Cryptography-Digest Digest #823, Volume #10 Sun, 2 Jan 00 08:13:01 EST
Contents:
Re: RFC1750: Randomness Recommendations for Security (2 of 2) (Guy Macon)
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From: [EMAIL PROTECTED] (Guy Macon)
Subject: Re: RFC1750: Randomness Recommendations for Security (2 of 2)
Date: 02 Jan 2000 07:10:12 EST
Second of two parts.
RFC 1750 Randomness Recommendations for Security December 1994
as follows:
e = 2 * e * e
output input 1 input 2
Since e is never greater than 1/2, the eccentricity is always
improved except in the case where at least one input is a totally
skewed constant. This is illustrated in the following table where
the top and left side values are the two input eccentricities and the
entries are the output eccentricity:
+--------+--------+--------+--------+--------+--------+--------+
| e | 0.00 | 0.10 | 0.20 | 0.30 | 0.40 | 0.50 |
+--------+--------+--------+--------+--------+--------+--------+
| 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 0.10 | 0.00 | 0.02 | 0.04 | 0.06 | 0.08 | 0.10 |
| 0.20 | 0.00 | 0.04 | 0.08 | 0.12 | 0.16 | 0.20 |
| 0.30 | 0.00 | 0.06 | 0.12 | 0.18 | 0.24 | 0.30 |
| 0.40 | 0.00 | 0.08 | 0.16 | 0.24 | 0.32 | 0.40 |
| 0.50 | 0.00 | 0.10 | 0.20 | 0.30 | 0.40 | 0.50 |
+--------+--------+--------+--------+--------+--------+--------+
However, keep in mind that the above calculations assume that the
inputs are not correlated. If the inputs were, say, the parity of
the number of minutes from midnight on two clocks accurate to a few
seconds, then each might appear random if sampled at random intervals
much longer than a minute. Yet if they were both sampled and
combined with xor, the result would be zero most of the time.
6.1.2 Stronger Mixing Functions
The US Government Data Encryption Standard [DES] is an example of a
strong mixing function for multiple bit quantities. It takes up to
120 bits of input (64 bits of "data" and 56 bits of "key") and
produces 64 bits of output each of which is dependent on a complex
non-linear function of all input bits. Other strong encryption
functions with this characteristic can also be used by considering
them to mix all of their key and data input bits.
Another good family of mixing functions are the "message digest" or
hashing functions such as The US Government Secure Hash Standard
[SHS] and the MD2, MD4, MD5 [MD2, MD4, MD5] series. These functions
all take an arbitrary amount of input and produce an output mixing
all the input bits. The MD* series produce 128 bits of output and SHS
produces 160 bits.
Eastlake, Crocker & Schiller [Page 16]
RFC 1750 Randomness Recommendations for Security December 1994
Although the message digest functions are designed for variable
amounts of input, DES and other encryption functions can also be used
to combine any number of inputs. If 64 bits of output is adequate,
the inputs can be packed into a 64 bit data quantity and successive
56 bit keys, padding with zeros if needed, which are then used to
successively encrypt using DES in Electronic Codebook Mode [DES
MODES]. If more than 64 bits of output are needed, use more complex
mixing. For example, if inputs are packed into three quantities, A,
B, and C, use DES to encrypt A with B as a key and then with C as a
key to produce the 1st part of the output, then encrypt B with C and
then A for more output and, if necessary, encrypt C with A and then B
for yet more output. Still more output can be produced by reversing
the order of the keys given above to stretch things. The same can be
done with the hash functions by hashing various subsets of the input
data to produce multiple outputs. But keep in mind that it is
impossible to get more bits of "randomness" out than are put in.
An example of using a strong mixing function would be to reconsider
the case of a string of 308 bits each of which is biased 99% towards
zero. The parity technique given in Section 5.2.1 above reduced this
to one bit with only a 1/1000 deviance from being equally likely a
zero or one. But, applying the equation for information given in
Section 2, this 308 bit sequence has 5 bits of information in it.
Thus hashing it with SHS or MD5 and taking the bottom 5 bits of the
result would yield 5 unbiased random bits as opposed to the single
bit given by calculating the parity of the string.
6.1.3 Diffie-Hellman as a Mixing Function
Diffie-Hellman exponential key exchange is a technique that yields a
shared secret between two parties that can be made computationally
infeasible for a third party to determine even if they can observe
all the messages between the two communicating parties. This shared
secret is a mixture of initial quantities generated by each of them
[D-H]. If these initial quantities are random, then the shared
secret contains the combined randomness of them both, assuming they
are uncorrelated.
6.1.4 Using a Mixing Function to Stretch Random Bits
While it is not necessary for a mixing function to produce the same
or fewer bits than its inputs, mixing bits cannot "stretch" the
amount of random unpredictability present in the inputs. Thus four
inputs of 32 bits each where there is 12 bits worth of
unpredicatability (such as 4,096 equally probable values) in each
input cannot produce more than 48 bits worth of unpredictable output.
The output can be expanded to hundreds or thousands of bits by, for
example, mixing with successive integers, but the clever adversary's
Eastlake, Crocker & Schiller [Page 17]
RFC 1750 Randomness Recommendations for Security December 1994
search space is still 2^48 possibilities. Furthermore, mixing to
fewer bits than are input will tend to strengthen the randomness of
the output the way using Exclusive Or to produce one bit from two did
above.
The last table in Section 6.1.1 shows that mixing a random bit with a
constant bit with Exclusive Or will produce a random bit. While this
is true, it does not provide a way to "stretch" one random bit into
more than one. If, for example, a random bit is mixed with a 0 and
then with a 1, this produces a two bit sequence but it will always be
either 01 or 10. Since there are only two possible values, there is
still only the one bit of original randomness.
6.1.5 Other Factors in Choosing a Mixing Function
For local use, DES has the advantages that it has been widely tested
for flaws, is widely documented, and is widely implemented with
hardware and software implementations available all over the world
including source code available by anonymous FTP. The SHS and MD*
family are younger algorithms which have been less tested but there
is no particular reason to believe they are flawed. Both MD5 and SHS
were derived from the earlier MD4 algorithm. They all have source
code available by anonymous FTP [SHS, MD2, MD4, MD5].
DES and SHS have been vouched for the the US National Security Agency
(NSA) on the basis of criteria that primarily remain secret. While
this is the cause of much speculation and doubt, investigation of DES
over the years has indicated that NSA involvement in modifications to
its design, which originated with IBM, was primarily to strengthen
it. No concealed or special weakness has been found in DES. It is
almost certain that the NSA modification to MD4 to produce the SHS
similarly strengthened the algorithm, possibly against threats not
yet known in the public cryptographic community.
DES, SHS, MD4, and MD5 are royalty free for all purposes. MD2 has
been freely licensed only for non-profit use in connection with
Privacy Enhanced Mail [PEM]. Between the MD* algorithms, some people
believe that, as with "Goldilocks and the Three Bears", MD2 is strong
but too slow, MD4 is fast but too weak, and MD5 is just right.
Another advantage of the MD* or similar hashing algorithms over
encryption algorithms is that they are not subject to the same
regulations imposed by the US Government prohibiting the unlicensed
export or import of encryption/decryption software and hardware. The
same should be true of DES rigged to produce an irreversible hash
code but most DES packages are oriented to reversible encryption.
Eastlake, Crocker & Schiller [Page 18]
RFC 1750 Randomness Recommendations for Security December 1994
6.2 Non-Hardware Sources of Randomness
The best source of input for mixing would be a hardware randomness
such as disk drive timing affected by air turbulence, audio input
with thermal noise, or radioactive decay. However, if that is not
available there are other possibilities. These include system
clocks, system or input/output buffers, user/system/hardware/network
serial numbers and/or addresses and timing, and user input.
Unfortunately, any of these sources can produce limited or
predicatable values under some circumstances.
Some of the sources listed above would be quite strong on multi-user
systems where, in essence, each user of the system is a source of
randomness. However, on a small single user system, such as a
typical IBM PC or Apple Macintosh, it might be possible for an
adversary to assemble a similar configuration. This could give the
adversary inputs to the mixing process that were sufficiently
correlated to those used originally as to make exhaustive search
practical.
The use of multiple random inputs with a strong mixing function is
recommended and can overcome weakness in any particular input. For
example, the timing and content of requested "random" user keystrokes
can yield hundreds of random bits but conservative assumptions need
to be made. For example, assuming a few bits of randomness if the
inter-keystroke interval is unique in the sequence up to that point
and a similar assumption if the key hit is unique but assuming that
no bits of randomness are present in the initial key value or if the
timing or key value duplicate previous values. The results of mixing
these timings and characters typed could be further combined with
clock values and other inputs.
This strategy may make practical portable code to produce good random
numbers for security even if some of the inputs are very weak on some
of the target systems. However, it may still fail against a high
grade attack on small single user systems, especially if the
adversary has ever been able to observe the generation process in the
past. A hardware based random source is still preferable.
6.3 Cryptographically Strong Sequences
In cases where a series of random quantities must be generated, an
adversary may learn some values in the sequence. In general, they
should not be able to predict other values from the ones that they
know.
Eastlake, Crocker & Schiller [Page 19]
RFC 1750 Randomness Recommendations for Security December 1994
The correct technique is to start with a strong random seed, take
cryptographically strong steps from that seed [CRYPTO2, CRYPTO3], and
do not reveal the complete state of the generator in the sequence
elements. If each value in the sequence can be calculated in a fixed
way from the previous value, then when any value is compromised, all
future values can be determined. This would be the case, for
example, if each value were a constant function of the previously
used values, even if the function were a very strong, non-invertible
message digest function.
It should be noted that if your technique for generating a sequence
of key values is fast enough, it can trivially be used as the basis
for a confidentiality system. If two parties use the same sequence
generating technique and start with the same seed material, they will
generate identical sequences. These could, for example, be xor'ed at
one end with data being send, encrypting it, and xor'ed with this
data as received, decrypting it due to the reversible properties of
the xor operation.
6.3.1 Traditional Strong Sequences
A traditional way to achieve a strong sequence has been to have the
values be produced by hashing the quantities produced by
concatenating the seed with successive integers or the like and then
mask the values obtained so as to limit the amount of generator state
available to the adversary.
It may also be possible to use an "encryption" algorithm with a
random key and seed value to encrypt and feedback some or all of the
output encrypted value into the value to be encrypted for the next
iteration. Appropriate feedback techniques will usually be
recommended with the encryption algorithm. An example is shown below
where shifting and masking are used to combine the cypher output
feedback. This type of feedback is recommended by the US Government
in connection with DES [DES MODES].
Eastlake, Crocker & Schiller [Page 20]
RFC 1750 Randomness Recommendations for Security December 1994
+---------------+
| V |
| | n |
+--+------------+
| | +---------+
| +---------> | | +-----+
+--+ | Encrypt | <--- | Key |
| +-------- | | +-----+
| | +---------+
V V
+------------+--+
| V | |
| n+1 |
+---------------+
Note that if a shift of one is used, this is the same as the shift
register technique described in Section 3 above but with the all
important difference that the feedback is determined by a complex
non-linear function of all bits rather than a simple linear or
polynomial combination of output from a few bit position taps.
It has been shown by Donald W. Davies that this sort of shifted
partial output feedback significantly weakens an algorithm compared
will feeding all of the output bits back as input. In particular,
for DES, repeated encrypting a full 64 bit quantity will give an
expected repeat in about 2^63 iterations. Feeding back anything less
than 64 (and more than 0) bits will give an expected repeat in
between 2**31 and 2**32 iterations!
To predict values of a sequence from others when the sequence was
generated by these techniques is equivalent to breaking the
cryptosystem or inverting the "non-invertible" hashing involved with
only partial information available. The less information revealed
each iteration, the harder it will be for an adversary to predict the
sequence. Thus it is best to use only one bit from each value. It
has been shown that in some cases this makes it impossible to break a
system even when the cryptographic system is invertible and can be
broken if all of each generated value was revealed.
6.3.2 The Blum Blum Shub Sequence Generator
Currently the generator which has the strongest public proof of
strength is called the Blum Blum Shub generator after its inventors
[BBS]. It is also very simple and is based on quadratic residues.
It's only disadvantage is that is is computationally intensive
compared with the traditional techniques give in 6.3.1 above. This
is not a serious draw back if it is used for moderately infrequent
purposes, such as generating session keys.
Eastlake, Crocker & Schiller [Page 21]
RFC 1750 Randomness Recommendations for Security December 1994
Simply choose two large prime numbers, say p and q, which both have
the property that you get a remainder of 3 if you divide them by 4.
Let n = p * q. Then you choose a random number x relatively prime to
n. The initial seed for the generator and the method for calculating
subsequent values are then
2
s = ( x )(Mod n)
0
2
s = ( s )(Mod n)
i+1 i
You must be careful to use only a few bits from the bottom of each s.
It is always safe to use only the lowest order bit. If you use no
more than the
log ( log ( s ) )
2 2 i
low order bits, then predicting any additional bits from a sequence
generated in this manner is provable as hard as factoring n. As long
as the initial x is secret, you can even make n public if you want.
An intersting characteristic of this generator is that you can
directly calculate any of the s values. In particular
i
( ( 2 )(Mod (( p - 1 ) * ( q - 1 )) ) )
s = ( s )(Mod n)
i 0
This means that in applications where many keys are generated in this
fashion, it is not necessary to save them all. Each key can be
effectively indexed and recovered from that small index and the
initial s and n.
7. Key Generation Standards
Several public standards are now in place for the generation of keys.
Two of these are described below. Both use DES but any equally
strong or stronger mixing function could be substituted.
Eastlake, Crocker & Schiller [Page 22]
RFC 1750 Randomness Recommendations for Security December 1994
7.1 US DoD Recommendations for Password Generation
The United States Department of Defense has specific recommendations
for password generation [DoD]. They suggest using the US Data
Encryption Standard [DES] in Output Feedback Mode [DES MODES] as
follows:
use an initialization vector determined from
the system clock,
system ID,
user ID, and
date and time;
use a key determined from
system interrupt registers,
system status registers, and
system counters; and,
as plain text, use an external randomly generated 64 bit
quantity such as 8 characters typed in by a system
administrator.
The password can then be calculated from the 64 bit "cipher text"
generated in 64-bit Output Feedback Mode. As many bits as are needed
can be taken from these 64 bits and expanded into a pronounceable
word, phrase, or other format if a human being needs to remember the
password.
7.2 X9.17 Key Generation
The American National Standards Institute has specified a method for
generating a sequence of keys as follows:
s is the initial 64 bit seed
0
g is the sequence of generated 64 bit key quantities
n
k is a random key reserved for generating this key sequence
t is the time at which a key is generated to as fine a resolution
as is available (up to 64 bits).
DES ( K, Q ) is the DES encryption of quantity Q with key K
Eastlake, Crocker & Schiller [Page 23]
RFC 1750 Randomness Recommendations for Security December 1994
g = DES ( k, DES ( k, t ) .xor. s )
n n
s = DES ( k, DES ( k, t ) .xor. g )
n+1 n
If g sub n is to be used as a DES key, then every eighth bit should
be adjusted for parity for that use but the entire 64 bit unmodified
g should be used in calculating the next s.
8. Examples of Randomness Required
Below are two examples showing rough calculations of needed
randomness for security. The first is for moderate security
passwords while the second assumes a need for a very high security
cryptographic key.
8.1 Password Generation
Assume that user passwords change once a year and it is desired that
the probability that an adversary could guess the password for a
particular account be less than one in a thousand. Further assume
that sending a password to the system is the only way to try a
password. Then the crucial question is how often an adversary can
try possibilities. Assume that delays have been introduced into a
system so that, at most, an adversary can make one password try every
six seconds. That's 600 per hour or about 15,000 per day or about
5,000,000 tries in a year. Assuming any sort of monitoring, it is
unlikely someone could actually try continuously for a year. In
fact, even if log files are only checked monthly, 500,000 tries is
more plausible before the attack is noticed and steps taken to change
passwords and make it harder to try more passwords.
To have a one in a thousand chance of guessing the password in
500,000 tries implies a universe of at least 500,000,000 passwords or
about 2^29. Thus 29 bits of randomness are needed. This can probably
be achieved using the US DoD recommended inputs for password
generation as it has 8 inputs which probably average over 5 bits of
randomness each (see section 7.1). Using a list of 1000 words, the
password could be expressed as a three word phrase (1,000,000,000
possibilities) or, using case insensitive letters and digits, six
would suffice ((26+10)^6 = 2,176,782,336 possibilities).
For a higher security password, the number of bits required goes up.
To decrease the probability by 1,000 requires increasing the universe
of passwords by the same factor which adds about 10 bits. Thus to
have only a one in a million chance of a password being guessed under
the above scenario would require 39 bits of randomness and a password
Eastlake, Crocker & Schiller [Page 24]
RFC 1750 Randomness Recommendations for Security December 1994
that was a four word phrase from a 1000 word list or eight
letters/digits. To go to a one in 10^9 chance, 49 bits of randomness
are needed implying a five word phrase or ten letter/digit password.
In a real system, of course, there are also other factors. For
example, the larger and harder to remember passwords are, the more
likely users are to write them down resulting in an additional risk
of compromise.
8.2 A Very High Security Cryptographic Key
Assume that a very high security key is needed for symmetric
encryption / decryption between two parties. Assume an adversary can
observe communications and knows the algorithm being used. Within
the field of random possibilities, the adversary can try key values
in hopes of finding the one in use. Assume further that brute force
trial of keys is the best the adversary can do.
8.2.1 Effort per Key Trial
How much effort will it take to try each key? For very high security
applications it is best to assume a low value of effort. Even if it
would clearly take tens of thousands of computer cycles or more to
try a single key, there may be some pattern that enables huge blocks
of key values to be tested with much less effort per key. Thus it is
probably best to assume no more than a couple hundred cycles per key.
(There is no clear lower bound on this as computers operate in
parallel on a number of bits and a poor encryption algorithm could
allow many keys or even groups of keys to be tested in parallel.
However, we need to assume some value and can hope that a reasonably
strong algorithm has been chosen for our hypothetical high security
task.)
If the adversary can command a highly parallel processor or a large
network of work stations, 2*10^10 cycles per second is probably a
minimum assumption for availability today. Looking forward just a
couple years, there should be at least an order of magnitude
improvement. Thus assuming 10^9 keys could be checked per second or
3.6*10^11 per hour or 6*10^13 per week or 2.4*10^14 per month is
reasonable. This implies a need for a minimum of 51 bits of
randomness in keys to be sure they cannot be found in a month. Even
then it is possible that, a few years from now, a highly determined
and resourceful adversary could break the key in 2 weeks (on average
they need try only half the keys).
Eastlake, Crocker & Schiller [Page 25]
RFC 1750 Randomness Recommendations for Security December 1994
8.2.2 Meet in the Middle Attacks
If chosen or known plain text and the resulting encrypted text are
available, a "meet in the middle" attack is possible if the structure
of the encryption algorithm allows it. (In a known plain text
attack, the adversary knows all or part of the messages being
encrypted, possibly some standard header or trailer fields. In a
chosen plain text attack, the adversary can force some chosen plain
text to be encrypted, possibly by "leaking" an exciting text that
would then be sent by the adversary over an encrypted channel.)
An oversimplified explanation of the meet in the middle attack is as
follows: the adversary can half-encrypt the known or chosen plain
text with all possible first half-keys, sort the output, then half-
decrypt the encoded text with all the second half-keys. If a match
is found, the full key can be assembled from the halves and used to
decrypt other parts of the message or other messages. At its best,
this type of attack can halve the exponent of the work required by
the adversary while adding a large but roughly constant factor of
effort. To be assured of safety against this, a doubling of the
amount of randomness in the key to a minimum of 102 bits is required.
The meet in the middle attack assumes that the cryptographic
algorithm can be decomposed in this way but we can not rule that out
without a deep knowledge of the algorithm. Even if a basic algorithm
is not subject to a meet in the middle attack, an attempt to produce
a stronger algorithm by applying the basic algorithm twice (or two
different algorithms sequentially) with different keys may gain less
added security than would be expected. Such a composite algorithm
would be subject to a meet in the middle attack.
Enormous resources may be required to mount a meet in the middle
attack but they are probably within the range of the national
security services of a major nation. Essentially all nations spy on
other nations government traffic and several nations are believed to
spy on commercial traffic for economic advantage.
8.2.3 Other Considerations
Since we have not even considered the possibilities of special
purpose code breaking hardware or just how much of a safety margin we
want beyond our assumptions above, probably a good minimum for a very
high security cryptographic key is 128 bits of randomness which
implies a minimum key length of 128 bits. If the two parties agree
on a key by Diffie-Hellman exchange [D-H], then in principle only
half of this randomness would have to be supplied by each party.
However, there is probably some correlation between their random
inputs so it is probably best to assume that each party needs to
Eastlake, Crocker & Schiller [Page 26]
RFC 1750 Randomness Recommendations for Security December 1994
provide at least 96 bits worth of randomness for very high security
if Diffie-Hellman is used.
This amount of randomness is beyond the limit of that in the inputs
recommended by the US DoD for password generation and could require
user typing timing, hardware random number generation, or other
sources.
It should be noted that key length calculations such at those above
are controversial and depend on various assumptions about the
cryptographic algorithms in use. In some cases, a professional with
a deep knowledge of code breaking techniques and of the strength of
the algorithm in use could be satisfied with less than half of the
key size derived above.
9. Conclusion
Generation of unguessable "random" secret quantities for security use
is an essential but difficult task.
We have shown that hardware techniques to produce such randomness
would be relatively simple. In particular, the volume and quality
would not need to be high and existing computer hardware, such as
disk drives, can be used. Computational techniques are available to
process low quality random quantities from multiple sources or a
larger quantity of such low quality input from one source and produce
a smaller quantity of higher quality, less predictable key material.
In the absence of hardware sources of randomness, a variety of user
and software sources can frequently be used instead with care;
however, most modern systems already have hardware, such as disk
drives or audio input, that could be used to produce high quality
randomness.
Once a sufficient quantity of high quality seed key material (a few
hundred bits) is available, strong computational techniques are
available to produce cryptographically strong sequences of
unpredicatable quantities from this seed material.
10. Security Considerations
The entirety of this document concerns techniques and recommendations
for generating unguessable "random" quantities for use as passwords,
cryptographic keys, and similar security uses.
Eastlake, Crocker & Schiller [Page 27]
RFC 1750 Randomness Recommendations for Security December 1994
References
[ASYMMETRIC] - Secure Communications and Asymmetric Cryptosystems,
edited by Gustavus J. Simmons, AAAS Selected Symposium 69, Westview
Press, Inc.
[BBS] - A Simple Unpredictable Pseudo-Random Number Generator, SIAM
Journal on Computing, v. 15, n. 2, 1986, L. Blum, M. Blum, & M. Shub.
[BRILLINGER] - Time Series: Data Analysis and Theory, Holden-Day,
1981, David Brillinger.
[CRC] - C.R.C. Standard Mathematical Tables, Chemical Rubber
Publishing Company.
[CRYPTO1] - Cryptography: A Primer, A Wiley-Interscience Publication,
John Wiley & Sons, 1981, Alan G. Konheim.
[CRYPTO2] - Cryptography: A New Dimension in Computer Data Security,
A Wiley-Interscience Publication, John Wiley & Sons, 1982, Carl H.
Meyer & Stephen M. Matyas.
[CRYPTO3] - Applied Cryptography: Protocols, Algorithms, and Source
Code in C, John Wiley & Sons, 1994, Bruce Schneier.
[DAVIS] - Cryptographic Randomness from Air Turbulence in Disk
Drives, Advances in Cryptology - Crypto '94, Springer-Verlag Lecture
Notes in Computer Science #839, 1984, Don Davis, Ross Ihaka, and
Philip Fenstermacher.
[DES] - Data Encryption Standard, United States of America,
Department of Commerce, National Institute of Standards and
Technology, Federal Information Processing Standard (FIPS) 46-1.
- Data Encryption Algorithm, American National Standards Institute,
ANSI X3.92-1981.
(See also FIPS 112, Password Usage, which includes FORTRAN code for
performing DES.)
[DES MODES] - DES Modes of Operation, United States of America,
Department of Commerce, National Institute of Standards and
Technology, Federal Information Processing Standard (FIPS) 81.
- Data Encryption Algorithm - Modes of Operation, American National
Standards Institute, ANSI X3.106-1983.
[D-H] - New Directions in Cryptography, IEEE Transactions on
Information Technology, November, 1976, Whitfield Diffie and Martin
E. Hellman.
Eastlake, Crocker & Schiller [Page 28]
RFC 1750 Randomness Recommendations for Security December 1994
[DoD] - Password Management Guideline, United States of America,
Department of Defense, Computer Security Center, CSC-STD-002-85.
(See also FIPS 112, Password Usage, which incorporates CSC-STD-002-85
as one of its appendices.)
[GIFFORD] - Natural Random Number, MIT/LCS/TM-371, September 1988,
David K. Gifford
[KNUTH] - The Art of Computer Programming, Volume 2: Seminumerical
Algorithms, Chapter 3: Random Numbers. Addison Wesley Publishing
Company, Second Edition 1982, Donald E. Knuth.
[KRAWCZYK] - How to Predict Congruential Generators, Journal of
Algorithms, V. 13, N. 4, December 1992, H. Krawczyk
[MD2] - The MD2 Message-Digest Algorithm, RFC1319, April 1992, B.
Kaliski
[MD4] - The MD4 Message-Digest Algorithm, RFC1320, April 1992, R.
Rivest
[MD5] - The MD5 Message-Digest Algorithm, RFC1321, April 1992, R.
Rivest
[PEM] - RFCs 1421 through 1424:
- RFC 1424, Privacy Enhancement for Internet Electronic Mail: Part
IV: Key Certification and Related Services, 02/10/1993, B. Kaliski
- RFC 1423, Privacy Enhancement for Internet Electronic Mail: Part
III: Algorithms, Modes, and Identifiers, 02/10/1993, D. Balenson
- RFC 1422, Privacy Enhancement for Internet Electronic Mail: Part
II: Certificate-Based Key Management, 02/10/1993, S. Kent
- RFC 1421, Privacy Enhancement for Internet Electronic Mail: Part I:
Message Encryption and Authentication Procedures, 02/10/1993, J. Linn
[SHANNON] - The Mathematical Theory of Communication, University of
Illinois Press, 1963, Claude E. Shannon. (originally from: Bell
System Technical Journal, July and October 1948)
[SHIFT1] - Shift Register Sequences, Aegean Park Press, Revised
Edition 1982, Solomon W. Golomb.
[SHIFT2] - Cryptanalysis of Shift-Register Generated Stream Cypher
Systems, Aegean Park Press, 1984, Wayne G. Barker.
[SHS] - Secure Hash Standard, United States of American, National
Institute of Science and Technology, Federal Information Processing
Standard (FIPS) 180, April 1993.
[STERN] - Secret Linear Congruential Generators are not
Cryptograhically Secure, Proceedings of IEEE STOC, 1987, J. Stern.
Eastlake, Crocker & Schiller [Page 29]
RFC 1750 Randomness Recommendations for Security December 1994
[VON NEUMANN] - Various techniques used in connection with random
digits, von Neumann's Collected Works, Vol. 5, Pergamon Press, 1963,
J. von Neumann.
Authors' Addresses
Donald E. Eastlake 3rd
Digital Equipment Corporation
550 King Street, LKG2-1/BB3
Littleton, MA 01460
Phone: +1 508 486 6577(w) +1 508 287 4877(h)
EMail: [EMAIL PROTECTED]
Stephen D. Crocker
CyberCash Inc.
2086 Hunters Crest Way
Vienna, VA 22181
Phone: +1 703-620-1222(w) +1 703-391-2651 (fax)
EMail: [EMAIL PROTECTED]
Jeffrey I. Schiller
Massachusetts Institute of Technology
77 Massachusetts Avenue
Cambridge, MA 02139
Phone: +1 617 253 0161(w)
EMail: [EMAIL PROTECTED]
Eastlake, Crocker & Schiller [Page 30]
(End of document)
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