Network Working Group Donald E. Eastlake, 3rd
OBSOLETES RFC 1750 Jeffrey I. Schiller
Steve Crocker
Expires October 2001 April 2001
Randomness Requirements for Security
---------- ------------ --- --------
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D. Eastlake, J. Schiller, S. Crocker [Page 1]
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Abstract
Security systems today are built on increasingly strong cryptographic
algorithms that foil pattern analysis attempts. However, the security
of these systems is dependent on generating secret quantities for
passwords, cryptographic keys, and similar quantities. The use of
pseudo-random processes to generate secret quantities can result in
pseudo-security. The sophisticated attacker of these security
systems may find it easier to reproduce the environment that produced
the secret quantities, searching the resulting small set of
possibilities, than to locate the quantities in the whole of the
number space.
Choosing random quantities to foil a resourceful and motivated
adversary is surprisingly difficult. This document points out many
pitfalls in using traditional pseudo-random number generation
techniques for choosing such quantities. It recommends the use of
truly random hardware techniques and shows that the existing hardware
on many systems can be used for this purpose. It provides
suggestions to ameliorate the problem when a hardware solution is not
available. And it gives examples of how large such quantities need
to be for some applications.
Acknowledgements
Special thanks to
(1) The authors of "Minimal Key Lengths for Symmetric Ciphers to
Provide Adequate Commercial Security" which is incorporated as
Appendix A.
(2) Peter Gutmann who has permitted the incorporation into this
replacement for RFC 1750 of materila from is paper "Software
Generation of Practially Strong Random Numbers".
The following other persons (in alphabetic order) contributed to this
document:
(tbd)
The following persons (in alpahbetic order) contributed to RFC 1750,
the predeceasor 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, Doug Tygar.
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Table of Contents
Status of This Document....................................1
Abstract...................................................2
Acknowledgements...........................................2
Table of Contents..........................................3
1. Introduction............................................5
2. Requirements............................................6
3. Traditional Pseudo-Random Sequences.....................8
4. Unpredictability.......................................10
4.1 Problems with Clocks and Serial Numbers...............10
4.2 Timing and Content of External Events.................11
4.3 The Fallacy of Complex Manipulation...................11
4.4 The Fallacy of Selection from a Large Database........12
5. Hardware for Randomness................................13
5.1 Volume Required.......................................13
5.2 Sensitivity to Skew...................................13
5.2.1 Using Stream Parity to De-Skew......................14
5.2.2 Using Transition Mappings to De-Skew................15
5.2.3 Using FFT to De-Skew................................16
5.2.4 Using Compression to De-Skew........................16
5.3 Existing Hardware Can Be Used For Randomness..........17
5.3.1 Using Existing Sound/Video Input....................17
5.3.2 Using Existing Disk Drives..........................17
6. Recommended Non-Hardware Strategy......................18
6.1 Mixing Functions......................................18
6.1.1 A Trivial Mixing Function...........................18
6.1.2 Stronger Mixing Functions...........................19
6.1.3 Diff-Hellman as a Mixing Function...................20
6.1.4 Using a Mixing Function to Stretch Random Bits......21
6.1.5 Other Factors in Choosing a Mixing Function.........21
6.2 Non-Hardware Sources of Randomness....................22
6.3 Cryptographically Strong Sequences....................23
6.3.1 Traditional Strong Sequences........................23
6.3.2 The Blum Blum Shub Sequence Generator...............24
7. Key Generation Standards and Examples..................26
7.1 US DoD Recommendations for Password Generation........26
7.2 X9.17 Key Generation..................................26
7.3 The /dev/random Device under Linux....................27
7.4 additional example....................................28
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More Table of Contents
8. Examples of Randomness Required........................29
8.1 Password Generation..................................29
8.2 A Very High Security Cryptographic Key................30
8.2.1 Effort per Key Trial................................30
8.2.2 Meet in the Middle Attacks..........................30
9. Conclusion.............................................32
10. Security Considerations...............................32
Appendix: Minimal Secure Key Lengths Study................33
A.0 Abstract..............................................33
A.1. Encryption Plays an Essential Role in Protecting.....34
A.1.1 There is a need for information security............34
A.1.2 Encryption to protect confidentiality...............35
A.1.3 There are a variety of attackers....................36
A.1.4 Strong encryption is not expensive..................37
A.2. Brute-Forece is becoming easier......................37
A.3. 40-Bit Key Lengths Offer Virtually No Protection.....39
A.4. Even DES with 56-Bit Keys Is Increasingly Inadequate.40
A.4.1 DES is no panacea today.............................40
A.4.2 There are smarter avenues of attack than brute force41
A.4.3 Other algorithms are similar........................41
A.5. Appropriate Key Lengths for the Future --- A Proposal42
A.6 About the Authors.....................................44
A.7 Acknowledgement.......................................45
References................................................46
Authors Addresses.........................................49
File Name and Expiration..................................49
D. Eastlake, J. Schiller, S. Crocker [Page 4]
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1. Introduction
Software cryptography is coming into wider use and is continuing to
spread, although there is a long way to go until it becomes
pervasive.
Systems like IPSEC, TLS, S/MIME, PGP, DNSSEC, Kerberos, etc. are
maturing and becoming a part of the network landscape [DNSSEC, IPSEC,
MAIL*, TLS]. By comparison, when the previous version of this
document [RFC 1750] was issued in 1994, about the only cryptographic
security specification in the IETF was the Privacy Enhanced Mail
protocol [MAIL PEM].
These systems provide substantial protection against snooping and
spoofing. However, there is a potential flaw. At the heart of all
cryptographic systems is the generation of secret, unguessable (i.e.,
random) numbers.
For the present, the lack of generally available facilities for
generating such unpredictable numbers is an open wound in the design
of cryptographic software. For the software developer who wants to
build a key or password generation procedure that runs on a wide
range of hardware, the only safe strategy so far has been to force
the local installation to supply a suitable routine to generate
random numbers. To say the least, this is an awkward, error-prone
and unpalatable solution.
It is important to keep in mind that the requirement is for data that
an adversary has a very low probability of guessing or determining.
This can easily fail if pseudo-random data is used which only meets
traditional statistical tests for randomness or which is based on
limited range sources, such as clocks. Frequently such random
quantities are determinable by an adversary searching through an
embarrassingly small space of possibilities.
This informational document suggests techniques for producing random
quantities that will be resistant to such attack. It recommends that
future systems include hardware random number generation or provide
access to existing hardware that can be used for this purpose. It
suggests methods for use if such hardware is not available. And it
gives some estimates of the number of random bits required for sample
applications.
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2. Requirements
Probably the most commonly encountered randomness requirement today
is the user password. This is usually a simple character string.
Obviously, if a password can be guessed, it does not provide
security. (For re-usable passwords, it is desirable that users be
able to remember the password. This may make it advisable to use
pronounceable character strings or phrases composed on ordinary
words. But this only affects the format of the password information,
not the requirement that the password be very hard to guess.)
Many other requirements come from the cryptographic arena.
Cryptographic techniques can be used to provide a variety of services
including confidentiality and authentication. Such services are
based on quantities, traditionally called "keys", that are unknown to
and unguessable by an adversary.
In some cases, such as the use of symmetric encryption with the one
time pads [CRYPTO*] or the US Data Encryption Standard [DES] or
Advanced Encryption Standard [AES], the parties who wish to
communicate confidentially and/or with authentication must all know
the same secret key. In other cases, using what are called
asymmetric or "public key" cryptographic techniques, keys come in
pairs. One key of the pair is private and must be kept secret by one
party, the other is public and can be published to the world. It is
computationally infeasible to determine the private key from the
public key. [ASYMMETRIC, CRYPTO*]
The frequency and volume of the requirement for random quantities
differs greatly for different cryptographic systems. Using pure RSA
[CRYPTO*], random quantities are required when the key pair is
generated, but thereafter any number of messages can be signed
without any further need for randomness. The public key Digital
Signature Algorithm devused by the US National Institute of Standards
and Technology (NIST) requires good random numbers for each
signature. And encrypting with a one time pad, in principle the
strongest possible encryption technique, requires a volume of
randomness equal to all the messages to be processed.
In most of these cases, an adversary can try to determine the
"secret" key by trial and error. (This is possible as long as the
key is enough smaller than the message that the correct key can be
uniquely identified.) The probability of an adversary succeeding at
this must be made acceptably low, depending on the particular
application. The size of the space the adversary must search is
related to the amount of key "information" present in the information
theoretic sense [SHANNON]. This depends on the number of different
secret values possible and the probability of each value as follows:
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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 initially ignored by an adversary, who will search through the
more probable values first.
For example, consider a cryptographic system that uses 128 bit keys.
If these 128 bit keys are derived by using a fixed pseudo-random
number generator that is seeded with an 8 bit seed, then an adversary
needs to search through only 256 keys (by running the pseudo-random
number generator with every possible seed), not the 2^128 keys that
may at first appear to be the case. Only 8 bits of "information" are
in these 128 bit keys.
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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 classic exposition on pseudo-random numbers.
Applications he mentions are simulation of natural phenomena,
sampling, numerical analysis, testing computer programs, decision
making, and games. None of these have the same characteristics as
the sort of security uses we are talking about. Only in the last two
could there be an adversary trying to find the random quantity.
However, in these cases, the adversary normally has only a single
chance to use a guessed value. In guessing passwords or attempting
to break an encryption scheme, the adversary normally has many,
perhaps unlimited, chances at guessing the correct value 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:
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+----+ +----+ +----+ +----+
| B | <-- | B | <-- | B | <-- . . . . . . <-- | B | <-+
| 0 | | 1 | | 2 | | n | |
+----+ +----+ +----+ +----+ |
| | | |
| | V +-----+
| V +----------------> | |
V +-----------------------------> | XOR |
+---------------------------------------------------> | |
+-----+
V = ( ( V * 2 ) + B .xor. B ... )(Mod 2^n)
N+1 N 0 2
The goodness of traditional pseudo-random number generator algorithms
is measured by statistical tests on such sequences. Carefully chosen
values of the initial V and a, b, and c or the placement of shift
register tap in the above simple processes can produce excellent
statistics.
These sequences may be adequate in simulations (Monte Carlo
experiments) as long as the sequence is orthogonal to the structure
of the space being explored. Even there, subtle patterns may cause
problems. However, such sequences are clearly bad for use in
security applications. They are fully predictable if the initial
state is known. Depending on the form of the pseudo-random number
generator, the sequence may be determinable from observation of a
short portion of the sequence [CRYPTO*, STERN]. For example, with
the generators above, one can determine V(n+1) given knowledge of
V(n). In fact, it has been shown that with these techniques, even if
only one bit of the pseudo-random values 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.
[KRAWCZYK]
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4. Unpredictability
Randomness in the traditional sense described in section 3 is NOT the
same as the unpredictability required for security use.
For example, use of a widely available constant sequence, such as
that from the CRC tables, is very weak against an adversary. Once
they learn of or guess it, they can easily break all security, future
and past, based on the sequence. [CRC] Yet the statistical properties
of these tables are good.
The following sections describe the limitations of some randomness
generation techniques and sources.
4.1 Problems with Clocks and Serial Numbers
Computer clocks, or similar operating system or hardware values,
provide significantly fewer real bits of unpredictability than might
appear from their specifications.
Tests have been done on clocks on numerous systems and it was found
that their behavior can vary widely and in unexpected ways. One
version of an operating system running on one set of hardware may
actually provide, say, microsecond resolution in a clock while a
different configuration of the "same" system may always provide the
same lower bits and only count in the upper bits at much lower
resolution. This means that successive reads on the clock may
produce identical values even if enough time has passed that the
value "should" change based on the nominal clock resolution. There
are also cases where frequently reading a clock can produce
artificial sequential values because of extra code that checks for
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 a company that manfactures both computers and Ethernet
adapters will, at least internally, use its own adapters, which
significantly limits the range of built in addresses.
Problems such as those described above related to clocks and serial
numbers make code to produce unpredictable quantities difficult if
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the code is to be ported across a variety of computer platforms and
systems.
4.2 Timing and Content of External Events
It is possible to measure the timing and content of mouse movement,
key strokes, and similar user events. This is a reasonable source of
unguessable data with some qualifications. On some machines, inputs
such as key strokes are buffered. Even though the user's inter-
keystroke timing may have sufficient variation and unpredictability,
there might not be an easy way to access that variation. Another
problem is that no standard method exists to sample timing details.
This makes it hard to build standard software intended for
distribution to a large range of machines based on this technique.
The amount of mouse movement or the keys actually hit are usually
easier to access than timings but may yield less unpredictability as
the user may provide highly repetitive input.
Other external events, such as network packet arrival times, can also
be used with care. In particular, the possibility of manipulation of
such times by an adversary and the lack of history on system start up
must be considered.
4.3 The Fallacy of Complex Manipulation
One strategy which may give a misleading appearance of
unpredictability is to take a very complex algorithm (or an excellent
traditional pseudo-random number generator with good statistical
properties) and calculate a cryptographic key by starting with the
current value of a computer system clock as the seed. An adversary
who knew roughly when the generator was started would have a
relatively small number of seed values to test as they would know
likely values of the system clock. Large numbers of pseudo-random
bits could be generated but the search space an adversary would need
to check could be quite small.
Thus very strong and/or complex manipulation of data will not help if
the adversary can learn what the manipulation is and there is not
enough unpredictability in the starting seed value. Even if they can
not learn what the manipulation is, they may be able to use the
limited number of results stemming from a limited number of seed
values to defeat security.
Another serious strategy error is to assume that a very complex
pseudo-random number generation algorithm will produce strong random
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numbers when there has been no theory behind or analysis of the
algorithm. There is a excellent example of this fallacy right near
the beginning of chapter 3 in [KNUTH] where the author describes a
complex algorithm. It was intended that the machine language program
corresponding to the algorithm would be so complicated that a person
trying to read the code without comments wouldn't know what the
program was doing. Unfortunately, actual use of this algorithm
showed that it almost immediately converged to a single repeated
value in one case and a small cycle of values in another case.
Not only does complex manipulation not help you if you have a limited
range of seeds but blindly chosen complex manipulation can destroy
the randomness in a good seed!
4.4 The Fallacy of Selection from a Large Database
Another strategy that can give a misleading appearance of
unpredictability is selection of a quantity randomly from a database
and assume that its strength is related to the total number of bits
in the database. For example, typical USENET servers process many
megabytes of information per day. Assume a random quantity was
selected by fetching 32 bytes of data from a random starting point in
this data. This does not yield 32*8 = 256 bits worth of
unguessability. Even after allowing that much of the data is human
language and probably has more like 2 or 3 bits of information per
byte, it doesn't yield 32*2.5 = 80 bits of unguessability. For an
adversary with access to the same 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
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 very little. However, if a selection
can be made from data to which the adversary has no access, such as
system buffers on an active multi-user system, it may be of help.
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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 directly [GIFFORD]. This is a trivial
amount of hardware, and could easily be included as a standard part
of a computer system's architecture. Furthermore, any system with a
spinning disk or the like has an adequate source of randomness
[DAVIS]. All that's needed is the common perception among computer
vendors that this small additional hardware and the software to
access it is necessary and useful.
5.1 Volume Required
How much unpredictability is needed? Is it possible to quantify the
requirement in, say, number of random bits per second?
The answer is not very much is needed. For AES, the key can be 128
bits and, as we show in an example in Section 8, even the highest
security system is unlikely to require a keying material of much over
200 bits. 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.
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5.2.1 Using Stream Parity to De-Skew
Consider taking a sufficiently long string of bits and map the string
to "zero" or "one". The mapping will not yield a perfectly uniform
distribution, but it can be as close as desired. One mapping that
serves the purpose is to take the parity of the string. This has the
advantages that it is robust across all degrees of skew up to the
estimated maximum skew and is absolutely trivial to implement in
hardware.
The following analysis gives the number of bits that must be sampled:
Suppose the ratio of ones to zeros is 0.5 + e : 0.5 - e, where e is
between 0 and 0.5 and is a measure of the "eccentricity" of the
distribution. Consider the distribution of the parity function of N
bit samples. The probabilities that the parity will be one or zero
will be the sum of the odd or even terms in the binomial expansion of
(p + q)^N, where p = 0.5 + e, the probability of a one, and q = 0.5 -
e, the probability of a zero.
These sums can be computed easily as
N N
1/2 * ( ( p + q ) + ( p - q ) )
and
N N
1/2 * ( ( p + q ) - ( p - q ) ).
(Which one corresponds to the probability the parity will be 1
depends on whether N is odd or even.)
Since p + q = 1 and p - q = 2e, these expressions reduce to
N
1/2 * [1 + (2e) ]
and
N
1/2 * [1 - (2e) ].
Neither of these will ever be exactly 0.5 unless e is zero, but we
can bring them arbitrarily close to 0.5. If we want the
probabilities to be within some delta d of 0.5, i.e. then
N
( 0.5 + ( 0.5 * (2e) ) ) < 0.5 + d.
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.)
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The following table gives the length of the string which must be
sampled for various degrees of skew in order to come within 0.001 of
a 50/50 distribution.
+---------+--------+-------+
| Prob(1) | e | N |
+---------+--------+-------+
| 0.5 | 0.00 | 1 |
| 0.6 | 0.10 | 4 |
| 0.7 | 0.20 | 7 |
| 0.8 | 0.30 | 13 |
| 0.9 | 0.40 | 28 |
| 0.95 | 0.45 | 59 |
| 0.99 | 0.49 | 308 |
+---------+--------+-------+
The last entry shows that even if the distribution is skewed 99% in
favor of ones, the parity of a string of 308 samples will be within
0.001 of a 50/50 distribution.
5.2.2 Using Transition Mappings to De-Skew
Another technique, originally due to von Neumann [VON NEUMANN], is to
examine a bit stream as a sequence of non-overlapping pairs. You
could then discard any 00 or 11 pairs found, interpret 01 as a 0 and
10 as a 1. Assume the probability of a 1 is 0.5+e and the
probability of a 0 is 0.5-e where e is the eccentricity of the source
and described in the previous section. Then the probability of each
pair is as follows:
+------+-----------------------------------------+
| pair | probability |
+------+-----------------------------------------+
| 00 | (0.5 - e)^2 = 0.25 - e + e^2 |
| 01 | (0.5 - e)*(0.5 + e) = 0.25 - e^2 |
| 10 | (0.5 + e)*(0.5 - e) = 0.25 - e^2 |
| 11 | (0.5 + e)^2 = 0.25 + e + e^2 |
+------+-----------------------------------------+
This technique will completely eliminate any bias but at the expense
of taking an indeterminate number of input bits for any particular
desired number of output bits. The probability of any particular
pair being discarded is 0.5 + 2e^2 so the expected number of input
bits to produce X output bits is X/(0.25 - e^2).
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
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identical independent distributions. If alternate bits were from two
correlated sources, for example, the above analysis breaks down.
The above technique also provides another illustration of how a
simple statistical analysis can mislead if one is not always on the
lookout for patterns that could be exploited by an adversary. If the
algorithm were mis-read slightly so that overlapping successive bits
pairs were used instead of non-overlapping pairs, the statistical
analysis given is the same; however, instead of providing an unbiased
uncorrelated series of random 1's and 0's, it instead produces a
totally predictable sequence of exactly alternating 1's and 0's.
5.2.3 Using FFT to De-Skew
When real world data consists of strongly biased or correlated bits,
it may still contain useful amounts of randomness. This randomness
can be extracted through use of the discrete Fourier transform or its
optimized variant, the FFT.
Using the Fourier transform of the data, strong correlations can be
discarded. If adequate data is processed and remaining correlations
decay, spectral lines approaching statistical independence and
normally distributed randomness can be produced [BRILLINGER].
5.2.4 Using Compression to De-Skew
Reversible compression techniques also provide a crude method of de-
skewing a skewed bit stream. This follows directly from the
definition of reversible compression and the formula in Section 2
above for the amount of information in a sequence. Since the
compression is reversible, the same amount of information must be
present in the shorter output than was present in the longer input.
By the Shannon information equation, this is only possible if, on
average, the probabilities of the different shorter sequences are
more uniformly distributed than were the probabilities of the longer
sequences. Thus the shorter sequences are de-skewed relative to the
input.
However, many compression techniques add a somewhat predictable
preface to their output stream and may insert such a sequence again
periodically in their output or otherwise introduce subtle patterns
of their own. They should be considered only a rough technique
compared with those described 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.
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5.3 Existing Hardware Can Be Used For Randomness
As described below, many computers come with hardware that can, with
care, be used to generate truly random quantities.
5.3.1 Using Existing Sound/Video Input
Increasingly computers are being built with inputs that digitize some
real world analog source, such as sound from a microphone or video
input from a camera. Under appropriate circumstances, such input can
provide reasonably high quality random bits. The "input" from a
sound digitizer with no source plugged in or a camera with the lens
cap on, if the system has enough gain to detect anything, is
essentially thermal noise.
For example, on a SPARCstation, one can read from the /dev/audio
device with nothing plugged into the microphone jack. Such data is
essentially random noise although it should not be trusted without
some checking in case of hardware failure. It will, in any case,
need to be de-skewed as described elsewhere.
Combining this with compression to de-skew one can, in UNIXese,
generate a huge amount of medium quality random data by doing
cat /dev/audio | compress - >random-bits-file
5.3.2 Using Existing Disk Drives
Disk drives have small random fluctuations in their rotational speed
due to chaotic air turbulence [DAVIS]. By adding low level disk seek
time instrumentation to a system, a series of measurements can be
obtained that include this randomness. Such data is usually highly
correlated so that significant processing is needed, including FFT
(see section 5.2.3). Nevertheless experimentation has shown that,
with such processing, disk drives easily produce 100 bits a minute or
more of excellent random data.
Partly offsetting this need for processing is the fact that disk
drive failure will normally be rapidly noticed. Thus, problems with
this method of random number generation due to hardware failure are
unlikely.
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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 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
complexity.
6.1 Mixing Functions
A strong mixing function is one which combines two or more inputs and
produces an output where each output bit is a different complex non-
linear function of all the input bits. On average, changing any
input bit will change about half the output bits. But because the
relationship is complex and non-linear, no particular output bit is
guaranteed to change when any particular input bit is changed.
Consider the problem of converting a stream of bits that is skewed
towards 0 or 1 to a shorter stream which is more random, as discussed
in Section 5.2 above. This is simply another case where a strong
mixing function is desired, mixing the input bits to produce a
smaller number of output bits. The technique given in Section 5.2.1
of using the parity of a number of bits is simply the result of
successively Exclusive Or'ing them which is examined as a trivial
mixing function immediately below. Use of stronger mixing functions
to extract more of the randomness in a stream of skewed bits is
examined in Section 6.1.2.
6.1.1 A Trivial Mixing Function
A trivial example for single bit inputs is the Exclusive Or function,
which is equivalent to addition without carry, as show in the table
below. This is a degenerate case in which the one output bit always
changes for a change in either input bit. But, despite its
simplicity, it will still provide a useful illustration.
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+-----------+-----------+----------+
| input 1 | input 2 | output |
+-----------+-----------+----------+
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
+-----------+-----------+----------+
If inputs 1 and 2 are uncorrelated and combined in this fashion then
the output will be an even better (less skewed) random bit than the
inputs. If we assume an "eccentricity" e as defined in Section 5.2
above, then the output eccentricity relates to the input eccentricity
as follows:
e = 2 * e * e
output input 1 input 2
Since e is never greater than 1/2, the eccentricity is always
improved except in the case where at least one input is a totally
skewed constant. This is illustrated in the following table where
the top and left side values are the two input eccentricities and the
entries are the output eccentricity:
+--------+--------+--------+--------+--------+--------+--------+
| e | 0.00 | 0.10 | 0.20 | 0.30 | 0.40 | 0.50 |
+--------+--------+--------+--------+--------+--------+--------+
| 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 0.10 | 0.00 | 0.02 | 0.04 | 0.06 | 0.08 | 0.10 |
| 0.20 | 0.00 | 0.04 | 0.08 | 0.12 | 0.16 | 0.20 |
| 0.30 | 0.00 | 0.06 | 0.12 | 0.18 | 0.24 | 0.30 |
| 0.40 | 0.00 | 0.08 | 0.16 | 0.24 | 0.32 | 0.40 |
| 0.50 | 0.00 | 0.10 | 0.20 | 0.30 | 0.40 | 0.50 |
+--------+--------+--------+--------+--------+--------+--------+
However, keep in mind that the above calculations assume that the
inputs are not correlated. If the inputs were, say, the parity of
the number of minutes from midnight on two clocks accurate to a few
seconds, then each might appear random if sampled at random intervals
much longer than a minute. Yet if they were both sampled and
combined with xor, the result would be zero most of the time.
6.1.2 Stronger Mixing Functions
The US Government Advanced Encryption Standard [AES] is an example of
a strong mixing function for multiple bit quantities. It takes up to
384 bits of input (128 bits of "data" and 256 bits of "key") and
produces 128 bits of output each of which is dependent on a complex
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non-linear function of all input bits. Other encryption functions
with this characteristic, such as [DES], can also be used by
considering them to mix all of their key and data input bits.
Another good family of mixing functions are the "message digest" or
hashing functions such as The US Government Secure Hash Standards
[SHA*] and the MD4, MD5 [MD4, MD5] series. These functions all take
an arbitrary amount of input and produce an output mixing all the
input bits. The MD* series produce 128 bits of output, SHA-1 produces
160 bits, and SHA-256 and SHA-512 produce 256 and 512 bits
respectively.
Although the message digest functions are designed for variable
amounts of input, AES and other encryption functions can also be used
to combine any number of inputs. If 128 bits of output is adequate,
the inputs can be packed into a 128 bit data quantity and successive
AES keys, padding with zeros if needed, which are then used to
successively encrypt using AES in Electronic Codebook Mode [DES
MODES]. If more than 128 bits of output are needed, use more complex
mixing. For example, if inputs are packed into three quantities, A,
B, and C, use AES to encrypt A with B as a key and then with C as a
key to produce the 1st part of the output, then encrypt B with C and
then A for more output and, if necessary, encrypt C with A and then B
for yet more output. Still more output can be produced by reversing
the order of the keys given above to stretch things. The same can be
done with the hash functions by hashing various subsets of the input
data 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 skewed sequence has over 5 bits of
information in it. Thus hashing it with SHA-1 and taking the bottom
5 bits of the result would yield 5 unbiased random bits as opposed to
the single bit given by calculating the parity of the string.
6.1.3 Diff-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
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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
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, AES has the advantages that it has been widely tested
for flaws, is reasonably efficient in software, and will be widely
documented and implemented with hardware and software implementations
available all over the world including source code available on the
Internet. The SHA* family are younger algorithms but there is no
particular reason to believe they are flawed. Both SHA* and MD5 were
derived from the earlier MD4 algorithm. Some signs of weakness have
been found in MD4 and MD5. They all have source code available [SHA*,
MD*].
AES and SHA* have been vouched for the the US National Security
Agency (NSA) on the basis of criteria that primarily remain secret,
as was DES. 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
modifications to MD4 to produce the SHA* similarly strengthened these
algorithms, possibly against threats not yet known in the public
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cryptographic community.
AES, DES, SHA*, MD4, and MD5 are royalty free for all purposes.
Continued advances in crypography and computing power have cast some
doubts on MD4 and MD5 so their use is NOT RECOMMENDED.
Another advantage of the SHA* or similar hashing algorithms over
encryption algorithms in the past was 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.
6.2 Non-Hardware Sources of Randomness
The best source of input for mixing would be a hardware randomness
such as disk drive timing effected 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, especially at
start up, it might be possible for an adversary to assemble a similar
configuration. This could give the adversary inputs to the mixing
process that were sufficiently correlated to those used originally as
to make exhaustive search practical.
The use of multiple random inputs with a strong mixing function is
recommended and can overcome weakness in any particular input. 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
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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.
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] but should be avoided for reasons
described below.
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+---------------+
| 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 major draw back if it is used for moderately infrequent
purposes, such as generating session keys.
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Simply choose two large prime numbers, say p and q, which both have
the property that you get a remainder of 3 if you divide them by 4.
Let n = p * q. Then you choose a random number x relatively prime to
n. The initial seed for the generator and the method for calculating
subsequent values are then
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.
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7. Key Generation Standards and Examples
Several public standards and widely deplyed examples are now in place
for the generation of keys without special hardware. Two standards
are described below. Both use DES but any equally strong or stronger
mixing function could be substituted. Then a few widely deployed
examples are described.
7.1 US DoD Recommendations for Password Generation
The United States Department of Defense has specific recommendations
for password generation [DoD]. They suggest using the US Data
Encryption Standard [DES] in Output Feedback Mode [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
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t is the time at which a key is generated to as fine a resolution
as is available (up to 64 bits).
DES ( K, Q ) is the DES encryption of quantity Q with key K
g = DES ( k, DES ( k, t ) .xor. s )
n n
s = DES ( k, DES ( k, t ) .xor. g )
n+1 n
If g sub n is to be used as a DES key, then every eighth bit should
be adjusted for parity for that use but the entire 64 bit unmodified
g should be used in calculating the next s.
7.3 The /dev/random Device under Linux
The Linux operating system provides a Kernel resident random number
generator. This generator makes use of events captured by the Kernel
during normal system operation.
The generator consists of a random pool of bytes, by default 512
bytes (represented as 128, 4 byte integers). When an event occurs,
such as a disk drive interrupt, the time of the event is xored into
the pool and the pool is stirred via a primitive polynomial of degree
128. The pool itself is treated as a ring buffer, with new data
being xored (after stirring with the polynomial) across the entire
pool.
Each call that adds entropy to the pool estimates the amount of
likely true entropy the input contains. The pool itself contains a
accumulator that estimates the total over all entropy of the pool.
Input events come from several sources:
1. Keyboard interrupts. The time of the interrupt as well as the scan
code are added to the pool. This in effect adds entropy from the
human operator by measuring inter-keystroke arrival times.
2. Disk completion and other interrupts. A system being used by a
person will likely have a hard to predict pattern of disk
accesses.
3. Mouse motion. The timing as well as mouse position is added in.
When random bytes are required, the pool is hashed with SHA-1 [SHA1]
to yield the returned bytes of randomness. If more bytes are required
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than the output of SHA-1 (20 bytes), then the hashed output is
stirred back into the pool and a new hash performed to obtain the
next 20 bytes. As bytes are removed from the pool, the estimate of
entropy is similarly decremented.
To ensure a reasonable random pool upon system startup, the standard
Linux startup scripts (and shutdown scripts) save the pool to a disk
file at shutdown and read this file at system startup.
There are two user exported interfaces. /dev/random returns bytes
from the pool, but blocks when the estimated entropy drops to zero.
As entropy is added to the pool from events, more data becomes
available via /dev/random. Random data obtained /dev/random is
suitable for key generation for long term keys.
/dev/urandom works like /dev/random, however it provides data even
when the entropy estimate for the random pool drops to zero. This
should be fine for session keys. The risk of continuing to take data
even when the pools entropy estimate is small is that past output may
be computable from current output provided an attacker can reverse
SHA-1. Given that SHA-1 should not be invertible, this is a
reasonable risk.
To obtain random numbers under Linux, all an application needs to do
is open either /dev/random or /dev/urandom and read the desired
number of bytes.
The Linux Random device was written by Theodore Ts'o. It is based
loosely on the random number generator in PGP 2.X and PGP 3.0 (aka
PGP 5.0).
7.4 additional example
(tba)
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8. Examples of Randomness Required
Below are two examples showing rough calculations of needed
randomness for security. The first is for moderate security
passwords while the second assumes a need for a very high security
cryptographic key.
In addition [ORMAN] provides information on the public key lengths
that should be used for exchanging symmetric keys.
8.1 Password Generation
Assume that user passwords change once a year and it is desired that
the probability that an adversary could guess the password for a
particular account be less than one in a thousand. Further assume
that sending a password to the system is the only way to try a
password. Then the crucial question is how often an adversary can
try possibilities. Assume that delays have been introduced into a
system so that, at most, an adversary can make one password try every
six seconds. That's 600 per hour or about 15,000 per day or about
5,000,000 tries in a year. Assuming any sort of monitoring, it is
unlikely someone could actually try continuously for a year. In
fact, even if log files are only checked monthly, 500,000 tries is
more plausible before the attack is noticed and steps taken to change
passwords and make it harder to try more passwords.
To have a one in a thousand chance of guessing the password in
500,000 tries implies a universe of at least 500,000,000 passwords or
about 2^29. Thus 29 bits of randomness are needed. This can probably
be achieved using the US DoD recommended inputs for password
generation as it has 8 inputs which probably average over 5 bits of
randomness each (see section 7.1). Using a list of 1000 words, the
password could be expressed as a three word phrase (1,000,000,000
possibilities) or, using case insensitive letters and digits, six
would suffice ((26+10)^6 = 2,176,782,336 possibilities).
For a higher security password, the number of bits required goes up.
To decrease the probability by 1,000 requires increasing the universe
of passwords by the same factor which adds about 10 bits. Thus to
have only a one in a million chance of a password being guessed under
the above scenario would require 39 bits of randomness and a password
that was a four word phrase from a 1000 word list or eight
letters/digits. To go to a one in 10^9 chance, 49 bits of randomness
are needed implying a five word phrase or ten letter/digit password.
In a real system, of course, there are also other factors. For
example, the larger and harder to remember passwords are, the more
likely users are to write them down resulting in an additional risk
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of compromise.
8.2 A Very High Security Cryptographic Key
Assume that a very high security key is needed for symmetric
encryption / decryption between two parties. Assume an adversary can
observe communications and knows the algorithm being used. Within
the field of random possibilities, the adversary can try key values
in hopes of finding the one in use. Assume further that brute force
trial of keys is the best the adversary can do.
8.2.1 Effort per Key Trial
How much effort will it take to try each key? For very high security
applications it is best to assume a low value of effort. This
questions is considered in detail in Appendix A. It concludes 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 very
much with they key size, recommends 90 bits. To update these
recommendations, just add 2/3 of a bit per year for Moore's law
[MOORE]. Thus, in the year 2001, this translates to a determination
that a reasonable key length is in 78 to 93 bit range.
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 very strong key to a minimum of 176 bits
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is required for the year 2001 based on the Appendix A analysis.
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.
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 deepthorough 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.
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
176 bit key size derived above.
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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, initialiazation vectors, sequence numbers, and
similar security uses.
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Appendix: Minimal Secure Key Lengths Study
Minimal Key Lengths for Symmetric Ciphers
to Provide Adequate Commercial Security
A Report by an Ad Hoc Group of
Cryptographers and Computer Scientists
Matt Blaze, AT&T Research, mab@research.att.com
Whitfield Diffie, Sun Microsystems, diffie@eng.sun.com
Ronald L. Rivest, MIT LCS, rivest@lcs.mit.edu
Bruce Schneier, Counterpane Systems, schneier@counterpane.com
Tsutomu Shimomura, San Diego Supercomputer Center, tsutomu@sdsc.edu
Eric Thompson Access Data, Inc., eric@accessdata.com
Michael Wiener, Bell Northern Research, wiener@bnr.ca
January 1996
A.0 Abstract
Encryption plays an essential role in protecting the privacy of
electronic information against threats from a variety of potential
attackers. In so doing, modern cryptography employs a combination of
_conventional_ or _symmetric_ cryptographic systems for encrypting
data and _public key_ or _asymmetric_ systems for managing the _keys_
used by the symmetric systems. Assessing the strength required of
the symmetric cryptographic systems is therefore an essential step in
employing cryptography for computer and communication security.
Technology readily available today (late 1995) makes _brute-
force_ attacks against cryptographic systems considered adequate for
the past several years both fast and cheap. General purpose
computers can be used, but a much more efficient approach is to
employ commercially available _Field Programmable Gate Array (FPGA)_
technology. For attackers prepared to make a higher initial
investment, custom-made, special-purpose chips make such calculations
much faster and significantly lower the amortized cost per solution.
As a result, cryptosystems with 40-bit keys offer virtually no
protection at this point against brute-force attacks. Even the U.S.
Data Encryption Standard with 56-bit keys is increasingly inadequate.
As cryptosystems often succumb to `smarter' attacks than brute-force
key search, it is also important to remember that the keylengths
discussed here are the minimum needed for security against the
computational threats considered.
Fortunately, the cost of very strong encryption is not
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significantly greater than that of weak encryption. Therefore, to
provide adequate protection against the most serious threats ---
well-funded commercial enterprises or government intelligence
agencies --- keys used to protect data today should be at least 75
bits long. To protect information adequately for the next 20 years
in the face of expected advances in computing power, keys in newly-
deployed systems should be at least 90 bits long.
A.1. Encryption Plays an Essential Role in Protecting
the Privacy of Electronic Information
A.1.1 There is a need for information security
As we write this paper in late 1995, the development of
electronic commerce and the Global Information Infrastructure is at a
critical juncture. The dirt paths of the middle ages only became
highways of business and culture after the security of travelers and
the merchandise they carried could be assured. So too the
information superhighway will be an ill-traveled road unless
information, the goods of the Information Age, can be moved, stored,
bought, and sold securely. Neither corporations nor individuals will
entrust their private business or personal data to computer networks
unless they can assure their information's security.
Today, most forms of information can be stored and processed
electronically. This means a wide variety of information, with
varying economic values and privacy aspects and with a wide variation
in the time over which the information needs to be protected, will be
found on computer networks. Consider the spectrum:
o Electronic Funds Transfers of millions or even billions of
dollars, whose short term security is essential but whose
exposure is brief;
o A company's strategic corporate plans, whose confidentiality
must be preserved for a small number of years;
o A proprietary product (Coke formula, new drug design) that
needs to be protected over its useful life, often decades;
and
o Information private to an individual (medical condition,
employment evaluation) that may need protection for the
lifetime of the individual.
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A.1.2 Encryption to protect confidentiality
Encryption Can Provide Strong Confidentiality Protection
Encryption is accomplished by scrambling data using mathematical
procedures that make it extremely difficult and time consuming for
anyone other than authorized recipients --- those with the correct
decryption _keys_ --- to recover the _plain text_. Proper encryption
guarantees that the information will be safe even if it falls into
hostile hands.
Encryption --- and decryption --- can be performed by either
computer software or hardware. Common approaches include writing the
algorithm on a disk for execution by a computer central processor;
placing it in ROM or PROM for execution by a microprocessor; and
isolating storage and execution in a computer accessory device (smart
card or PCMCIA card).
The degree of protection obtained depends on several factors.
These include: the quality of the cryptosystem; the way it is
implemented in software or hardware (especially its reliability and
the manner in which the keys are chosen); and the total number of
possible keys that can be used to encrypt the information. A
cryptographic algorithm is considered strong if:
1. There is no shortcut that allows the opponent to recover the
plain text without using brute force to test keys until the
correct one is found; and
2. The number of possible keys is sufficiently large to make
such an attack infeasible.
The principle here is similar to that of a combination lock on a
safe. If the lock is well designed so that a burglar cannot hear or
feel its inner workings, a person who does not know the combination
can open it only by dialing one set of numbers after another until it
yields.
The sizes of encryption keys are measured in bits and the
difficulty of trying all possible keys grows exponentially with the
number of bits used. Adding one bit to the key doubles the number of
possible keys; adding ten increases it by a factor of more than a
thousand.
There is no definitive way to look at a cipher and determine
whether a shortcut exists. Nonetheless, several encryption
algorithms --- most notably the U.S Data Encryption Standard (DES)
--- have been extensively studied in the public literature and are
widely believed to be of very high quality. An essential element in
cryptographic algorithm design is thus the length of the key, whose
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size places an upper bound on the system's strength.
Throughout this paper, we will assume that there are no shortcuts
and treat the length of the key as representative of the
cryptosystem's _workfactor_ --- the minimum amount of effort required
to break the system. It is important to bear in mind, however, that
cryptographers regard this as a rash assumption and many would
recommend keys two or more times as long as needed to resist brute-
force attacks. Prudent cryptographic designs not only employ longer
keys than might appear to be needed, but devote more computation to
encrypting and decrypting. A good example of this is the popular
approach of using _triple-DES_: encrypting the output of DES twice
more, using a total of three distinct keys.
Encryption systems fall into two broad classes. Conventional or
symmetric cryptosystems --- those in which an entity with the ability
to encrypt also has the ability to decrypt and vice versa --- are the
systems under consideration in this paper. The more recent public
key or asymmetric cryptosystems have the property that the ability to
encrypt does not imply the ability to decrypt. In contemporary
cryptography, public-key systems are indispensable for managing the
keys of conventional cryptosystems. All known public key
cryptosystems, however, are subject to shortcut attacks and must
therefore use keys ten or more times the lengths of those discussed
here to achieve the an equivalent level of security.
Although computers permit electronic information to be encrypted
using very large keys, advances in computing power keep pushing up
the size of keys that can be considered large and thus keep making it
easier for individuals and organizations to attack encrypted
information without the expenditure of unreasonable resources.
A.1.3 There are a variety of attackers
There Are Threats from a Variety of Potential Attackers.
Threats to confidentiality of information come from a number of
directions and their forms depend on the resources of the attackers.
`Hackers,' who might be anything from high school students to
commercial programmers, may have access to mainframe computers or
networks of workstations. The same people can readily buy
inexpensive, off-the-shelf, boards, containing _Field Programmable
Gate Array (FPGA)_ chips that function as `programmable hardware' and
vastly increase the effectiveness of a cryptanalytic effort. A
startup company or even a well-heeled individual could afford large
numbers of these chips. A major corporation or organized crime
operation with `serious money' to spend could acquire custom computer
chips specially designed for decryption. An intelligence agency,
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engaged in espionage for national economic advantage, could build a
machine employing millions of such chips.
A.1.4 Strong encryption is not expensive
Current Technology Permits Very Strong Encryption for Effectively the
Same Cost As Weaker Encryption.
It is a property of computer encryption that modest increases in
computational cost can produce vast increases in security.
Encrypting information very securely (e.g., with 128-bit keys)
typically requires little more computing than encrypting it weakly
(e.g., with 40-bit keys). In many applications, the cryptography
itself accounts for only a small fraction of the computing costs,
compared to such processes as voice or image compression required to
prepare material for encryption.
One consequence of this uniformity of costs is that there is
rarely any need to tailor the strength of cryptography to the
sensitivity of the information being protected. Even if most of the
information in a system has neither privacy implications nor monetary
value, there is no practical or economic reason to design computer
hardware or software to provide differing levels of encryption for
different messages. It is simplest, most prudent, and thus
fundamentally most economical, to employ a uniformly high level of
encryption: the strongest encryption required for any information
that might be stored or transmitted by a secure system.
A.2. Brute-Forece is becoming easier
Readily Available Technology Makes Brute-Force Decryption Attacks
Faster and Cheaper.
The kind of hardware used to mount a brute-force attack against
an encryption algorithm depends on the scale of the cryptanalytic
operation and the total funds available to the attacking enterprise.
In the analysis that follows, we consider three general classes of
technology that are likely to be employed by attackers with differing
resources available to them. Not surprisingly, the cryptanalytic
technologies that require larger up-front investments yield the
lowest cost per recovered key, amortized over the life of the
hardware.
It is the nature of brute-force attacks that they can be
parallelized indefinitely. It is possible to use as many machines as
are available, assigning each to work on a separate part of the
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problem. Thus regardless of the technology employed, the search time
can be reduced by adding more equipment; twice as much hardware can
be expected to find the right key in half the time. The total
investment will have doubled, but if the hardware is kept constantly
busy finding keys, the cost per key recovered is unchanged.
At the low end of the technology spectrum is the use of
conventional personal computers or workstations programmed to test
keys. Many people, by virtue of already owning or having access to
the machines, are in a position use such resources at little or no
cost. However, general purpose computers --- laden with such
ancillary equipment as video controllers, keyboards, interfaces,
memory, and disk storage --- make expensive search engines. They are
therefore likely to be employed only by casual attackers who are
unable or unwilling to invest in more specialized equipment.
A more efficient technological approach is to take advantage of
commercially available Field Programmable Gate Arrays. FPGAs
function as programmable hardware and allow faster implementations of
such tasks as encryption and decryption than conventional processors.
FPGAs are a commonly used tool for simple computations that need to
be done very quickly, particularly simulating integrated circuits
during development.
FPGA technology is fast and cheap. The cost of an AT&T ORCA chip
that can test 30 million DES keys per second is $200. This is 1,000
times faster than a PC at about one-tenth the cost! FPGAs are widely
available and, mounted on cards, can be installed in standard PCs
just like sound cards, modems, or extra memory.
FPGA technology may be optimal when the same tool must be used
for attacking a variety of different cryptosystems. Often, as with
DES, a cryptosystem is sufficiently widely used to justify the
construction of more specialized facilities. In these circumstances,
the most cost-effective technology, but the one requiring the largest
initial investment, is the use of _Application-Specific Integrated
Circuits (ASICs)_. A $10 chip can test 200 million keys per second.
This is seven times faster than an FPGA chip at one-twentieth the
cost.
Because ASICs require a far greater engineering investment than
FPGAs and must be fabricated in quantity before they are economical,
this approach is only available to serious, well-funded operations
such as dedicated commercial (or criminal) enterprises and government
intelligence agencies.
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A.3. 40-Bit Key Lengths Offer Virtually No Protection
Current U.S. Government policy generally limits exportable mass
market software that incorporates encryption for confidentiality to
using the RC2 or RC4 algorithms with 40-bit keys. A 40-bit key
length means that there are 2^40 possible keys. On average, half of
these (2^39) must be tried to find the correct one. Export of other
algorithms and key lengths must be approved on a case by case basis.
For example, DES with a 56-bit key has been approved for certain
applications such as financial transactions.
The recent successful brute-force attack by two French graduate
students on Netscape's 40-bit RC4 algorithm demonstrates the dangers
of such short keys. These students at the Ecole Polytechnique in
Paris used `idle time' on the school's computers, incurring no cost
to themselves or their school. Even with these limited resources,
they were able to recover the 40-bit key in a few days.
There is no need to have the resources of an institution of
higher education at hand, however. Anyone with a modicum of computer
expertise and a few hundred dollars would be able to attack 40-bit
encryption much faster. An FPGA chip --- costing approximately $400
mounted on a card --- would on average recover a 40-bit key in five
hours. Assuming the FPGA lasts three years and is used continuously
to find keys, the average cost per key is eight cents.
A more determined commercial predator, prepared to spend $10,000
for a set-up with 25 ORCA chips, can find 40-bit keys in an average
of 12 minutes, at the same average eight cent cost. Spending more
money to buy more chips reduces the time accordingly: $300,000
results in a solution in an average of 24 seconds; $10,000,000
results in an average solution in 0.7 seconds.
As already noted, a corporation with substantial resources can
design and commission custom chips that are much faster. By doing
this, a company spending $300,000 could find the right 40-bit key in
an average of 0.18 seconds at 1/10th of a cent per solution; a larger
company or government agency willing to spend $10,000,000 could find
the right key on average in 0.005 seconds (again at 1/10th of a cent
per solution). (Note that the cost per solution remains constant
because we have conservatively assumed constant costs for chip
acquisition --- in fact increasing the quantities purchased of a
custom chip reduces the average chip cost as the initial design and
set-up costs are spread over a greater number of chips.)
These results are summarized in Table I (below).
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A.4. Even DES with 56-Bit Keys Is Increasingly Inadequate
A.4.1 DES is no panacea today
The Data Encryption Standard (DES) was developed in the 1970s by
IBM and NSA and adopted by the U.S. Government as a Federal
Information Processing Standard for data encryption. It was intended
to provide strong encryption for the government's sensitive but
unclassified information. It was recognized by many, even at the
time DES was adopted, that technological developments would make
DES's 56-bit key exceedingly vulnerable to attack before the end of
the century.
Today, DES may be the most widely employed encryption algorithm
and continues to be a commonly cited benchmark. Yet DES-like
encryption strength is no panacea. Calculations show that DES is
inadequate against a corporate or government attacker committing
serious resources. The bottom line is that DES is cheaper and easier
to break than many believe.
As explained above, 40-bit encryption provides inadequate
protection against even the most casual of intruders, content to
scavenge time on idle machines or to spend a few hundred dollars.
Against such opponents, using DES with a 56-bit key will provide a
substantial measure of security. At present, it would take a year
and a half for someone using $10,000 worth of FPGA technology to
search out a DES key. In ten years time an investment of this size
would allow one to find a DES key in less than a week.
The real threat to commercial transactions and to privacy on the
Internet is from individuals and organizations willing to invest
substantial time and money. As more and more business and personal
information becomes electronic, the potential rewards to a dedicated
commercial predator also increase significantly and may justify the
commitment of adequate resources.
A serious effort --- on the order of $300,000 --- by a legitimate
or illegitimate business could find a DES key in an average of 19
days using off-the-shelf technology and in only 3 hours using a
custom developed chip. In the latter case, it would cost $38 to find
each key (again assuming a 3 year life to the chip and continual
use). A business or government willing to spend $10,000,000 on
custom chips, could recover DES keys in an average of 6 minutes, for
the same $38 per key.
At the very high end, an organization --- presumably a government
intelligence agency --- willing to spend $300,000,000 could recover
DES keys in 12 seconds each! The investment required is large but
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not unheard of in the intelligence community. It is less than the
cost of the Glomar Explorer, built to salvage a single Russian
submarine, and far less than the cost of many spy satellites. Such
an expense might be hard to justify in attacking a single target, but
seems entirely appropriate against a cryptographic algorithm, like
DES, enjoying extensive popularity around the world.
There is ample evidence of the danger presented by government
intelligence agencies seeking to obtain information not only for
military purposes but for commercial advantage. Congressional
hearings in 1993 highlighted instances in which the French and
Japanese governments spied on behalf of their countries' own
businesses. Thus, having to protect commercial information against
such threats is not a hypothetical proposition.
A.4.2 There are smarter avenues of attack than brute force
It is easier to walk around a tree than climb up and down it.
There is no need to break the window of a house to get in if the
front door is unlocked.
Calculations regarding the strength of encryption against brute-
force attack are _worst case_ scenarios. They assume that the
ciphers are in a sense perfect and that attempts to find shortcuts
have failed. One important point is that the crudest approach ---
searching through the keys --- is entirely feasible against many
widely used systems. Another is that the keylengths we discuss are
always minimal. As discussed earlier, prudent designs might use keys
twice or three times as long to provide a margin of safety.
A.4.3 Other algorithms are similar
The Analysis for Other Algorithms Is Roughly Comparable.
The above analysis has focused on the time and money required to
find a key to decrypt information using the RC4 algorithm with a 40-
bit key or the DES algorithm with its 56-bit key, but the results are
not peculiar to these ciphers. Although each algorithm has its own
particular characteristics, the effort required to find the keys of
other ciphers is comparable. There may be some differences as the
result of implementation procedures, but these do not materially
affect the brute-force breakability of algorithms with roughly
comparable key lengths.
Specifically, it has been suggested at times that differences in
set-up procedures, such as the long key-setup process in RC4, result
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in some algorithms having effectively longer keys than others. For
the purpose of our analysis, such factors appear to vary the
effective key length by no more than about eight bits.
A.5. Appropriate Key Lengths for the Future --- A Proposal
Table I summarizes the costs of carrying out brute-force attacks
against symmetric cryptosystems with 40-bit and 56-bit keys using
networks of general purpose computers, Field Programmable Gate
Arrays, and special-purpose chips.
It shows that 56 bits provides a level of protection --- about a
year and a half --- that would be adequate for many commercial
purposes against an opponent prepared to invest $10,000. Against an
opponent prepared to invest $300,000, the period of protection has
dropped to the barest minimum of 19 days. Above this, the protection
quickly declines to negligible. A very large, but easily imaginable,
investment by an intelligence agency would clearly allow it to
recover keys in real time.
What workfactor would be required for security today? For an
opponent whose budget lay in the $10 to 300 million range, the time
required to search out keys in a 75-bit keyspace would be between 6
years and 70 days. Although the latter figure may seem comparable to
the `barest minimum' 19 days mentioned earlier, it represents ---
under our amortization assumptions --- a cost of $19 million and a
recovery rate of only five keys a year. The victims of such an
attack would have to be fat targets indeed.
Because many kinds of information must be kept confidential for
long periods of time, assessment cannot be limited to the protection
required today. Equally important, cryptosystems --- especially if
they are standards --- often remain in use for years or even decades.
DES, for example, has been in use for more than 20 years and will
probably continue to be employed for several more. In particular,
the lifetime of a cryptosystem is likely to exceed the lifetime of
any individual product embodying it.
A rough estimate of the minimum strength required as a function
of time can be obtained by applying an empirical rule, popularly
called `Moore's Law,' which holds that the computing power available
for a given cost doubles every 18 months. Taking into account both
the lifetime of cryptographic equipment and the lifetime of the
secrets it protects, we believe it is prudent to require that
encrypted data should still be secure in 20 years. Moore's Law thus
predicts that the keys should be approximately 14 bits longer than
required to protect against an attack today.
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*Bearing in mind that the additional computational costs of
stronger encryption are modest, we strongly recommend a minimum key-
length of 90 bits for symmetric cryptosystems.*
It is instructive to compare this recommendation with both
Federal Information Processing Standard 46, The Data Encryption
Standard (DES), and Federal Information Processing Standard 185, The
Escrowed Encryption Standard (EES). DES was proposed 21 years ago
and used a 56-bit key. Applying Moore's Law and adding 14 bits, we
see that the strength of DES when it was proposed in 1975 was
comparable to that of a 70-bit system today. Furthermore, it was
estimated at the time that DES was not strong enough and that keys
could be recovered at a rate of one per day for an investment of
about twenty-million dollars. Our 75-bit estimate today corresponds
to 61 bits in 1975, enough to have moved the cost of key recovery
just out of reach. The Escrowed Encryption Standard, while
unacceptable to many potential users for other reasons, embodies a
notion of appropriate key length that is similar to our own. It uses
80-bit keys, a number that lies between our figures of 75 and 90
bits.
Table I
Time and cost Length Needed
Type of Budget Tool per key recovered for protection
Attacker 40bits 56bits in Late 1995
Pedestrian Hacker
tiny scavenged 1 week infeasible 45
computer
time
$400 FPGA 5 hours 38 years 50
($0.08) ($5,000)
Small Business
$10,000 FPGA 12 minutes 556 days 55
($0.08) ($5,000)
Corporate Department
$300K FPGA 24 seconds 19 days 60
or ($0.08) ($5,000)
ASIC .18 seconds 3 hours
($0.001) ($38)
Big Company
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$10M FPGA .7 seconds 13 hours 70
or ($0.08) ($5,000)
ASIC .005 seconds 6 minutes
($0.001) ($38)
Intellegence Agency
$300M ASIC .0002 seconds 12 seconds 75
($0.001) ($38)
A.6 About the Authors
*Matt Blaze* is a senior research scientist at AT&T Research in the
area of computer security and cryptography. Recently Blaze
demonstrated weaknesses in the U.S. government's `Clipper Chip' key
escrow encryption system. His current interests include large-scale
trust management and the applications of smartcards.
*Whitfield Diffie* is a distinguished Engineer at Sun Microsystems
specializing in security. In 1976 Diffie and Martin Hellman created
public key cryptography, which solved the problem of sending coded
information between individuals with no prior relationship and is the
basis for widespread encryption in the digital information age.
*Ronald L. Rivest* is a professor of computer science at the
Massachusetts Institute of Technology, and is Associate Director of
MIT's Laboratory for Computer Science. Rivest, together with Leonard
Adleman and Adi Shamir, invented the RSA public-key cryptosystem that
is used widely throughout industry. Ron Rivest is one of the
founders of RSA Data Security Inc. and is the creator of variable key
length symmetric key ciphers (e.g., RC4).
*Bruce Schneier* is president of Counterpane Systems, a consulting
firm specializing in cryptography and computer security. Schneier
writes and speaks frequently on computer security and privacy and is
the author of a leading cryptography textbook, Applied Cryptography,
and is the creator of the symmetric key cipher Blowfish.
*Tsutomu Shimomura* is a computational physicist employed by the San
Diego Supercomputer Center who is an expert in designing software
security tools. Last year, Shimomura was responsible for tracking
down the computer outlaw Kevin Mitnick, who electronically stole and
altered valuable electronic information around the country.
*Eric Thompson* heads AccessData Corporation's cryptanalytic team and
is a frequent lecturer on applied crytography. AccessData
specializes in data recovery and decrypting information utilizing
brute force as well as `smarter' attacks. Regular clients include
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the FBI and other law enforcement agencies as well as corporations.
*Michael Wiener* is a cryptographic advisor at Bell-Northern Research
where he focuses on cryptanalysis, security architectures, and
public-key infrastructures. His influential 1993 paper, Efficient
DES Key Search, describes in detail how to construct a machine to
brute force crack DES coded information (and provides cost estimates
as well).
A.7 Acknowledgement
The [Appendix] authors would like to thank the Business Software
Alliance, which provided support for a one-day meeting, held in
Chicago on 20 November 1995.
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[AES] - "Advanced Encryption Standard", United States of America,
Department of Commerce, National Institute of Standards and
Technology, Federal Information Processing Standard (FIPS) xxx.
[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
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[CRYPTO1] - "Cryptography: A Primer", A Wiley-Interscience
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[CRYPTO2] - "Cryptography: A New Dimension in Computer Data
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Carl H. Meyer & Stephen M. Matyas.
[CRYPTO3] - "Applied Cryptography: Protocols, Algorithsm, and Source
Code in C", Second Edition, John Wiley & Sons, 1996, Bruce Schneier.
[DAVIS] - "Cryptographic Randomness from Air Turbulence in Disk
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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
D. Eastlake, J. Schiller, S. Crocker [Page 46]
INTERNET DRAFT Randomness Requirements for Security April 2001
E. Hellman.
[DNSSEC] - RFC 2535, "Domain Name System Security Extensions", D.
Eastlake, March 1999.
[DoD] - "Password Management Guideline", United States of America,
Department of Defense, Computer Security Center, CSC-STD-002-85.
(See also FIPS 112, Password Usage, which incorporates CSC-STD-002-85
as one of its appendices.)
[GIFFORD] - "Natural Random Number", MIT/LCS/TM-371, September 1988,
David K. Gifford
[IPSEC] - RFC 2401, "Security Architecture for the Internet
Protocol", S. Kent, R. Atkinson, November 1998
[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
[MAIL 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
[MAIL PGP] - RFC 2440, "OpenPGP Message Format", J. Callas, L.
Donnerhacke, H. Finney, R. Thayer", November 1998
[MAIL S/MIME] - RFC 2633, "S/MIME Version 3 Message Specification",
B. Ramsdell, Ed., June 1999.
[MD4] - "The MD4 Message-Digest Algorithm", RFC1320, April 1992, R.
Rivest
[MD5] - "The MD5 Message-Digest Algorithm", RFC1321, April 1992, R.
Rivest
[MOORE] - Moore's Law: the exponential increase the logic density of
silicon circuts. 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. Raymondm 1996.
D. Eastlake, J. Schiller, S. Crocker [Page 47]
INTERNET DRAFT Randomness Requirements for Security April 2001
[ORMAN] - "Determining Strengths For Public Keys Used For Exchanging
Symmetric Keys", draft-orman-public-key-lengths-*.txt, Hilarie Orman,
Paul Hoffman, work in progress.
[RFC 1750] - "Randomness Requirements for Security", D. Eastlake, S.
Crocker, J. Schiller, December 1994.
[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.
[SHA-1] - "Secure Hash Standard", United States of American, National
Institute of Science and Technology, Federal Information Processing
Standard (FIPS) 180-1, April 1993.
- "US Secure Hash Algorithm 1 (SHA1)", D. Eastlake, P. Jones, draft-
eastlake-sha1-01.txt, work in progress.
[SHA-256] -
[SHA-512] -
[STERN] - "Secret Linear Congruential Generators are not
Cryptograhically Secure", Proceedings of IEEE STOC, 1987, J. Stern.
[TLS] - RFC 2246, "The TLS Protocol Version 1.0", T. Dierks, C.
Allen, January 1999.
[VON NEUMANN] - "Various techniques used in connection with random
digits", von Neumann's Collected Works, Vol. 5, Pergamon Press, 1963,
J. von Neumann.
D. Eastlake, J. Schiller, S. Crocker [Page 48]
INTERNET DRAFT Randomness Requirements for Security April 2001
Authors Addresses
Donald E. Eastlake 3rd
Motorola
155 Beaver Street
Milford, MA 01757 USA
Telephone: +1 508-261-5434 (w)
+1 508-634-2066 (h)
FAX: +1 508-261-4447 (w)
EMail: Donald.Eastlake@motorola.com
Jeffrey I. Schiller
MIT Room E40-311
77 Massachusetts Avenue
Cambridge, MA 02139-4307 USA
Telephone: +1 617-253-0161
E-mail: jis@mit.edu
Steve Crocker
Longitude Systems, Inc.
Suite 100
1319 Shepard Drive
Sterling, VA 20164 USA
Telephone: +1 703-433-0808 x206
FAX: +1 202-478-0458
EMail: steve@stevecrocker.com
File Name and Expiration
This is file draft-eastlake-randomness2-02.txt.
It expires October 2001.
D. Eastlake, J. Schiller, S. Crocker [Page 49]