< draft-eastlake-randomness2-08.txt   draft-eastlake-randomness2-09.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 February 2005 August 2004 Expires April 2005 October 2004
Randomness Requirements for Security Randomness Requirements for Security
---------- ------------ --- -------- ---------- ------------ --- --------
<draft-eastlake-randomness2-08.txt> <draft-eastlake-randomness2-09.txt>
Status of This Document Status of This Document
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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.
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Copyright (C) The Internet Society 2004. 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
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pitfalls in using traditional pseudo-random number generation pitfalls in using traditional pseudo-random number generation
techniques for choosing such quantities. It recommends the use of techniques for choosing such quantities. It recommends the use of
truly random hardware techniques and shows that the existing hardware truly random hardware techniques and shows that the existing hardware
on many systems can be used for this purpose. It provides suggestions on many systems can be used for this purpose. It provides suggestions
to ameliorate the problem when a hardware solution is not available. to ameliorate the problem when a hardware solution is not available.
And it gives examples of how large such quantities need to be for And it gives examples of how large such quantities need to be for
some applications. some applications.
Acknowledgements Acknowledgements
Special thanks to Peter Gutmann, who has permitted the incorporation Special thanks to Paul Hoffman and John Kelsey for their extensive
of material from his paper "Software Generation of Practically Strong comments and to Peter Gutmann, who has permitted the incorporation of
Random Numbers", and to Paul Hoffman for his extensive comments. material from his paper "Software Generation of Practically Strong
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:
Tony Hansen, Sandy Harris, Russ Housley Daniel Brown, Don Davis, Peter Gutmann, Tony Hansen, Sandy
Harris, Paul Hoffman, Scott Hollenback, Russ Housley, Christian
Huitema, John Kelsey, 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...........................................3 Table of Contents..........................................3
Table of Contents..........................................4
1. Introduction............................................6 1. Introduction............................................5
2. General Requirements....................................7 2. General Requirements....................................6
3. Traditional Pseudo-Random Sequences.....................9 3. Traditional Pseudo-Random Sequences.....................9
4. Unpredictability.......................................11 4. Unpredictability.......................................11
4.1 Problems with Clocks and Serial Numbers...............11 4.1 Problems with Clocks and Serial Numbers...............11
4.2 Timing and Content of External Events.................12 4.2 Timing and Value of External Events...................12
4.3 The Fallacy of Complex Manipulation...................12 4.3 The Fallacy of Complex Manipulation...................12
4.4 The Fallacy of Selection from a Large Database........13 4.4 The Fallacy of Selection from a Large Database........13
5. Hardware for Randomness................................14 5. Hardware for Randomness................................15
5.1 Volume Required.......................................14 5.1 Volume Required.......................................15
5.2 Sensitivity to Skew...................................14 5.2 Sensitivity to Skew...................................15
5.2.1 Using Stream Parity to De-Skew......................15 5.2.1 Using Stream Parity to De-Skew......................16
5.2.2 Using Transition Mappings to De-Skew................16 5.2.2 Using Transition Mappings to De-Skew................17
5.2.3 Using FFT to De-Skew................................17 5.2.3 Using FFT to De-Skew................................18
5.2.4 Using Compression to De-Skew........................17 5.2.4 Using Compression to De-Skew........................18
5.3 Existing Hardware Can Be Used For Randomness..........18 5.3 Existing Hardware Can Be Used For Randomness..........19
5.3.1 Using Existing Sound/Video Input....................18 5.3.1 Using Existing Sound/Video Input....................19
5.3.2 Using Existing Disk Drives..........................18 5.3.2 Using Existing Disk Drives..........................19
5.4 Ring Oscillator Sources...............................19 5.4 Ring Oscillator Sources...............................20
6. Recommended Software Strategy..........................21 6. Recommended Software Strategy..........................22
6.1 Mixing Functions......................................21 6.1 Mixing Functions......................................22
6.1.1 A Trivial Mixing Function...........................21 6.1.1 A Trivial Mixing Function...........................22
6.1.2 Stronger Mixing Functions...........................22 6.1.2 Stronger Mixing Functions...........................23
6.1.3 Using S-Boxes for Mixing............................24 6.1.3 Using S-Boxes for Mixing............................25
6.1.4 Diffie-Hellman as a Mixing Function.................24 6.1.4 Diffie-Hellman as a Mixing Function.................25
6.1.5 Using a Mixing Function to Stretch Random Bits......24 6.1.5 Using a Mixing Function to Stretch Random Bits......25
6.1.6 Other Factors in Choosing a Mixing Function.........25 6.1.6 Other Factors in Choosing a Mixing Function.........26
6.2 Non-Hardware Sources of Randomness....................26 6.2 Non-Hardware Sources of Randomness....................27
6.3 Cryptographically Strong Sequences....................27 6.3 Cryptographically Strong Sequences....................28
6.3.1 Traditional Strong Sequences........................27 6.3.1 OFB and CTR Sequences...............................28
6.3.2 The Blum Blum Shub Sequence Generator...............28 6.3.2 The Blum Blum Shub Sequence Generator...............29
6.3.3 Entropy Pool Techniques.............................29 6.3.3 Entropy Pool Techniques.............................30
7. Key Generation Standards and Examples..................31 7. Key Generation Examples and Standards..................32
7.1 US DoD Recommendations for Password Generation........31 7.1 US DoD Recommendations for Password Generation........32
7.2 X9.17 Key Generation..................................31 7.2 X9.17 Key Generation..................................32
7.3 DSS Pseudo-Random Number Generation...................32 7.3 DSS Pseudo-Random Number Generation...................33
7.4 X9.82 Pseudo-Random Number Generation.................33 7.4 X9.82 Pseudo-Random Number Generation.................34
7.5 The /dev/random Device................................33 7.5 The /dev/random Device................................34
7.6 Windows CryptGenRandom................................36
8. Examples of Randomness Required........................35 8. Examples of Randomness Required........................37
8.1 Password Generation..................................35 8.1 Password Generation..................................37
8.2 A Very High Security Cryptographic Key................36 8.2 A Very High Security Cryptographic Key................38
8.2.1 Effort per Key Trial................................36 8.2.1 Effort per Key Trial................................38
8.2.2 Meet in the Middle Attacks..........................37 8.2.2 Meet in the Middle Attacks..........................39
8.2.3 Other Considerations................................38 8.2.3 Other Considerations................................40
9. Conclusion.............................................39 9. Conclusion.............................................41
10. Security Considerations...............................40 10. Security Considerations...............................42
11. Copyright and Disclaimer..............................40 11. Copyright and Disclaimer..............................42
12. Appendix A: Changes from RFC 1750.....................41 12. Appendix A: Changes from RFC 1750.....................43
14. Informative References................................42 14. Informative References................................44
Authors Addresses.........................................46 Author's Addresses........................................48
File Name and Expiration..................................46 File Name and Expiration..................................48
1. Introduction 1. Introduction
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
Enhanced Mail protocol [MAIL PEM]. Enhanced Mail protocol [MAIL PEM *].
These systems provide substantial protection against snooping and These systems provide substantial protection against snooping and
spoofing. However, there is a potential flaw. At the heart of all spoofing. However, there is a potential flaw. At the heart of all
cryptographic systems is the generation of secret, unguessable (i.e., cryptographic systems is the generation of secret, unguessable (i.e.,
random) numbers. random) numbers.
The lack of generally available facilities for generating such The lack of generally available facilities for generating such random
unpredictable numbers is an open wound in the design of cryptographic numbers, that is the lack of general availability of truly
software. For the software developer who wants to build a key or unpredictable sources, forms an open wound in the design of
password generation procedure that runs on a wide range of hardware, cryptographic software. For the software developer who wants to build
the only safe strategy so far has been to force the local a key or password generation procedure that runs on a wide range of
installation to supply a suitable routine to generate random numbers. hardware, this is a very real problem.
This is an awkward, error-prone and unpalatable solution.
It is important to keep in mind that the requirement is for data that It is important to keep in mind that the requirement is for data that
an adversary has a very low probability of guessing or determining. an adversary has a very low probability of guessing or determining.
This can easily fail if pseudo-random data is used which only meets This can easily fail if pseudo-random data is used which only meets
traditional statistical tests for randomness or which is based on traditional statistical tests for randomness or which is based on
limited range sources, such as clocks. Frequently such random limited range sources, such as clocks. Sometimes such pseudo-random
quantities are determinable by an adversary searching through an quantities are determinable by an adversary searching through an
embarrassingly small space of possibilities. embarrassingly small space of possibilities.
This Best Current Practice describes techniques for producing random This Best Current Practice describes techniques for producing random
quantities that will be resistant to such attack. It recommends that quantities that will be resistant to such attack. It recommends that
future systems include hardware random number generation or provide future systems include hardware random number generation or provide
access to existing hardware that can be used for this purpose. It access to existing hardware that can be used for this purpose. It
suggests methods for use if such hardware is not available. And it suggests methods for use if such hardware is not available. And it
gives some estimates of the number of random bits required for sample gives some estimates of the number of random bits required for sample
applications. applications.
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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
different types of random quantities that may be wanted. In the case
of human 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
quantities that are indistinguishable from truly random, that is, all
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
published to the world. It is computationally infeasible to determine published to the world. It is computationally infeasible to determine
the private key from the public key and knowledge of the public is of the private key from the public key and knowledge of the public is of
no help to an adversary [ASYMMETRIC]. [SCHNEIER, FERGUSON, KAUFMAN] no help to an adversary [ASYMMETRIC]. [SCHNEIER, FERGUSON, KAUFMAN]
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more probable values first. more probable values first.
For example, consider a cryptographic system that uses 128 bit keys. For example, consider a cryptographic system that uses 128 bit keys.
If these 128 bit keys are derived by using a fixed pseudo-random If these 128 bit keys are derived by using a fixed pseudo-random
number generator that is seeded with an 8 bit seed, then an adversary 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
in some cases for cryptographic analysis where what is really
important is the work factor for an adversary. For example, assume
that there was a pseudo-random number generator generating 128 bit
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 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,
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
also useful to look at other measures, such as min-entropy, defined
as
Min-entropy = - log ( maximum ( p ) )
i
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
of classical Shannon entropy.
A continuous spectrum of entropies, sometimes called Renyi entropy,
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 =
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
min-entropy is always the most conservative measure of entropy and
usually the best to use for cryptographic evaluation. [LUBY]
3. Traditional Pseudo-Random Sequences 3. Traditional Pseudo-Random Sequences
Most traditional sources of random numbers use deterministic sources This section talks about traditional sources of deterministic of
of "pseudo-random" numbers. These typically start with a "seed" "pseudo-random" numbers. These typically start with a "seed" quantity
quantity and use numeric or logical operations to produce a sequence and use numeric or logical operations to produce a sequence of
of values. 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. [KNUTH] has a classic exposition on pseudo-random numbers.
Applications he mentions are simulation of natural phenomena, Applications he mentions are simulation of natural phenomena,
sampling, numerical analysis, testing computer programs, decision sampling, numerical analysis, testing computer programs, decision
making, and games. None of these have the same characteristics as the 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 sort of security uses we are talking about. Only in the last two
could there be an adversary trying to find the random quantity. could there be an adversary trying to find the random quantity.
However, in these cases, the adversary normally has only a single However, in these cases, the adversary normally has only a single
chance to use a guessed value. In guessing passwords or attempting to chance to use a guessed value. In guessing passwords or attempting to
break an encryption scheme, the adversary normally has many, perhaps break an encryption scheme, the adversary normally has many, perhaps
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4. Unpredictability 4. Unpredictability
Statistically tested randomness in the traditional sense described in Statistically tested randomness in the traditional sense described in
section 3 is NOT the same as the unpredictability required for section 3 is NOT the same as the unpredictability required for
security use. security use.
For example, use of a widely available constant sequence, such as For example, use of a widely available constant sequence, such as
that from the CRC tables, is very weak against an adversary. Once that from the CRC tables, is very weak against an adversary. Once
they learn of or guess it, they can easily break all security, future they learn of or guess it, they can easily break all security, future
and past, based on the sequence. [CRC] Yet the statistical properties and past, based on the sequence. [CRC] Yet the statistical properties
of these tables are good. of these tables are good. So you should keep in mind that passing
statistical tests doesn't tell you that something is unpredictable.
The following sections describe the limitations of some randomness The following sections describe the limitations of some randomness
generation techniques and sources. generation techniques and sources. Much better sources are described
in Section 5.
4.1 Problems with Clocks and Serial Numbers 4.1 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
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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 Content of External Events 4.2 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どﾖ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.
The amount of mouse movement or the keys actually hit are usually The amount of mouse movement or the keys actually hit are usually
easier to access than timings but may yield less unpredictability as easier to access than timings but may yield less unpredictability as
the user may provide highly repetitive input. the user may provide highly repetitive input.
Other external events, such as network packet arrival times, can also Other external events, such as network packet arrival times and
be used, with care. In particular, the possibility of manipulation of lengths, can also be used, but only with great care. In particular,
such times by an adversary and the lack of history at system start up the possibility of manipulation of such network traffic measurements
must be considered. by an adversary and the lack of history at system start up must be
carefully considered. If this input is subject to manipulation, it
must not be trusted as a source of entropy.
Indeed, almost any external sensor, such as raw radio reception or
temperature sensing in appropriately equipped computers, can be used
in principle. But in each case careful consideration must be given to
how much such data is subject to adversarial manipulation and to how
much entropy it can actually provide.
The above techniques are quite powerful against any attackers having
no access to the quantities being measured. For example, they would
be powerful against offline attackers who had no access to your
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
of an external sensor, the more rapidly you can generate bits.
4.3 The Fallacy of Complex Manipulation 4.3 The Fallacy of Complex Manipulation
One strategy which may give a misleading appearance of One strategy which may give a misleading appearance of
unpredictability is to take a very complex algorithm (or an excellent unpredictability is to take a very complex algorithm (or an excellent
traditional pseudo-random number generator with good statistical traditional pseudo-random number generator with good statistical
properties) and calculate a cryptographic key by starting with properties) and calculate a cryptographic key by starting with
limited data such as the computer system clock value as the seed. An limited data such as the computer system clock value as the seed. An
adversary who knew roughly when the generator was started would have adversary who knew roughly when the generator was started would have
a relatively small number of seed values to test as they would know a relatively small number of seed values to test as they would know
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use the limited number of results stemming from a limited number of use the limited number of results stemming from a limited number of
seed values to defeat security. seed values to defeat security.
Another serious strategy error is to assume that a very complex Another serious strategy error is to assume that a very complex
pseudo-random number generation algorithm will produce strong random pseudo-random number generation algorithm will produce strong random
numbers when there has been no theory behind or analysis of the numbers when there has been no theory behind or analysis of the
algorithm. There is a excellent example of this fallacy right near algorithm. There is a excellent example of this fallacy right near
the beginning of Chapter 3 in [KNUTH] where the author describes a the beginning of Chapter 3 in [KNUTH] where the author describes a
complex algorithm. It was intended that the machine language program complex algorithm. It was intended that the machine language program
corresponding to the algorithm would be so complicated that a person corresponding to the algorithm would be so complicated that a person
trying to read the code without comments wouldnどﾖt know what the trying to read the code without comments wouldn't know what the
program was doing. Unfortunately, actual use of this algorithm showed program was doing. Unfortunately, actual use of this algorithm showed
that it almost immediately converged to a single repeated value in that it almost immediately converged to a single repeated value in
one case and a small cycle of values in another case. one case and a small cycle of values in another case.
Not only does complex manipulation not help you if you have a limited Not only does complex manipulation not help you if you have a limited
range of seeds but blindly chosen complex manipulation can destroy range of seeds but blindly chosen complex manipulation can destroy
the randomness in a good seed! the randomness in a good seed!
4.4 The Fallacy of Selection from a Large Database 4.4 The Fallacy of Selection from a Large Database
Another strategy that can give a misleading appearance of Another strategy that can give a misleading appearance of
unpredictability is selection of a quantity randomly from a database unpredictability is selection of a quantity randomly from a database
and assume that its strength is related to the total number of bits and assume that its strength is related to the total number of bits
in the database. For example, typical USENET servers process many in the database. For example, typical USENET servers process many
megabytes of information per day [USENET]. Assume a random quantity megabytes of information per day [USENET]. Assume a random quantity
was selected by fetching 32 bytes of data from a random starting 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 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 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 language and probably has no more than 2 or 3 bits of information per
byte, it doesnどﾖt yield 32*2 = 64 bits of unguessability. For an byte, it doesn't yield 32*2 = 64 bits of unguessability. For an
adversary with access to the same usenet database the unguessability adversary with access to the same usenet database the unguessability
rests only on the starting point of the selection. That is perhaps a rests only on the starting point of the selection. That is perhaps a
little over a couple of dozen bits of unguessability. little over a couple of dozen bits of unguessability.
The same argument applies to selecting sequences from the data on a The same argument applies to selecting sequences from the data on a
publicly available CD/DVD recording or any other large public publicly available CD/DVD recording or any other large public
database. If the adversary has access to the same database, this database. If the adversary has access to the same database, this
"selection from a large volume of data" step buys little. However, "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 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 access, such as system buffers on an active multi-user system, it may
be of help. be of help.
5. Hardware for Randomness 5. Hardware for Randomness
Is there any hope for true strong portable randomness in the future? Is there any hope for true strong portable randomness in the future?
There might be. All thatどﾖs needed is a physical source of There might be. All that's needed is a physical source of
unpredictable numbers. unpredictable numbers.
A thermal noise (sometimes called Johnson noise in integrated A thermal noise (sometimes called Johnson noise in integrated
circuits) or radioactive decay source and a fast, free-running circuits) or radioactive decay source and a fast, free-running
oscillator would do the trick directly [GIFFORD]. This is a trivial oscillator would do the trick directly [GIFFORD]. This is a trivial
amount of hardware, and could easily be included as a standard part amount of hardware, and could easily be included as a standard part
of a computer systemどﾖs architecture. Most audio (or video) input of a computer system's architecture. Most audio (or video) input
devices are useable [TURBID]. Furthermore, any system with a devices are useable [TURBID]. Furthermore, any system with a
spinning disk or ring oscillator and a stable (crystal) time source spinning disk or ring oscillator and a stable (crystal) time source
or the like has an adequate source of randomness ([DAVIS] and Section or the like has an adequate source of randomness ([DAVIS] and Section
5.4). All thatどﾖs needed is the common perception among computer 5.4). All that's needed is the common perception among computer
vendors that this small additional hardware and the software to vendors that this small additional hardware and the software to
access it is necessary and useful. access it is necessary and useful.
5.1 Volume Required 5.1 Volume Required
How much unpredictability is needed? Is it possible to quantify the How much unpredictability is needed? Is it possible to quantify the
requirement in, say, number of random bits per second? requirement in, say, number of random bits per second?
The answer is not very much is needed. For AES, the key can be 128 The answer is not very much is needed. For AES, the key can be 128
bits and, as we show in an example in Section 8, even the highest bits and, as we show in an example in Section 8, even the highest
skipping to change at page 15, line 8 skipping to change at page 16, line 8
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 the random numbers? The good news is the distribution need not be
uniform. All that is needed is a conservative estimate of how non- uniform. All that is needed is a conservative estimate of how non-
uniform it is to bound performance. Simple techniques to de-skew the uniform it is to bound performance. Simple techniques to de-skew the
bit stream are given below and stronger cryptographic techniques are bit stream are given below and stronger cryptographic techniques are
described in Section 6.1.2 below. described in Section 6.1.2 below.
5.2.1 Using Stream Parity to De-Skew 5.2.1 Using Stream Parity to De-Skew
Consider taking a sufficiently long string of bits and map the string As a simple but not particularly practical example, consider taking a
to "zero" or "one". The mapping will not yield a perfectly uniform sufficiently long string of bits and map the string to "zero" or
distribution, but it can be as close as desired. One mapping that "one". The mapping will not yield a perfectly uniform distribution,
serves the purpose is to take the parity of the string. This has the but it can be as close as desired. One mapping that serves the
advantages that it is robust across all degrees of skew up to the purpose is to take the parity of the string. This has the advantages
estimated maximum skew and is absolutely trivial to implement in that it is robust across all degrees of skew up to the estimated
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.
skipping to change at page 16, line 27 skipping to change at page 17, line 27
| 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. 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
entropy.
5.2.2 Using Transition Mappings to De-Skew 5.2.2 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:
skipping to change at page 17, line 17 skipping to change at page 18, line 20
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 1どﾖs and 0どﾖs, it instead produces a uncorrelated series of random 1s and 0s, it instead produces a
totally predictable sequence of exactly alternating 1どﾖs and 0どﾖs. totally predictable sequence of exactly alternating 1s and 0s.
5.2.3 Using FFT to De-Skew 5.2.3 Using FFT to De-Skew
When real world data consists of strongly biased or correlated bits, When real world data consists of strongly biased or correlated bits,
it may still contain useful amounts of randomness. This randomness it may still contain useful amounts of randomness. This randomness
can be extracted through use of the discrete Fourier transform or its can be extracted through use of various transforms, the most powerful
optimized variant, the FFT. of which are described in section 6.1.2 below.
Using the Fourier transform of the data, strong correlations can be Using the Fourier transform of the data or its optimized variant, the
discarded. If adequate data is processed and remaining correlations FFT, is an technique interesting primarily for theoretical reasons.
decay, spectral lines approaching statistical independence and It can be show that this will discard strong correlations. If
normally distributed randomness can be produced [BRILLINGER]. adequate data is processed and remaining correlations decay, spectral
lines approaching statistical independence and normally distributed
randomness can be produced [BRILLINGER].
5.2.4 Using Compression to De-Skew 5.2.4 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 above or in Section 6.1.2. At a compared with those described in Section 6.1.2. At a minimum, the
minimum, the beginning of the compressed sequence should be skipped beginning of the compressed sequence should be skipped and only later
and only later bits used for applications requiring random bits. bits used for applications requiring roughly random bits.
5.3 Existing Hardware Can Be Used For Randomness 5.3 Existing Hardware Can Be Used For Randomness
As described below, many computers come with hardware that can, with As described below, many computers come with hardware that can, with
care, be used to generate truly random quantities. care, be used to generate truly random quantities.
5.3.1 Using Existing Sound/Video Input 5.3.1 Using Existing Sound/Video Input
Many computers are built with inputs that digitize some real world Many computers are built with inputs that digitize some real world
analog source, such as sound from a microphone or video input from a analog source, such as sound from a microphone or video input from a
camera. Under appropriate circumstances, such input can provide camera. Under appropriate circumstances, such input can provide
reasonably high quality random bits. The "input" from a sound reasonably high quality random bits. The "input" from a sound
digitizer with no source plugged in or a camera with the lens cap on, 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 if the system has enough gain to detect anything, is essentially
thermal noise. thermal noise. This method is extremely hardware and implementation
dependent.
For example, on some UNIX based systems, one can read from the For example, on some UNIX based systems, one can read from the
/dev/audio device with nothing plugged into the microphone jack or /dev/audio device with nothing plugged into the microphone jack or
the microphone receiving only low level background noise. Such data the microphone receiving only low level background noise. Such data
is essentially random noise although it should not be trusted without is essentially random noise although it should not be trusted without
some checking in case of hardware failure. It will, in any case, need some checking in case of hardware failure. It will, in any case, need
to be de-skewed as described elsewhere. to be de-skewed as described elsewhere.
Combining this with compression to de-skew one can, in UNIXese, Combining this with compression to de-skew one can, in UNIXese,
generate a huge amount of medium quality random data by doing generate a huge amount of medium quality random data by doing
cat /dev/audio | compress - >random-bits-file cat /dev/audio | compress - >random-bits-file
A detailed examination of this type of randomness source appears in A detailed examination of this type of randomness source appears in
[TURBID]. [TURBID].
5.3.2 Using Existing Disk Drives 5.3.2 Using Existing Disk Drives
Disk drives have small random fluctuations in their rotational speed Disk drives have small random fluctuations in their rotational speed
due to chaotic air turbulence [DAVIS]. By adding low level disk seek due to chaotic air turbulence [DAVIS, Jakobsson]. By adding low
time instrumentation to a system, a series of measurements can be level disk seek time instrumentation to a system, a series of
obtained that include this randomness. Such data is usually highly measurements can be obtained that include this randomness. Such data
correlated so that significant processing is needed, such as FFT (see is usually highly correlated so that significant processing is
section 5.2.3). Nevertheless experimentation has shown that, with needed, such as described in 6.1.2 below. Nevertheless
such processing, most disk drives easily produce 100 bits a minute or experimentation a decade ago showed that, with such processing, even
more of excellent random data. slow disk drives on the slower computers of that day could easily
produce 100 bits a minute or more of excellent random data.
Partly offsetting this need for processing is the fact that disk Every increase in processor speed, which increases the resolution
drive failure will normally be rapidly noticed. Thus, problems with with which disk motion can be timed, or increase in the rate of disk
this method of random number generation due to hardware failure are seeks, increases the rate of random bit generation possible with this
unlikely. 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 5.4 Ring Oscillator Sources
If an integrated circuit is being designed or field programmed, an If an integrated circuit is being designed or field programmed, an
odd number of gates can be connected in series to produce a free- 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 running ring oscillator. By sampling a point in the ring at a fixed
frequency, say one determined by a stable crystal oscillator, some frequency, say one determined by a stable crystal oscillator, some
amount of entropy can be extracted due to variations in the free- amount of entropy can be extracted due to variations in the free-
running oscillator timing. It is possible to increase the rate of running oscillator timing. It is possible to increase the rate of
entropy by xorどﾖing sampled values from a few ring oscillators with entropy by xor'ing sampled values from a few ring oscillators with
relatively prime lengths. It is sometimes recommended that an odd relatively prime lengths. It is sometimes recommended that an odd
number of rings be used so that, even if the rings somehow become number of rings be used so that, even if the rings somehow become
synchronously locked to each other, there will still be sampled bit synchronously locked to each other, there will still be sampled bit
transitions. Another possibility source to sample is the output of a transitions. Another possibility source to sample is the output of a
noisy diode. noisy diode.
Sampled bits from such sources will have to be heavily de-skewed, as 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 disk rotation timings must be (Section 5.3.2). An engineering study
would be needed to determine the amount of entropy being produced would be needed to determine the amount of entropy being produced
depending on the particular design. In any case, these can be good depending on the particular design. In any case, these can be good
skipping to change at page 21, line 20 skipping to change at page 22, line 20
sources and to mix them with a strong mixing function. Such a sources and to mix them with a strong mixing function. Such a
function will preserve the randomness present in any of the sources function will preserve the randomness present in any of the sources
even if other quantities being combined happen to be fixed or easily even if other quantities being combined happen to be fixed or easily
guessable. This may be advisable even with a good hardware source, as guessable. This may be advisable even with a good hardware source, as
hardware can also fail, though this should be weighed against any hardware can also fail, though this should be weighed against any
increase in the chance of overall failure due to added software increase in the chance of overall failure due to added software
complexity. complexity.
6.1 Mixing Functions 6.1 Mixing Functions
A strong mixing function is one which combines two or more inputs and A strong mixing function is one which combines inputs and produces an
produces an output where each output bit is a different complex non- output where each output bit is a different complex non-linear
linear function of all the input bits. On average, changing any input function of all the input bits. On average, changing any input bit
bit will change about half the output bits. But because the will change about half the output bits. But because the relationship
relationship is complex and non-linear, no particular output bit is is complex and non-linear, no particular output bit is guaranteed to
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 5.2
above. This is simply another case where a strong mixing function is above. 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 5.2.1 of using the parity of a
number of bits is simply the result of successively Exclusive Orどﾖ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 6.1.2.
6.1.1 A Trivial Mixing Function 6.1.1 A Trivial Mixing Function
A trivial example for single bit inputs is the Exclusive Or function, A trivial example for single bit inputs described only for expository
which is equivalent to addition without carry, as show in the table purposes is the Exclusive Or function, which is equivalent to
below. This is a degenerate case in which the one output bit always addition without carry, as show in the table below. This is a
changes for a change in either input bit. But, despite its degenerate case in which the one output bit always changes for a
simplicity, it provides a useful illustration. change in either input bit. But, despite its simplicity, it provides
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 |
+-----------+-----------+----------+ +-----------+-----------+----------+
skipping to change at page 23, line 25 skipping to change at page 24, line 25
Although the message digest functions are designed for variable Although the message digest functions are designed for variable
amounts of input, AES and other encryption functions can also be used amounts of input, AES and other encryption functions can also be used
to combine any number of inputs. If 128 bits of output is adequate, to combine any number of inputs. If 128 bits of output is adequate,
the inputs can be packed into a 128-bit data quantity and successive the inputs can be packed into a 128-bit data quantity and successive
AES keys, padding with zeros if needed, which are then used to AES keys, padding with zeros if needed, which are then used to
successively encrypt using AES in Electronic Codebook Mode. Or the successively encrypt using AES in Electronic Codebook Mode. Or the
input could be packed into one 128-bit key and multiple data blocks input could be packed into one 128-bit key and multiple data blocks
and a CBC-MAC calculated [MODES]. and a CBC-MAC calculated [MODES].
If more than 128 bits of output are needed, use more complex mixing. If more than 128 bits of output are needed and you want to employ
But keep in mind that it is absolutely impossible to get more bits of AES, use more complex mixing. But keep in mind that it is absolutely
"randomness" out than are put in. For example, if inputs are packed impossible to get more bits of "randomness" out than are put in. For
into three quantities, A, B, and C, use AES to encrypt A with B as a example, if inputs are packed into three quantities, A, B, and C, use
key and then with C as a key to produce the 1st part of the output, AES to encrypt A with B as a key and then with C as a key to produce
then encrypt B with C and then A for more output and, if necessary, the 1st part of the output, then encrypt B with C and then A for more
encrypt C with A and then B for yet more output. Still more output output and, if necessary, encrypt C with A and then B for yet more
can be produced by reversing the order of the keys given above to output. Still more output can be produced by reversing the order of
stretch things. The same can be done with the hash functions by the keys given above to stretch things. The same can be done with the
hashing various subsets of the input data or different copies of the hash functions by hashing various subsets of the input data or
input data with different prefixes and/or suffixes to produce different copies of the input data with different prefixes and/or
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 5.2.1 above reduced this
to one bit with only a 1/1000 deviance from being equally likely a to one bit with only a 1/1000 deviance from being equally likely a
zero or one. But, applying the equation for information given in zero or one. But, applying the equation for information given in
Section 2, this 308 bit skewed sequence has over 5 bits of Section 2, this 308 bit skewed sequence has over 5 bits of
information in it. Thus hashing it with SHA-1 and taking the bottom 5 information in it. Thus hashing it with SHA-1 and taking the bottom 5
bits of the result would yield 5 unbiased random bits as opposed to bits of the result would yield 5 unbiased random bits as opposed to
the single bit given by calculating the parity of the string. the single bit given by calculating the parity of the string.
skipping to change at page 24, line 14 skipping to change at page 25, line 14
6.1.3 Using S-Boxes for Mixing 6.1.3 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*]
skipping to change at page 25, line 8 skipping to change at page 26, line 8
6.1.5 Using a Mixing Function to Stretch Random Bits 6.1.5 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どﾖ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 way using Exclusive Or to produce one bit from two did the output.
above.
The last table in Section 6.1.1 shows that mixing a random bit with a The last table in Section 6.1.1 shows that mixing a random bit with a
constant bit with Exclusive Or will produce a random bit. While this 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 6.1.6 Other Factors in Choosing a Mixing Function
skipping to change at page 27, line 26 skipping to change at page 28, line 26
way from the previous value, then when any value is compromised, all way from the previous value, then when any value is compromised, all
future values can be determined. This would be the case, for example, future values can be determined. This would be the case, for example,
if each value were a constant function of the previously used values, if each value were a constant function of the previously used values,
even if the function were a very strong, non-invertible message even if the function were a very strong, non-invertible message
digest function. digest function.
(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どﾖed at generate identical sequences. These could, for example, be xor'ed at
one end with data being send, encrypting it, and xorどﾖ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 Traditional Strong Sequences 6.3.1 OFB and CTR Sequences
A traditional way to achieve a strong sequence has been to have the One way to achieve a strong sequence is to have the values be
values be produced by hashing the quantities produced by produced by taking a seed value and hashing the quantities produced
concatenating the seed with successive integers or the like and then by concatenating the seed with successive integers or the like and
mask the values obtained so as to limit the amount of generator state then mask the values obtained so as to limit the amount of generator
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 and feedback some or all of the random key and seed value to encrypt successive integers as in
output encrypted value into the value to be encrypted for the next counter (CTR) mode encryption. Alternatively, you can feedback all of
iteration. Appropriate feedback techniques will usually be the output encrypted value into the value to be encrypted for the
recommended with the encryption algorithm. An example is shown below next iteration. This is a particular example of output feedback mode
where shifting and masking are used to combine the cypher output (OFB). [MODES]
feedback. This type of feedback is defined by the US Government in
connection with AES and DES [MODES] as Output Feedback Mode (OFM) but An example is shown below where shifting and masking are used to
should be avoided for reasons described below. combine part of the output feedback with part of the old input. This
type of partial feedback should be avoided for reasons described
below.
+---------------+ +---------------+
| V | | V |
| | n |--+ | | n |--+
+--+------------+ | +--+------------+ |
| | +---------+ | | +---------+
shift| +---> | | +-----+ shift| +---> | | +-----+
+--+ | Encrypt | <--- | Key | +--+ | Encrypt | <--- | Key |
| +-------- | | +-----+ | +-------- | | +-----+
| | +---------+ | | +---------+
skipping to change at page 28, line 49 skipping to change at page 29, line 49
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 6.3.2 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.
Itどﾖs 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 6.3.1 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
skipping to change at page 29, line 45 skipping to change at page 30, line 45
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 6.3.3 Entropy Pool Techniques
Many modern pseudo-random number sources utilize the technique of Many modern pseudo-random number sources utilize the technique of
maintaining a "pool" of bits and providing operations for strongly maintaining a "pool" of bits and providing operations for strongly
mixing input with some randomness into the pool and extracting psuedo mixing input with some randomness into the pool and extracting pseudo
random bits from the pool. This is illustrated in the figure below. 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 Section 7.5 for more
details on an example implementation and [RSA BULL1] for similar details on an example implementation and [RSA BULL1] for similar
suggestions. suggestions.
7. Key Generation Standards and Examples 7. Key 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 without special hardware. Three
standards are described below. The two older standards use DES, with standards are described below. The two older standards use DES, with
its 64-bit block and key size limit, but any equally strong or its 64-bit block and key size limit, but any equally strong or
stronger mixing function could be substituted. The third is a more stronger mixing function could be substituted [DES]. The third is a
modern and stronger standard based on SHA-1. Lastly the widely more modern and stronger standard based on SHA-1 [SHA*]. Lastly the
deployed modern UNIX random number generators are described. widely deployed modern UNIX random number generators are described.
7.1 US DoD Recommendations for Password Generation 7.1 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,
skipping to change at page 32, line 24 skipping to change at page 33, line 24
s = DES ( k, DES ( k, t ) .xor. g ) s = DES ( k, DES ( k, t ) .xor. g )
n+1 n n+1 n
If g sub n is to be used as a DES key, then every eighth bit should 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 be adjusted for parity for that use but the entire 64 bit unmodified
g should be used in calculating the next s. g should be used in calculating the next s.
7.3 DSS Pseudo-Random Number Generation 7.3 DSS Pseudo-Random Number Generation
Appendix 3 of the NIST Digital Signature Standard [DSS] provides an Appendix 3 of the NIST Digital Signature Standard [DSS] provides a
approved method of producing a sequence of pseudo-random 160 bit method of producing a sequence of pseudo-random 160 bit quantities
quantities for use as private keys or the like. A subset of that for use as private keys or the like. This has been modified by Change
algorithm is as follows: Notice 1 [DSS CN1] to produce the following algorithm for generating
general purpose pseudorandom numbers:
t = 0x 67452301 EFCDAB89 98BADCFE 10325476 C3D2E1F0 t = 0x 67452301 EFCDAB89 98BADCFE 10325476 C3D2E1F0
q = a 160-bit prime number
XKEY = initial seed XKEY = initial seed
0 0
For j = 0 to ... For j = 0 to ...
XVAL = ( XKEY + optional user input ) (Mod 2^512) XVAL = ( XKEY + optional user input ) (Mod 2^512)
j j
X = G( t, XVAL ) (Mod q) X = G( t, XVAL )
j j
XKEY = ( 1 + XKEY + X ) (Mod 2^512) XKEY = ( 1 + XKEY + X ) (Mod 2^512)
j+1 j j j+1 j j
The quantities X thus produced are the pseudo-random sequence of The quantities X thus produced are the pseudo-random sequence of 160
values in the rang 0 to q. Two functions can be used for "G" above. bit values. Two functions can be used for "G" above. Each produces
Each produces a 160-bit value and takes two arguments, the first a a 160-bit value and takes two arguments, the first argument a 160-bit
160-bit value and the second a 512 bit value. value and the second a 512 bit value.
The first is based on SHA-1 and works by setting the 5 linking 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 variables, denoted H with subscripts in the SHA-1 specification, to
the first argument divided into fifths. Then steps (a) through (e) of 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 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 argument as if it were a 512-bit data block. The values of the
linking variable after those steps are then concatenated to produce linking variable after those steps are then concatenated to produce
the output of G. [SHA-1] the output of G. [SHA*]
As an alternative second method, NIST also defined an alternate G As an alternative second method, NIST also defined an alternate G
function based on multiple applications of the DES encryption function based on multiple applications of the DES encryption
function [DSS]. function [DSS].
7.4 X9.82 Pseudo-Random Number Generation 7.4 X9.82 Pseudo-Random Number Generation
The National Institute for Standards and Technology (NIST) and the The National Institute for Standards and Technology (NIST) and the
American National Standards Institutes (ANSI) X9F1 committee are in American National Standards Institutes (ANSI) X9F1 committee are in
the final stages of creating a standard for random number generation. the final stages of creating a standard for random number generation
This standard includes a number of random number generators for use covering both true randomness generators and pseudo-random number
with AES and other block ciphers. It also includes random number generators. It includes a number of pseudo-random number generators
generators based on hash functions and the arithmetic of elliptic for use with AES and other block ciphers. It also includes random
curves [X9.82]. number generators based on hash functions and the arithmetic of
elliptic curves [X9.82].
7.5 The /dev/random Device 7.5 The /dev/random Device
Several versions of the UNIX operating system provides a kernel- Several versions of the UNIX operating system provides 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 makes 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どﾖed into the pool and the pool is stirred via a primitive event is xor'ed into the pool and the pool is stirred via a primitive
polynomial of degree 128. The pool itself is treated as a ring 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
skipping to change at page 34, line 16 skipping to change at page 35, line 16
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 [SHA1] 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 scripts (and shutdown scripts) save the pool to a disk file
at shutdown and read this file at system startup. at shutdown and read this file at system startup.
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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どﾖs entropy estimate risk of continuing to take data even when the pool's entropy estimate
is small in that past output may be computable from current output is small in that past output may be computable from current output
provided an attacker can reverse SHA-1. Given that SHA-1 is designed provided an attacker can reverse SHA-1. Given that SHA-1 is designed
to be non-invertible, this is a reasonable risk. 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どﾖ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
Microsoft's recommendation to users of the widely deployed Windows
operating system is generally to use the CryptGenRandom pseudo-random
number generation call with the CryptAPI cryptographic service
provider. This takes a handle to a cryptographic service provider
library, a pointer to a buffer by which the caller can provide
entropy and into which the generated pseudo-randomness is returned,
and an indication of how many octets of randomness are desired.
The Windows CryptAPI cryptographic service provider stores a seed
state variable with every user. When CryptGenRandom is called, this
is combined with any randomness provided in the call and various
system and user data such as the process ID, thread ID, system clock,
system time, system counter, memory status, free disk clusters, and
hashed user environment block. This data is all feed to SHA-1 and the
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
seed state variable.
Users of Windows ".NET" will probably find it easier to use the
RNGCryptoServiceProvider.GetBytes method interface.
For further information, see [WSC].
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.
8.1 Password Generation 8.1 Password Generation
Assume that user passwords change once a year and it is desired that Assume that user passwords change once a year and it is desired that
the probability that an adversary could guess the password for a the probability that an adversary could guess the password for a
particular account be less than one in a thousand. Further assume particular account be less than one in a thousand. Further assume
that sending a password to the system is the only way to try a that sending a password to the system is the only way to try a
password. Then the crucial question is how often an adversary can try password. Then the crucial question is how often an adversary can try
possibilities. Assume that delays have been introduced into a system possibilities. Assume that delays have been introduced into a system
so that, at most, an adversary can make one password try every six so that, at most, an adversary can make one password try every six
seconds. Thatどﾖs 600 per hour or about 15,000 per day or about seconds. That's 600 per hour or about 15,000 per day or about
5,000,000 tries in a year. Assuming any sort of monitoring, it is 5,000,000 tries in a year. Assuming any sort of monitoring, it is
unlikely someone could actually try continuously for a year. In fact, unlikely someone could actually try continuously for a year. In fact,
even if log files are only checked monthly, 500,000 tries is more even if log files are only checked monthly, 500,000 tries is more
plausible before the attack is noticed and steps taken to change plausible before the attack is noticed and steps taken to change
passwords and make it harder to try more passwords. passwords and make it harder to try more passwords.
To have a one in a thousand chance of guessing the password in To have a one in a thousand chance of guessing the password in
500,000 tries implies a universe of at least 500,000,000 passwords or 500,000 tries implies a universe of at least 500,000,000 passwords or
about 2^29. Thus 29 bits of randomness are needed. This can probably about 2^29. Thus 29 bits of randomness are needed. This can probably
be achieved using the US DoD recommended inputs for password be achieved using the US DoD recommended inputs for password
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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どﾖ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 determination [MOORE]. Thus, in the year 2004, this translates to a determination
that a reasonable key length is in the 81 to 96 bit range. In fact, that a reasonable key length is in the 81 to 96 bit range. In fact,
today, it is increasingly common to use keys longer than 96 bits, today, it is increasingly common to use keys longer than 96 bits,
such as 128-bit (or longer) keys with AES and keys with effective such as 128-bit (or longer) keys with AES and keys with 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
skipping to change at page 37, line 41 skipping to change at page 39, line 41
amount of randomness in the very strong key to a minimum of 192 bits amount of randomness in the very strong key to a minimum of 192 bits
(96*2) is required for the year 2004 based on the [KeyStudy] (96*2) is required for the year 2004 based on the [KeyStudy]
analysis. analysis.
This amount of randomness is well beyond the limit of that in the This amount of randomness is well beyond the limit of that in the
inputs recommended by the US DoD for password generation and could inputs recommended by the US DoD for password generation and could
require user typing timing, hardware random number generation, or require user typing timing, hardware random number generation, or
other sources. other sources.
The meet in the middle attack assumes that the cryptographic The meet in the middle attack assumes that the cryptographic
algorithm can be decomposed in this way but we can not rule that out algorithm can be decomposed in this way. Hopefully no modern
without a deep knowledge of the algorithm. Even if a basic algorithm algorithm has this weakness but there may be cases where we are not
is not subject to a meet in the middle attack, an attempt to produce sure of that or even of what algorithm a key will be used with. Even
a stronger algorithm by applying the basic algorithm twice (or two if a basic algorithm is not subject to a meet in the middle attack,
different algorithms sequentially) with different keys may gain less an attempt to produce a stronger algorithm by applying the basic
added security than would be expected. Such a composite algorithm algorithm twice (or two different algorithms sequentially) with
would be subject to a meet in the middle attack. different keys will gain less added security than would be expected.
Such a composite algorithm would be subject to a meet in the middle
attack.
Enormous resources may be required to mount a meet in the middle Enormous resources may be required to mount a meet in the middle
attack but they are probably within the range of the national attack but they are probably within the range of the national
security services of a major nation. Essentially all nations spy on security services of a major nation. Essentially all nations spy on
other nations traffic. other nations traffic.
8.2.3 Other Considerations 8.2.3 Other Considerations
[KeyStudy] also considers the possibilities of special purpose code [KeyStudy] also considers the possibilities of special purpose code
breaking hardware and having an adequate safety margin. breaking hardware and having an adequate safety margin.
skipping to change at page 41, line 20 skipping to change at page 43, line 20
3. Addition of section 5.4 on Ring Oscillator randomness sources. 3. Addition of section 5.4 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 6.3.3 on entropy pool techniques.
6. Addition of section 7.3 on the pseudo-random number generation 6. Addition of section 7.3 on the pseudo-random number generation
techniques given in FIPS 186-2, 7.4 on those given in X9.82, and techniques given in FIPS 186-2 (with Change Notice 1), 7.4 on
section 7.5 on the random number generation techniques of the those given in X9.82, section 7.5 on the random number generation
/dev/random device in Linux and other UNIX systems. techniques of the /dev/random device in Linux and other UNIX
systems, and section 7.6 on random number generation 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].
8. Added caveats to using Diffie-Hellman as a mixing function. 8. Added caveats to using Diffie-Hellman as a mixing function.
9. Addition of references to the [TURBID] paper and system. 9. Addition of references to the [TURBID] paper and system.
10. Minor wording changes and reference updates. 10. Addition of discussion of min-entropy and Renyi entropy and
references to the [LUBY] book.
11. Minor wording changes and reference updates.
14. Informative References 14. 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.
skipping to change at page 42, line 25 skipping to change at page 44, line 25
[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 どﾖ94, Springer-Verlag Lecture Drives", Advances in Cryptology - Crypto '94, Springer-Verlag Lecture
Notes in Computer Science #839, 1984, Don Davis, Ross Ihaka, and Notes in Computer Science #839, 1984, Don Davis, Ross Ihaka, and
Philip Fenstermacher. 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.)
skipping to change at page 42, line 50 skipping to change at page 44, line 50
Eastlake, March 1999. Eastlake, March 1999.
[DoD] - "Password Management Guideline", United States of America, [DoD] - "Password Management Guideline", United States of America,
Department of Defense, Computer Security Center, CSC-STD-002-85. Department of Defense, Computer Security Center, CSC-STD-002-85.
(See also FIPS 112, Password Usage, which incorporates CSC-STD-002-85 (See also FIPS 112, Password Usage, which incorporates CSC-STD-002-85
as one of its appendices.) as one of its appendices.)
[DSS] - "Digital Signature Standard (DSS)", US National Institute of [DSS] - "Digital Signature Standard (DSS)", US National Institute of
Standards and Technology, FIPS 186-2, January 2000. Standards and Technology, FIPS 186-2, January 2000.
[DSS CN1] - "Digital Signature Standard Change Notice 1", US National
Institute of Standards and Technology, FIPS 186-2 Change Notice 1, 5
October 2001.
[FERGUSON] - "Practical Cryptography", Niels Ferguson and Bruce [FERGUSON] - "Practical Cryptography", Niels Ferguson and Bruce
Schneier, Wiley Publishing Inc., ISBN 047122894X, April 2003. Schneier, Wiley Publishing Inc., ISBN 047122894X, April 2003.
[GIFFORD] - "Natural Random Number", MIT/LCS/TM-371, David K. [GIFFORD] - "Natural Random Number", MIT/LCS/TM-371, David K.
Gifford, September 1988. Gifford, September 1988.
[IEEE 802.11i] - "Amendment to Standard for Telecommunications and [IEEE 802.11i] - "Amendment to Standard for Telecommunications and
Information Exchange Between Systems - LAN/MAN Specific Requirements Information Exchange Between Systems - LAN/MAN Specific Requirements
- Part 11: Wireless Medium Access Control (MAC) and physical layer - Part 11: Wireless Medium Access Control (MAC) and physical layer
(PHY) specifications: Medium Access Control (MAC) Security (PHY) specifications: Medium Access Control (MAC) Security
Enhancements", The Institute for Electrical and Electronics Enhancements", The Institute for Electrical and Electronics
Engineers, January 2004. Engineers, January 2004.
[IPSEC] - RFC 2401, "Security Architecture for the Internet [IPSEC] - RFC 2401, "Security Architecture for the Internet
Protocol", S. Kent, R. Atkinson, November 1998. Protocol", S. Kent, R. Atkinson, November 1998.
[Jakobsson] - M. Jakobsson, E. Shriver, B. K. Hillyer, and A. Juels,
"A practical secure random bit generator", Proceedings of the Fifth
ACM Conference on Computer and Communications Security, 1998. See
also http://citeseer.ist.psu.edu/article/jakobsson98practical.html.
[KAUFMAN] - "Network Security: Private Communication in a Public [KAUFMAN] - "Network Security: Private Communication in a Public
World", Charlie Kaufman, Radia Perlman, and Mike Speciner, Prentis World", Charlie Kaufman, Radia Perlman, and Mike Speciner, Prentis
Hall PTR, ISBN 0-13-046019-2, 2nd Edition 2002. Hall PTR, ISBN 0-13-046019-2, 2nd Edition 2002.
[KeyStudy] - "Minimal Key Lengths for Symmetric Ciphers to Provide [KeyStudy] - "Minimal Key Lengths for Symmetric Ciphers to Provide
Adequate Commercial Security: A Report by an Ad Hoc Group of Adequate Commercial Security: A Report by an Ad Hoc Group of
Cryptographers and Computer Scientists", M. Blaze, W. Diffie, R. Cryptographers and Computer Scientists", M. Blaze, W. Diffie, R.
Rivest, B. Schneier, T. Shimomura, E. Thompson, and M. Weiner, Rivest, B. Schneier, T. Shimomura, E. Thompson, and M. Weiner,
January 1996, <www.counterpane.com/keylength.html>. January 1996, <www.counterpane.com/keylength.html>.
[KNUTH] - "The Art of Computer Programming", Volume 2: Seminumerical [KNUTH] - "The Art of Computer Programming", Volume 2: Seminumerical
Algorithms, Chapter 3: Random Numbers. Addison Wesley Publishing Algorithms, Chapter 3: Random Numbers, Donald E. Knuth, Addison
Company, 3rd Edition November 1997, Donald E. Knuth. Wesley Publishing Company, 3rd Edition November 1997.
[KRAWCZYK] - "How to Predict Congruential Generators", Journal of [KRAWCZYK] - "How to Predict Congruential Generators", H. Krawczyk,
Algorithms, V. 13, N. 4, December 1992, H. Krawczyk Journal of Algorithms, V. 13, N. 4, December 1992.
[MAIL PEM] - RFCs 1421 through 1424: [LUBY] - "Pseudorandomness and Cryptographic Applications", Michael
- RFC 1421, Privacy Enhancement for Internet Electronic Mail: Luby, Princeton University Press, ISBN 0691025460, 8 January 1996.
Part I: Message Encryption and Authentication Procedures, 02/10/1993,
J. Linn [MAIL PEM 1] - RFC 1421, "Privacy Enhancement for Internet Electronic
- RFC 1422, Privacy Enhancement for Internet Electronic Mail: Mail: Part I: Message Encryption and Authentication Procedures", J.
Part II: Certificate-Based Key Management, 02/10/1993, S. Kent Linn, 02/10/1993.
- RFC 1423, Privacy Enhancement for Internet Electronic Mail: [MAIL PEM 2] - RFC 1422, "Privacy Enhancement for Internet
Part III: Algorithms, Modes, and Identifiers, 02/10/1993, D. Balenson Electronic Mail: Part II: Certificate-Based Key Management", S. Kent,
- RFC 1424, Privacy Enhancement for Internet Electronic Mail: 02/10/1993.
Part IV: Key Certification and Related Services, 02/10/1993, B. [MAIL PEM 3] - RFC 1423, "Privacy Enhancement for Internet
Kaliski Electronic Mail: Part III: Algorithms, Modes, and Identifiers", D.
Balenson, 02/10/1993.
[MAIL PEM 4] - RFC 1424, "Privacy Enhancement for Internet
Electronic Mail: Part IV: Key Certification and Related Services", B.
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.
skipping to change at page 44, line 4 skipping to change at page 46, line 19
- 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.
[MD4] - "The MD4 Message-Digest Algorithm", RFC1320, April 1992, R. [MD4] - "The MD4 Message-Digest Algorithm", RFC1320, April 1992, R.
Rivest Rivest
[MD5] - "The MD5 Message-Digest Algorithm", RFC1321, April 1992, R. [MD5] - "The MD5 Message-Digest Algorithm", RFC1321, April 1992, R.
Rivest Rivest
[MODES] - "DES Modes of Operation", US National Institute of [MODES] - "DES Modes of Operation", US National Institute of
Standards and Technology, FIPS 81, December 1980. Standards and Technology, FIPS 81, December 1980.
- "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どﾖ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どﾖ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.
[ORMAN] - "Determining Strengths For Public Keys Used For Exchanging [ORMAN] - "Determining Strengths For Public Keys Used For Exchanging
Symmetric Keys", draft-orman-public-key-lengths-*.txt, Hilarie Orman, Symmetric Keys", RFC 3766, Hilarie Orman, Paul Hoffman, April 2004.
Paul Hoffman, work in progress.
[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.
[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.
[SBOX2] - "Perfect Non-linear S-boxes", K. Nyberg, Advances in [SBOX2] - "Perfect Non-linear S-boxes", K. Nyberg, Advances in
Cryptography - Eurocrypt どﾖ91 Proceedings, Springer-Verland, 1991. Cryptography - Eurocrypt '91 Proceedings, Springer-Verland, 1991.
[SCHNEIER] - "Applied Cryptography: Protocols, Algorithms, and Source [SCHNEIER] - "Applied Cryptography: Protocols, Algorithms, and Source
Code in C", 2nd Edition, John Wiley & Sons, 1996, Bruce Schneier. Code in C", Bruce Schneier, 2nd Edition, John Wiley & Sons, 1996.
[SHANNON] - "The Mathematical Theory of Communication", University of [SHANNON] - "The Mathematical Theory of Communication", University of
Illinois Press, 1963, Claude E. Shannon. (originally from: Bell Illinois Press, 1963, Claude E. Shannon. (originally from: Bell
System Technical Journal, July and October 1948) System Technical Journal, July and October 1948)
[SHIFT1] - "Shift Register Sequences", Aegean Park Press, Revised [SHIFT1] - "Shift Register Sequences", Solomon W. Golomb, Aegean Park
Edition 1982, Solomon W. Golomb. Press, Revised Edition 1982.
[SHIFT2] - "Cryptanalysis of Shift-Register Generated Stream Cypher [SHIFT2] - "Cryptanalysis of Shift-Register Generated Stream Cypher
Systems", Aegean Park Press, 1984, Wayne G. Barker. Systems", Wayne G. Barker, Aegean Park Press, 1984.
[SHA-1] - "Secure Hash Standard (SHA-1)", US National Institute of [SHA] - "Secure Hash Standard", US National Institute of Science and
Science and Technology, FIPS 180-1, April 1993. Technology, FIPS 180-2, 1 August 2002.
- RFC 3174, "US Secure Hash Algorithm 1 (SHA1)", D. Eastlake,
P. Jones, September 2001.
[SHA-2] - "Secure Hash Standard", Draft (SHA-2156/384/512), US [SHA RFC] - RFC 3174, "US Secure Hash Algorithm 1 (SHA1)", D.
National Institute of Science and Technology, FIPS 180-2, not yet Eastlake, P. Jones, September 2001.
issued.
[SSH] - draft-ietf-secsh-*, work in progress. [SSH] - draft-ietf-secsh-*, work in progress.
[STERN] - "Secret Linear Congruential Generators are not [STERN] - "Secret Linear Congruential Generators are not
Cryptographically Secure", Proceedings of IEEE STOC, 1987, J. Stern. Cryptographically Secure", J. Stern, Proceedings of IEEE STOC, 1987.
[TLS] - RFC 2246, "The TLS Protocol Version 1.0", T. Dierks, C. [TLS] - RFC 2246, "The TLS Protocol Version 1.0", T. Dierks, C.
Allen, January 1999. Allen, January 1999.
[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どﾖs Collected Works, Vol. 5, Pergamon Press, 1963, digits", von Neumann's Collected Works, Vol. 5, Pergamon Press, 1963,
J. von Neumann. J. von Neumann.
[WSC] - "Writing Secure Code, Second Edition", Michael Howard, David.
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", ANSI X9F1, work in progress. [X9.82] - "Random Number Generation", American National Standards
Institute, ANSI X9F1, work in progress.
Authors 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)
EMail: Donald.Eastlake@motorola.com EMail: Donald.Eastlake@motorola.com
skipping to change at page 46, line 30 skipping to change at page 48, 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-randomness2-08.txt. This is file draft-eastlake-randomness2-09.txt.
It expires February 2005. It expires April 2005.
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