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RFC 4086

Randomness Requirements for Security

Pages: 48
BCP 106
Obsoletes:  1750
Part 2 of 2 – Pages 21 to 48
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Top   ToC   Page 21   prevText
6.  Pseudo-random Number Generators

   When a seed has sufficient entropy, from input as described in
   Section 3 and possibly de-skewed and mixed as described in Sections 4
   and 5, one can algorithmically extend that seed to produce a large
   number of cryptographically-strong random quantities.  Such
   algorithms are platform independent and can operate in the same
   fashion on any computer.  For the algorithms to be secure, their
   input and internal workings must be protected from adversarial

   The design of such pseudo-random number generation algorithms, like
   the design of symmetric encryption algorithms, is not a task for
   amateurs.  Section 6.1 below lists a number of bad ideas that failed
   algorithms have used.  To learn what works, skip Section 6.1 and just
   read the remainder of this section and Section 7, which describes and
   references some standard pseudo random number generation algorithms.
   See Section 7 and Part 3 of [X9.82].

6.1.  Some Bad Ideas

   The subsections below describe a number of ideas that might seem
   reasonable but that lead to insecure pseudo-random number generation.

6.1.1.  The Fallacy of Complex Manipulation

   One approach that 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 to calculate a cryptographic key by starting with
   limited data such as the computer system clock value as the seed.
   Adversaries 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-
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   random bits could be generated, but the search space that an
   adversary would need to check could be quite small.

   Thus, very strong or complex manipulation of data will not help if
   the adversary can learn what the manipulation is and if there is not
   enough entropy in the starting seed value.  They can usually use the
   limited number of results stemming from a limited number of seed
   values to defeat security.

   Another serious strategic 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 near the
   beginning of Chapter 3 in [KNUTH], where the author describes a
   complex algorithm.  It was intended that the machine language program
   corresponding to the algorithm would be so complicated that a person
   trying to read the code without comments wouldn't know what the
   program was doing.  Unfortunately, actual use of this algorithm
   showed that it almost immediately converged to a single repeated
   value in one case and a small cycle of values in another case.

   Not only does complex manipulation not help you if you have a limited
   range of seeds, but blindly-chosen complex manipulation can destroy
   the entropy in a good seed!

6.1.2.  The Fallacy of Selection from a Large Database

   Another approach that can give a misleading appearance of
   unpredictability is to randomly select a quantity from a database and
   to assume that its strength is related to the total number of bits in
   the database.  For example, typical USENET servers process many
   megabytes of information per day [USENET_1, USENET_2].  Assume that 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 if much of the data is human
   language that contains no more than 2 or 3 bits of information per
   byte, it doesn't yield 32*2 = 64 bits of unguessability.  For an
   adversary with access to the same Usenet database, the unguessability
   rests only on the starting point of the selection.  That is perhaps a
   little over a couple of dozen bits of unguessability.

   The same argument applies to selecting sequences from the data on a
   publicly available CD/DVD recording or any other large public
   database.  If the adversary has access to the same database, this
   "selection from a large volume of data" step buys little.  However,
   if a selection can be made from data to which the adversary has no
   access, such as system buffers on an active multi-user system, it may
   be of help.
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6.1.3.  Traditional Pseudo-random Sequences

   This section talks about traditional sources of deterministic or
   "pseudo-random" numbers.  These typically start with a "seed"
   quantity and use simple numeric or logical operations to produce a
   sequence of values.  Note that none of the techniques discussed in
   this section is suitable for cryptographic use.  They are presented
   for general information.

   [KNUTH] has a classic exposition on pseudo-random numbers.
   Applications he mentions are simulations of natural phenomena,
   sampling, numerical analysis, testing computer programs, decision
   making, and games.  None of these have the same characteristics as
   the sorts of security uses we are talking about.  Only in the last
   two could there be an adversary trying to find the random quantity.
   However, in these cases, the adversary normally has only a single
   chance to use a guessed value.  In guessing passwords or attempting
   to break an encryption scheme, the adversary normally has many,
   perhaps unlimited, chances at guessing the correct value.  Sometimes
   the adversary can store the message to be broken and repeatedly
   attack it.  Adversaries are also be assumed to be aided by a

   For testing the "randomness" of numbers, Knuth suggests a variety of
   measures, including statistical and spectral.  These tests check
   things like autocorrelation between different parts of a "random"
   sequence or distribution of its values.  But these tests could be met
   by a constant stored random sequence, such as the "random" sequence
   printed in the CRC Standard Mathematical Tables [CRC].  Despite
   meeting all the tests suggested by Knuth, that sequence is unsuitable
   for cryptographic us, as adversaries must be assumed to have copies
   of all commonly published "random" sequences and to be able to spot
   the source and predict future values.

   A typical pseudo-random number generation technique is the linear
   congruence pseudo-random number generator.  This technique uses
   modular arithmetic, where the value numbered N+1 is calculated from
   the value numbered N by

        V    = ( V  * a + b )(Mod c)
         N+1      N

   The above technique has a strong relationship to linear shift
   register pseudo-random number generators, which are well understood
   cryptographically [SHIFT*].  In such generators, bits are introduced
   at one end of a shift register as the Exclusive Or (binary sum
   without carry) of bits from selected fixed taps into the register.
   For example, consider the following:
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      +----+     +----+     +----+                      +----+
      | B  | <-- | B  | <-- | B  | <--  . . . . . . <-- | B  | <-+
      |  0 |     |  1 |     |  2 |                      |  n |   |
      +----+     +----+     +----+                      +----+   |
        |                     |            |                     |
        |                     |            V                  +-----+
        |                     V            +----------------> |     |
        V                     +-----------------------------> | XOR |
        +---------------------------------------------------> |     |

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

   The quality of traditional pseudo-random number generator algorithms
   is measured by statistical tests on such sequences.  Carefully-chosen
   values a, b, c, and initial V or carefully-chosen placement of the
   shift register tap in the above simple process 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.  However, such sequences are clearly bad for use in
   security applications.  They are fully predictable if the initial
   state is known.  Depending on the form of the pseudo-random number
   generator, the sequence may be determinable from observation of a
   short portion of the sequence [SCHNEIER, STERN].  For example, with
   the generators above, one can determine V(n+1) given knowledge of
   V(n).  In fact, it has been shown that with these techniques, even if
   only one bit of the pseudo-random values are released, the seed can
   be determined from short sequences.

   Not only have linear congruent generators been broken, but techniques
   are now known for breaking all polynomial congruent generators

6.2.  Cryptographically Strong Sequences

   In cases where a series of random quantities must be generated, an
   adversary may learn some values in the sequence.  In general,
   adversaries should not be able to predict other values from the ones
   that they know.

   The correct technique is to start with a strong random seed, to take
   cryptographically strong steps from that seed [FERGUSON, SCHNEIER],
   and not to reveal the complete state of the generator in the sequence
   elements.  If each value in the sequence can be calculated in a fixed
Top   ToC   Page 25
   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.

   (Note that if a 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
   generation 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 sent to encrypt it, and XOR'ed with this data
   as received to decrypt it, due to the reversible properties of the
   XOR operation.  This is commonly referred to as a simple stream

6.2.1.  OFB and CTR Sequences

   One way to produce a strong sequence is to take a seed value and hash
   the quantities produced by concatenating the seed with successive
   integers, or the like, and then to mask the values obtained so as to
   limit the amount of generator state available to the adversary.

   It may also be possible to use an "encryption" algorithm with a
   random key and seed value to encrypt successive integers, as in
   counter (CTR) mode encryption.  Alternatively, one can feedback all
   of the output value from encryption into the value to be encrypted
   for the next iteration.  This is a particular example of output
   feedback mode (OFB) [MODES].

   An example is shown below in which shifting and masking are used to
   combine part of the output feedback with part of the old input.  This
   type of partial feedback should be avoided for reasons described
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            |       V       |
            |  |     n      |--+
            +--+------------+  |
                  |            |     +---------+
             shift|            +---> |         |      +-----+
               +--+                  | Encrypt | <--- | Key |
               |           +-------- |         |      +-----+
               |           |         +---------+
               V           V
            |      V     |  |
            |       n+1     |

   Note that if a shift of one is used, this is the same as the shift
   register technique described in Section 6.1.3, but with the all-
   important difference that the feedback is determined by a complex
   non-linear function of all bits rather than by a simple linear or
   polynomial combination of output from a few bit position taps.

   Donald W. Davies showed that this sort of shifted partial output
   feedback significantly weakens an algorithm, compared to feeding all
   the output bits back as input.  In particular, for DES, repeatedly
   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

   To predict values of a sequence from others when the sequence was
   generated by these techniques is equivalent to breaking the
   cryptosystem or to inverting the "non-invertible" hashing with only
   partial information available.  The less information revealed in 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 could be
   broken if all of each generated value were revealed.

6.2.2.  The Blum Blum Shub Sequence Generator

   Currently the generator which has the strongest public proof of
   strength is called the Blum Blum Shub generator, named after its
   inventors [BBS].  It is also very simple and is based on quadratic
   residues.  Its only disadvantage is that it is computationally
   intensive compared to the traditional techniques given in Section
   6.1.3.  This is not a major drawback if it is used for moderately-
   infrequent purposes, such as generating session keys.
Top   ToC   Page 27
   Simply choose two large prime numbers (say, p and q) that each gives
   a remainder of 3 when divided by 4.  Let n = p * q.  Then choose a
   random number, x, that is relatively prime to n.  The initial seed
   for the generator and the method for calculating subsequent values
   are then:

         s    =  ( x  )(Mod n)
         s    = ( s   )(Mod n)
          i+1      i

   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 one uses 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 provably as hard as factoring n.  As long
   as the initial x is secret, n can be made public if desired.

   An interesting characteristic of this generator is that any of the s
   values can be directly calculated.  In particular,

               ( (2^i) (Mod ((p-1)*(q-1)) ) )
      s  = ( s                                )(Mod n)
       i      0

   This means that in applications where many keys are generated in this
   fashion, it is not necessary to save them all.  Each key can be
   effectively indexed and recovered from that small index and the
   initial s and n.

6.3.  Entropy Pool Techniques

   Many modern pseudo-random number sources, such as those described in
   Sections 7.1.2 and 7.1.3 utilize the technique of maintaining a
   "pool" of bits and providing operations for strongly mixing input
   with some randomness into the pool and extracting pseudo-random bits
   from the pool.  This is illustrated in the figure below.
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             +--------+    +------+    +---------+
         --->| Mix In |--->| POOL |--->| Extract |--->
             |  Bits  |    |      |    |   Bits  |
             +--------+    +------+    +---------+
                               ^           V
                               |           |

   Bits to be fed into the pool can come from any of the various
   hardware, environmental, or user input sources discussed above.  It
   is also common to save the state of the pool on system shutdown and
   to restore it on re-starting, when stable storage is available.

   Care must be taken that enough entropy has been added to the pool to
   support particular output uses desired.  See [RSA_BULL1] for similar

7.  Randomness Generation Examples and Standards

   Several public standards and widely deployed examples are now in
   place for the generation of keys or other cryptographically random
   quantities.  Some, in section 7.1, include an entropy source.
   Others, described in section 7.2, provide the pseudo-random number
   strong-sequence generator but assume the input of a random seed or
   input from a source of entropy.

7.1.  Complete Randomness Generators

   Three standards are described below.  The two older standards use
   DES, with its 64-bit block and key size limit, but any equally strong
   or stronger mixing function could be substituted [DES].  The third is
   a more modern and stronger standard based on SHA-1 [SHA*].  Lastly,
   the widely deployed modern UNIX and Windows random number generators
   are described.

7.1.1.  US DoD Recommendations for Password Generation

   The United States Department of Defense has specific recommendations
   for password generation [DoD].  It suggests using the US Data
   Encryption Standard [DES] in Output Feedback Mode [MODES] as follows:
Top   ToC   Page 29
         Use an initialization vector determined from
              the system clock,
              system ID,
              user ID, and
              date and time;
         use a key determined from
              system interrupt registers,
              system status registers, and
              system counters; and,
         as plain text, use an external randomly generated 64-bit
         quantity such as the ASCII bytes for 8 characters typed
         in by a system administrator.

   The password can then be calculated from the 64 bit "cipher text"
   generated by DES in 64-bit Output Feedback Mode.  As many bits as are
   needed can be taken from these 64 bits and expanded into a
   pronounceable word, phrase, or other format if a human being needs to
   remember the password.

7.1.2.  The /dev/random Device

   Several versions of the UNIX operating system provide a kernel-
   resident random number generator.  Some of these generators use
   events captured by the Kernel during normal system operation.

   For example, on some versions of Linux, the generator consists of a
   random pool of 512 bytes represented as 128 words of 4 bytes each.
   When an event occurs, such as a disk drive interrupt, the time of the
   event is XOR'ed into the pool, and the pool is stirred via a
   primitive polynomial of degree 128.  The pool itself is treated as a
   ring buffer, with new data being XOR'ed (after stirring with the
   polynomial) across the entire pool.

   Each call that adds entropy to the pool estimates the amount of
   likely true entropy the input contains.  The pool itself contains a
   accumulator that estimates the total over all entropy of the pool.

   Input events come from several sources, as listed below.
   Unfortunately, for server machines without human operators, the first
   and third are not available, and entropy may be added slowly in that

   1. Keyboard interrupts.  The time of the interrupt and the scan code
      are added to the pool.  This in effect adds entropy from the human
      operator by measuring inter-keystroke arrival times.

   2. Disk completion and other interrupts.  A system being used by a
      person will likely have a hard-to-predict pattern of disk
Top   ToC   Page 30
      accesses.  (But not all disk drivers support capturing this timing
      information with sufficient accuracy to be useful.)

   3. Mouse motion.  The timing and mouse position are added in.

   When random bytes are required, the pool is hashed with SHA-1 [SHA*]
   to yield the returned bytes of randomness.  If more bytes are
   required than the output of SHA-1 (20 bytes), then the hashed output
   is stirred back into the pool and a new hash is performed to obtain
   the next 20 bytes.  As bytes are removed from the pool, the estimate
   of entropy is correspondingly decremented.

   To ensure a reasonably random pool upon system startup, the standard
   startup and shutdown scripts save the pool to a disk file at shutdown
   and read this file at system startup.

   There are two user-exported interfaces. /dev/random returns bytes
   from the pool but blocks when the estimated entropy drops to zero.
   As entropy is added to the pool from events, more data becomes
   available via /dev/random.  Random data obtained from such a
   /dev/random device is suitable for key generation for long term keys,
   if enough random bits are in the pool or are added in a reasonable
   amount of time.

   /dev/urandom works like /dev/random; however, it provides data even
   when the entropy estimate for the random pool drops to zero.  This
   may be adequate for session keys or for other key generation tasks
   for which blocking to await more random bits is not acceptable.  The
   risk of continuing to take data even when the pool's entropy estimate
   is small in that past output may be computable from current output,
   provided that an attacker can reverse SHA-1.  Given that SHA-1 is
   designed to be non-invertible, this is a reasonable risk.

   To obtain random numbers under Linux, Solaris, or other UNIX systems
   equipped with code as described above, all an application has to do
   is open either /dev/random or /dev/urandom and read the desired
   number of bytes.

   (The Linux Random device was written by Theodore Ts'o.  It was based
   loosely on the random number generator in PGP 2.X and PGP 3.0 (aka
   PGP 5.0).)

7.1.3.  Windows CryptGenRandom

   Microsoft's recommendation to users of the widely deployed Windows
   operating system is generally to use the CryptGenRandom pseudo-random
   number generation call with the CryptAPI cryptographic service
   provider.  This takes a handle to a cryptographic service provider
Top   ToC   Page 31
   library, a pointer to a buffer by which the caller can provide
   entropy and into which the generated pseudo-randomness is returned,
   and an indication of how many octets of randomness are desired.

   The Windows CryptAPI cryptographic service provider stores a seed
   state variable with every user.  When CryptGenRandom is called, this
   is combined with any randomness provided in the call and with various
   system and user data such as the process ID, thread ID, system clock,
   system time, system counter, memory status, free disk clusters, and
   hashed user environment block.  This data is all fed to SHA-1, and
   the output is used to seed an RC4 key stream.  That key stream is
   used to produce the pseudo-random data requested and to update the
   user's seed state variable.

   Users of Windows ".NET" will probably find it easier to use the
   RNGCryptoServiceProvider.GetBytes method interface.

   For further information, see [WSC].

7.2.  Generators Assuming a Source of Entropy

   The pseudo-random number generators described in the following three
   sections all assume that a seed value with sufficient entropy is
   provided to them.  They then generate a strong sequence (see Section
   6.2) from that seed.

7.2.1.  X9.82 Pseudo-Random Number Generation

   The ANSI X9F1 committee is in the final stages of creating a standard
   for random number generation covering both true randomness generators
   and pseudo-random number generators.  It includes a number of
   pseudo-random number generators based on hash functions, one of which
   will probably be based on HMAC SHA hash constructs [RFC2104].  The
   draft version of this generator is described below, omitting a number
   of optional features [X9.82].

   In the subsections below, the HMAC hash construct is simply referred
   to as HMAC but, of course, a particular standard SHA function must be
   selected in an particular use.  Generally speaking, if the strength
   of the pseudo-random values to be generated is to be N bits, the SHA
   function chosen must generate N or more bits of output, and a source
   of at least N bits of input entropy will be required.  The same hash
   function must be used throughout an instantiation of this generator.
Top   ToC   Page 32  Notation

   In the following sections, the notation give below is used:

      hash_length is the output size of the underlying hash function in

      input_entropy is the input bit string that provides entropy to the

      K is a bit string of size hash_length that is part of the state of
         the generator and is updated at least once each time random
         bits are generated.

      V is a bit string of size hash_length and is part of the state of
         the generator.  It is updated each time hash_length bits of
         output are generated.

      "|" represents concatenation.  Initializing the Generator

   Set V to all zero bytes, except the low-order bit of each byte is set
      to one.

   Set K to all zero bytes, then set:

         K = HMAC ( K, V | 0x00 | input_entropy )

         V = HMAC ( K, V )

         K = HMAC ( K, V | 0x01 | input_entropy )

         V = HMAC ( K, V )

   Note: All SHA algorithms produce an integral number of bytes, so the
   lengths of K and V will be integral numbers of bytes.  Generating Random Bits

   When output is called for, simply set:

         V = HMAC ( K, V )

   and use the leading bits from V.  If more bits are needed than the
   length of V, set "temp" to a null bit string and then repeatedly
Top   ToC   Page 33
         V = HMAC ( K, V )
         temp = temp | V

   stopping as soon as temp is equal to or longer than the number of
   random bits requested.  Use the requested number of leading bits from
   temp.  The definition of the algorithm prohibits requesting more than
   2^35 bits.

   After extracting and saving the pseudo-random output bits as
   described above, before returning you must also perform two more
   HMACs as follows:

         K = HMAC ( K, V | 0x00 )
         V = HMAC ( K, V )

7.2.2.  X9.17 Key Generation

         The American National Standards Institute has specified the
         following method for generating a sequence of keys [X9.17]:

      s  is the initial 64 bit seed.

      g  is the sequence of generated 64-bit key quantities

      k is a random key reserved for generating this key sequence.

      t is the time at which a key is generated, to as fine a resolution
         as is available (up to 64 bits).

      DES ( K, Q ) is the DES encryption of quantity Q with key K.


         g    = DES ( k, DES ( k, t ) XOR s  )
          n                                n

         s    = DES ( k, DES ( k, t ) XOR  g  )
          n+1                               n

   If g sub n is to be used as a DES key, then every eighth bit should
   be adjusted for parity for that use, but the entire 64 bit unmodified
   g should be used in calculating the next s.
Top   ToC   Page 34
7.2.3.  DSS Pseudo-random Number Generation

   Appendix 3 of the NIST Digital Signature Standard [DSS] provides a
   method of producing a sequence of pseudo-random 160 bit quantities
   for use as private keys or the like.  This has been modified by
   Change Notice 1 [DSS_CN1] to produce the following algorithm for
   generating general-purpose pseudo-random numbers:

         t = 0x 67452301 EFCDAB89 98BADCFE 10325476 C3D2E1F0

         XKEY  = initial seed

         For j = 0 to ...

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

             X  = G( t, XVAL )

             XKEY   = ( 1 + XKEY  + X  ) (Mod 2^512)
                 j+1            j    j

   The quantities X thus produced are the pseudo-random sequence of
   160-bit values.  Two functions can be used for "G" above.  Each
   produces a 160-bit value and takes two arguments, a 160-bit value and
   a 512 bit value.

   The first is based on SHA-1 and works by setting the 5 linking
   variables, denoted H with subscripts in the SHA-1 specification, to
   the first argument divided into fifths.  Then steps (a) through (e)
   of section 7 of the NIST SHA-1 specification are run over the second
   argument as if it were a 512-bit data block.  The values of the
   linking variable after those steps are then concatenated to produce
   the output of G [SHA*].

   As an alternative method, NIST also defined an alternate G function
   based on multiple applications of the DES encryption function [DSS].

8.  Examples of Randomness Required

   Below are two examples showing rough calculations of randomness
   needed for security.  The first is for moderate security passwords,
   while the second assumes a need for a very high-security
   cryptographic key.
Top   ToC   Page 35
   In addition, [ORMAN] and [RSA_BULL13] provide information on the
   public key lengths that should be used for exchanging symmetric keys.

8.1.  Password Generation

   Assume that user passwords change once a year and that 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 an adversary can make at most 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 that someone could actually try continuously for a year.
   Even if log files are only checked monthly, 500,000 tries is more
   plausible before the attack is noticed and steps are 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 by using the US DoD-recommended inputs for
   password generation, as it has 8 inputs that probably average over 5
   bits of randomness each (see section 7.1).  Using a list of 1,000
   words, the password could be expressed as a three-word phrase
   (1,000,000,000 possibilities).  By using case-insensitive letters and
   digits, six characters would suffice ((26+10)^6 = 2,176,782,336

   For a higher-security password, the number of bits required goes up.
   To decrease the probability by 1,000 requires increasing the universe
   of passwords by the same factor, which adds about 10 bits.  Thus, to
   have only a one in a million chance of a password being guessed under
   the above scenario would require 39 bits of randomness and a password
   that was a four-word phrase from a 1,000 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 a ten-letter/digit

   In a real system, of course, there are other factors.  For example,
   the larger and harder to remember passwords are, the more likely
   users will bed to write them down, resulting in an additional risk of
Top   ToC   Page 36
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 also that 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 of hundred cycles
   per key.  (There is no clear lower bound on this, as computers
   operate in parallel on a number of bits and a poor encryption
   algorithm could allow many keys or even groups of keys to be tested
   in parallel.  However, we need to assume some value and can hope that
   a reasonably strong algorithm has been chosen for our hypothetical
   high-security task.)

   If the adversary can command a highly parallel processor or a large
   network of work stations, 10^11 cycles per second is probably a
   minimum assumption today.  Looking forward a few years, there should
   be at least an order of magnitude improvement.  Thus, it is
   reasonable to assume that 10^10 keys could be checked per second, or
   3.6*10^12 per hour or 6*10^14 per week, or 2.4*10^15 per month.  This
   implies a need for a minimum of 63 bits of randomness in keys, to be
   sure that they cannot be found in a month.  Even then it is possible
   that, a few years from now, a highly determined and resourceful
   adversary could break the key in 2 weeks; on average, they need try
   only half the keys.

   These questions are considered in detail in "Minimal Key Lengths for
   Symmetric Ciphers to Provide Adequate Commercial Security: A Report
   by an Ad Hoc Group of Cryptographers and Computer Scientists"
   [KeyStudy] that was sponsored by the Business Software Alliance.  It
   concluded that a reasonable key length in 1995 for very high security
   is in the range of 75 to 90 bits and, since the cost of cryptography
   does not vary much with the key size, it recommends 90 bits.  To
   update these recommendations, just add 2/3 of a bit per year for
   Moore's law [MOORE].  This translates to a determination, in the year
   2004, a reasonable key length is in the 81- to 96-bit range.  In
   fact, today, it is increasingly common to use keys longer than 96
Top   ToC   Page 37
   bits, such as 128-bit (or longer) keys with AES and keys with
   effective lengths of 112-bits with triple-DES.

8.2.2.  Meet-in-the-Middle Attacks

   If chosen or known plain text and the resulting encrypted text are
   available, a "meet-in-the-middle" attack is possible if the structure
   of the encryption algorithm allows it.  (In a known plain text
   attack, the adversary knows all or part (possibly some standard
   header or trailer fields) of the messages being encrypted.  In a
   chosen plain text attack, the adversary can force some chosen plain
   text to be encrypted, possibly by "leaking" an exciting text that is
   sent by the adversary over an encrypted channel because the text is
   so interesting.

   The following is an oversimplified explanation of the meet-in-the-
   middle attack:  the adversary can half-encrypt the known or chosen
   plain text with all possible first half-keys, sort the output, and
   then half-decrypt the encoded text with all the second half-keys.  If
   a match is found, the full key can be assembled from the halves and
   used to decrypt other parts of the message or other messages.  At its
   best, this type of attack can halve the exponent of the work required
   by the adversary while adding a very large but roughly constant
   factor of effort.  Thus, if this attack can be mounted, a doubling of
   the amount of randomness in the very strong key to a minimum of 192
   bits (96*2) is required for the year 2004, based on the [KeyStudy]

   This amount of randomness is well beyond the limit of that in the
   inputs recommended by the US DoD for password generation and could
   require user-typing timing, hardware random number generation, or
   other sources of randomness.

   The meet-in-the-middle attack assumes that the cryptographic
   algorithm can be decomposed in this way.  Hopefully no modern
   algorithm has this weakness, but there may be cases where we are not
   sure of that or even of what algorithm a key will be used with.  Even
   if a basic algorithm is not subject to a meet-in-the-middle attack,
   an attempt to produce a stronger algorithm by applying the basic
   algorithm twice (or two different algorithms sequentially) with
   different keys will gain less added security than would be expected.
   Such a composite algorithm would be subject to a meet-in-the-middle

   Enormous resources may be required to mount a meet-in-the-middle
   attack, but they are probably within the range of the national
   security services of a major nation.  Essentially all nations spy on
   other nations' traffic.
Top   ToC   Page 38
8.2.3.  Other Considerations

   [KeyStudy] also considers the possibilities of special-purpose code-
   breaking hardware and having an adequate safety margin.

   Note that key length calculations such as 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 algorithm-breaking techniques and of the strength
   of the algorithm in use could be satisfied with less than half of the
   192 bit key size derived above.

   For further examples of conservative design principles, see

9.  Conclusion

   Generation of unguessable "random" secret quantities for security use
   is an essential but difficult task.

   Hardware techniques for producing the needed entropy would be
   relatively simple.  In particular, the volume and quality would not
   need to be high, and existing computer hardware, such as audio input
   or disk drives, can be used.

   Widely-available computational techniques can process low-quality
   random quantities from multiple sources, or a larger quantity of such
   low-quality input from one source, to produce a smaller quantity of
   higher-quality keying material.  In the absence of hardware sources
   of randomness, a variety of user and software sources can frequently,
   with care, be used instead.  However, most modern systems already
   have hardware, such as disk drives or audio input, that could be used
   to produce high-quality randomness.

   Once a sufficient quantity of high-quality seed key material (a
   couple of hundred bits) is available, computational techniques are
   available to produce cryptographically-strong sequences of
   computationally-unpredictable quantities from this seed material.

10.  Security Considerations

   The entirety of this document concerns techniques and recommendations
   for generating unguessable "random" quantities for use as passwords,
   cryptographic keys, initialization vectors, sequence numbers, and
   similar security applications.
Top   ToC   Page 39
11.  Acknowledgements

   Special thanks to Paul Hoffman and John Kelsey for their extensive
   comments and to Peter Gutmann, who has permitted the incorporation of
   material from his paper "Software Generation of Practically Strong
   Random Numbers".

   The following people (in alphabetic order) have contributed
   substantially to this document:

      Steve Bellovin, Daniel Brown, Don Davis, Peter Gutmann, Tony
      Hansen, Sandy Harris, Paul Hoffman, Scott Hollenback, Russ
      Housley, Christian Huitema, John Kelsey, Mats Naslund, and Damir

   The following people (in alphabetic order) contributed to RFC 1750,
   the predecessor of this document:

      David M.  Balenson, Don T.  Davis, Carl Ellison, Marc Horowitz,
      Christian Huitema, Charlie Kaufman, Steve Kent, Hal Murray, Neil
      Haller, Richard Pitkin, Tim Redmond, and Doug Tygar.
Top   ToC   Page 40
Appendix A: Changes from RFC 1750

   1. Additional acknowledgements have been added.

   2. Insertion of section 5.3 on mixing with S-boxes.

   3. Addition of section 3.3 on Ring Oscillator randomness sources.

   4. Addition of AES and the members of the SHA series producing more
      than 160 bits.  Use of AES has been emphasized and the use of DES

   5. Addition of section 6.3 on entropy pool techniques.

   6. Addition of section 7.2.3 on the pseudo-random number generation
      techniques given in FIPS 186-2 (with Change Notice 1), 7.2.1 on
      those given in X9.82, section 7.1.2 on the random number
      generation techniques of the /dev/random device in Linux and other
      UNIX systems, and section 7.1.3 on random number generation
      techniques in the Windows operating system.

   7. Addition of references to the "Minimal Key Lengths for Symmetric
      Ciphers to Provide Adequate Commercial Security" study published
      in January 1996 [KeyStudy] and to [RFC1948].

   8. Added caveats to using Diffie-Hellman as a mixing function and,
      because of those caveats and its computationally intensive nature,
      recommend against its use.

   9. Addition of references to the X9.82 effort and the [TURBID] and
      [NASLUND] papers.

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

  11. Major restructuring, minor wording changes, and a variety of
      reference updates.
Top   ToC   Page 41
Informative References

   [AES]          "Specification of the Advanced Encryption Standard
                   (AES)", United States of America, US National
                   Institute of Standards and Technology, FIPS 197,
                   November 2001.

   [ASYMMETRIC]    Simmons, G., Ed., "Secure Communications and
                   Asymmetric Cryptosystems", AAAS Selected Symposium
                   69, ISBN 0-86531-338-5, Westview Press, 1982.

   [BBS]           Blum, L., Blum, M., and M. Shub, "A Simple
                   Unpredictable Pseudo-Random Number Generator", SIAM
                   Journal on Computing, v. 15, n. 2, 1986.

   [BRILLINGER]    Brillinger, D., "Time Series: Data Analysis and
                   Theory", Holden-Day, 1981.

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

   [DAVIS]         Davis, D., Ihaka, R., and P. Fenstermacher,
                   "Cryptographic Randomness from Air Turbulence in Disk
                   Drives", Advances in Cryptology - Crypto '94,
                   Springer-Verlag Lecture Notes in Computer Science
                   #839, 1984.

   [DES]           "Data Encryption Standard", US National Institute of
                   Standards and Technology, FIPS 46-3, October 1999.
                   Also, "Data Encryption Algorithm", American National
                   Standards Institute, ANSI X3.92-1981.  See also FIPS
                   112, "Password Usage", which includes FORTRAN code
                   for performing DES.

   [D-H]           Rescorla, E., "Diffie-Hellman Key Agreement Method",
                   RFC 2631, June 1999.

   [DNSSEC1]       Arends, R., Austein, R., Larson, M., Massey, D., and
                   S. Rose, "DNS Security Introduction and
                   Requirements", RFC 4033, March 2005.

   [DNSSEC2]       Arends, R., Austein, R., Larson, M., Massey, D., and
                   S. Rose, "Resource Records for the DNS Security
                   Extensions", RFC 4034, March 2005.

   [DNSSEC3]       Arends, R., Austein, R., Larson, M., Massey, D., and
                   S. Rose, "Protocol Modifications for the DNS Security
                   Extensions", RFC 4035, March 2005.
Top   ToC   Page 42
   [DoD]           "Password Management Guideline", United States of
                   America, Department of Defense, Computer Security
                   Center, CSC-STD-002-85, April 1885.

                   (See also "Password Usage", FIPS 112, which
                   incorporates CSC-STD-002-85 as one of its appendices.
                   FIPS 112 is currently available at:

   [DSS]           "Digital Signature Standard (DSS)", US National
                   Institute of Standards and Technology, FIPS 186-2,
                   January 2000.

   [DSS_CN1]       "Digital Signature Standard Change Notice 1", US
                   National Institute of Standards and Technology, FIPS
                   186-2 Change Notice 1, 5, October 2001.

   [FERGUSON]      Ferguson, N. and B. Schneier, "Practical
                   Cryptography",  Wiley Publishing Inc., ISBN
                   047122894X, April 2003.

   [GIFFORD]       Gifford, D., "Natural Random Number", MIT/LCS/TM-371,
                   September 1988.

   [IEEE_802.11i]  "Amendment to Standard for Telecommunications and
                   Information Exchange Between Systems - LAN/MAN
                   Specific Requirements - Part 11: Wireless Medium
                   Access Control (MAC) and physical layer (PHY)
                   specifications: Medium Access Control (MAC) Security
                   Enhancements", IEEE, January 2004.

   [IPSEC]         Kent, S. and R. Atkinson, "Security Architecture for
                   the Internet Protocol", RFC 2401, November 1998.

   [Jakobsson]     Jakobsson, M., Shriver, E., Hillyer, B., and A.
                   Juels, "A practical secure random bit generator",
                   Proceedings of the Fifth ACM Conference on Computer
                   and Communications Security, 1998.

   [KAUFMAN]       Kaufman, C., Perlman, R., and M. Speciner, "Network
                   Security:  Private Communication in a Public World",
                   Prentis Hall PTR, ISBN 0-13-046019-2, 2nd Edition
Top   ToC   Page 43
   [KeyStudy]      Blaze, M., Diffie, W., Riverst, R., Schneier, B.
                   Shimomura, T., Thompson, E., and M.  Weiner, "Minimal
                   Key Lengths for Symmetric Ciphers to Provide Adequate
                   Commercial Security: A Report by an Ad Hoc Group of
                   Cryptographers and Computer Scientists", January
                   1996.  Currently available at:

   [KNUTH]         Knuth, D., "The Art of Computer Programming", Volume
                   2:  Seminumerical Algorithms, Chapter 3: Random
                   Numbers, Addison-Wesley Publishing Company, 3rd
                   Edition, November 1997.

   [KRAWCZYK]      Krawczyk, H., "How to Predict Congruential
                   Generators", Journal of Algorithms, V. 13, N. 4,
                   December 1992.

   [LUBY]          Luby, M., "Pseudorandomness and Cryptographic
                   Applications", Princeton University Press, ISBN
                   0691025460, 8 January 1996.

   [MAIL_PEM1]     Linn, J., "Privacy Enhancement for Internet
                   Electronic Mail: Part I: Message Encryption and
                   Authentication Procedures", RFC 1421, February 1993.

   [MAIL_PEM2]     Kent, S., "Privacy Enhancement for Internet
                   Electronic Mail: Part II: Certificate-Based Key
                   Management", RFC 1422, February 1993.

   [MAIL_PEM3]     Balenson, D., "Privacy Enhancement for Internet
                   Electronic Mail: Part III: Algorithms, Modes, and
                   Identifiers", RFC 1423, February 1993.

   [MAIL_PEM4]     Kaliski, B., "Privacy Enhancement for Internet
                   Electronic Mail: Part IV: Key Certification and
                   Related Services", RFC 1424, February 1993.

   [MAIL_PGP1]     Callas, J., Donnerhacke, L., Finney, H., and R.
                   Thayer, "OpenPGP Message Format", RFC 2440, November

   [MAIL_PGP2]     Elkins, M., Del Torto, D., Levien, R., and T.
                   Roessler, "MIME Security with OpenPGP", RFC 3156,
                   August 2001.
Top   ToC   Page 44
   [S/MIME]        RFCs 2632 through 2634:

                   Ramsdell, B., "S/MIME Version 3 Certificate
                   Handling", RFC 2632, June 1999.

                   Ramsdell, B., "S/MIME Version 3 Message
                   Specification", RFC 2633, June 1999.

                   Hoffman, P., "Enhanced Security Services for S/MIME",
                   RFC 2634, June 1999.

   [MD4]           Rivest, R., "The MD4 Message-Digest Algorithm", RFC
                   1320, April 1992.

   [MD5]           Rivest, R., "The MD5 Message-Digest Algorithm ", RFC
                   1321, April 1992.

   [MODES]         "DES Modes of Operation", US National Institute of
                   Standards and Technology, FIPS 81, December 1980.
                   Also:  "Data Encryption Algorithm - Modes of
                   Operation", American National Standards Institute,
                   ANSI X3.106-1983.

   [MOORE]         Moore's Law: the exponential increase in the logic
                   density of silicon circuits.  Originally formulated
                   by Gordon Moore in 1964 as a doubling every year
                   starting in 1962, in the late 1970s the rate fell to
                   a doubling every 18 months and has remained there
                   through the date of this document.  See "The New
                   Hacker's Dictionary", Third Edition, MIT Press, ISBN
                   0-262-18178-9, Eric S.  Raymond, 1996.

   [NASLUND]       Naslund, M. and A. Russell, "Extraction of Optimally
                   Unbiased Bits from a Biased Source", IEEE
                   Transactions on Information Theory. 46(3), May 2000.

   [ORMAN]         Orman, H. and P. Hoffman, "Determining Strengths For
                   Public Keys Used For Exchanging Symmetric Keys", BCP
                   86, RFC 3766, April 2004.

   [RFC1750]       Eastlake 3rd, D., Crocker, S., and J. Schiller,
                   "Randomness Recommendations for Security", RFC 1750,
                   December 1994.

   [RFC1948]       Bellovin, S., "Defending Against Sequence Number
                   Attacks", RFC 1948, May 1996.
Top   ToC   Page 45
   [RFC2104]       Krawczyk, H., Bellare, M., and R. Canetti, "HMAC:
                   Keyed-Hashing for Message Authentication", RFC 2104,
                   February 1997.

   [RSA_BULL1]     "Suggestions for Random Number Generation in
                   Software", RSA Laboratories Bulletin #1, January

   [RSA_BULL13]    Silverman, R., "A Cost-Based Security Analysis of
                   Symmetric and Asymmetric Key Lengths", RSA
                   Laboratories Bulletin #13, April 2000 (revised
                   November 2001).

   [SBOX1]         Mister, S. and C. Adams, "Practical S-box Design",
                   Selected Areas in Cryptography, 1996.

   [SBOX2]         Nyberg, K., "Perfect Non-linear S-boxes", Advances in
                   Cryptography, Eurocrypt '91 Proceedings, Springer-
                   Verland, 1991.

   [SCHNEIER]      Schneier, B., "Applied Cryptography: Protocols,
                   Algorithms, and Source Code in C", 2nd Edition, John
                   Wiley & Sons, 1996.

   [SHANNON]       Shannon, C., "The Mathematical Theory of
                   Communication", University of Illinois Press, 1963.
                   Originally from:  Bell System Technical Journal, July
                   and October, 1948.

   [SHIFT1]        Golub, S., "Shift Register Sequences", Aegean Park
                   Press, Revised Edition, 1982.

   [SHIFT2]        Barker, W., "Cryptanalysis of Shift-Register
                   Generated Stream Cypher Systems", Aegean Park Press,

   [SHA]           "Secure Hash Standard", US National Institute of
                   Science and Technology, FIPS 180-2, 1 August 2002.

   [SHA_RFC]       Eastlake 3rd, D. and P. Jones, "US Secure Hash
                   Algorithm 1 (SHA1)", RFC 3174, September 2001.

   [SSH]           Products of the SECSH Working Group, Works in
                   Progress, 2005.

   [STERN]         Stern, J., "Secret Linear Congruential Generators are
                   not Cryptographically Secure", Proc. IEEE STOC, 1987.
Top   ToC   Page 46
   [TLS]           Dierks, T. and C. Allen, "The TLS Protocol Version
                   1.0", RFC 2246, January 1999.

   [TURBID]        Denker, J., "High Entropy Symbol Generator",
                   <>, 2003.

   [USENET_1]      Kantor, B. and P. Lapsley, "Network News Transfer
                   Protocol", RFC 977, February 1986.

   [USENET_2]      Barber, S., "Common NNTP Extensions", RFC 2980,
                   October 2000.

   [VON_NEUMANN]   Von Nuemann, J., "Various techniques used in
                   connection with random digits", Von Neumann's
                   Collected Works, Vol. 5, Pergamon Press, 1963.

   [WSC]           Howard, M. and D. LeBlanc, "Writing Secure Code,
                   Second Edition", Microsoft Press, ISBN 0735617228,
                   December 2002.

   [X9.17]         "American National Standard for Financial Institution
                   Key Management (Wholesale)", American Bankers
                   Association, 1985.

   [X9.82]         "Random Number Generation", American National
                   Standards Institute, ANSI X9F1, Work in Progress.
                      Part 1 - Overview and General Principles.
                      Part 2 - Non-Deterministic Random Bit Generators
                      Part 3 - Deterministic Random Bit Generators
Top   ToC   Page 47
Authors' Addresses

   Donald E. Eastlake 3rd
   Motorola Laboratories
   155 Beaver Street
   Milford, MA 01757 USA

   Phone: +1 508-786-7554 (w)
          +1 508-634-2066 (h)

   Jeffrey I. Schiller
   MIT, Room E40-311
   77 Massachusetts Avenue
   Cambridge, MA 02139-4307 USA

   Phone: +1 617-253-0161

   Steve Crocker

Top   ToC   Page 48
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