< draft-eastlake-randomness2-09.txt   draft-eastlake-randomness2-10.txt >
Network Working Group Donald E. Eastlake, 3rd Network Working Group Donald E. Eastlake, 3rd
OBSOLETES RFC 1750 Jeffrey I. Schiller OBSOLETES RFC 1750 Jeffrey I. Schiller
Steve Crocker Steve Crocker
Expires April 2005 October 2004 Expires July 2005 January 2005
Randomness Requirements for Security Randomness Requirements for Security
---------- ------------ --- -------- ---------- ------------ --- --------
<draft-eastlake-randomness2-09.txt> <draft-eastlake-randomness2-10.txt>
Status of This Document Status of This Document
By submitting this Internet-Draft, I certify that any applicable By submitting this Internet-Draft, I certify that any applicable
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or will be disclosed, and any of which I become aware will be or will be disclosed, and any of which I become aware will be
disclosed, in accordance with RFC 3668. disclosed, in accordance with RFC 3668.
This document is intended to become a Best Current Practice. This document is intended to become a Best Current Practice.
Comments should be sent to the authors. Distribution is unlimited. Comments should be sent to the authors. Distribution is unlimited.
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Copyright (C) The Internet Society 2004. All Rights Reserved. Copyright (C) The Internet Society 2005. All Rights Reserved.
Abstract Abstract
Security systems are built on strong cryptographic algorithms that Security systems are built on strong cryptographic algorithms that
foil pattern analysis attempts. However, the security of these foil pattern analysis attempts. However, the security of these
systems is dependent on generating secret quantities for passwords, systems is dependent on generating secret quantities for passwords,
cryptographic keys, and similar quantities. The use of pseudo-random cryptographic keys, and similar quantities. The use of pseudo-random
processes to generate secret quantities can result in pseudo- processes to generate secret quantities can result in pseudo-
security. The sophisticated attacker of these security systems may security. The sophisticated attacker of these security systems may
find it easier to reproduce the environment that produced the secret find it easier to reproduce the environment that produced the secret
quantities, searching the resulting small set of possibilities, than quantities, searching the resulting small set of possibilities, than
to locate the quantities in the whole of the potential number space. to locate the quantities in the whole of the potential number space.
Choosing random quantities to foil a resourceful and motivated Choosing random quantities to foil a resourceful and motivated
adversary is surprisingly difficult. This document points out many adversary is surprisingly difficult. This document points out many
pitfalls in using traditional pseudo-random number generation pitfalls in using poor entropy sources or traditional pseudo-random
techniques for choosing such quantities. It recommends the use of number generation techniques for generating such quantities. It
truly random hardware techniques and shows that the existing hardware recommends the use of truly random hardware techniques and shows that
on many systems can be used for this purpose. It provides suggestions the existing hardware on many systems can be used for this purpose.
to ameliorate the problem when a hardware solution is not available. It provides suggestions to ameliorate the problem when a hardware
And it gives examples of how large such quantities need to be for solution is not available. And it gives examples of how large such
some applications. quantities need to be for some applications.
Acknowledgements Acknowledgements
Special thanks to Paul Hoffman and John Kelsey for their extensive Special thanks to Paul Hoffman and John Kelsey for their extensive
comments and to Peter Gutmann, who has permitted the incorporation of comments and to Peter Gutmann, who has permitted the incorporation of
material from his paper "Software Generation of Practically Strong material from his paper "Software Generation of Practically Strong
Random Numbers". Random Numbers".
The following other persons (in alphabetic order) have also The following other persons (in alphabetic order) have also
contributed substantially to this document: contributed substantially to this document:
Daniel Brown, Don Davis, Peter Gutmann, Tony Hansen, Sandy Steve Bellovin, Daniel Brown, Don Davis, Peter Gutmann, Tony
Harris, Paul Hoffman, Scott Hollenback, Russ Housley, Christian Hansen, Sandy Harris, Paul Hoffman, Scott Hollenback, Russ
Huitema, John Kelsey, and Damir Rajnovic. Housley, Christian Huitema, John Kelsey, Mats Naslund, and Damir
Rajnovic.
The following persons (in alphabetic order) contributed to RFC 1750, The following persons (in alphabetic order) contributed to RFC 1750,
the predecessor of this document: the predecessor of this document:
David M. Balenson, Don T. Davis, Carl Ellison, Marc Horowitz, David M. Balenson, Don T. Davis, Carl Ellison, Marc Horowitz,
Christian Huitema, Charlie Kaufman, Steve Kent, Hal Murray, Neil Christian Huitema, Charlie Kaufman, Steve Kent, Hal Murray, Neil
Haller, Richard Pitkin, Tim Redmond, and Doug Tygar. Haller, Richard Pitkin, Tim Redmond, and Doug Tygar.
Table of Contents Table of Contents
Status of This Document....................................1 Status of This Document....................................1
Abstract...................................................1 Abstract...................................................1
Acknowledgements...........................................2 Acknowledgements...........................................2
Table of Contents..........................................3 Table of Contents..........................................3
1. Introduction............................................5 1. Introduction and Overview...............................5
2. General Requirements....................................6 2. General Requirements....................................6
3. Traditional Pseudo-Random Sequences.....................9 3. Entropy Sources.........................................9
3.1 Volume Required........................................9
3.2 Existing Hardware Can Be Used For Randomness..........10
3.2.1 Using Existing Sound/Video Input....................10
3.2.2 Using Existing Disk Drives..........................10
3.3 Ring Oscillator Sources...............................11
3.4 Problems with Clocks and Serial Numbers...............12
3.5 Timing and Value of External Events...................13
3.6 Non-Hardware Sources of Randomness....................14
4. Unpredictability.......................................11 4. De-skewing.............................................15
4.1 Problems with Clocks and Serial Numbers...............11 4.1 Using Stream Parity to De-Skew........................15
4.2 Timing and Value of External Events...................12 4.2 Using Transition Mappings to De-Skew..................16
4.3 The Fallacy of Complex Manipulation...................12 4.3 Using FFT to De-Skew..................................17
4.4 The Fallacy of Selection from a Large Database........13 4.4 Using Compression to De-Skew..........................18
5. Hardware for Randomness................................15 5. Mixing.................................................19
5.1 Volume Required.......................................15 5.1 A Trivial Mixing Function.............................19
5.2 Sensitivity to Skew...................................15 5.2 Stronger Mixing Functions.............................20
5.2.1 Using Stream Parity to De-Skew......................16 5.3 Using S-Boxes for Mixing..............................22
5.2.2 Using Transition Mappings to De-Skew................17 5.4 Diffie-Hellman as a Mixing Function...................22
5.2.3 Using FFT to De-Skew................................18 5.5 Using a Mixing Function to Stretch Random Bits........23
5.2.4 Using Compression to De-Skew........................18 5.6 Other Factors in Choosing a Mixing Function...........23
5.3 Existing Hardware Can Be Used For Randomness..........19
5.3.1 Using Existing Sound/Video Input....................19
5.3.2 Using Existing Disk Drives..........................19
5.4 Ring Oscillator Sources...............................20
6. Recommended Software Strategy..........................22 6. Pseudo Random Number Generators........................25
6.1 Mixing Functions......................................22 6.1 Some Bad Ideas........................................25
6.1.1 A Trivial Mixing Function...........................22 6.1.1 The Fallacy of Complex Manipulation.................25
6.1.2 Stronger Mixing Functions...........................23 6.1.2 The Fallacy of Selection from a Large Database......26
6.1.3 Using S-Boxes for Mixing............................25 6.1.3. Traditional Pseudo-Random Sequences................26
6.1.4 Diffie-Hellman as a Mixing Function.................25 6.2 Cryptographically Strong Sequences....................28
6.1.5 Using a Mixing Function to Stretch Random Bits......25 6.2.1 OFB and CTR Sequences...............................29
6.1.6 Other Factors in Choosing a Mixing Function.........26 6.2.2 The Blum Blum Shub Sequence Generator...............30
6.2 Non-Hardware Sources of Randomness....................27 6.3 Entropy Pool Techniques...............................31
6.3 Cryptographically Strong Sequences....................28
6.3.1 OFB and CTR Sequences...............................28
6.3.2 The Blum Blum Shub Sequence Generator...............29
6.3.3 Entropy Pool Techniques.............................30
7. Key Generation Examples and Standards..................32 7. Randomness Generation Examples and Standards...........33
7.1 US DoD Recommendations for Password Generation........32 7.1 Complete Randomness Generators........................33
7.2 X9.17 Key Generation..................................32 7.1.1 US DoD Recommendations for Password Generation......33
7.3 DSS Pseudo-Random Number Generation...................33 7.1.2 The /dev/random Device..............................34
7.4 X9.82 Pseudo-Random Number Generation.................34 7.1.3 Windows CryptGenRandom..............................35
7.5 The /dev/random Device................................34 7.2 Generators Assuming a Source of Entropy...............36
7.6 Windows CryptGenRandom................................36 7.2.1 X9.82 Pseudo-Random Number Generation...............36
7,2.1.1 Notation..........................................36
7.1.2.2 Initializing the Generator........................37
7.1.2.5 Generating Random Bits............................37
7.2.2 X9.17 Key Generation................................37
7.2.3 DSS Pseudo-Random Number Generation.................38
8. Examples of Randomness Required........................37 8. Examples of Randomness Required........................40
8.1 Password Generation..................................37 8.1 Password Generation..................................40
8.2 A Very High Security Cryptographic Key................38 8.2 A Very High Security Cryptographic Key................41
8.2.1 Effort per Key Trial................................38 8.2.1 Effort per Key Trial................................41
8.2.2 Meet in the Middle Attacks..........................39 8.2.2 Meet in the Middle Attacks..........................42
8.2.3 Other Considerations................................40 8.2.3 Other Considerations................................43
9. Conclusion.............................................41 9. Conclusion.............................................44
10. Security Considerations...............................42 10. Security Considerations...............................45
11. Copyright and Disclaimer..............................42 11. Copyright and Disclaimer..............................45
12. Appendix A: Changes from RFC 1750.....................43 12. Appendix A: Changes from RFC 1750.....................46
14. Informative References................................44 13. Informative References................................47
Author's Addresses........................................48 Author's Addresses........................................52
File Name and Expiration..................................48 File Name and Expiration..................................52
1. Introduction 1. Introduction and Overview
Software cryptography is coming into wider use and is continuing to Software cryptography is coming into wider use and is continuing to
spread, although there is a long way to go until it becomes spread, although there is a long way to go until it becomes
pervasive. pervasive.
Systems like SSH, IPSEC, TLS, S/MIME, PGP, DNSSEC, Kerberos, etc. are Systems like SSH, IPSEC, TLS, S/MIME, PGP, DNSSEC, Kerberos, etc. are
maturing and becoming a part of the network landscape [SSH, IPSEC, maturing and becoming a part of the network landscape [SSH, IPSEC,
MAIL*, TLS, DNSSEC]. By comparison, when the previous version of this MAIL*, TLS, DNSSEC]. By comparison, when the previous version of this
document [RFC 1750] was issued in 1994, about the only Internet document [RFC 1750] was issued in 1994, about the only Internet
cryptographic security specification in the IETF was the Privacy cryptographic security specification in the IETF was the Privacy
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strings or phrases composed on ordinary words. But this only affects strings or phrases composed on ordinary words. But this only affects
the format of the password information, not the requirement that the the format of the password information, not the requirement that the
password be very hard to guess.) password be very hard to guess.)
Many other requirements come from the cryptographic arena. Many other requirements come from the cryptographic arena.
Cryptographic techniques can be used to provide a variety of services Cryptographic techniques can be used to provide a variety of services
including confidentiality and authentication. Such services are based including confidentiality and authentication. Such services are based
on quantities, traditionally called "keys", that are unknown to and on quantities, traditionally called "keys", that are unknown to and
unguessable by an adversary. unguessable by an adversary.
Generally speaking, the above two examples also illustrate two There are even TCP/IP protocol uses for randomness in picking initial
different types of random quantities that may be wanted. In the case sequence numbers [RFC 1948].
of human usable passwords, the only important characteristic is that
it be unguessable; it is not important that they may be composed of Generally speaking, the above examples also illustrate two different
ASCII characters, for example, so the top bit of every byte is zero. types of random quantities that may be wanted. In the case of human
On the other hand, for fixed length keys and the like, you normally usable passwords, the only important characteristic is that it be
unguessable; it is not important that they may be composed of ASCII
characters, for example, so the top bit of every byte is zero. On the
other hand, for fixed length keys and the like, you normally want
quantities that are indistinguishable from truly random, that is, all quantities that are indistinguishable from truly random, that is, all
bits will pass statistical randomness tests. bits will pass statistical randomness tests.
In some cases, such as the use of symmetric encryption with the one In some cases, such as the use of symmetric encryption with the one
time pads or an algorithm like the US Advanced Encryption Standard time pads or an algorithm like the US Advanced Encryption Standard
[AES], the parties who wish to communicate confidentially and/or with [AES], the parties who wish to communicate confidentially and/or with
authentication must all know the same secret key. In other cases, authentication must all know the same secret key. In other cases,
using what are called asymmetric or "public key" cryptographic using what are called asymmetric or "public key" cryptographic
techniques, keys come in pairs. One key of the pair is private and 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 must be kept secret by one party, the other is public and can be
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The frequency and volume of the requirement for random quantities The frequency and volume of the requirement for random quantities
differs greatly for different cryptographic systems. Using pure RSA, differs greatly for different cryptographic systems. Using pure RSA,
random quantities are required only when a new key pair is generated; random quantities are required only when a new key pair is generated;
thereafter any number of messages can be signed without a further thereafter any number of messages can be signed without a further
need for randomness. The public key Digital Signature Algorithm need for randomness. The public key Digital Signature Algorithm
devised by the US National Institute of Standards and Technology devised by the US National Institute of Standards and Technology
(NIST) requires good random numbers for each signature [DSS]. And (NIST) requires good random numbers for each signature [DSS]. And
encrypting with a one time pad, in principle the strongest possible encrypting with a one time pad, in principle the strongest possible
encryption technique, requires a volume of randomness equal to all encryption technique, requires a volume of randomness equal to all
the messages to be processed. [SCHNEIER, FERGUSON, KAUFMAN] the messages to be processed. [SCHNEIER, FERGUSON, KAUFMAN]
In most of these cases, an adversary can try to determine the 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 "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 is enough smaller than the message that the correct key can be
uniquely identified.) The probability of an adversary succeeding at uniquely identified.) The probability of an adversary succeeding at
this must be made acceptably low, depending on the particular this must be made acceptably low, depending on the particular
application. The size of the space the adversary must search is application. The size of the space the adversary must search is
related to the amount of key "information" present in the information related to the amount of key "information" present in an information
theoretic sense [SHANNON]. This depends on the number of different theoretic sense [SHANNON]. This depends on the number of different
secret values possible and the probability of each value as follows: secret values possible and the probability of each value as follows:
----- -----
\ \
Bits-of-information = \ - p * log ( p ) Bits-of-information = \ - p * log ( p )
/ i 2 i / i 2 i
/ /
----- -----
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number generator that is seeded with an 8 bit seed, then an adversary 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 needs to search through only 256 keys (by running the pseudo-random
number generator with every possible seed), not the 2^128 keys that 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 may at first appear to be the case. Only 8 bits of "information" are
in these 128 bit keys. in these 128 bit keys.
While the above analysis is correct on average, it can be misleading While the above analysis is correct on average, it can be misleading
in some cases for cryptographic analysis where what is really in some cases for cryptographic analysis where what is really
important is the work factor for an adversary. For example, assume important is the work factor for an adversary. For example, assume
that there was a pseudo-random number generator generating 128 bit that there was a pseudo-random number generator generating 128 bit
keys, as in the previous paragraph, but that it generated 0 half of keys, as in the previous paragraph, but that it generated 0 half of
the time and a random selection from the remaining 2**128 - 1 values the time and a random selection from the remaining 2**128 - 1 values
the rest of the time. The Shannon equation above says that there are the rest of the time. The Shannon equation above says that there are
64 bits of information in one of these key values but an adversary, 64 bits of information in one of these key values but an adversary,
by simply trying the values 0, can break the security of half of the by simply trying the values 0, can break the security of half of the
uses, albeit a random half. Thus for cryptographic purposes, it is uses, albeit a random half. Thus for cryptographic purposes, it is
also useful to look at other measures, such as min-entropy, defined also useful to look at other measures, such as min-entropy, defined
as as
Min-entropy = - log ( maximum ( p ) ) Min-entropy = - log ( maximum ( p ) )
i i
where i is as above. Using this equation, we get 1 bit of min- where i is as above. Using this equation, we get 1 bit of min-
entropy for our new hypothetical distribution as opposed to 64 bits entropy for our new hypothetical distribution as opposed to 64 bits
of classical Shannon entropy. of classical Shannon entropy.
A continuous spectrum of entropies, sometimes called Renyi entropy, A continuous spectrum of entropies, sometimes called Renyi entropy,
have been defined, specified by a parameter r. When r = 1, it is have been defined, specified by a parameter r. When r = 1, it is
Shannon entropy, and with r = infinity, it is min-entropy. When r = Shannon entropy, and with r = infinity, it is min-entropy. When r =
0, it is just log (n) where n is the number of non-zero 0, it is just log (n) where n is the number of non-zero
probabilities. Renyi entropy is a non-increasing function of r, so probabilities. Renyi entropy is a non-increasing function of r, so
min-entropy is always the most conservative measure of entropy and min-entropy is always the most conservative measure of entropy and
usually the best to use for cryptographic evaluation. [LUBY] usually the best to use for cryptographic evaluation. [LUBY]
3. Traditional Pseudo-Random Sequences Statistically tested randomness in the traditional sense is NOT the
same as the unpredictability required for security use.
This section talks about traditional sources of deterministic of For example, use of a widely available constant sequence, such as
"pseudo-random" numbers. These typically start with a "seed" quantity that from the CRC tables, is very weak against an adversary. Once
and use numeric or logical operations to produce a sequence of they learn of or guess it, they can easily break all security, future
values. Note that none of the techniques discussed in this section is and past, based on the sequence. [CRC] As another example, using AES
suitable for cryptographic use. They are presented for general to encrypt successive integers such as 1, 2, 3 ... will also produce
information. output that has excellent statistical randomness properties but is
also predictable. On the other hand, taking successive rolls of a
six-sided die and encoding the resulting values in ASCII would
produce statistically poor output with a substantial unpredictable
component. So you should keep in mind that passing or failing
statistical tests doesn't tell you that something is unpredictable
or predictable.
[KNUTH] has a classic exposition on pseudo-random numbers. 3. Entropy Sources
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. They can store the
message they are trying to break and repeatedly attack it. They are
also be assumed to be aided by a computer.
For testing the "randomness" of numbers, Knuth suggests a variety of Entropy sources tend to be very implementation dependent. Once one
measures including statistical and spectral. These tests check things has gathered sufficient entropy it can be used as the seed to produce
like autocorrelation between different parts of a "random" sequence the required amount of cryptographically strong pseudo-randomness, as
or distribution of its values. But they could be met by a constant described in Sections 6 and 7, after being de-skewed and/or mixed if
stored random sequence, such as the "random" sequence printed in the necessary as described in Sections 4 and 5.
CRC Standard Mathematical Tables [CRC]. Despite meeting all the tests
suggested by Knuth, that sequence is unsuitable for cryptographic use
as adversaries must be assumed to have copies of all common published
"random" sequences and will able to spot the source and predict
future values.
A typical pseudo-random number generation technique, known as a Is there any hope for true strong portable randomness in the future?
linear congruence pseudo-random number generator, is modular There might be. All that's needed is a physical source of
arithmetic where the value numbered N+1 is calculated from the value unpredictable numbers.
numbered N by
V = ( V * a + b )(Mod c) A thermal noise (sometimes called Johnson noise in integrated
N+1 N circuits) 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. Most audio (or video) input
devices are useable [TURBID]. Furthermore, any system with a
spinning disk or ring oscillator and a stable (crystal) time source
or the like has an adequate source of randomness ([DAVIS] and Section
3.3). 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.
The above technique has a strong relationship to linear shift ANSI X9 is currently developing a standard which includes a part
register pseudo-random number generators, which are well understood devoted to entropy sources. See [X9.82 - Part 2].
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:
+----+ +----+ +----+ +----+ 3.1 Volume Required
| B | <-- | B | <-- | B | <-- . . . . . . <-- | B | <-+
| 0 | | 1 | | 2 | | n | |
+----+ +----+ +----+ +----+ |
| | | |
| | V +-----+
| V +----------------> | |
V +-----------------------------> | XOR |
+---------------------------------------------------> | |
+-----+
V = ( ( V * 2 ) + B .xor. B ... )(Mod 2^n) How much unpredictability is needed? Is it possible to quantify the
N+1 N 0 2 requirement in, say, number of random bits per second?
The goodness of traditional pseudo-random number generator algorithms The answer is not very much is needed. For AES, the key can be 128
is measured by statistical tests on such sequences. Carefully chosen bits and, as we show in an example in Section 8, even the highest
values a, b, c, and initial V or the placement of shift register tap security system is unlikely to require strong keying material of much
in the above simple processes can produce excellent statistics. over 200 bits. If a series of keys are needed, they can be generated
from a strong random seed (starting value) using a cryptographically
strong sequence as explained in Section 6.2. A few hundred random
bits generated at start up or 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 most high security applications to wait 200
seconds occasionally.
These sequences may be adequate in simulations (Monte Carlo These numbers are trivial to achieve. It could be done by a person
experiments) as long as the sequence is orthogonal to the structure repeatedly tossing a coin. Almost any hardware based process is
of the space being explored. Even there, subtle patterns may cause likely to be much faster.
problems. However, such sequences are clearly bad for use in security
applications. They are fully predictable if the initial state is
known. Depending on the form of the pseudo-random number generator,
the sequence may be determinable from observation of a short portion
of the sequence [SCHNEIER, STERN]. For example, with the generators
above, one can determine V(n+1) given knowledge of V(n). In fact, it
has been shown that with these techniques, even if only one bit of
the pseudo-random values are released, the seed can be determined
from short sequences.
Not only have linear congruent generators been broken, but techniques 3.2 Existing Hardware Can Be Used For Randomness
are now known for breaking all polynomial congruent generators.
[KRAWCZYK]
4. Unpredictability As described below, many computers come with hardware that can, with
care, be used to generate truly random quantities.
Statistically tested randomness in the traditional sense described in 3.2.1 Using Existing Sound/Video Input
section 3 is NOT the same as the unpredictability required for
security use.
For example, use of a widely available constant sequence, such as Many computers are built with inputs that digitize some real world
that from the CRC tables, is very weak against an adversary. Once analog source, such as sound from a microphone or video input from a
they learn of or guess it, they can easily break all security, future camera. Under appropriate circumstances, such input can provide
and past, based on the sequence. [CRC] Yet the statistical properties reasonably high quality random bits. The "input" from a sound
of these tables are good. So you should keep in mind that passing digitizer with no source plugged in or a camera with the lens cap on,
statistical tests doesn't tell you that something is unpredictable. if the system has enough gain to detect anything, is essentially
thermal noise. This method is extremely hardware and implementation
dependent.
The following sections describe the limitations of some randomness For example, on some UNIX based systems, one can read from the
generation techniques and sources. Much better sources are described /dev/audio device with nothing plugged into the microphone jack or
in Section 5. the microphone receiving only low level background noise. 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.
4.1 Problems with Clocks and Serial Numbers Combining this with compression to de-skew (see Section 4) one can,
in UNIXese, generate a huge amount of medium quality random data by
doing
cat /dev/audio | compress - >random-bits-file
A detailed examination of this type of randomness source appears in
[TURBID].
3.2.2 Using Existing Disk Drives
Disk drives have small random fluctuations in their rotational speed
due to chaotic air turbulence [DAVIS, Jakobsson]. 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, such as described in 5.2 below. Nevertheless experimentation
a decade ago showed that, with such processing, even slow disk drives
on the slower computers of that day could easily produce 100 bits a
minute or more of excellent random data.
Every increase in processor speed, which increases the resolution
with which disk motion can be timed, or increase in the rate of disk
seeks, increases the rate of random bit generation possible with this
technique. At the time of this paper and using modern hardware, a
more typical rate of random bit production would be in excess of
10,000 bits a second. This technique is used in many operating system
library random number generators.
Note: the inclusion of cache memories in disk controllers has little
effect on this technique if very short seek times, which represent
cache hits, are simply ignored.
3.3 Ring Oscillator Sources
If an integrated circuit is being designed or field programmed, an
odd number of gates can be connected in series to produce a free-
running ring oscillator. By sampling a point in the ring at a fixed
frequency, say one determined by a stable crystal oscillator, some
amount of entropy can be extracted due to variations in the free-
running oscillator timing. It is possible to increase the rate of
entropy by xor'ing sampled values from a few ring oscillators with
relatively prime lengths. It is sometimes recommended that an odd
number of rings be used so that, even if the rings somehow become
synchronously locked to each other, there will still be sampled bit
transitions. Another possibility source to sample is the output of a
noisy diode.
Sampled bits from such sources will have to be heavily de-skewed, as
disk rotation timings must be (see Section 4). An engineering study
would be needed to determine the amount of entropy being produced
depending on the particular design. In any case, these can be good
sources whose cost is a trivial amount of hardware by modern
standards.
As an example, IEEE 802.11i suggests that the circuit below be
considered, with due attention in the design to isolation of the
rings from each other and from clocked circuits to avoid undesired
synchronization, etc., and extensive post processing. [IEEE 802.11i]
|\ |\ |\
+-->| >0-->| >0-- 19 total --| >0--+-------+
| |/ |/ |/ | |
| | |
+----------------------------------+ V
+-----+
|\ |\ |\ | | output
+-->| >0-->| >0-- 23 total --| >0--+--->| XOR |------>
| |/ |/ |/ | | |
| | +-----+
+----------------------------------+ ^ ^
| |
|\ |\ |\ | |
+-->| >0-->| >0-- 29 total --| >0--+------+ |
| |/ |/ |/ | |
| | |
+----------------------------------+ |
|
other randomness if available--------------+
3.4 Problems with Clocks and Serial Numbers
Computer clocks, or similar operating system or hardware values, Computer clocks, or similar operating system or hardware values,
provide significantly fewer real bits of unpredictability than might provide significantly fewer real bits of unpredictability than might
appear from their specifications. appear from their specifications.
Tests have been done on clocks on numerous systems and it was found Tests have been done on clocks on numerous systems and it was found
that their behavior can vary widely and in unexpected ways. One that their behavior can vary widely and in unexpected ways. One
version of an operating system running on one set of hardware may version of an operating system running on one set of hardware may
actually provide, say, microsecond resolution in a clock while a actually provide, say, microsecond resolution in a clock while a
different configuration of the "same" system may always provide the different configuration of the "same" system may always provide the
skipping to change at page 11, line 45 skipping to change at page 12, line 47
identical values even if enough time has passed that the value identical values even if enough time has passed that the value
"should" change based on the nominal clock resolution. There are also "should" change based on the nominal clock resolution. There are also
cases where frequently reading a clock can produce artificial cases where frequently reading a clock can produce artificial
sequential values because of extra code that checks for the clock sequential values because of extra code that checks for the clock
being unchanged between two reads and increases it by one! Designing being unchanged between two reads and increases it by one! Designing
portable application code to generate unpredictable numbers based on portable application code to generate unpredictable numbers based on
such system clocks is particularly challenging because the system such system clocks is particularly challenging because the system
designer does not always know the properties of the system clocks designer does not always know the properties of the system clocks
that the code will execute on. that the code will execute on.
Use of hardware serial numbers such as an Ethernet addresses may also Use of hardware serial numbers such as an Ethernet MAC addresses may
provide fewer bits of uniqueness than one would guess. Such also provide fewer bits of uniqueness than one would guess. Such
quantities are usually heavily structured and subfields may have only quantities are usually heavily structured and subfields may have only
a limited range of possible values or values easily guessable based a limited range of possible values or values easily guessable based
on approximate date of manufacture or other data. For example, it is on approximate date of manufacture or other data. For example, it is
likely that a company that manufactures both computers and Ethernet likely that a company that manufactures both computers and Ethernet
adapters will, at least internally, use its own adapters, which adapters will, at least internally, use its own adapters, which
significantly limits the range of built-in addresses. significantly limits the range of built-in addresses.
Problems such as those described above related to clocks and serial Problems such as those described above related to clocks and serial
numbers make code to produce unpredictable quantities difficult if numbers make code to produce unpredictable quantities difficult if
the code is to be ported across a variety of computer platforms and the code is to be ported across a variety of computer platforms and
systems. systems.
4.2 Timing and Value of External Events 3.5 Timing and Value of External Events
It is possible to measure the timing and content of mouse movement, It is possible to measure the timing and content of mouse movement,
key strokes, and similar user events. This is a reasonable source of key strokes, and similar user events. This is a reasonable source of
unguessable data with some qualifications. On some machines, inputs unguessable data with some qualifications. On some machines, inputs
such as key strokes are buffered. Even though the user's inter- such as key strokes are buffered. Even though the user's inter-
keystroke timing may have sufficient variation and unpredictability, keystroke timing may have sufficient variation and unpredictability,
there might not be an easy way to access that variation. Another there might not be an easy way to access that variation. Another
problem is that no standard method exists to sample timing details. problem is that no standard method exists to sample timing details.
This makes it hard to build standard software intended for This makes it hard to build standard software intended for
distribution to a large range of machines based on this technique. distribution to a large range of machines based on this technique.
skipping to change at page 12, line 46 skipping to change at page 14, line 5
how much such data is subject to adversarial manipulation and to how how much such data is subject to adversarial manipulation and to how
much entropy it can actually provide. much entropy it can actually provide.
The above techniques are quite powerful against any attackers having The above techniques are quite powerful against any attackers having
no access to the quantities being measured. For example, they would no access to the quantities being measured. For example, they would
be powerful against offline attackers who had no access to your be powerful against offline attackers who had no access to your
environment and were trying to crack your random seed after the fact. environment and were trying to crack your random seed after the fact.
In all cases, the more accurately you can measure the timing or value In all cases, the more accurately you can measure the timing or value
of an external sensor, the more rapidly you can generate bits. of an external sensor, the more rapidly you can generate bits.
4.3 The Fallacy of Complex Manipulation 3.6 Non-Hardware Sources of Randomness
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
limited data such as the computer system clock value 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. They can usually
use the limited number of results stemming from a limited number of
seed values to defeat security.
Another serious strategy error is to assume that a very complex
pseudo-random number generation algorithm will produce strong random
numbers when there has been no theory behind or analysis of the
algorithm. There is a excellent example of this fallacy right near
the beginning of Chapter 3 in [KNUTH] where the author describes a
complex algorithm. It was intended that the machine language program
corresponding to the algorithm would be so complicated that a person
trying to read the code without comments wouldn't know what the
program was doing. Unfortunately, actual use of this algorithm showed
that it almost immediately converged to a single repeated value in
one case and a small cycle of values in another case.
Not only does complex manipulation not help you if you have a limited
range of seeds but blindly chosen complex manipulation can destroy
the randomness in a good seed!
4.4 The Fallacy of Selection from a Large Database
Another strategy that can give a misleading appearance of
unpredictability is selection of a quantity randomly from a database
and assume that its strength is related to the total number of bits
in the database. For example, typical USENET servers process many
megabytes of information per day [USENET]. 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 no more than 2 or 3 bits of information per
byte, it doesn't yield 32*2 = 64 bits of unguessability. For an
adversary with access to the same usenet database the unguessability
rests only on the starting point of the selection. That is perhaps a
little over a couple of dozen bits of unguessability.
The same argument applies to selecting sequences from the data on a
publicly available CD/DVD recording or any other large public
database. If the adversary has access to the same database, this
"selection from a large volume of data" step buys little. However,
if a selection can be made from data to which the adversary has no
access, such as system buffers on an active multi-user system, it may
be of help.
5. Hardware for Randomness
Is there any hope for true strong portable randomness in the future?
There might be. All that's needed is a physical source of
unpredictable numbers.
A thermal noise (sometimes called Johnson noise in integrated
circuits) 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. Most audio (or video) input
devices are useable [TURBID]. Furthermore, any system with a
spinning disk or ring oscillator and a stable (crystal) time source
or the like has an adequate source of randomness ([DAVIS] and Section
5.4). 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 The best source of input entropy would be a hardware randomness such
as ring oscillators, disk drive timing, 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, each of these sources can
produce very limited or predictable values under some circumstances.
How much unpredictability is needed? Is it possible to quantify the Some of the sources listed above would be quite strong on multi-user
requirement in, say, number of random bits per second? systems where, in essence, each user of the system is a source of
randomness. However, on a small single user or embedded 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 answer is not very much is needed. For AES, the key can be 128 The use of multiple random inputs with a strong mixing function is
bits and, as we show in an example in Section 8, even the highest recommended and can overcome weakness in any particular input. The
security system is unlikely to require strong keying material of much timing and content of requested "random" user keystrokes can yield
over 200 bits. If a series of keys are needed, they can be generated hundreds of random bits but conservative assumptions need to be made.
from a strong random seed (starting value) using a cryptographically For example, assuming at most a few bits of randomness if the inter-
strong sequence as explained in Section 6.3. A few hundred random keystroke interval is unique in the sequence up to that point and a
bits generated at start up or once a day would be enough using such similar assumption if the key hit is unique but assuming that no bits
techniques. Even if the random bits are generated as slowly as one of randomness are present in the initial key value or if the timing
per second and it is not possible to overlap the generation process, or key value duplicate previous values. The results of mixing these
it should be tolerable in most high security applications to wait 200 timings and characters typed could be further combined with clock
seconds occasionally. values and other inputs.
These numbers are trivial to achieve. It could be done by a person This strategy may make practical portable code to produce good random
repeatedly tossing a coin. Almost any hardware based process is numbers for security even if some of the inputs are very weak on some
likely to be much faster. of the target systems. However, it may still fail against a high
grade attack on small, single user or embedded systems, especially if
the adversary has ever been able to observe the generation process in
the past. A hardware based random source is still preferable.
5.2 Sensitivity to Skew 4. De-skewing
Is there any specific requirement on the shape of the distribution of Is there any specific requirement on the shape of the distribution of
the random numbers? The good news is the distribution need not be quantities gathered for the entropy to produce the random numbers?
uniform. All that is needed is a conservative estimate of how non- The good news is the distribution need not be uniform. All that is
uniform it is to bound performance. Simple techniques to de-skew the needed is a conservative estimate of how non-uniform it is to bound
bit stream are given below and stronger cryptographic techniques are performance. Simple techniques to de-skew a bit stream are given
described in Section 6.1.2 below. below and stronger cryptographic techniques are described in Section
5.2 below.
5.2.1 Using Stream Parity to De-Skew 4.1 Using Stream Parity to De-Skew
As a simple but not particularly practical example, consider taking a As a simple but not particularly practical example, consider taking a
sufficiently long string of bits and map the string to "zero" or sufficiently long string of bits and map the string to "zero" or
"one". The mapping will not yield a perfectly uniform distribution, "one". The mapping will not yield a perfectly uniform distribution,
but it can be as close as desired. One mapping that serves the 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 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 that it is robust across all degrees of skew up to the estimated
maximum skew and is absolutely trivial to implement in hardware. maximum skew and is absolutely trivial to implement in hardware.
The following analysis gives the number of bits that must be sampled: The following analysis gives the number of bits that must be sampled:
Suppose the ratio of ones to zeros is ( 0.5 + e ) to ( 0.5 - e ), Suppose the ratio of ones to zeros is ( 0.5 + E ) to ( 0.5 - E ),
where e is between 0 and 0.5 and is a measure of the "eccentricity" 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 the distribution. Consider the distribution of the parity function
of N bit samples. The probabilities that the parity will be one or 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 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, 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. and q = 0.5 - E, the probability of a zero.
These sums can be computed easily as These sums can be computed easily as
N N N N
1/2 * ( ( p + q ) + ( p - q ) ) 1/2 * ( ( p + q ) + ( p - q ) )
and and
N N N N
1/2 * ( ( p + q ) - ( p - q ) ). 1/2 * ( ( p + q ) - ( p - q ) ).
(Which one corresponds to the probability the parity will be 1 (Which one corresponds to the probability the parity will be 1
depends on whether N is odd or even.) depends on whether N is odd or even.)
Since p + q = 1 and p - q = 2e, these expressions reduce to Since p + q = 1 and p - q = 2e, these expressions reduce to
N N
1/2 * [1 + (2e) ] 1/2 * [1 + (2E) ]
and and
N N
1/2 * [1 - (2e) ]. 1/2 * [1 - (2E) ].
Neither of these will ever be exactly 0.5 unless e is zero, but we 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 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 to be within some delta d of 0.5, i.e. then
N N
( 0.5 + ( 0.5 * (2e) ) ) < 0.5 + d. ( 0.5 + ( 0.5 * (2E) ) ) < 0.5 + d.
Solving for N yields N > log(2d)/log(2e). (Note that 2e is less than 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 1, so its log is negative. Division by a negative number reverses the
sense of an inequality.) sense of an inequality.)
The following table gives the length of the string which must be 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 sampled for various degrees of skew in order to come within 0.001 of
a 50/50 distribution. a 50/50 distribution.
+---------+--------+-------+ +---------+--------+-------+
| Prob(1) | e | N | | Prob(1) | E | N |
+---------+--------+-------+ +---------+--------+-------+
| 0.5 | 0.00 | 1 | | 0.5 | 0.00 | 1 |
| 0.6 | 0.10 | 4 | | 0.6 | 0.10 | 4 |
| 0.7 | 0.20 | 7 | | 0.7 | 0.20 | 7 |
| 0.8 | 0.30 | 13 | | 0.8 | 0.30 | 13 |
| 0.9 | 0.40 | 28 | | 0.9 | 0.40 | 28 |
| 0.95 | 0.45 | 59 | | 0.95 | 0.45 | 59 |
| 0.99 | 0.49 | 308 | | 0.99 | 0.49 | 308 |
+---------+--------+-------+ +---------+--------+-------+
The last entry shows that even if the distribution is skewed 99% in 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 favor of ones, the parity of a string of 308 samples will be within
0.001 of a 50/50 distribution. But, as we shall see in section 6.1.2, 0.001 of a 50/50 distribution. But, as we shall see in section 6.1.2,
there are much stronger techniques that extract more of the available there are much stronger techniques that extract more of the available
entropy. entropy.
5.2.2 Using Transition Mappings to De-Skew 4.2 Using Transition Mappings to De-Skew
Another technique, originally due to von Neumann [VON NEUMANN], is to Another technique, originally due to von Neumann [VON NEUMANN], is to
examine a bit stream as a sequence of non-overlapping pairs. You 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 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 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 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 described in the previous section. Then the probability of each pair
is as follows: is as follows:
+------+-----------------------------------------+ +------+-----------------------------------------+
| pair | probability | | pair | probability |
+------+-----------------------------------------+ +------+-----------------------------------------+
| 00 | (0.5 - e)^2 = 0.25 - e + e^2 | | 00 | (0.5 - E)^2 = 0.25 - E + E^2 |
| 01 | (0.5 - e)*(0.5 + e) = 0.25 - e^2 | | 01 | (0.5 - E)*(0.5 + E) = 0.25 - E^2 |
| 10 | (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 | | 11 | (0.5 + E)^2 = 0.25 + E + E^2 |
+------+-----------------------------------------+ +------+-----------------------------------------+
This technique will completely eliminate any bias but at the expense This technique will completely eliminate any bias but at the expense
of taking an indeterminate number of input bits for any particular of taking an indeterminate number of input bits for any particular
desired number of output bits. The probability of any particular pair 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 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). produce X output bits is X/(0.25 - E^2).
This technique assumes that the bits are from a stream where each bit 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 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 stream and that bits are not correlated, i.e., that the bits are
identical independent distributions. If alternate bits were from two identical independent distributions. If alternate bits were from two
correlated sources, for example, the above analysis breaks down. correlated sources, for example, the above analysis breaks down.
The above technique also provides another illustration of how a The above technique also provides another illustration of how a
simple statistical analysis can mislead if one is not always on the simple statistical analysis can mislead if one is not always on the
lookout for patterns that could be exploited by an adversary. If the lookout for patterns that could be exploited by an adversary. If the
algorithm were mis-read slightly so that overlapping successive bits algorithm were mis-read slightly so that overlapping successive bits
pairs were used instead of non-overlapping pairs, the statistical pairs were used instead of non-overlapping pairs, the statistical
analysis given is the same; however, instead of providing an unbiased analysis given is the same; however, instead of providing an unbiased
uncorrelated series of random 1s and 0s, it instead produces a uncorrelated series of random 1s and 0s, it instead produces a
totally predictable sequence of exactly alternating 1s and 0s. totally predictable sequence of exactly alternating 1s and 0s.
5.2.3 Using FFT to De-Skew 4.3 Using FFT to De-Skew
When real world data consists of strongly biased or correlated bits, When real world data consists of strongly correlated bits, it may
it may still contain useful amounts of randomness. This randomness still contain useful amounts of entropy. This entropy can be
can be extracted through use of various transforms, the most powerful extracted through use of various transforms, the most powerful of
of which are described in section 6.1.2 below. which are described in section 5.2 below.
Using the Fourier transform of the data or its optimized variant, the Using the Fourier transform of the data or its optimized variant, the
FFT, is an technique interesting primarily for theoretical reasons. FFT, is an technique interesting primarily for theoretical reasons.
It can be show that this will discard strong correlations. If It can be show that this will discard strong correlations. If
adequate data is processed and remaining correlations decay, spectral adequate data is processed and remaining correlations decay, spectral
lines approaching statistical independence and normally distributed lines approaching statistical independence and normally distributed
randomness can be produced [BRILLINGER]. randomness can be produced [BRILLINGER].
5.2.4 Using Compression to De-Skew 4.4 Using Compression to De-Skew
Reversible compression techniques also provide a crude method of de- Reversible compression techniques also provide a crude method of de-
skewing a skewed bit stream. This follows directly from the skewing a skewed bit stream. This follows directly from the
definition of reversible compression and the formula in Section 2 definition of reversible compression and the formula in Section 2
above for the amount of information in a sequence. Since the above for the amount of information in a sequence. Since the
compression is reversible, the same amount of information must be compression is reversible, the same amount of information must be
present in the shorter output than was present in the longer input. present in the shorter output than was present in the longer input.
By the Shannon information equation, this is only possible if, on By the Shannon information equation, this is only possible if, on
average, the probabilities of the different shorter sequences are average, the probabilities of the different shorter sequences are
more uniformly distributed than were the probabilities of the longer more uniformly distributed than were the probabilities of the longer
sequences. Therefore the shorter sequences must be de-skewed relative sequences. Therefore the shorter sequences must be de-skewed relative
to the input. to the input.
However, many compression techniques add a somewhat predictable However, many compression techniques add a somewhat predictable
preface to their output stream and may insert such a sequence again preface to their output stream and may insert such a sequence again
periodically in their output or otherwise introduce subtle patterns periodically in their output or otherwise introduce subtle patterns
of their own. They should be considered only a rough technique of their own. They should be considered only a rough technique
compared with those described in Section 6.1.2. At a minimum, the compared with those described in Section 5.2. At a minimum, the
beginning of the compressed sequence should be skipped and only later beginning of the compressed sequence should be skipped and only later
bits used for applications requiring roughly random bits. bits used for applications requiring roughly random bits.
5.3 Existing Hardware Can Be Used For Randomness 5. Mixing
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
Many computers are 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. This method is extremely hardware and implementation
dependent.
For example, on some UNIX based systems, one can read from the
/dev/audio device with nothing plugged into the microphone jack or
the microphone receiving only low level background noise. 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
A detailed examination of this type of randomness source appears in
[TURBID].
5.3.2 Using Existing Disk Drives
Disk drives have small random fluctuations in their rotational speed
due to chaotic air turbulence [DAVIS, Jakobsson]. 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, such as described in 6.1.2 below. Nevertheless
experimentation a decade ago showed that, with such processing, even
slow disk drives on the slower computers of that day could easily
produce 100 bits a minute or more of excellent random data.
Every increase in processor speed, which increases the resolution
with which disk motion can be timed, or increase in the rate of disk
seeks, increases the rate of random bit generation possible with this
technique. At the time of this paper and using modern hardware, a
more typical rate of random bit production would be in excess of
10,000 bits a second. This technique is used in many operating system
library random number generators.
Note: the inclusion of cache memories in disk controllers has little
effect on this technique if very short seek times, which represent
cache hits, are simply ignored.
5.4 Ring Oscillator Sources
If an integrated circuit is being designed or field programmed, an
odd number of gates can be connected in series to produce a free-
running ring oscillator. By sampling a point in the ring at a fixed
frequency, say one determined by a stable crystal oscillator, some
amount of entropy can be extracted due to variations in the free-
running oscillator timing. It is possible to increase the rate of
entropy by xor'ing sampled values from a few ring oscillators with
relatively prime lengths. It is sometimes recommended that an odd
number of rings be used so that, even if the rings somehow become
synchronously locked to each other, there will still be sampled bit
transitions. Another possibility source to sample is the output of a
noisy diode.
Sampled bits from such sources will have to be heavily de-skewed, as
disk rotation timings must be (Section 5.3.2). An engineering study
would be needed to determine the amount of entropy being produced
depending on the particular design. In any case, these can be good
sources whose cost is a trivial amount of hardware by modern
standards.
As an example, IEEE 802.11i suggests that the circuit below be
considered, with due attention in the design to isolation of the
rings from each other and from clocked circuits to avoid undesired
synchronization, etc., and extensive post processing. [IEEE 802.11i]
|\ |\ |\
+-->| >0-->| >0-- 19 total --| >0--+-------+
| |/ |/ |/ | |
| | |
+----------------------------------+ V
+-----+
|\ |\ |\ | | output
+-->| >0-->| >0-- 23 total --| >0--+--->| XOR |------>
| |/ |/ |/ | | |
| | +-----+
+----------------------------------+ ^ ^
| |
|\ |\ |\ | |
+-->| >0-->| >0-- 29 total --| >0--+------+ |
| |/ |/ |/ | |
| | |
+----------------------------------+ |
|
other randomness if available--------------+
6. Recommended Software Strategy
What is the best overall strategy for meeting the requirement for What is the best overall strategy for meeting the requirement for
unguessable random numbers in the absence of a reliable hardware unguessable random numbers in the absence of a strong reliable
source? It is to obtain random input from a number of uncorrelated hardware entropy source? It is to obtain input from a number of
sources and to mix them with a strong mixing function. Such a uncorrelated sources and to mix them with a strong mixing function.
function will preserve the randomness present in any of the sources Such a function will preserve the entropy present in any of the
even if other quantities being combined happen to be fixed or easily sources even if other quantities being combined happen to be fixed or
guessable. This may be advisable even with a good hardware source, as easily guessable (low entropy). This may be advisable even with a
hardware can also fail, though this should be weighed against any good hardware source, as hardware can also fail, though this should
increase in the chance of overall failure due to added software be weighed against any increase in the chance of overall failure due
complexity. to added software complexity.
6.1 Mixing Functions Once you have used good sources, such as some of those listed in
Section 3, and mixed them as described in this section, you have a
strong seed. This can then be used to produce large quantities of
cryptographically strong material as described in Sections 6 and 7.
A strong mixing function is one which combines inputs and produces an A strong mixing function is one which combines inputs and produces an
output where each output bit is a different complex non-linear output where each output bit is a different complex non-linear
function of all the input bits. On average, changing any input bit function of all the input bits. On average, changing any input bit
will change about half the output bits. But because the relationship will change about half the output bits. But because the relationship
is complex and non-linear, no particular output bit is guaranteed to is complex and non-linear, no particular output bit is guaranteed to
change when any particular input bit is changed. change when any particular input bit is changed.
Consider the problem of converting a stream of bits that is skewed Consider the problem of converting a stream of bits that is skewed
towards 0 or 1 or which has a somewhat predictable pattern to a towards 0 or 1 or which has a somewhat predictable pattern to a
shorter stream which is more random, as discussed in Section 5.2 shorter stream which is more random, as discussed in Section 4 above.
above. This is simply another case where a strong mixing function is This is simply another case where a strong mixing function is
desired, mixing the input bits to produce a smaller number of output 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 bits. The technique given in Section 4.1 of using the parity of a
number of bits is simply the result of successively Exclusive Or'ing number of bits is simply the result of successively Exclusive Or'ing
them which is examined as a trivial mixing function immediately them which is examined as a trivial mixing function immediately
below. Use of stronger mixing functions to extract more of the 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. randomness in a stream of skewed bits is examined in Section 5.2. See
also [NASLUND].
6.1.1 A Trivial Mixing Function 5.1 A Trivial Mixing Function
A trivial example for single bit inputs described only for expository A trivial example for single bit inputs described only for expository
purposes is the Exclusive Or function, which is equivalent to purposes is the Exclusive Or function, which is equivalent to
addition without carry, as show in the table below. This is a addition without carry, as show in the table below. This is a
degenerate case in which the one output bit always changes for a degenerate case in which the one output bit always changes for a
change in either input bit. But, despite its simplicity, it provides change in either input bit. But, despite its simplicity, it provides
a useful illustration. a useful illustration.
+-----------+-----------+----------+ +-----------+-----------+----------+
| input 1 | input 2 | output | | input 1 | input 2 | output |
+-----------+-----------+----------+ +-----------+-----------+----------+
| 0 | 0 | 0 | | 0 | 0 | 0 |
| 0 | 1 | 1 | | 0 | 1 | 1 |
| 1 | 0 | 1 | | 1 | 0 | 1 |
| 1 | 1 | 0 | | 1 | 1 | 0 |
+-----------+-----------+----------+ +-----------+-----------+----------+
If inputs 1 and 2 are uncorrelated and combined in this fashion then 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 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 inputs. If we assume an "eccentricity" E as defined in Section 5.2
above, then the output eccentricity relates to the input eccentricity above, then the output eccentricity relates to the input eccentricity
as follows: as follows:
e = 2 * e * e E = 2 * E * E
output input 1 input 2 output input 1 input 2
Since e is never greater than 1/2, the eccentricity is always 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 improved except in the case where at least one input is a totally
skewed constant. This is illustrated in the following table where the skewed constant. This is illustrated in the following table where the
top and left side values are the two input eccentricities and the top and left side values are the two input eccentricities and the
entries are the output eccentricity: entries are the output eccentricity:
+--------+--------+--------+--------+--------+--------+--------+ +--------+--------+--------+--------+--------+--------+--------+
| e | 0.00 | 0.10 | 0.20 | 0.30 | 0.40 | 0.50 | | 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 | | 0.00 | 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 | | 0.10 | 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 | | 0.20 | 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 | | 0.30 | 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 | | 0.40 | 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 | | 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 However, keep in mind that the above calculations assume that the
inputs are not correlated. If the inputs were, say, the parity of 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 number of minutes from midnight on two clocks accurate to a few
seconds, then each might appear random if sampled at random intervals seconds, then each might appear random if sampled at random intervals
much longer than a minute. Yet if they were both sampled and combined much longer than a minute. Yet if they were both sampled and combined
with xor, the result would be zero most of the time. with xor, the result would be zero most of the time.
6.1.2 Stronger Mixing Functions 5.2 Stronger Mixing Functions
The US Government Advanced Encryption Standard [AES] is an example of The US Government Advanced Encryption Standard [AES] is an example of
a strong mixing function for multiple bit quantities. It takes up to 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 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 produces 128 bits of output each of which is dependent on a complex
non-linear function of all input bits. Other encryption functions non-linear function of all input bits. Other encryption functions
with this characteristic, such as [DES], can also be used by with this characteristic, such as [DES], can also be used by
considering them to mix all of their key and data input bits. considering them to mix all of their key and data input bits.
Another good family of mixing functions are the "message digest" or Another good family of mixing functions are the "message digest" or
skipping to change at page 24, line 40 skipping to change at page 21, line 40
the 1st part of the output, then encrypt B with C and then A for more 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 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 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 the keys given above to stretch things. The same can be done with the
hash functions by hashing various subsets of the input data or hash functions by hashing various subsets of the input data or
different copies of the input data with different prefixes and/or different copies of the input data with different prefixes and/or
suffixes to produce multiple outputs. suffixes to produce multiple outputs.
An example of using a strong mixing function would be to reconsider 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 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 zero. The parity technique given in Section 4.1 above reduced this to
to one bit with only a 1/1000 deviance from being equally likely a one bit with only a 1/1000 deviance from being equally likely a zero
zero or one. But, applying the equation for information given in or one. But, applying the equation for information given in Section
Section 2, this 308 bit skewed sequence has over 5 bits of 2, this 308 bit skewed sequence has over 5 bits of information in it.
information in it. Thus hashing it with SHA-1 and taking the bottom 5 Thus hashing it with SHA-1 and taking the bottom 5 bits of the result
bits of the result would yield 5 unbiased random bits as opposed to would yield 5 unbiased random bits as opposed to the single bit given
the single bit given by calculating the parity of the string. by calculating the parity of the string. Alternatively, for some
Alternatively, for some applications, you could use the entire hash applications, you could use the entire hash output to retain almost
output to retain almost all of the entropy. all of the 5+ bits of entropy in a 160 bit quantity.
6.1.3 Using S-Boxes for Mixing 5.3 Using S-Boxes for Mixing
Many modern block encryption functions, including DES and AES, Many modern block encryption functions, including DES and AES,
incorporate modules known as S-Boxes (substitution boxes). These incorporate modules known as S-Boxes (substitution boxes). These
produce a smaller number of outputs from a larger number of inputs produce a smaller number of outputs from a larger number of inputs
through a complex non-linear mixing function which would have the through a complex non-linear mixing function which would have the
effect of concentrating limited entropy in the inputs into the effect of concentrating limited entropy in the inputs into the
output. output.
S-Boxes sometimes incorporate bent Boolean functions (functions of an S-Boxes sometimes incorporate bent Boolean functions (functions of an
even number of bits producing one output bit with maximum non- even number of bits producing one output bit with maximum non-
linearity). Looking at the output for all input pairs differing in linearity). Looking at the output for all input pairs differing in
any particular bit position, exactly half the outputs are different. any particular bit position, exactly half the outputs are different.
An S-Box in which each output bit is produced by a bent function such An S-Box in which each output bit is produced by a bent function such
that any linear combination of these functions is also a bent that any linear combination of these functions is also a bent
function is called a "perfect S-Box". function is called a "perfect S-Box".
S-boxes and various repeated application or cascades of such boxes S-boxes and various repeated application or cascades of such boxes
can be used for mixing. [SBOX*] can be used for mixing. [SBOX*]
6.1.4 Diffie-Hellman as a Mixing Function 5.4 Diffie-Hellman as a Mixing Function
Diffie-Hellman exponential key exchange is a technique that yields a Diffie-Hellman exponential key exchange is a technique that yields a
shared secret between two parties that can be made computationally shared secret between two parties that can be made computationally
infeasible for a third party to determine even if they can observe infeasible for a third party to determine even if they can observe
all the messages between the two communicating parties. This shared all the messages between the two communicating parties. This shared
secret is a mixture of initial quantities generated by each of the secret is a mixture of initial quantities generated by each of the
parties [D-H]. parties [D-H].
If these initial quantities are random and uncorrelated, then the If these initial quantities are random and uncorrelated, then the
shared secret combines that randomness, but, of course, can not shared secret combines their entropy, but, of course, can not produce
produce more randomness than the size of the shared secret generated. more randomness than the size of the shared secret generated.
While this is true if the Diffie-Hellman computation is performed While this is true if the Diffie-Hellman computation is performed
privately, if an adversary can observe either of the public keys and privately, an adversary that can observe either of the public keys
knows the modulus being used, they need only search through the space and knows the modulus being used need only search through the space
of the other secret key in order to be able to calculate the shared of the other secret key in order to be able to calculate the shared
secret [D-H]. So, conservatively, it would be best to consider public secret [D-H]. So, conservatively, it would be best to consider public
Diffie-Hellman to produce a quantity whose guessability corresponds Diffie-Hellman to produce a quantity whose guessability corresponds
to the worst of the two inputs. to the worst of the two inputs. Because of this and the fact that
Diffie-Hellman is computationally intensive, its use as a mixing
function is not recommended.
6.1.5 Using a Mixing Function to Stretch Random Bits 5.5 Using a Mixing Function to Stretch Random Bits
While it is not necessary for a mixing function to produce the same While it is not necessary for a mixing function to produce the same
or fewer bits than its inputs, mixing bits cannot "stretch" the or fewer bits than its inputs, mixing bits cannot "stretch" the
amount of random unpredictability present in the inputs. Thus four amount of random unpredictability present in the inputs. Thus four
inputs of 32 bits each where there is 12 bits worth of inputs of 32 bits each where there is 12 bits worth of
unpredictability (such as 4,096 equally probable values) in each unpredictability (such as 4,096 equally probable values) in each
input cannot produce more than 48 bits worth of unpredictable output. input cannot produce more than 48 bits worth of unpredictable output.
The output can be expanded to hundreds or thousands of bits by, for The output can be expanded to hundreds or thousands of bits by, for
example, mixing with successive integers, but the clever adversary's example, mixing with successive integers, but the clever adversary's
search space is still 2^48 possibilities. Furthermore, mixing to search space is still 2^48 possibilities. Furthermore, mixing to
fewer bits than are input will tend to strengthen the randomness of fewer bits than are input will tend to strengthen the randomness of
the output. the output.
The last table in Section 6.1.1 shows that mixing a random bit with a The last table in Section 5.1 shows that mixing a random bit with a
constant bit with Exclusive Or will produce a random bit. While this 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 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 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 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 either 01 or 10. Since there are only two possible values, there is
still only the one bit of original randomness. still only the one bit of original randomness.
6.1.6 Other Factors in Choosing a Mixing Function 5.6 Other Factors in Choosing a Mixing Function
For local use, AES has the advantages that it has been widely tested For local use, AES has the advantages that it has been widely tested
for flaws, is reasonably efficient in software, and is widely for flaws, is reasonably efficient in software, and is widely
documented and implemented with hardware and software implementations documented and implemented with hardware and software implementations
available all over the world including open source code. The SHA* available all over the world including open source code. The SHA*
family have had a little less study and tend to require more CPU family have had a little less study and tend to require more CPU
cycles than AES but there is no reason to believe they are flawed. cycles than AES but there is no reason to believe they are flawed.
Both SHA* and MD5 were derived from the earlier MD4 algorithm. They Both SHA* and MD5 were derived from the earlier MD4 algorithm. They
all have source code available [SHA*, MD*]. Some signs of weakness all have source code available [SHA*, MD*]. Some signs of weakness
have been found in MD4 and MD5. In particular, MD4 has only three have been found in MD4 and MD5. In particular, MD4 has only three
rounds and there are several independent breaks of the first two or rounds and there are several independent breaks of the first two or
last two rounds. And some collisions have been found in MD5 output. last two rounds. And some collisions have been found in MD5 output.
AES was selected by a robust, public, and international process. It AES was selected by a robust, public, and international process. It
and SHA* have been vouched for by the US National Security Agency and SHA* have been vouched for by the US National Security Agency
(NSA) on the basis of criteria that mostly remain secret, as was DES. (NSA) on the basis of criteria that mostly remain secret, as was DES.
While this has been the cause of much speculation and doubt, While this has been the cause of much speculation and doubt,
investigation of DES over the years has indicated that NSA investigation of DES over the years has indicated that NSA
involvement in modifications to its design, which originated with involvement in modifications to its design, which originated with
IBM, was primarily to strengthen it. No concealed or special weakness IBM, was primarily to strengthen it. There has been no announcement
has been found in DES. It is likely that the NSA modifications to MD4 of a concealed or special weakness being found in DES. It is likely
to produce the SHA algorithms similarly strengthened these that the NSA modifications to MD4 to produce the SHA algorithms
algorithms, possibly against threats not yet known in the public similarly strengthened these algorithms, possibly against threats not
cryptographic community. yet known in the public cryptographic community.
Where input lengths are unpredictable, hash algorithms are a little Where input lengths are unpredictable, hash algorithms are more
more convenient to use than block encryption algorithms since they convenient to use than block encryption algorithms since they are
are generally designed to accept variable length inputs. Block generally designed to accept variable length inputs. Block encryption
encryption algorithms generally require an additional padding algorithms generally require an additional padding algorithm to
algorithm to accommodate inputs that are not an even multiple of the accommodate inputs that are not an even multiple of the block size.
block size.
As of the time of this document, the authors know of no patent claims As of the time of this document, the authors know of no patent claims
to the basic AES, DES, SHA*, MD4, and MD5 algorithms other than to the basic AES, DES, SHA*, MD4, and MD5 algorithms other than
patents for which an irrevocable royalty free license has been patents for which an irrevocable royalty free license has been
granted to the world. There may, of course, be basic patents of which granted to the world. There may, of course, be essential patents of
the authors are unaware or patents on implementations or uses or which the authors are unaware or patents on implementations or uses
other relevant patents issued or to be issued. or other relevant patents issued or to be issued.
6.2 Non-Hardware Sources of Randomness 6. Pseudo Random Number Generators
The best source of input for mixing would be a hardware randomness When you have a seed with sufficient entropy, from input as described
such as ring oscillators, disk drive timing, thermal noise, or in Section 3 possibly de-skewed and mixed as described in Sections 4
radioactive decay. However, if that is not available, there are other and 5, you can algorithmically extend that seed to produce a large
possibilities. These include system clocks, system or input/output quantity of cryptographically strong random quantities. Such
buffers, user/system/hardware/network serial numbers and/or addresses algorithms are platform independent and can operate in the same
and timing, and user input. Unfortunately, each of these sources can fashion on any computer. To be secure their input and internal
produce very limited or predictable values under some circumstances. workings must be protected from adversarial observation.
Some of the sources listed above would be quite strong on multi-user The design of such pseudo random number generation algorithms, like
systems where, in essence, each user of the system is a source of the design of symmetric encryption algorithms, is not a task for
randomness. However, on a small single user or embedded system, amateurs. Section 6.1 below lists a number of bad ideas which failed
especially at start up, it might be possible for an adversary to algorithms have used. If you are interested in what works, you can
assemble a similar configuration. This could give the adversary skip 6.1 and just read the remainder of this section and Section 7
inputs to the mixing process that were sufficiently correlated to below which describes and gives references for some standard pseudo
those used originally as to make exhaustive search practical. random number generation algorithms. See Section 7 and [X9.82 - Part
3].
The use of multiple random inputs with a strong mixing function is 6.1 Some Bad Ideas
recommended and can overcome weakness in any particular input. 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 at most 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 The subsections below describe a number of idea which might seem
numbers for security even if some of the inputs are very weak on some reasonable but which lead to insecure pseudo random number
of the target systems. However, it may still fail against a high generation.
grade attack on small, single user or embedded systems, especially if
the adversary has ever been able to observe the generation process in
the past. A hardware based random source is still preferable.
6.3 Cryptographically Strong Sequences 6.1.1 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
limited data such as the computer system clock value 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 entropy in the starting seed value. They can usually use the
limited number of results stemming from a limited number of seed
values to defeat security.
Another serious strategy error is to assume that a very complex
pseudo-random number generation algorithm will produce strong random
numbers when there has been no theory behind or analysis of the
algorithm. There is a excellent example of this fallacy right near
the beginning of Chapter 3 in [KNUTH] where the author describes a
complex algorithm. It was intended that the machine language program
corresponding to the algorithm would be so complicated that a person
trying to read the code without comments wouldn't know what the
program was doing. Unfortunately, actual use of this algorithm showed
that it almost immediately converged to a single repeated value in
one case and a small cycle of values in another case.
Not only does complex manipulation not help you if you have a limited
range of seeds but blindly chosen complex manipulation can destroy
the entropy in a good seed!
6.1.2 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 [USENET]. 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 no more than 2 or 3 bits of information per
byte, it doesn't yield 32*2 = 64 bits of unguessability. For an
adversary with access to the same Usenet database the unguessability
rests only on the starting point of the selection. That is perhaps a
little over a couple of dozen bits of unguessability.
The same argument applies to selecting sequences from the data on a
publicly available CD/DVD recording or any other large public
database. If the adversary has access to the same database, this
"selection from a large volume of data" step buys little. However,
if a selection can be made from data to which the adversary has no
access, such as system buffers on an active multi-user system, it may
be of help.
6.1.3. Traditional Pseudo-Random Sequences
This section talks about traditional sources of deterministic of
"pseudo-random" numbers. These typically start with a "seed" quantity
and use simple numeric or logical operations to produce a sequence of
values. Note that none of the techniques discussed in this section is
suitable for cryptographic use. They are presented for general
information.
[KNUTH] has a classic exposition on pseudo-random numbers.
Applications he mentions are 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. Sometimes they can
store the message they are trying to break and repeatedly attack it.
They are also 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. But they could be met by a constant
stored random sequence, such as the "random" sequence printed in the
CRC Standard Mathematical Tables [CRC]. Despite meeting all the tests
suggested by Knuth, that sequence is unsuitable for cryptographic use
as adversaries must be assumed to have copies of all common published
"random" sequences and will able to spot the source and predict
future values.
A typical pseudo-random number generation technique, known as a
linear congruence pseudo-random number generator, is modular
arithmetic where the value numbered N+1 is calculated from the value
numbered N by
V = ( V * a + b )(Mod c)
N+1 N
The above technique has a strong relationship to linear shift
register pseudo-random number generators, which are well understood
cryptographically [SHIFT*]. In such generators bits are introduced at
one end of a shift register as the Exclusive Or (binary sum without
carry) of bits from selected fixed taps into the register. For
example:
+----+ +----+ +----+ +----+
| 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 a, b, c, and initial V 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 [SCHNEIER, STERN]. For example, with the generators
above, one can determine V(n+1) given knowledge of V(n). In fact, it
has been shown that with these techniques, even if only one bit of
the pseudo-random values are released, the seed can be determined
from short sequences.
Not only have linear congruent generators been broken, but techniques
are now known for breaking all polynomial congruent generators.
[KRAWCZYK]
6.2 Cryptographically Strong Sequences
In cases where a series of random quantities must be generated, an In cases where a series of random quantities must be generated, an
adversary may learn some values in the sequence. In general, they adversary may learn some values in the sequence. In general, they
should not be able to predict other values from the ones that they should not be able to predict other values from the ones that they
know. know.
The correct technique is to start with a strong random seed, take The correct technique is to start with a strong random seed, take
cryptographically strong steps from that seed [FERGUSON, SCHNEIER], cryptographically strong steps from that seed [FERGUSON, SCHNEIER],
and do not reveal the complete state of the generator in the sequence 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 elements. If each value in the sequence can be calculated in a fixed
skipping to change at page 28, line 32 skipping to change at page 29, line 20
(It should be noted that if your technique for generating a sequence (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 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 for a confidentiality system. If two parties use the same sequence
generating technique and start with the same seed material, they will generating technique and start with the same seed material, they will
generate identical sequences. These could, for example, be xor'ed at generate identical sequences. These could, for example, be xor'ed at
one end with data being send, encrypting it, and xor'ed with this one end with data being send, encrypting it, and xor'ed with this
data as received, decrypting it due to the reversible properties of data as received, decrypting it due to the reversible properties of
the xor operation. This is commonly referred to as a simple stream the xor operation. This is commonly referred to as a simple stream
cipher.) cipher.)
6.3.1 OFB and CTR Sequences 6.2.1 OFB and CTR Sequences
One way to achieve a strong sequence is to have the values be One way to achieve a strong sequence is to have the values be
produced by taking a seed value and hashing the quantities produced produced by taking a seed value and hashing the quantities produced
by concatenating the seed with successive integers or the like and by concatenating the seed with successive integers or the like and
then mask the values obtained so as to limit the amount of generator then mask the values obtained so as to limit the amount of generator
state available to the adversary. state available to the adversary.
It may also be possible to use an "encryption" algorithm with a It may also be possible to use an "encryption" algorithm with a
random key and seed value to encrypt successive integers as in random key and seed value to encrypt successive integers as in
counter (CTR) mode encryption. Alternatively, you can feedback all of counter (CTR) mode encryption. Alternatively, you can feedback all of
the output encrypted value into the value to be encrypted for the the output value from encryption into the value to be encrypted for
next iteration. This is a particular example of output feedback mode the next iteration. This is a particular example of output feedback
(OFB). [MODES] mode (OFB). [MODES]
An example is shown below where shifting and masking are used to An example is shown below where shifting and masking are used to
combine part of the output feedback with part of the old input. This combine part of the output feedback with part of the old input. This
type of partial feedback should be avoided for reasons described type of partial feedback should be avoided for reasons described
below. below.
+---------------+ +---------------+
| V | | V |
| | n |--+ | | n |--+
+--+------------+ | +--+------------+ |
skipping to change at page 29, line 44 skipping to change at page 30, line 44
To predict values of a sequence from others when the sequence was To predict values of a sequence from others when the sequence was
generated by these techniques is equivalent to breaking the generated by these techniques is equivalent to breaking the
cryptosystem or inverting the "non-invertible" hashing involved with cryptosystem or inverting the "non-invertible" hashing involved with
only partial information available. The less information revealed only partial information available. The less information revealed
each iteration, the harder it will be for an adversary to predict the 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 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 been shown that in some cases this makes it impossible to break a
system even when the cryptographic system is invertible and can be system even when the cryptographic system is invertible and can be
broken if all of each generated value was revealed. broken if all of each generated value was revealed.
6.3.2 The Blum Blum Shub Sequence Generator 6.2.2 The Blum Blum Shub Sequence Generator
Currently the generator which has the strongest public proof of Currently the generator which has the strongest public proof of
strength is called the Blum Blum Shub generator after its inventors strength is called the Blum Blum Shub generator after its inventors
[BBS]. It is also very simple and is based on quadratic residues. [BBS]. It is also very simple and is based on quadratic residues.
Its only disadvantage is that it is computationally intensive Its only disadvantage is that it is computationally intensive
compared with the traditional techniques give in 6.3.1 above. This is compared with the traditional techniques give in 6.1.3 above. This is
not a major draw back if it is used for moderately infrequent not a major draw back if it is used for moderately infrequent
purposes, such as generating session keys. purposes, such as generating session keys.
Simply choose two large prime numbers, say p and q, which both have 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. 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 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 n. The initial seed for the generator and the method for calculating
subsequent values are then subsequent values are then
2 2
skipping to change at page 30, line 41 skipping to change at page 31, line 41
i i
( ( 2 )(Mod (( p - 1 ) * ( q - 1 )) ) ) ( ( 2 )(Mod (( p - 1 ) * ( q - 1 )) ) )
s = ( s )(Mod n) s = ( s )(Mod n)
i 0 i 0
This means that in applications where many keys are generated in this 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 fashion, it is not necessary to save them all. Each key can be
effectively indexed and recovered from that small index and the effectively indexed and recovered from that small index and the
initial s and n. initial s and n.
6.3.3 Entropy Pool Techniques 6.3 Entropy Pool Techniques
Many modern pseudo-random number sources utilize the technique of Many modern pseudo-random number sources, such as those describe in
maintaining a "pool" of bits and providing operations for strongly Sections 7.1.2 and 7.1.3, utilize the technique of maintaining a
mixing input with some randomness into the pool and extracting pseudo "pool" of bits and providing operations for strongly mixing input
random bits from the pool. This is illustrated in the figure below. with some randomness into the pool and extracting pseudo random bits
from the pool. This is illustrated in the figure below.
+--------+ +------+ +---------+ +--------+ +------+ +---------+
--->| Mix In |--->| POOL |--->| Extract |---> --->| Mix In |--->| POOL |--->| Extract |--->
| Bits | | | | Bits | | Bits | | | | Bits |
+--------+ +------+ +---------+ +--------+ +------+ +---------+
^ V ^ V
| | | |
+-----------+ +-----------+
Bits to be feed into the pool can be any of the various hardware, Bits to be feed into the pool can be any of the various hardware,
environmental, or user input sources discussed above. It is also environmental, or user input sources discussed above. It is also
common to save the state of the pool on system shut down and restore common to save the state of the pool on system shut down and restore
it on re-starting, if stable storage is available. it on re-starting, if stable storage is available.
Care must be taken that enough entropy has been added to the pool to Care must be taken that enough entropy has been added to the pool to
support particular output uses desired. See Section 7.5 for more support particular output uses desired. See [RSA BULL1] for similar
details on an example implementation and [RSA BULL1] for similar
suggestions. suggestions.
7. Key Generation Examples and Standards 7. Randomness Generation Examples and Standards
Several public standards and widely deployed examples are now in Several public standards and widely deployed examples are now in
place for the generation of keys without special hardware. Three place for the generation of keys or other cryptographically random
standards are described below. The two older standards use DES, with quantities. Some, in section 7.1 below, include an entropy source.
its 64-bit block and key size limit, but any equally strong or Others, described in section 7.2, provide the pseudo-random number
stronger mixing function could be substituted [DES]. The third is a strong sequence generator but assume the input of a random seed or
more modern and stronger standard based on SHA-1 [SHA*]. Lastly the input from a source of entropy.
widely deployed modern UNIX random number generators are described.
7.1 US DoD Recommendations for Password Generation 7.1 Complete Randomness Generators
Three standards are described below. The two older standards use
DES, with its 64-bit block and key size limit, but any equally strong
or stronger mixing function could be substituted [DES]. The third is
a more modern and stronger standard based on SHA-1 [SHA*]. Lastly
the widely deployed modern UNIX and Windows random number generators
are described.
7.1.1 US DoD Recommendations for Password Generation
The United States Department of Defense has specific recommendations The United States Department of Defense has specific recommendations
for password generation [DoD]. They suggest using the US Data for password generation [DoD]. They suggest using the US Data
Encryption Standard [DES] in Output Feedback Mode [MODES] as follows: Encryption Standard [DES] in Output Feedback Mode [MODES] as follows:
use an initialization vector determined from use an initialization vector determined from
the system clock, the system clock,
system ID, system ID,
user ID, and user ID, and
date and time; date and time;
use a key determined from use a key determined from
system interrupt registers, system interrupt registers,
system status registers, and system status registers, and
system counters; and, system counters; and,
as plain text, use an external randomly generated 64 bit as plain text, use an external randomly generated 64 bit
quantity such as 8 characters typed in by a system quantity such as the ASCII bytes for 8 characters typed in by a
administrator. system administrator.
The password can then be calculated from the 64 bit "cipher text" The password can then be calculated from the 64 bit "cipher text"
generated by DES in 64-bit Output Feedback Mode. As many bits as are generated by DES in 64-bit Output Feedback Mode. As many bits as are
needed can be taken from these 64 bits and expanded into a needed can be taken from these 64 bits and expanded into a
pronounceable word, phrase, or other format if a human being needs to pronounceable word, phrase, or other format if a human being needs to
remember the password. remember the password.
7.2 X9.17 Key Generation 7.1.2 The /dev/random Device
The American National Standards Institute has specified a method for
generating a sequence of keys as follows [X9.17]:
s is the initial 64 bit seed
0
g is the sequence of generated 64 bit key quantities
n
k is a random key reserved for generating this key sequence
t is the time at which a key is generated to as fine a resolution
as is available (up to 64 bits).
DES ( K, Q ) is the DES encryption of quantity Q with key K
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 DSS Pseudo-Random Number Generation
Appendix 3 of the NIST Digital Signature Standard [DSS] provides a
method of producing a sequence of pseudo-random 160 bit quantities
for use as private keys or the like. This has been modified by Change
Notice 1 [DSS CN1] to produce the following algorithm for generating
general purpose pseudorandom numbers:
t = 0x 67452301 EFCDAB89 98BADCFE 10325476 C3D2E1F0
XKEY = initial seed
0
For j = 0 to ...
XVAL = ( XKEY + optional user input ) (Mod 2^512)
j
X = G( t, XVAL )
j
XKEY = ( 1 + XKEY + X ) (Mod 2^512)
j+1 j j
The quantities X thus produced are the pseudo-random sequence of 160
bit values. Two functions can be used for "G" above. Each produces
a 160-bit value and takes two arguments, the first argument a 160-bit
value and the second a 512 bit value.
The first is based on SHA-1 and works by setting the 5 linking
variables, denoted H with subscripts in the SHA-1 specification, to
the first argument divided into fifths. Then steps (a) through (e) of
section 7 of the NIST SHA-1 specification are run over the second
argument as if it were a 512-bit data block. The values of the
linking variable after those steps are then concatenated to produce
the output of G. [SHA*]
As an alternative second method, NIST also defined an alternate G
function based on multiple applications of the DES encryption
function [DSS].
7.4 X9.82 Pseudo-Random Number Generation
The National Institute for Standards and Technology (NIST) and the
American National Standards Institutes (ANSI) X9F1 committee are in
the final stages of creating a standard for random number generation
covering both true randomness generators and pseudo-random number
generators. It includes a number of pseudo-random number generators
for use with AES and other block ciphers. It also includes random
number generators based on hash functions and the arithmetic of
elliptic curves [X9.82].
7.5 The /dev/random Device
Several versions of the UNIX operating system provides a kernel- Several versions of the UNIX operating system provide a kernel-
resident random number generator. In some cases, these generators resident random number generator. In some cases, these generators
makes use of events captured by the Kernel during normal system make use of events captured by the Kernel during normal system
operation. operation.
For example, on some versions of Linux, the generator consists of a For example, on some versions of Linux, the generator consists of a
random pool of 512 bytes represented as 128 words of 4-bytes each. random pool of 512 bytes represented as 128 words of 4-bytes each.
When an event occurs, such as a disk drive interrupt, the time of the When an event occurs, such as a disk drive interrupt, the time of the
event is xor'ed into the pool and the pool is stirred via a primitive 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 polynomial of degree 128. The pool itself is treated as a ring
buffer, with new data being XORed (after stirring with the buffer, with new data being XORed (after stirring with the
polynomial) across the entire pool. polynomial) across the entire pool.
Each call that adds entropy to the pool estimates the amount of Each call that adds entropy to the pool estimates the amount of
likely true entropy the input contains. The pool itself contains a likely true entropy the input contains. The pool itself contains a
accumulator that estimates the total over all entropy of the pool. accumulator that estimates the total over all entropy of the pool.
Input events come from several sources as listed below. Input events come from several sources as listed below.
Unfortunately, for server machines without human operators, the first Unfortunately, for server machines without human operators, the first
and third are not available and entropy may be added very slowly in and third are not available and entropy may be added slowly in that
that case. case.
1. Keyboard interrupts. The time of the interrupt as well as the scan 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 code are added to the pool. This in effect adds entropy from the
human operator by measuring inter-keystroke arrival times. human operator by measuring inter-keystroke arrival times.
2. Disk completion and other interrupts. A system being used by a 2. Disk completion and other interrupts. A system being used by a
person will likely have a hard to predict pattern of disk person will likely have a hard to predict pattern of disk
accesses. (But not all disk drivers support capturing this timing accesses. (But not all disk drivers support capturing this timing
information with sufficient accuracy to be useful.) information with sufficient accuracy to be useful.)
3. Mouse motion. The timing as well as mouse position is added in. 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 [SHA*] When random bytes are required, the pool is hashed with SHA-1 [SHA*]
to yield the returned bytes of randomness. If more bytes are required to yield the returned bytes of randomness. If more bytes are required
than the output of SHA-1 (20 bytes), then the hashed output is 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 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 next 20 bytes. As bytes are removed from the pool, the estimate of
entropy is similarly decremented. entropy is similarly decremented.
To ensure a reasonable random pool upon system startup, the standard To ensure a reasonable random pool upon system startup, the standard
startup scripts (and shutdown scripts) save the pool to a disk file startup and shutdown scripts save the pool to a disk file at shutdown
at shutdown and read this file at system startup. and read this file at system startup.
There are two user exported interfaces. /dev/random returns bytes There are two user exported interfaces. /dev/random returns bytes
from the pool, but blocks when the estimated entropy drops to zero. from the pool, but blocks when the estimated entropy drops to zero.
As entropy is added to the pool from events, more data becomes As entropy is added to the pool from events, more data becomes
available via /dev/random. Random data obtained from such a available via /dev/random. Random data obtained from such a
/dev/random device is suitable for key generation for long term keys, /dev/random device is suitable for key generation for long term keys,
if enough random bits are in the pool or are added in a reasonable if enough random bits are in the pool or are added in a reasonable
amount of time. amount of time.
/dev/urandom works like /dev/random, however it provides data even /dev/urandom works like /dev/random, however it provides data even
when the entropy estimate for the random pool drops to zero. This may when the entropy estimate for the random pool drops to zero. This may
be adequate for session keys or for other key generation tasks where be adequate for session keys or for other key generation tasks where
blocking while waiting for more random bits is not acceptable. The blocking while waiting for more random bits is not acceptable. The
risk of continuing to take data even when the pool's entropy estimate risk of continuing to take data even when the pool's entropy
is small in that past output may be computable from current output estimate is small in that past output may be computable from current
provided an attacker can reverse SHA-1. Given that SHA-1 is designed output provided an attacker can reverse SHA-1. Given that SHA-1 is
to be non-invertible, this is a reasonable risk. designed to be non-invertible, this is a reasonable risk.
To obtain random numbers under Linux, Solaris, or other UNIX systems To obtain random numbers under Linux, Solaris, or other UNIX systems
equipped with code as described above, all an application needs to do equipped with code as described above, all an application needs to do
is open either /dev/random or /dev/urandom and read the desired is open either /dev/random or /dev/urandom and read the desired
number of bytes. number of bytes.
(The Linux Random device was written by Theodore Ts'o. It was based (The Linux Random device was written by Theodore Ts'o. It was based
loosely on the random number generator in PGP 2.X and PGP 3.0 (aka loosely on the random number generator in PGP 2.X and PGP 3.0 (aka
PGP 5.0).) PGP 5.0).)
7.6 Windows CryptGenRandom 7.1.3 Windows CryptGenRandom
Microsoft's recommendation to users of the widely deployed Windows Microsoft's recommendation to users of the widely deployed Windows
operating system is generally to use the CryptGenRandom pseudo-random operating system is generally to use the CryptGenRandom pseudo-random
number generation call with the CryptAPI cryptographic service number generation call with the CryptAPI cryptographic service
provider. This takes a handle to a cryptographic service provider provider. This takes a handle to a cryptographic service provider
library, a pointer to a buffer by which the caller can provide library, a pointer to a buffer by which the caller can provide
entropy and into which the generated pseudo-randomness is returned, entropy and into which the generated pseudo-randomness is returned,
and an indication of how many octets of randomness are desired. and an indication of how many octets of randomness are desired.
The Windows CryptAPI cryptographic service provider stores a seed The Windows CryptAPI cryptographic service provider stores a seed
skipping to change at page 37, line 5 skipping to change at page 36, line 5
hashed user environment block. This data is all feed to SHA-1 and the hashed user environment block. This data is all feed to SHA-1 and the
output used to seed an RC4 key stream. That key stream is used to output used to seed an RC4 key stream. That key stream is used to
produce the pseudo-random data requested and to update the user's produce the pseudo-random data requested and to update the user's
seed state variable. seed state variable.
Users of Windows ".NET" will probably find it easier to use the Users of Windows ".NET" will probably find it easier to use the
RNGCryptoServiceProvider.GetBytes method interface. RNGCryptoServiceProvider.GetBytes method interface.
For further information, see [WSC]. For further information, see [WSC].
7.2 Generators Assuming a Source of Entropy
The pseudo-random number generators described in the following three
sections all assume that a seed value with sufficient entropy is
provided to them. They then generate a strong sequence (see Section
6.2) from that seed.
7.2.1 X9.82 Pseudo-Random Number Generation
The ANSI X9F1 committee is in the final stages of creating a standard
for random number generation covering both true randomness generators
and pseudo-random number generators. It includes a number of pseudo-
random number generators based on hash functions one of which will
probably be based on HMAC SHA hash constructs [HMAC]. The draft
version of this generated is as described below omitting a number of
optional features [X9.82].
In the description in the subsections below, the HMAC hash construct
is simply referred to as HMAC but, of course, in an particular use, a
particular standard SHA function must be selected. Generally
speaking, if the strength of the pseudo-random values to be generated
is to be N bits, the SHA function chosen must be one generating N or
more bits of output and a source of at least N bits of input entropy
will be required. The same hash function must be used throughout an
instantiation of this generator.
7,2.1.1 Notation
In the following sections the notation give below is used:
hash_length is the output size of the underlying hash function in
use.
input_entropy is the input bit string that provides entropy to the
generator.
K is a bit string of size hash_length that is part of the state of
the generator and is updated at least once each time random
bits are generated.
V is a bit string of size hash_length and is part of the state of
the generator which is updated each time hash_length bits of
output are generated.
| represents concatenation
7.1.2.2 Initializing the Generator
Set V to all zero bytes except that the low order bit of each byte is
set to one.
Set K to all zero bytes.
K = HMAC ( K, V | 0x00 | input_entropy )
V = HMAC ( K, V )
K = HMAC ( K, V | 0x01 | input_entropy )
V = HMAC ( K, V )
Note: all SHA algorithms produce an integral number of bytes of the
length of K and V will be an integral number of bytes.
7.1.2.5 Generating Random Bits
When output is called for simply set
V = HMAC ( K, V )
and use leading bits from V. If more bits are needed than the length
of V, set "temp" to a null bit string and then repeatedly perform
V = HMAC ( K, V )
temp = temp | V
stopping as soon a temp is equal to or longer than the number of
random bits called for and use the called for number of leading bits
from temp. The definition of the algorithm prohibits calling from
more than 2**35 bits.
7.2.2 X9.17 Key Generation
The American National Standards Institute has specified a method for
generating a sequence of keys as follows [X9.17]:
s is the initial 64 bit seed
0
g is the sequence of generated 64 bit key quantities
n
k is a random key reserved for generating this key sequence
t is the time at which a key is generated to as fine a resolution
as is available (up to 64 bits).
DES ( K, Q ) is the DES encryption of quantity Q with key K
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.
7.2.3 DSS Pseudo-Random Number Generation
Appendix 3 of the NIST Digital Signature Standard [DSS] provides a
method of producing a sequence of pseudo-random 160 bit quantities
for use as private keys or the like. This has been modified by Change
Notice 1 [DSS CN1] to produce the following algorithm for generating
general purpose pseudorandom numbers:
t = 0x 67452301 EFCDAB89 98BADCFE 10325476 C3D2E1F0
XKEY = initial seed
0
For j = 0 to ...
XVAL = ( XKEY + optional user input ) (Mod 2^512)
j
X = G( t, XVAL )
j
XKEY = ( 1 + XKEY + X ) (Mod 2^512)
j+1 j j
The quantities X thus produced are the pseudo-random sequence of 160
bit values. Two functions can be used for "G" above. Each produces
a 160-bit value and takes two arguments, the first argument a 160-bit
value and the second a 512 bit value.
The first is based on SHA-1 and works by setting the 5 linking
variables, denoted H with subscripts in the SHA-1 specification, to
the first argument divided into fifths. Then steps (a) through (e) of
section 7 of the NIST SHA-1 specification are run over the second
argument as if it were a 512-bit data block. The values of the
linking variable after those steps are then concatenated to produce
the output of G. [SHA*]
As an alternative second method, NIST also defined an alternate G
function based on multiple applications of the DES encryption
function [DSS].
8. Examples of Randomness Required 8. Examples of Randomness Required
Below are two examples showing rough calculations of needed Below are two examples showing rough calculations of needed
randomness for security. The first is for moderate security passwords randomness for security. The first is for moderate security passwords
while the second assumes a need for a very high security while the second assumes a need for a very high security
cryptographic key. cryptographic key.
In addition [ORMAN] and [RSA BULL13] provide information on the In addition [ORMAN] and [RSA BULL13] provide information on the
public key lengths that should be used for exchanging symmetric keys. public key lengths that should be used for exchanging symmetric keys.
skipping to change at page 38, line 49 skipping to change at page 41, line 49
could break the key in 2 weeks (on average they need try only half could break the key in 2 weeks (on average they need try only half
the keys). the keys).
These questions are considered in detail in "Minimal Key Lengths for These questions are considered in detail in "Minimal Key Lengths for
Symmetric Ciphers to Provide Adequate Commercial Security: A Report Symmetric Ciphers to Provide Adequate Commercial Security: A Report
by an Ad Hoc Group of Cryptographers and Computer Scientists" by an Ad Hoc Group of Cryptographers and Computer Scientists"
[KeyStudy] which was sponsored by the Business Software Alliance. It [KeyStudy] which was sponsored by the Business Software Alliance. It
concluded that a reasonable key length in 1995 for very high security concluded that a reasonable key length in 1995 for very high security
is in the range of 75 to 90 bits and, since the cost of cryptography is in the range of 75 to 90 bits and, since the cost of cryptography
does not vary much with they key size, recommends 90 bits. To update does not vary 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 these recommendations, just add 2/3 of a bit per year for Moore's
law [MOORE]. Thus, in the year 2004, this translates to a
[MOORE]. Thus, in the year 2004, this translates to a determination determination that a reasonable key length is in the 81 to 96 bit
that a reasonable key length is in the 81 to 96 bit range. In fact, range. In fact, today, it is increasingly common to use keys longer
today, it is increasingly common to use keys longer than 96 bits, than 96 bits, such as 128-bit (or longer) keys with AES and keys with
such as 128-bit (or longer) keys with AES and keys with effective effective lengths of 112-bits using triple-DES.
lengths of 112-bits using triple-DES.
8.2.2 Meet in the Middle Attacks 8.2.2 Meet in the Middle Attacks
If chosen or known plain text and the resulting encrypted text are If chosen or known plain text and the resulting encrypted text are
available, a "meet in the middle" attack is possible if the structure available, a "meet in the middle" attack is possible if the structure
of the encryption algorithm allows it. (In a known plain text attack, of the encryption algorithm allows it. (In a known plain text attack,
the adversary knows all or part of the messages being encrypted, the adversary knows all or part of the messages being encrypted,
possibly some standard header or trailer fields. In a chosen plain possibly some standard header or trailer fields. In a chosen plain
text attack, the adversary can force some chosen plain text to be text attack, the adversary can force some chosen plain text to be
encrypted, possibly by "leaking" an exciting text that would then be encrypted, possibly by "leaking" an exciting text that would then be
skipping to change at page 41, line 10 skipping to change at page 44, line 10
bit key size derived above. bit key size derived above.
For further examples of conservative design principles see For further examples of conservative design principles see
[FERGUSON]. [FERGUSON].
9. Conclusion 9. Conclusion
Generation of unguessable "random" secret quantities for security use Generation of unguessable "random" secret quantities for security use
is an essential but difficult task. is an essential but difficult task.
Hardware techniques to produce such randomness would be relatively Hardware techniques to produce the needed entropy would be relatively
simple. In particular, the volume and quality would not need to be simple. In particular, the volume and quality would not need to be
high and existing computer hardware, such as audio input or disk high and existing computer hardware, such as audio input or disk
drives, can be used. drives, can be used.
Widely available computational techniques are available to process Widely available computational techniques are available to process
low quality random quantities from multiple sources or a larger low quality random quantities from multiple sources or a larger
quantity of such low quality input from one source and produce a quantity of such low quality input from one source and produce a
smaller quantity of higher quality keying material. In the absence of smaller quantity of higher quality keying material. In the absence of
hardware sources of randomness, a variety of user and software hardware sources of randomness, a variety of user and software
sources can frequently, with care, be used instead; however, most sources can frequently, with care, be used instead; however, most
skipping to change at page 42, line 14 skipping to change at page 45, line 14
10. Security Considerations 10. Security Considerations
The entirety of this document concerns techniques and recommendations The entirety of this document concerns techniques and recommendations
for generating unguessable "random" quantities for use as passwords, for generating unguessable "random" quantities for use as passwords,
cryptographic keys, initialization vectors, sequence numbers, and cryptographic keys, initialization vectors, sequence numbers, and
similar security uses. similar security uses.
11. Copyright and Disclaimer 11. Copyright and Disclaimer
Copyright (C) The Internet Society 2004. This document is subject to Copyright (C) The Internet Society 2005. This document is subject to
the rights, licenses and restrictions contained in BCP 78 and except the rights, licenses and restrictions contained in BCP 78 and except
as set forth therein, the authors retain all their rights. as set forth therein, the authors retain all their rights.
This document and the information contained herein are provided on an This document and the information contained herein are provided on an
"AS IS" basis and THE CONTRIBUTOR, THE ORGANIZATION HE/SHE REPRESENTS "AS IS" basis and THE CONTRIBUTOR, THE ORGANIZATION HE/SHE REPRESENTS
OR IS SPONSORED BY (IF ANY), THE INTERNET SOCIETY AND THE INTERNET OR IS SPONSORED BY (IF ANY), THE INTERNET SOCIETY AND THE INTERNET
ENGINEERING TASK FORCE DISCLAIM ALL WARRANTIES, EXPRESS OR IMPLIED, ENGINEERING TASK FORCE DISCLAIM ALL WARRANTIES, EXPRESS OR IMPLIED,
INCLUDING BUT NOT LIMITED TO ANY WARRANTY THAT THE USE OF THE INCLUDING BUT NOT LIMITED TO ANY WARRANTY THAT THE USE OF THE
INFORMATION HEREIN WILL NOT INFRINGE ANY RIGHTS OR ANY IMPLIED INFORMATION HEREIN WILL NOT INFRINGE ANY RIGHTS OR ANY IMPLIED
WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE.
12. Appendix A: Changes from RFC 1750 12. Appendix A: Changes from RFC 1750
1. Additional acknowledgements have been added. 1. Additional acknowledgements have been added.
2. Insertion of section 5.2.4 on de-skewing with S-boxes. 2. Insertion of section 5.3 on mixing with S-boxes.
3. Addition of section 5.4 on Ring Oscillator randomness sources. 3. Addition of section 3.3 on Ring Oscillator randomness sources.
4. AES and the members of the SHA series producing more than 160 4. AES and the members of the SHA series producing more than 160
bits have been added. Use of AES has been emphasized and the use bits have been added. Use of AES has been emphasized and the use
of DES de-emphasized. of DES de-emphasized.
5. Addition of section 6.3.3 on entropy pool techniques. 5. Addition of section 6.3 on entropy pool techniques.
6. Addition of section 7.3 on the pseudo-random number generation 6. Addition of section 7.2.3 on the pseudo-random number generation
techniques given in FIPS 186-2 (with Change Notice 1), 7.4 on techniques given in FIPS 186-2 (with Change Notice 1), 7.2.1 on
those given in X9.82, section 7.5 on the random number generation those given in X9.82, section 7.1.2 on the random number
techniques of the /dev/random device in Linux and other UNIX generation techniques of the /dev/random device in Linux and
systems, and section 7.6 on random number generation techniques other UNIX systems, and section 7.1.3 on random number generation
in the Windows operating system. techniques in the Windows operating system.
7. Addition of references to the "Minimal Key Lengths for Symmetric 7. Addition of references to the "Minimal Key Lengths for Symmetric
Ciphers to Provide Adequate Commercial Security" study published Ciphers to Provide Adequate Commercial Security" study published
in January 1996 [KeyStudy]. in January 1996 [KeyStudy] and to [RFC 1948].
8. Added caveats to using Diffie-Hellman as a mixing function. 8. Added caveats to using Diffie-Hellman as a mixing function and,
because of those caveats and its computationally intensive
nature, recommend against its use.
9. Addition of references to the [TURBID] paper and system. 9. Addition of references to the X9.82 effort and the [TURBID] and
[NASLUND] papers.
10. Addition of discussion of min-entropy and Renyi entropy and 10. Addition of discussion of min-entropy and Renyi entropy and
references to the [LUBY] book. references to the [LUBY] book.
11. Minor wording changes and reference updates. 11. Major restructuring, minor wording changes, and a variety of
reference updates.
14. Informative References 13. Informative References
[AES] - "Specification of the Advanced Encryption Standard (AES)", [AES] - "Specification of the Advanced Encryption Standard (AES)",
United States of America, US National Institute of Standards and United States of America, US National Institute of Standards and
Technology, FIPS 197, November 2001. Technology, FIPS 197, November 2001.
[ASYMMETRIC] - "Secure Communications and Asymmetric Cryptosystems", [ASYMMETRIC] - "Secure Communications and Asymmetric Cryptosystems",
edited by Gustavus J. Simmons, AAAS Selected Symposium 69, Westview edited by Gustavus J. Simmons, AAAS Selected Symposium 69, Westview
Press, Inc. Press, Inc.
[BBS] - "A Simple Unpredictable Pseudo-Random Number Generator", SIAM [BBS] - "A Simple Unpredictable Pseudo-Random Number Generator", SIAM
Journal on Computing, v. 15, n. 2, 1986, L. Blum, M. Blum, & M. Shub. Journal on Computing, v. 15, n. 2, 1986, L. Blum, M. Blum, & M. Shub.
[BRILLINGER] - "Time Series: Data Analysis and Theory", Holden-Day, [BRILLINGER] - "Time Series: Data Analysis and Theory", Holden-Day,
1981, David Brillinger. 1981, David Brillinger.
[CRC] - "C.R.C. Standard Mathematical Tables", Chemical Rubber [CRC] - "C.R.C. Standard Mathematical Tables", Chemical Rubber
Publishing Company. Publishing Company.
[DAVIS] - "Cryptographic Randomness from Air Turbulence in Disk [DAVIS] - "Cryptographic Randomness from Air Turbulence in Disk
Drives", Advances in Cryptology - Crypto '94, Springer-Verlag Lecture Drives", Advances in Cryptology - Crypto '94, Springer-Verlag
Notes in Computer Science #839, 1984, Don Davis, Ross Ihaka, and Lecture Notes in Computer Science #839, 1984, Don Davis, Ross Ihaka,
Philip Fenstermacher. and Philip Fenstermacher.
[DES] - "Data Encryption Standard", US National Institute of [DES] - "Data Encryption Standard", US National Institute of
Standards and Technology, FIPS 46-3, October 1999. Standards and Technology, FIPS 46-3, October 1999.
- "Data Encryption Algorithm", American National Standards - "Data Encryption Algorithm", American National Standards
Institute, ANSI X3.92-1981. Institute, ANSI X3.92-1981.
(See also FIPS 112, Password Usage, which includes FORTRAN (See also FIPS 112, Password Usage, which includes FORTRAN
code for performing DES.) code for performing DES.)
[D-H] - RFC 2631, "Diffie-Hellman Key Agreement Method", Eric [D-H] - RFC 2631, "Diffie-Hellman Key Agreement Method", Eric
Rescrola, June 1999. Rescrola, June 1999.
skipping to change at page 46, line 10 skipping to change at page 49, line 10
[MAIL PEM 3] - RFC 1423, "Privacy Enhancement for Internet [MAIL PEM 3] - RFC 1423, "Privacy Enhancement for Internet
Electronic Mail: Part III: Algorithms, Modes, and Identifiers", D. Electronic Mail: Part III: Algorithms, Modes, and Identifiers", D.
Balenson, 02/10/1993. Balenson, 02/10/1993.
[MAIL PEM 4] - RFC 1424, "Privacy Enhancement for Internet [MAIL PEM 4] - RFC 1424, "Privacy Enhancement for Internet
Electronic Mail: Part IV: Key Certification and Related Services", B. Electronic Mail: Part IV: Key Certification and Related Services", B.
Kaliski, 02/10/1993. Kaliski, 02/10/1993.
[MAIL PGP] [MAIL PGP]
- RFC 2440, "OpenPGP Message Format", J. Callas, L. - RFC 2440, "OpenPGP Message Format", J. Callas, L.
Donnerhacke, H. Finney, R. Thayer", November 1998. Donnerhacke, H. Finney, R. Thayer, November 1998.
- RFC 3156, "MIME Security with OpenPGP" M. Elkins, D. Del - RFC 3156, "MIME Security with OpenPGP" M. Elkins, D. Del
Torto, R. Levien, T. Roessler, August 2001. Torto, R. Levien, T. Roessler, August 2001.
[MAIL S/MIME] - RFCs 2632 through 2634: [MAIL S/MIME] - RFCs 2632 through 2634:
- RFC 2632, "S/MIME Version 3 Certificate Handling", B. - RFC 2632, "S/MIME Version 3 Certificate Handling", B.
Ramsdell, Ed., June 1999. Ramsdell, Ed., June 1999.
- RFC 2633, "S/MIME Version 3 Message Specification", B. - RFC 2633, "S/MIME Version 3 Message Specification", B.
Ramsdell, Ed., June 1999. Ramsdell, Ed., June 1999.
- RFC 2634, "Enhanced Security Services for S/MIME" P. - RFC 2634, "Enhanced Security Services for S/MIME" P.
Hoffman, Ed., June 1999. Hoffman, Ed., June 1999.
skipping to change at page 46, line 40 skipping to change at page 49, line 40
- "Data Encryption Algorithm - Modes of Operation", American - "Data Encryption Algorithm - Modes of Operation", American
National Standards Institute, ANSI X3.106-1983. National Standards Institute, ANSI X3.106-1983.
[MOORE] - Moore's Law: the exponential increase in the logic density [MOORE] - Moore's Law: the exponential increase in the logic density
of silicon circuits. Originally formulated by Gordon Moore in 1964 as of silicon circuits. Originally formulated by Gordon Moore in 1964 as
a doubling every year starting in 1962, in the late 1970s the rate 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 fell to a doubling every 18 months and has remained there through the
date of this document. See "The New Hacker's Dictionary", Third date of this document. See "The New Hacker's Dictionary", Third
Edition, MIT Press, ISBN 0-262-18178-9, Eric S. Raymond, 1996. Edition, MIT Press, ISBN 0-262-18178-9, Eric S. Raymond, 1996.
[NASLUND] - "Extraction of Optimally Unbiased Bits from a Biased
Source", M. Naslund and A. Russell, IEEE Transactions on Information
Theory. 46(3), May 2000.
<http://www.engr.uconn.edu/~acr/Papers/biasIEEEjour.ps>
[ORMAN] - "Determining Strengths For Public Keys Used For Exchanging [ORMAN] - "Determining Strengths For Public Keys Used For Exchanging
Symmetric Keys", RFC 3766, Hilarie Orman, Paul Hoffman, April 2004. Symmetric Keys", RFC 3766, Hilarie Orman, Paul Hoffman, April 2004.
[RFC 1750] - "Randomness Requirements for Security", D. Eastlake, S. [RFC 1750] - "Randomness Requirements for Security", D. Eastlake, S.
Crocker, J. Schiller, December 1994. Crocker, J. Schiller, December 1994.
[RFC 1948] - "Defending Against Sequence Number Attacks", S.
Bellovin, May 1986.
[RSA BULL1] - "Suggestions for Random Number Generation in Software", [RSA BULL1] - "Suggestions for Random Number Generation in Software",
RSA Laboratories Bulletin #1, January 1996. RSA Laboratories Bulletin #1, January 1996.
[RSA BULL13] - "A Cost-Based Security Analysis of Symmetric and [RSA BULL13] - "A Cost-Based Security Analysis of Symmetric and
Asymmetric Key Lengths", RSA Laboratories Bulletin #13, Robert Asymmetric Key Lengths", RSA Laboratories Bulletin #13, Robert
Silverman, April 2000 (revised November 2001). Silverman, April 2000 (revised November 2001).
[SBOX1] - "Practical s-box design", S. Mister, C. Adams, Selected [SBOX1] - "Practical s-box design", S. Mister, C. Adams, Selected
Areas in Cryptography, 1996. Areas in Cryptography, 1996.
skipping to change at page 47, line 44 skipping to change at page 50, line 52
[TURBID] - "High Entropy Symbol Generator", John S. Denker, [TURBID] - "High Entropy Symbol Generator", John S. Denker,
<http://www.av8n.com/turbid/paper/turbid.htm>, 2003. <http://www.av8n.com/turbid/paper/turbid.htm>, 2003.
[USENET] - RFC 977, "Network News Transfer Protocol", B. Kantor, P. [USENET] - RFC 977, "Network News Transfer Protocol", B. Kantor, P.
Lapsley, February 1986. Lapsley, February 1986.
- RFC 2980, "Common NNTP Extensions", S. Barber, October - RFC 2980, "Common NNTP Extensions", S. Barber, October
2000. 2000.
[VON NEUMANN] - "Various techniques used in connection with random [VON NEUMANN] - "Various techniques used in connection with random
digits", von Neumann's Collected Works, Vol. 5, Pergamon Press, 1963, digits", von Neumann's Collected Works, Vol. 5, Pergamon Press,
J. von Neumann. 1963, J. von Neumann.
[WSC] - "Writing Secure Code, Second Edition", Michael Howard, David. [WSC] - "Writing Secure Code, Second Edition", Michael Howard, David.
C. LeBlanc, Microsoft Press, ISBN 0735617228, December 2002. C. LeBlanc, Microsoft Press, ISBN 0735617228, December 2002.
[X9.17] - "American National Standard for Financial Institution Key [X9.17] - "American National Standard for Financial Institution Key
Management (Wholesale)", American Bankers Association, 1985. Management (Wholesale)", American Bankers Association, 1985.
[X9.82] - "Random Number Generation", American National Standards [X9.82] - "Random Number Generation", American National Standards
Institute, ANSI X9F1, work in progress. Institute, ANSI X9F1, work in progress.
Part 1 - Overview and General Principles.
Part 2 - Non-Deterministic Random Bit Generators
Part 3 - Deterministic Random Bit Generators
Author's Addresses Author's Addresses
Donald E. Eastlake 3rd Donald E. Eastlake 3rd
Motorola Laboratories Motorola Laboratories
155 Beaver Street 155 Beaver Street
Milford, MA 01757 USA Milford, MA 01757 USA
Telephone: +1 508-786-7554 (w) Telephone: +1 508-786-7554 (w)
+1 508-634-2066 (h) +1 508-634-2066 (h)
skipping to change at page 48, line 30 skipping to change at page 52, line 30
Telephone: +1 617-253-0161 Telephone: +1 617-253-0161
E-mail: jis@mit.edu E-mail: jis@mit.edu
Steve Crocker Steve Crocker
EMail: steve@stevecrocker.com EMail: steve@stevecrocker.com
File Name and Expiration File Name and Expiration
This is file draft-eastlake-randomness2-09.txt. This is file draft-eastlake-randomness2-10.txt.
It expires April 2005. It expires July 2005.
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