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INTERNET-DRAFT                      Randomness Requirements for Security
                                                         01 October 1993
                                                   Expires 31 March 1994

                  Randomness Requirements for Security
                  ---------- ------------ --- --------
   Donald E. Eastlake 3rd, Stephen D. Crocker, & Jeffrey I. Schiller

Status of This Document

   This draft, file name draft-ietf-security-randomness-01.txt, is
   intended to be submitted to the RFC editor as an Informational RFC.
   Distribution of this document is unlimited.

   This document is an Internet Draft.  Internet Drafts are working
   documents of the Internet Engineering Task Force (IETF), its Areas,
   its Working Groupsd and other organizations or individuals.

   Internet Drafts are draft documents valid for a maximum of six
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   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 will often 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
   attacker is surprisingly difficult.  This paper points out many
   pitfalls in using traditional pseudo-random number generation
   techniques for choosing such quantities, recommends the use of truly
   random hardware techniques, provides suggestions to ameliorate the
   problem when a hardware solution is not available, and gives examples
   of how large such quantities need to be for some particular

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

INTERNET-DRAFT                      Randomness Requirements for Security


   Useful comments on this document that have been incorporated were
   received from (in alphabetic order) the following:
        David M. Balenson (TIS)
        Carl Ellison (Stratus)
        Marc Horowitz (MIT)
        Charlie Kaufman (DEC)
        Steve Kent (BBN)
        Hal Murray (DEC)
        Neil Haller (Bellcore)
        Richard Pitkin (DEC)
        Tim Redmond (TIS)
        Doug Tygar (CMU)

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

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Table of Contents

      Status of This Document....................................1
      Table of Contents..........................................3
      1. Introduction............................................4
      2. Requirements............................................5
      3. Traditional Pseudo-Random Sequences.....................7
      4. Unpredictability........................................9
      4.1 Problems with Clocks and Serial Numbers................9
      4.2 Timing and Content of External Events.................10
      4.3 The Fallacy of Complex Manipulation...................10
      4.4 The Fallacy of Selection from a Large Database........11
      5. Hardware for Randomness................................12
      5.1 Volume Required.......................................12
      5.2 Sensitivity to Skew...................................12
      5.2.1 Using Stream Parity to De-Skew......................13
      5.2.2 Using Transition Mappings to De-Skew................14
      5.2.3 Using Compression to De-Skew........................15
      5.3 Using Sound/Video Input...............................15
      6. Recommended Non-Hardware Strategy......................17
      6.1 Mixing Functions......................................17
      6.1.1 A Trivial Mixing Function...........................17
      6.1.2 Stronger Mixing Functions...........................18
      6.1.3 Using a Mixing Function to Stretch Random Bits......19
      6.1.4 Other Factors in Choosing a Mixing Function.........20
      6.2 Non-Hardware Sources of Randomness....................20
      6.3 Cryptographically Strong Sequences....................21
      7. US DoD Recommendations for Password Generation.........23
      8. Examples of Randomness Required........................24
      8.1  Password Generation..................................24
      8.2 A Very High Security Cryptographic Key................24
      8.2.1 Effort per Key Trial................................25
      8.2.2 Meet in the Middle Attacks..........................25
      8.2.3 Other Considerations................................26
      9. Security Considerations................................27
      Authors Addresses.........................................29
      Expiration and File Name..................................29

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

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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.  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 random

   For the present, the lack of generally available facilities for
   generating 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
   unpredictable 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.  This will fail
   if pseudo-random data, which only meets traditional statistical tests
   for randomness or which is based on guessable range sources, such as
   clocks, is used.  Frequently such random quantities are guessable by
   an adversary searching through an embarrassingly small space of

   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, suggests
   methods for use if such hardware is not available, and gives some
   estimates of the number of random bits required for sample

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

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2. Requirements

   Probably the most commonly encountered randomness requirement is the
   typical user password character string.  Obviously, if a password can
   be guessed, it does not provide security.  (For this particular
   application 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 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 all of these cases, an adversary may 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 secret values
   possible and the probability of each value as follows:

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

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        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 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 pseudo-random number
   generator that is seeded with an 8 bit seed, then an attacker 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.

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

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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 general 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 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:

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

INTERNET-DRAFT                      Randomness Requirements for Security

      +----+     +----+     +----+                      +----+
      | B  | <-- | B  | <-- | B  | <--  . . . . . . <-- | B  | <-+
      |  0 |     |  1 |     |  2 |                      |  n |   |
      +----+     +----+     +----+                      +----+   |
        |                     |            |                     |
        |                     |            V                  +-----+
        |                     V            +----------------> |     |
        V                     +-----------------------------> | XOR |
        +---------------------------------------------------> |     |

       V    = ( ( V  * 2 ) + B .xor. B ...B )(Mod 2^n)
        N+1         N         0       1    n

   The goodness of traditional pseudo-random number generator algorithm
   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

   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.  [ref to come - Marsaglia] 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.  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 them even if only one
   bit of the pseudo-random values are released, the seed can be
   determined from short sequences.  [ref to come - Frieze, Hastad,
   Kannan, Lagaris, & Shamir]

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

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4. Unpredictability

   Randomness in the traditional sense described in the previous section
   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]

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 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 possible serial

   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

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

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4.2 Timing and Content of External Events

   It is possible to measure the timing of mouse movement, key strokes,
   and the like.  This is a reasonable source of unguessable data with
   two exceptions.  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.  The other 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 keys actually hit are usually easier
   to access than timings but may yield less unpredictability as the
   user may provide highly repetative input.

4.3 The Fallacy of Complex Manipulation

   One strategy which may give a misleading appearance of strength 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 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

   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.

   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 [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!

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

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4.4 The Fallacy of Selection from a Large Database

   Another strategy that can give a misleading appearance of strength is
   selection of a quantity randomly from a database and the assumption
   that its strength is related to the total number of bits in the
   database.  For example, typical NNTP servers as of this date process
   over 30 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 3 bits of information per byte,
   it doesn't yield 32*3 = 96 bits of unguessability.  For an adversary
   with access to the same 30 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
   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
   active system buffers on an active multi-user system, it may be of
   some help.

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

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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.  This is a trivial amount of hardware,
   and could easily be included as a standard part of a computer
   system's architecture.  All that's needed is the common perception
   among computer vendors that this small addition is necessary and

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.
   Even if a series of keys are needed, they 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.

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

INTERNET-DRAFT                      Randomness Requirements for Security

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

   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

        1/2 * ( ( p + q )^N + ( p - q )^N )
        1/2 * ( ( p + q )^N - ( p - q )^N ).

   (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)^N]
        1/2 * [1 - (2e)^N].

   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)^N ) ) < 0.5 + d.

   Solving for N yields N > log(2d)/log(2e).  (Note that 2e is less than
   1, so its log is negative.  Division by a negative number reverses
   the sense of an inequality.)

   The following table gives the length of the string which must be
   sampled for various degrees of skew in order to come within 0.001 of
   a 50/50 distribution.

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

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                       | 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 possible technique 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).

   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
   different 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

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

INTERNET-DRAFT                      Randomness Requirements for Security

   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 would instead produce a
   totally predictable sequence of exactly alternating 1's and 0's.

5.2.3 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

   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.

5.3 Using 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 is high enough gain to detect anything, is
   essentially thermal noise.

   For example, on a Sparkstation, 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.

   Thus, combining this with compression to de-skew, one can in UNIXese

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

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   generate a hugh amount of relatively random data by doing

        cat /dev/audio | compress - >random-bits-file

D. Eastlake, S. Crocker, & J. Schiller                         [Page 16]

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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.
   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

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.

   Note that 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, is simply another case where a strong mixing
   function is desired.  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

6.1.1 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 it will still provide 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 usually be a constant zero.

6.1.2 Stronger Mixing Functions

   The US Government Data Encryption Standard [DES] is a good 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

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   non-linear function of all input bits.  Another good family of mixing
   functions are the "message digest" or hashing functions such as MD2,
   MD4, or MD5 that take an arbitrary amount of input and produce an
   output, frequently 128 bits, mixing all the input bits. [MD2, MD4,

   Although message digest functions like MD5 are designed for variable
   amounts of input, DES 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, 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 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.

   Other functions besides DES and the MD* family should serve well as
   mixing functions.  This is an advantage of Diffie-Hellman exponential
   key exchange.  Diffie-Hellman yields a shared secret between two
   parties that is a mixture of initial random quantities generated by
   each of them [D-H, ref to come - Odlyzko].

6.1.3 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
   search space is still 2^48 possibilities.  Furthermore, mixing to

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   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

   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.

6.1.4 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 MD* family are
   younger algorithms which have been less tested but there is no
   particular reason to believe they are flawed.  They also have source
   code available by anonymous FTP [MD2, MD4, MD5].  DES, MD4, and MD5
   are royalty free for all purposes but MD2 has been freely licensed
   only for non-profit use in connection with Privacy Enhanced Mail.
   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

   Another advantage of the MD* or similar hashing algorithms is that
   they are not subject to the regulations imposed by the US Government
   prohibiting the export or import of encryption/decryption software
   (or hardware).  The same should be true of DES rigged to produce an
   irreversible hash code but most DES packages are oriented to
   reversible encryption.

6.2 Non-Hardware Sources of Randomness

   The best source of input for mixing would be a hardware random number
   generator based on some fundamentally random physical process such as
   thermal noise or radioactive decay.  However, if that is not
   available, other possibilities include system clocks, system or
   input/output buffers, user/system/hardware/network serial numbers
   and/or addresses, user input, and timings of input/output operations.
   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

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   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

   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 fail against a high grade
   attack on small single user systems, especially if the adversary has
   even been able to observe the generation process in the past.  A
   hardware 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

   The correct technique is to start with a strong random seed, take
   cryptographically strong steps from that seed [CRYPTO2], 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 previous values, even if
   the function were a very strong, non-invertible message digest

   A good way to achieve a strong sequence is 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

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   the adversary.  It may also be possible to use an encryption
   algorithm with a random key and seed value to encrypt and feedback
   some 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 in connection with

         |       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
   combination of output from a few bit position taps.

   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 some cases this makes it impossible to break a
   system even when the cryptographic system is invertible and can be
   broken if all of the generated values were revealed.

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7. 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

        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 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.

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8. Examples of Randomness Required

   Below are two examples showing rough calculations of needed
   randomness for security.

8.1  Password Generation

   Assume that user passwords change once a year and a probability of
   less than one in a thousand that an adversary could guess the
   password for a particular account is desired.  The key question is
   how often they 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.  (All this assumes that sending a password to the system
   is the only way to try a password.)

   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.  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 12 bits.  Thus to
   have only a one in a million chance of a password being guessed under
   the above scenario would require 31 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, 43 bits of randomness
   are needed implying a five word phrase or ten letter/digit password.

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 exhaustively try

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   key values.  Assume further that there is no systematic weakness in
   the cryptographic system so 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 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, 10^10 cycles per second is probably a
   minimum assumption for availability today.  Looking forward just a
   few 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 48 bits of
   randomness in keys to be sure they cannot be found in a week.  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).

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 known 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 attack
   is as follows: the adversary can half-encrypt the know or chosen
   plain text with all possible first half-keys, sort these, then half-
   decrypt the encoded text with all the second half-keys.  If a match

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   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 moderate constant factor.  To be assured
   of safety against this, a doubling of the amount of randomness in the
   key to a minimum of 96 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 with
   different keys may gain much less than would be expected.  Such a
   composite algorithm would be subject to this type of 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.  Almost all nations spy on other
   nations government traffic and some nations are known to spy on
   commercial traffic and give the information to their domestic
   companies to assist them against foreign competition.

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
   provide at least 96 bits worth of randomness for very high security.

   This amount of randomness is probably 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.

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9. Security Considerations

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


   [ASYMMETRIC] - Secure Communications and Asymmetric Cryptosystems,
   edited by Gustavus J. Simmons, AAAS Selected Symposium 69, Westview
   Press, Inc.

   [CRC] - C.R.C. Standard Mathematical Tables, Chemical Rubber
   Publishing Company.

   [CRYPTO1] - Cryptography:  A Primer, by Alan G. Konheim, 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.

   [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.

   [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 appendicies.)

   [KNUTH] - The Art of Computer Programming, Volume 2: Seminumerical
   Algorithms, Chapter 3: Random Numbers. Addison Wesley Publishing
   Company, Second Edition 1982, Donald E. Knuth.

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INTERNET-DRAFT                      Randomness Requirements for Security

   [MD2] - The MD2 Message-Digest Algorithm, RFC1319, April 1992, B.
   [MD4] - The MD4 Message-Digest Algorithm, RFC1320, April 1992, R.
   [MD5] - The MD5 Message-Digest Algorithm, RFC1321, April 1992, R.

   [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.

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Authors Addresses

   Donald E. Eastlake 3rd
   Digital Equipment Corporation
   550 King Street, LKG2-1/BB3
   Littleton, MA 01460

   Telephone:   +1 508 486 6577(w)  +1 508 287 4877(h)
   NIC Handle:  [DEE]

   Stephen D. Crocker
   Trusted Information Systems, Inc.
   3060 Washington Road
   Glenwood, MD 21738

   Telephone:   +1 301 854 6889
   NIC Handle:  [SDC1]

   Jeffrey I. Schiller
   Massachusetts Institute of Technology
   77 Massachusetts Avenue
   Cambridge, MA 02139

   Telephone:   +1 617 253 0161
   NIC Handle:  [JIS]

Expiration and File Name

   This draft expires 31 March 1994.

   Its file name is draft-ietf-security-randomness-01.txt.

D. Eastlake, S. Crocker, & J. Schiller                         [Page 29]

%%% overflow headers %%%
Cc: David M. Balenson <>, Stephen D. Crocker <>,
        Beast (Donald E. Eastlake,III) <>,
        Carl Ellison <>,
        Neil Haller <>, Marc Horowitz <>,
        Charlie Kaufman <>,
        Steve Kent <>, Hal Murray <>,
        Richard Pitkin <>,
        Tim Redmond <>, Jeffrey I. Schiller <>,
        Doug Tygar <>,
        Eirikur Hallgrimsson <>,
        Al Kent <>,
        Jim Burrows (Brons) <>
%%% end overflow headers %%%