e, these expressions reduce to
                     N
      1/2 * [1 + (2e) ]
 and
                     N
      1/2 * [1 - (2e) ].
 Neither of these will ever be exactly 0.5 unless e is zero, but we
 can bring them arbitrarily close to 0.5.  If we want the
 probabilities to be within some delta d of 0.5, i.e. then
                          N
      ( 0.5 + ( 0.5 * (2e)  ) )  <  0.5 + d.

Eastlake, Crocker & Schiller [Page 11] RFC 1750 Randomness Recommendations for Security December 1994

 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.

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
 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).

Eastlake, Crocker & Schiller [Page 12] RFC 1750 Randomness Recommendations for Security December 1994

 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 provided an unbiased
 uncorrelated series of random 1's and 0's, it instead produces a
 totally predictable sequence of exactly alternating 1's and 0's.

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 the discrete Fourier transform or its
 optimized variant, the FFT.
 Using the Fourier transform of the data, strong correlations can be
 discarded.  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.  Thus the shorter sequences are de-skewed relative to the
 input.
 However, many compression techniques add a somewhat predicatable
 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 above or 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 random bits.

Eastlake, Crocker & Schiller [Page 13] RFC 1750 Randomness Recommendations for Security December 1994

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

 Increasingly computers are being 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.
 For example, on a SPARCstation, one can read from the /dev/audio
 device with nothing plugged into the microphone jack.  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

5.3.2 Using Existing Disk Drives

 Disk drives have small random fluctuations in their rotational speed
 due to chaotic air turbulence [DAVIS].  By adding low level disk seek
 time instrumentation to a system, a series of measurements can be
 obtained that include this randomness. Such data is usually highly
 correlated so that significant processing is needed, including FFT
 (see section 5.2.3).  Nevertheless experimentation has shown that,
 with such processing, disk drives easily produce 100 bits a minute or
 more of excellent random data.
 Partly offsetting this need for processing is the fact that disk
 drive failure will normally be rapidly noticed.  Thus, problems with
 this method of random number generation due to hardware failure are
 very unlikely.

6. Recommended Non-Hardware 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 large number of
 uncorrelated sources and to mix them with a strong mixing function.

Eastlake, Crocker & Schiller [Page 14] RFC 1750 Randomness Recommendations for Security December 1994

 Such a function will preserve the randomness present in any of the
 sources even if other quantities being combined are 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
 complexity.

6.1 Mixing Functions

 A strong mixing function is one which combines two or more 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 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 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 will still provide a useful illustration.
                 +-----------+-----------+----------+
                 |  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

Eastlake, Crocker & Schiller [Page 15] RFC 1750 Randomness Recommendations for Security December 1994

 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 Data Encryption Standard [DES] is an example of a
 strong mixing function for multiple bit quantities.  It takes up to
 120 bits of input (64 bits of "data" and 56 bits of "key") and
 produces 64 bits of output each of which is dependent on a complex
 non-linear function of all input bits.  Other strong encryption
 functions with this characteristic 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 Standard
 [SHS] and the MD2, MD4, MD5 [MD2, MD4, MD5] series.  These functions
 all take an arbitrary amount of input and produce an output mixing
 all the input bits. The MD* series produce 128 bits of output and SHS
 produces 160 bits.

Eastlake, Crocker & Schiller [Page 16] RFC 1750 Randomness Recommendations for Security December 1994

 Although the message digest functions are designed for variable
 amounts of input, DES and other encryption functions can also be used
 to combine any number of inputs.  If 64 bits of output is adequate,
 the inputs can be packed into a 64 bit data quantity and successive
 56 bit keys, padding with zeros if needed, which are then used to
 successively encrypt using DES in Electronic Codebook Mode [DES
 MODES].  If more than 64 bits of output are needed, use more complex
 mixing.  For example, if inputs are packed into three quantities, A,
 B, and C, use DES 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 to produce multiple outputs.  But keep in mind that it is
 impossible to get more bits of "randomness" out than are put in.
 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 sequence has 5 bits of information in it.
 Thus hashing it with SHS or MD5 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.

6.1.3 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 them
 [D-H].  If these initial quantities are random, then the shared
 secret contains the combined randomness of them both, assuming they
 are uncorrelated.

6.1.4 Using a Mixing Function to Stretch Random Bits

 While it is not necessary for a mixing function to produce the same
 or fewer bits than its inputs, mixing bits cannot "stretch" the
 amount of random unpredictability present in the inputs.  Thus four
 inputs of 32 bits each where there is 12 bits worth of
 unpredicatability (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

Eastlake, Crocker & Schiller [Page 17] RFC 1750 Randomness Recommendations for Security December 1994

 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 way using Exclusive Or to produce one bit from two did
 above.
 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.

6.1.5 Other Factors in Choosing a Mixing Function

 For local use, DES has the advantages that it has been widely tested
 for flaws, is widely documented, and is widely implemented with
 hardware and software implementations available all over the world
 including source code available by anonymous FTP.  The SHS and MD*
 family are younger algorithms which have been less tested but there
 is no particular reason to believe they are flawed.  Both MD5 and SHS
 were derived from the earlier MD4 algorithm.  They all have source
 code available by anonymous FTP [SHS, MD2, MD4, MD5].
 DES and SHS have been vouched for the the US National Security Agency
 (NSA) on the basis of criteria that primarily remain secret.  While
 this is 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
 almost certain that the NSA modification to MD4 to produce the SHS
 similarly strengthened the algorithm, possibly against threats not
 yet known in the public cryptographic community.
 DES, SHS, MD4, and MD5 are royalty free for all purposes.  MD2 has
 been freely licensed only for non-profit use in connection with
 Privacy Enhanced Mail [PEM].  Between the MD* algorithms, some people
 believe that, as with "Goldilocks and the Three Bears", MD2 is strong
 but too slow, MD4 is fast but too weak, and MD5 is just right.
 Another advantage of the MD* or similar hashing algorithms over
 encryption algorithms is that they are not subject to the same
 regulations imposed by the US Government prohibiting the unlicensed
 export or import of encryption/decryption software and hardware.  The
 same should be true of DES rigged to produce an irreversible hash
 code but most DES packages are oriented to reversible encryption.

Eastlake, Crocker & Schiller [Page 18] RFC 1750 Randomness Recommendations for Security December 1994

6.2 Non-Hardware Sources of Randomness

 The best source of input for mixing would be a hardware randomness
 such as disk drive timing affected by air turbulence, audio input
 with 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, any of these sources can produce limited or
 predicatable 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 system, such as a
 typical IBM PC or Apple Macintosh, 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.  For
 example, 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 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 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.

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
 know.

Eastlake, Crocker & Schiller [Page 19] RFC 1750 Randomness Recommendations for Security December 1994

 The correct technique is to start with a strong random seed, take
 cryptographically strong steps from that seed [CRYPTO2, CRYPTO3], 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.

6.3.1 Traditional Strong Sequences

 A traditional way to achieve a strong sequence has been to have the
 values be produced by 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 and feedback some or all of the
 output encrypted value into the value to be encrypted for the next
 iteration.  Appropriate feedback techniques will usually be
 recommended with the encryption algorithm.  An example is shown below
 where shifting and masking are used to combine the cypher output
 feedback.  This type of feedback is recommended by the US Government
 in connection with DES [DES MODES].

Eastlake, Crocker & Schiller [Page 20] RFC 1750 Randomness Recommendations for Security December 1994

    +---------------+
    |       V       |
    |  |     n      |
    +--+------------+
          |      |           +---------+
          |      +---------> |         |      +-----+
       +--+                  | 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
 will 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.
 It's only disadvantage is that is is computationally intensive
 compared with the traditional techniques give in 6.3.1 above.  This
 is not a serious draw back if it is used for moderately infrequent
 purposes, such as generating session keys.

Eastlake, Crocker & Schiller [Page 21] RFC 1750 Randomness Recommendations for Security December 1994

 Simply choose two large prime numbers, say p and q, which both have
 the property that you get a remainder of 3 if you divide them by 4.
 Let n = p * q.  Then you choose a random number x relatively prime to
 n.  The initial seed for the generator and the method for calculating
 subsequent values are then
                 2
      s    =  ( x  )(Mod n)
       0
                 2
      s    = ( s   )(Mod n)
       i+1      i
 You must be careful to use only a few bits from the bottom of each s.
 It is always safe to use only the lowest order bit.  If you use no
 more than the
                log  ( log  ( s  ) )
                   2      2    i
 low order bits, then predicting any additional bits from a sequence
 generated in this manner is provable as hard as factoring n.  As long
 as the initial x is secret, you can even make n public if you want.
 An intersting characteristic of this generator is that you can
 directly calculate any of the s values.  In particular
                   i
             ( ( 2  )(Mod (( p - 1 ) * ( q - 1 )) ) )
    s  = ( s                                          )(Mod n)
     i      0
 This means that in applications where many keys are generated in this
 fashion, it is not necessary to save them all.  Each key can be
 effectively indexed and recovered from that small index and the
 initial s and n.

7. Key Generation Standards

 Several public standards are now in place for the generation of keys.
 Two of these are described below.  Both use DES but any equally
 strong or stronger mixing function could be substituted.

Eastlake, Crocker & Schiller [Page 22] RFC 1750 Randomness Recommendations for Security December 1994

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 [DES 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
      administrator.
 The password can then be calculated from the 64 bit "cipher text"
 generated 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:
      s  is the initial 64 bit seed
       0
      g  is the sequence of generated 64 bit key quantities
       n
      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

Eastlake, Crocker & Schiller [Page 23] RFC 1750 Randomness Recommendations for Security December 1994

      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.

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.

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

Eastlake, Crocker & Schiller [Page 24] RFC 1750 Randomness Recommendations for Security December 1994

 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
 of compromise.

8.2 A Very High Security Cryptographic Key

 Assume that a very high security key is needed for symmetric
 encryption / decryption between two parties.  Assume an adversary can
 observe communications and knows the algorithm being used.  Within
 the field of random possibilities, the adversary can try key values
 in hopes of finding the one in use.  Assume further that brute force
 trial of keys is the best the adversary can do.

8.2.1 Effort per Key Trial

 How much effort will it take to try each key?  For very high security
 applications it is best to assume a low value of effort.  Even if it
 would clearly take tens of thousands of computer cycles or more to
 try a single key, there may be some pattern that enables huge blocks
 of key values to be tested with much less effort per key.  Thus it is
 probably best to assume no more than a couple hundred cycles per key.
 (There is no clear lower bound on this as computers operate in
 parallel on a number of bits and a poor encryption algorithm could
 allow many keys or even groups of keys to be tested in parallel.
 However, we need to assume some value and can hope that a reasonably
 strong algorithm has been chosen for our hypothetical high security
 task.)
 If the adversary can command a highly parallel processor or a large
 network of work stations, 2*10^10 cycles per second is probably a
 minimum assumption for availability today.  Looking forward just a
 couple years, there should be at least an order of magnitude
 improvement.  Thus assuming 10^9 keys could be checked per second or
 3.6*10^11 per hour or 6*10^13 per week or 2.4*10^14 per month is
 reasonable.  This implies a need for a minimum of 51 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).

Eastlake, Crocker & Schiller [Page 25] RFC 1750 Randomness Recommendations for Security December 1994

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 large but roughly constant factor of
 effort.  To be assured of safety against this, a doubling of the
 amount of randomness in the key to a minimum of 102 bits is required.
 The meet in the middle attack assumes that the cryptographic
 algorithm can be decomposed in this way but we can not rule that out
 without a deep knowledge of the algorithm.  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 may gain less
 added security than would be expected.  Such a composite algorithm
 would be subject to a meet in the middle attack.
 Enormous resources may be required to mount a meet in the middle
 attack but they are probably within the range of the national
 security services of a major nation.  Essentially all nations spy on
 other nations government traffic and several nations are believed to
 spy on commercial traffic for economic advantage.

8.2.3 Other Considerations

 Since we have not even considered the possibilities of special
 purpose code breaking hardware or just how much of a safety margin we
 want beyond our assumptions above, probably a good minimum for a very
 high security cryptographic key is 128 bits of randomness which
 implies a minimum key length of 128 bits.  If the two parties agree
 on a key by Diffie-Hellman exchange [D-H], then in principle only
 half of this randomness would have to be supplied by each party.
 However, there is probably some correlation between their random
 inputs so it is probably best to assume that each party needs to

Eastlake, Crocker & Schiller [Page 26] RFC 1750 Randomness Recommendations for Security December 1994

 provide at least 96 bits worth of randomness for very high security
 if Diffie-Hellman is used.
 This amount of randomness is 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.
 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
 key size derived above.

9. Conclusion

 Generation of unguessable "random" secret quantities for security use
 is an essential but difficult task.
 We have shown that 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
 disk drives, can be used.  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, less predictable key material.
 In the absence of hardware sources of randomness, a variety of user
 and software sources can frequently be used instead with care;
 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 few
 hundred bits) is available, strong computational techniques are
 available to produce cryptographically strong sequences of
 unpredicatable quantities from this seed material.

10. Security Considerations

 The entirety of this document concerns techniques and recommendations
 for generating unguessable "random" quantities for use as passwords,
 cryptographic keys, and similar security uses.

Eastlake, Crocker & Schiller [Page 27] RFC 1750 Randomness Recommendations for Security December 1994

References

 [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.
 [CRYPTO1] - Cryptography: A Primer, A Wiley-Interscience Publication,
 John Wiley & Sons, 1981, Alan G. Konheim.
 [CRYPTO2] - Cryptography:  A New Dimension in Computer Data Security,
 A Wiley-Interscience Publication, John Wiley & Sons, 1982, Carl H.
 Meyer & Stephen M. Matyas.
 [CRYPTO3] - Applied Cryptography: Protocols, Algorithms, and Source
 Code in C, John Wiley & Sons, 1994, Bruce Schneier.
 [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, United States of America,
 Department of Commerce, National Institute of Standards and
 Technology, Federal Information Processing Standard (FIPS) 46-1.
 - Data Encryption Algorithm, American National Standards Institute,
 ANSI X3.92-1981.
 (See also FIPS 112, Password Usage, which includes FORTRAN code for
 performing DES.)
 [DES MODES] - DES Modes of Operation, United States of America,
 Department of Commerce, National Institute of Standards and
 Technology, Federal Information Processing Standard (FIPS) 81.
 - Data Encryption Algorithm - Modes of Operation, American National
 Standards Institute, ANSI X3.106-1983.
 [D-H] - New Directions in Cryptography, IEEE Transactions on
 Information Technology, November, 1976, Whitfield Diffie and Martin
 E. Hellman.

Eastlake, Crocker & Schiller [Page 28] RFC 1750 Randomness Recommendations for Security December 1994

 [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.)
 [GIFFORD] - Natural Random Number, MIT/LCS/TM-371, September 1988,
 David K. Gifford
 [KNUTH] - The Art of Computer Programming, Volume 2: Seminumerical
 Algorithms, Chapter 3: Random Numbers. Addison Wesley Publishing
 Company, Second Edition 1982, Donald E. Knuth.
 [KRAWCZYK] - How to Predict Congruential Generators, Journal of
 Algorithms, V. 13, N. 4, December 1992, H. Krawczyk
 [MD2] - The MD2 Message-Digest Algorithm, RFC1319, April 1992, B.
 Kaliski
 [MD4] - The MD4 Message-Digest Algorithm, RFC1320, April 1992, R.
 Rivest
 [MD5] - The MD5 Message-Digest Algorithm, RFC1321, April 1992, R.
 Rivest
 [PEM] - RFCs 1421 through 1424:
 - RFC 1424, Privacy Enhancement for Internet Electronic Mail: Part
 IV: Key Certification and Related Services, 02/10/1993, B. Kaliski
 - RFC 1423, Privacy Enhancement for Internet Electronic Mail: Part
 III: Algorithms, Modes, and Identifiers, 02/10/1993, D. Balenson
 - RFC 1422, Privacy Enhancement for Internet Electronic Mail: Part
 II: Certificate-Based Key Management, 02/10/1993, S. Kent
 - RFC 1421, Privacy Enhancement for Internet Electronic Mail: Part I:
 Message Encryption and Authentication Procedures, 02/10/1993, J. Linn
 [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, Aegean Park Press, Revised
 Edition 1982, Solomon W. Golomb.
 [SHIFT2] - Cryptanalysis of Shift-Register Generated Stream Cypher
 Systems, Aegean Park Press, 1984, Wayne G. Barker.
 [SHS] - Secure Hash Standard, United States of American, National
 Institute of Science and Technology, Federal Information Processing
 Standard (FIPS) 180, April 1993.
 [STERN] - Secret Linear Congruential Generators are not
 Cryptograhically Secure, Proceedings of IEEE STOC, 1987, J. Stern.

Eastlake, Crocker & Schiller [Page 29] RFC 1750 Randomness Recommendations for Security December 1994

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

Authors' Addresses

 Donald E. Eastlake 3rd
 Digital Equipment Corporation
 550 King Street, LKG2-1/BB3
 Littleton, MA 01460
 Phone:   +1 508 486 6577(w)  +1 508 287 4877(h)
 EMail:   dee@lkg.dec.com
 Stephen D. Crocker
 CyberCash Inc.
 2086 Hunters Crest Way
 Vienna, VA 22181
 Phone:   +1 703-620-1222(w)  +1 703-391-2651 (fax)
 EMail:   crocker@cybercash.com
 Jeffrey I. Schiller
 Massachusetts Institute of Technology
 77 Massachusetts Avenue
 Cambridge, MA 02139
 Phone:   +1 617 253 0161(w)
 EMail:   jis@mit.edu

Eastlake, Crocker & Schiller [Page 30]

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