Internet DRAFT - draft-security-randomness
draft-security-randomness
INTERNET-DRAFT Randomness Requirements for Security
25 March 1993
Expires 24 September 1993
Randomness Requirements for Security
---------- ------------ --- --------
Donald E. Eastlake 3rd, Stephen D. Crocker, & Jeffrey I. Schiller
Status of This Document
This draft is intended to be submitted to the RFC editor as an
Informational RFC. Distribution of this document is unlimited.
This document is an Internet Draft. Internet Drafts are working
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This draft expires 24 September 1993.
Abstract
At the heart of many security systems is the assumption that it is
possible to generate secret quantities that are very hard for an
adversary to guess. These include passwords, cryptographic keys, and
similar quantities. Choosing such quantities so as to foil a
resourceful and motivated adversary is surprisingly difficult. This
paper points out many pitfalls in using traditional pseudo-random
number generation techniques for choosing such secrets, recommends
the use of truly random hardware techniques, provides suggestions to
ameliorate the problem when a hardware solution is not available, and
gives examples of how large such quantities need to be for some
particular applications.
Acknowledgements
Substantive comments on this draft, or parts thereof, were received
from Charlie Kaufman, Dave Balenson, Tim Redmond, and Whitfield
Diffie.
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Table of Contents
Status of This Document....................................1
Abstract...................................................1
Acknowledgements...........................................1
Table of Contents..........................................2
1. Introduction............................................3
2. Requirements............................................3
3. Traditional Pseudo-Random Sequences.....................5
4. Unpredictability........................................6
4.1 Problems with Clocks and Serial Numbers................6
4.2 Timing External Events.................................7
4.3 The Fallacy of Complex Manipulation....................7
4.4 The Fallacy of Selection from a Large Database.........7
5. Hardware for Randomness.................................8
5.1 Volume Required........................................8
5.2 Sensitivity to Skew....................................9
5.2.1 Using Stream Parity to De-Skew.......................9
5.2.2 Using Transition Mappings to De-Skew................10
6. Recommended Non-Hardware Strategy......................11
6.1 Mixing Functions......................................11
6.1.1 A Trivial Mixing Function...........................12
6.1.2 Stronger Mixing Functions...........................13
6.1.3 Using a Mixing Function to Stretch Random Bits......14
6.1.4 Other Factors in Choosing a Mixing Function.........14
6.2 Non-Hardware Sources of Randomness....................15
6.3 Cryptographically Strong Sequences....................15
7. US DoD Recommendations for Password Generation.........16
8. Examples of Randomness Required........................16
8.1 A Low Security Password...............................16
8.2 A Very High Security Cryptographic Key................17
9. Security Considerations................................19
References................................................20
Authors Addresses.........................................21
Expiration................................................21
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1. Introduction
Software cryptography is coming into wider use. Systems like
Kerberos, PEM, PGP, etc. are maturing and becoming a part of the
network landscape. These systems provide substantial protection
against snooping and spoofing. However, there is a potential flaw.
At the heart of all cryptographic systems is the generation of random
numbers.
For the present, the lack of generally available facilities for
generating unpredictable numbers is an open wound in the design of
cryptographic software. For the software developer who wants to
build a key or password generation procedure that runs on a wide
range of hardware, the only safe strategy is to force the local
installation to supply a suitable routine to generate unpredictable
numbers. To say the least, this is an awkward, error-prone and
unpalatable solution.
It is important to keep in mind that the requirement is for data that
an adversary has a very low probability of guessing. This will fail
if pseudo-random data, which only meets traditional statistical tests
for randomness or which is based on guessable range sources, such as
clocks, is used. Frequently such random quantities are guessable by
an adversary searching through an embarrassingly small space of
possibilities.
This informational document suggests techniques for producing random
quantities that will be resistant to such attack. It recommends that
future systems include hardware random number generation, suggests
methods for use if such hardware is not available, and gives some
estimates of the number of random bits required for some sample
applications.
2. Requirements
Probably the most commonly encountered randomness requirement is the
typical user password character string. Obviously, if a password can
be guessed, it does not provide security. (For this particular
application it is desirable that users be able to remember the
password. This may make it advisable to use pronounceable character
strings or phrases composed on ordinary words. But this only affects
the format of the password information, not the requirement that the
password be hard to guess.)
Many other requirements come from the cryptographic arena.
Cryptographic techniques can be used to provide a variety of services
including confidentiality and authentication. Such services are
based on quantities, traditionally called "keys", that are unknown to
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and unguessable by an adversary.
In some cases, such as the use of symmetric encryption with the one
time pads [CRYPTO*] or the US Data Encryption Standard [DES], the
parties who wish to communicate confidentially and/or with
authentication must all know the same secret key. In other cases,
using what are called asymmetric or "public key" cryptographic
techniques, keys come in pairs. One key of the pair is private and
must be kept secret by one party, the other is public and can be
published to the world. It is computationally infeasible to
determine the private key from the public key. [ASYMMETRIC, CRYPTO*]
The frequency and volume of the requirement for random quantities
differs greatly for different cryptographic systems. Using RSA
[CRYPTO*], random quantities are required when the key pair is
generated, but thereafter any number of messages can be signed
without any further need for randomness. The public key Digital
Signature Algorithm that has been proposed by the US National
Institute of Standards and Technology requires good random numbers
for each signature. And encrypting with a one time pad, in principle
the strongest possible encryption technique, requires a volume of
randomness equal to all the messages to be processed.
In all of these cases, an adversary may try to determine the "secret"
key by trial and error as long as the key is enough smaller than the
message that the actual key can be uniquely identified. The
probability of an adversary succeeding at this must be made
acceptably low, depending on the particular application. The size of
the space the adversary must search is related to the amount of key
"information" present in the information theoretic sense [SHANNON].
This depends on the number of different secret values possible and
the probability of each value as follows:
-----
\
Bits-of-info = \ - p * log ( p )
/ i 2 i
/
-----
where i varies from 1 to the number of possible secret values and p
sub i is the probability of the value numbered i. (Since p sub i is
less than one, the log will be negative so each term in the sum will
be non-negative.)
If there are 2^n different values of equal probability, then n bits
of information are present and an adversary would, on the average,
have to try half of the values, or 2^(n-1) , before guessing the
secret quantity. If the probability of different values is unequal,
then there is less information present and fewer guesses will, on
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average, be required by an adversary. In particular, any values that
the adversary can know are impossible, or are of low probability, can
be ignored by an adversary, who will search through the more probable
values first.
For example, consider a cryptographic system that uses 56 bit keys.
If these 56 bit keys are derived by using a pseudo-random number
generator that is seeded with an 8 bit seed, then an attacker needs
to search through only 256 keys (by running the pseudo-random number
generator with every possible seed), not the 2^56 keys that may at
first appear to be the case. Only 8 bits of "information" are in
these 56 bit keys.
3. Traditional Pseudo-Random Sequences
Most traditional sources of random numbers use deterministic sources
of "pseudo-random numbers" . These typically start with a "seed"
quantity and use numeric operations to produce a sequence of values.
A typical technique is modular arithmetic where the N+1th value is
calculated from the Nth value by
V = ( V * a + b )(Mod c)
N+1 N
The goodness of traditional pseudo-random number generator algorithm
is measured by statistical tests on this sequence. Carefully chosen
values of a, b, and c in even the above simple iteration can produce
excellent statistics. These numbers work well in simulations (Monte
Carlo experiments) as long as the sequence is orthogonal to the
structure of the space being explored. However, such sequences are
bad for use in security applications. They are fully predictable if
the initial state is known. Depending on the form of the pseudo-
random number generator, the sequence may even be determinable from
observation of a short portion of the sequence. For example, with
the generator above, one can determine V(n+1) given knowledge of
V(n).
[KNUTH] has a good exposition on pseudo-random numbers. Applications
he mentions are simulation of natural phenomena, sampling, numerical
analysis, testing computer programs, decision making, and games.
None of these have the same characteristics as the sort of security
uses we are talking about. Only in the last two could there be an
adversary trying to find the random quantity. However, in these
cases, the adversary normally has only a single chance to use a
guessed value. In guessing passwords or attempting to break an
encryption scheme, the adversary normally has many, perhaps
unlimited, chances at guessing the correct value and should be
assumed to be aided by a computer.
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For testing the "randomness" of numbers, Knuth suggests a variety of
measures including statistical and spectral. These tests check
things like autocorrelation between different parts of a "random"
sequence or distribution of its values. They could be met by a
constant stored random sequence, such as the "random" sequence
printed in the CRC Standard Mathematical Tables [CRC].
4. Unpredictability
Randomness in the traditional sense described in the previous section
is NOT the same as the unpredictability required for security use.
For example, use of a widely available constant sequence, such as
that from the CRC tables, is very weak against an adversary. Once
they learn of or guess it, they can easily break all security, future
and past, based on the sequence. [CRC]
4.1 Problems with Clocks and Serial Numbers
Computer clocks, or similar operating system or hardware values, may
provide significantly fewer real bits of unpredictability than might
appear from their specifications.
Tests have been done on clocks on numerous systems and it was found
that their behavior can vary widely and in unexpected ways. One
version of an operating system running on one set of hardware may
actually provide, say, microsecond resolution in a clock while a
different configuration of the "same" system may always provide the
same lower bits and only count in the upper bits at much lower
resolution. This means that successive reads on the clock may
produce identical values even if enough time has passed that the
value "should" change based on the nominal clock resolution. There
are also cases where frequently reading a clock can produce
artificial sequential values because of extra code that checks for
the clock being unchanged between two reads and increases it by one!
Designing portable application code to generate unpredictable numbers
based on system clocks is particularly challenging because the system
designer does not always know the properties of the system clocks
that the code will execute on.
Use of a hardware serial number such as an ethernet address may also
provide fewer bits of uniqueness than one would guess. Such
quantities are usually heavily structured and subfields may have only
a limited range of possible values or values easily guessable based
on approximate date of manufacture or other data. For example, it is
likely that most of the ethernet cards installed on Digital Equipment
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Corporation (DEC) hardware within DEC were manufactured by DEC
itself, which significantly limits the range of possible serial
numbers.
Problems such as those described above related to clocks and serial
numbers make code to produce unpredictable quantities difficult if
the code is to be ported across a variety of computer platforms and
systems.
4.2 Timing External Events
It is possible to measure the timing of mouse movement, key strokes,
and the like. This a reasonable source of unguessable data with two
exceptions. On some machines, inputs such as key strokes are
buffered. Even though the user's inter-keystroke timing may have
sufficient variation and unpredictability, there might not be an easy
way to access that variation. The other problem is that no standard
method exists to sample timing details. This makes it very hard to
build standard software intended for distribution to a large range of
machines based on this technique.
4.3 The Fallacy of Complex Manipulation
One strategy which may give a misleading appearance of strength is to
take a very complex algorithm (or an excellent traditional pseudo-
random number generator with good statistical properties) and
calculate a cryptographic key by starting with the current value of a
computer system clock as the seed. An adversary who knew roughly
when the generator was started would have a relatively small number
of seed values to test as they would know likely values of the system
clock. Large numbers of pseudo-random bits could be generated but
the search space an adversary would need to check could be quite
small.
Thus very strong and/or complex manipulation of data will not help if
the adversary can learn what the manipulation is and there is not
enough unpredictability in the starting value.
4.4 The Fallacy of Selection from a Large Database
Another strategy that can give a misleading appearance of strength is
selection of a quantity randomly from a database and the assumption
that its strength is related to the total number of bits in the
database. For example, typical NNTP servers as of this date process
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over 30 megabytes of information per day. Assume a random quantity
was selected by fetching 32 bytes of data from a random starting
point in this data. This does not yield 32*8 = 256 bits worth of
unguessability. Even after allowing that much of the data is human
language and probably has more like 3 bits of information per byte,
it doesn't yield 32*3 = 96 bits of unguessability. For an adversary
with access to the same 30 megabytes the unguessability rests only on
the starting point of the selection. That is, at best, about 25 bits
of unguessability in this case.
The same argument applies to selecting sequences from the data on a
CD ROM or Audio CD recording or any other large public database. If
the adversary has access to the same database, this "selection from a
large volume of data" step buys very little. However, if a selection
can be made from data to which the adversary has no access, such as
active system buffers on an active multi-user system, it may be of
some help.
5. Hardware for Randomness
Is there any hope for strong portable randomness in the future?
There might be. All that's needed is a physical source of
unpredictable numbers. A thermal noise or radioactive decay source
and a fast, free-running oscillator would do the trick. This is a
trivial amount of hardware, and could easily be included as a
standard part of a computer system's architecture. All that's needed
is the common perception among computer vendors that this small
addition is necessary and useful.
5.1 Volume Required
How much unpredictability is needed? Is it possible to quantify the
requirement in, say, number of random bits per second?
The answer is not very much is needed. For DES, the key is 56 bits
and, as we show in an example below, even the highest security system
is unlikely to require a keying material of over 200 bits. Even if a
series of keys are needed, they can be generated from a strong random
seed using a cryptographically strong sequence as explained in
Section 6.3. A couple of hundred random bits generated once a day
may well be enough using such techniques. Even if the random bits
are generated as slowly as one per second, it should be tolerable in
high security applications to wait 200 seconds occasionally.
These numbers are trivial to achieve. It could be done by a person
repeatedly tossing a coin. Almost any hardware process is likely to
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be much faster.
5.2 Sensitivity to Skew
Is there any specific requirement on the shape of the distribution of
the random numbers? The good news is the distribution need not be
uniform. All that is needed is a conservative estimate of how non-
uniform it is to bound performance. Two simple techniques to de-skew
the bit stream are given below and stronger techniques are mentioned
in Section 6.1.2 below.
5.2.1 Using Stream Parity to De-Skew
Consider taking a sufficiently long string of bits and map the string
to "zero" or "one". The mapping will not yield a perfectly uniform
distribution, but it can be as close as desired. One mapping that
serves the purpose is to take the parity of the string. This has the
advantages that it is robust across all degrees of skew up to the
estimated maximum skew and is absolutely trivial to implement in
hardware.
The following analysis gives the number of bits that must be sampled:
Suppose the ratio of ones to zeros is 0.5 + e : 0.5 - e, where e is
between 0 and 0.5 and is a measure of the "eccentricity" of the
distribution. Consider the distribution of the parity function of N
bit samples. The probabilities that the parity will be one or zero
will be the sum of the odd or even terms in the binomial expansion of
(p + q)^N, where p = 0.5 + e, the probability of a one, and q = 0.5 -
e, the probability of a zero.
These sums can be computed easily as
1/2 * [(p + q)^N + (p - q)^N]
and
1/2 * [(p + q)^N - (p - q)^N].
(Which one corresponds to the probability the parity will be 1
depends on whether N is odd or even.)
Since p + q = 1 and p - q = 2e, these expressions reduce to
1/2 * [1 + (2e)^N]
and
1/2 * [1 - (2e)^N].
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Neither of these will ever be exactly 0.5 unless e is zero, but we
can bring them arbitrarily close to 0.5. If we want the
probabilities to be within some delta d of 0.5, i.e. then
0.5 + 0.5 * (2e)^N < 0.5 + d.
Solving for N yields N > log(2d)/log(2e). (Note that 2e is less than
1, so its log is negative. Division by a negative number reverses
the sense of an inequality.)
The following table gives the length of the string which must be
sampled for various degrees of skew in order to come within 0.001 of
a 50/50 distribution.
+---------+--------+-------+
| Prob(1) | e | N |
+---------+--------+-------+
| 0.5 | 0.00 | 1 |
| 0.6 | 0.10 | 4 |
| 0.7 | 0.20 | 7 |
| 0.8 | 0.30 | 13 |
| 0.9 | 0.40 | 28 |
| 0.95 | 0.45 | 59 |
| 0.99 | 0.49 | 308 |
+---------+--------+-------+
The last entry shows that even if the distribution is skewed 99% in
favor of ones, the parity of a string of 308 samples will be within
0.001 of a 50/50 distribution.
5.2.2 Using Transition Mappings to De-Skew
Another possible technique is to examine a bit stream as a sequence
of non-overlapping pairs. You could then discard any 00 or 11 pairs
found, interpret 01 as a 0 and 10 as a 1. Assume the probability of
a 1 is 0.5+e and the probability of a 0 is 0.5-e where e is the
eccentricity of the source and described in the previous section.
Then the probability of each pair is as follows:
+------+-----------------------------------------+
| pair | probability |
+------+-----------------------------------------+
| 00 | (0.5 - e)^2 = 0.25 - e + e^2 |
| 01 | (0.5 - e)*(0.5 + e) = 0.25 - e^2 |
| 10 | (0.5 + e)*(0.5 - e) = 0.25 - e^2 |
| 11 | (0.5 + e)^2 = 0.25 + e + e^2 |
+------+-----------------------------------------+
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This technique will completely eliminate any bias but at the expense
of taking an indeterminate number of input bits for any particular
desired number of output bits. The probability of any particular
pair being discarded is 0.5 + 2e^2 so the expected number of input
bits to produce X output bits is X/(0.25 - e^2).
This technique assume that the bits are from a stream where each bit
has the same probability of being a 0 or 1 as any other bit in the
stream and that bits are not correlated, i.e., that the bits are
identical independent distributions. If alternate bits were from two
different sources, for example, the above analysis breaks down.
The above technique provides another illustration of how a simple
statistical analysis can mislead if one is not always on the lookout
for patterns that could be exploited by an adversary. If the
algorithm were mis-read slightly so that overlapping successive bits
pairs were used instead of non-overlapping pairs, the statistical
analysis given is the same; however, instead of provided an unbiased
uncorrelated series of random 1's and 0's, it would instead produce a
totally predictable sequence of exactly alternating 1's and 0's.
6. Recommended Non-Hardware Strategy
What is the best overall strategy for meeting the requirement for
unguessable random numbers in the absence of a reliable hardware
source? It is to obtain random input from a large number of
uncorrelated sources and to mix them with a strong mixing function.
Such a function will preserve the randomness present in any of the
sources even if other quantities being combined are fixed or easily
guessable. This may be advisable even with a good hardware source as
hardware can also fail, though this should be weighed against any
increase in the chance of overall failure due to added software
complexity.
6.1 Mixing Functions
A strong mixing function is one which combines two or more inputs and
produces an output where each output bit is a complex non-linear
function of all the input bits. On average, changing any input bit
will change about half the output bits. But because the relationship
is complex and non-linear, no particular output bit is guaranteed to
change when any particular input bit is changed.
Note that the problem of converting a stream of bits that is skewed
towards 0 or 1 to a shorter stream which is more random, as discussed
in Section 5.2 above, is simply another case where a strong mixing
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function is desired. The technique given in Section 5.2.1 of using
the parity of a number of bits is simply the result of successively
xor'ing them which is examined as a trivial example immediately
below. Use of stronger mixing functions to extract more of the
randomness in a stream of skewed bits is mentioned in 6.1.2 below.
6.1.1 A Trivial Mixing Function
A trivial example for single bit inputs is the exclusive or (xor)
function, which is equivalent to addition without carry, as show in
the table below. This is a degenerate case in which the one output
bit always changes for a change in either input bit but it will still
provide a useful illustration.
+-----------+-----------+----------+
| input 1 | input 2 | output |
+-----------+-----------+----------+
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
+-----------+-----------+----------+
If inputs 1 and 2 are uncorrelated and combined in this fashion then
the output will be an even better (less skewed) random bit than the
inputs. If we assume an "eccentricity" e as defined in section 5.2
above, then the output eccentricity relates to the input eccentricity
as follows:
e = 2 * e * e
output input 1 input 2
Since e is never greater than 1/2, the eccentricity is always
improved except in the case where one input is a totally skewed
constant. This is illustrated in the following table where the top
and left side values are the two input eccentricities and the entries
are the output eccentricity:
+--------+--------+--------+--------+--------+--------+--------+
| e | 0.00 | 0.10 | 0.20 | 0.30 | 0.40 | 0.50 |
+--------+--------+--------+--------+--------+--------+--------+
| 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 0.10 | 0.00 | 0.02 | 0.04 | 0.06 | 0.08 | 0.10 |
| 0.20 | 0.00 | 0.04 | 0.08 | 0.12 | 0.16 | 0.20 |
| 0.30 | 0.00 | 0.06 | 0.12 | 0.18 | 0.24 | 0.30 |
| 0.40 | 0.00 | 0.08 | 0.16 | 0.24 | 0.32 | 0.40 |
| 0.50 | 0.00 | 0.10 | 0.20 | 0.30 | 0.40 | 0.50 |
+--------+--------+--------+--------+--------+--------+--------+
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However, keep in mind that the above calculations assume that the
inputs are not correlated. If the inputs were, say, the parity of
the number of minutes from midnight on two clocks accurate to a few
seconds, then each might appear random if sampled at random intervals
much longer than a minute. Yet if they were both sampled and
combined with xor, the result would normally be a constant zero.
6.1.2 Stronger Mixing Functions
The US Government Data Encryption Standard [DES] is a good example of
a strong mixing function for multiple bit quantities. It takes up to
120 bits of input (64 bits of "data" and 56 bits of "key") and
produces 64 bits of output each of which is dependent on a complex
function of all input bits. Another good family of mixing functions
are the "message digest" or hashing functions such as MD2, MD4, or
MD5 that take an arbitrary amount of input and produce an output,
frequently 128 bits, mixing all the input bits. [MD2, MD4, MD5]
Although message digest functions like MD5 are designed for variable
amounts of input, DES can also be used to combine any number of
inputs. If 64 bits of output is adequate, the inputs can be packed
into a 64 bit data quantity and successive 56 bit keys, padding with
zeros if needed, which are then used to successively encrypt using
DES in Electronic Codebook Mode [DES MODES]. If more than 64 bits of
output are needed, use more complex mixing. For example, if inputs
are packed into three quantities, A, B, and C, use DES to encrypt A
with B as a key and then with C as a key to produce the 1st part of
the output, then encrypt B with C and then A for more output and, if
necessary, encrypt C with A and then B for yet more output. Still
more output can be produced by reversing the order of the keys given
above to stretch things, but keep in mind that it is impossible to
get more bits of "randomness" out than are put in.
An example of using a strong mixing function would be to reconsider
the case of a string of 308 bits each of which is biased 99% towards
zero. The parity technique given in Section 5.2.1 above reduced this
to one bit with only a 1/1000 deviance from being equally likely a
zero or one. But, applying the equation for information given in
Section 2, this 308 bit sequence has 5 bits of information in it.
Thus hashing it with MD5 and taking the bottom 5 bits of the result
would yield 5 unbiased random bits as opposed to the single bit given
by calculating the parity of the string.
Other strong encryption functions besides DES and the MD* family
should serve well as mixing functions. This is an advantage of
Diffie-Hellman exponential key exchange. Diffie-Hellman yields a
shared secret between two parties that is a mixture of initial random
quantities generated by each of them [D-H].
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6.1.3 Using a Mixing Function to Stretch Random Bits
While it is not necessary for a mixing function to produce the same
or fewer bits than its inputs, mixing bits cannot "stretch" the
amount of random unpredictability present in the inputs. Thus four
inputs of 32 bits each where there is 12 bits worth of
unpredicatability (such as 4,096 equally probable values) in each
input cannot produce more than 48 bits worth of unpredictable output.
The output can be expanded to hundreds or thousands of bits by, for
example, mixing with successive integers, but the clever adversary's
search space is still 2^48 possibilities. Furthermore, mixing to
fewer bits than are input will tend to strengthen the randomness of
the output the way using xor to produce one bit from two did above.
The last table in Section 6.1.1 shows that mixing a random bit with a
constant bit with xor will produce a random bit. While this is true,
it does not provide a way to "stretch" one random bit into more than
one. If, for example, a random bit is mixed with a 0 and then with a
1, this produces a two bit sequence but it will always be either 01
or 10. Since there are only two possible values, there is still only
the one bit of original randomness.
6.1.4 Other Factors in Choosing a Mixing Function
For local use, DES has the advantages that it has been widely tested
for flaws, is widely documented, and is widely implemented with
hardware and software implementations available all over the world
including source code available by anonymous FTP. The MD* family are
younger algorithms which has been less tested but there is no
particular reason to believe they are flawed. They also have source
code available by anonymous FTP [MD2, MD4, MD5]. DES, MD4, and MD5
are royalty free for all purposes but MD2 has been freely licensed
only for non-profit use in connection with Privacy Enhanced Mail.
(Some people believe that, as with Goldilocks and the Three Bears,
MD2 is strong but too slow, MD4 is fast but too weak, and MD5 is just
right.)
Another advantage of the MD* or similar hashing algorithms is that
they are not subject to the regulations imposed by the US Government
prohibiting the export or import of encryption/decryption software
(or hardware). The same should be true of DES rigged to produce an
irreversible hash code but most DES packages are oriented to
reversible encryption.
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6.2 Non-Hardware Sources of Randomness
The best source of input for mixing would be a hardware random number
generator based on some fundamentally random physical process such as
thermal emission or radioactive decay. However, if that is not
available, other possibilities include system clocks, system or
input/output buffers, user/system/hardware/network serial numbers
and/or addresses, user input, and timings of input/output operations.
Any of these sources can produce limited or predicatable values under
some circumstances.
Most of the sources listed above would be quite strong on multi-user
system where, in essence, each user of the system is a source of
randomness. However, on a small single user system, such as a
typical IBM PC or Apple Macintosh, it might be possible for an
adversary to assemble a similar configuration. This could give the
adversary inputs to the mixing process that were sufficiently
correlated to those used originally as to make exhaustive search
practical.
The use of multiple random inputs with a strong mixing function is
recommended and can overcome weakness in any particular input. This
strategy may make practical portable code to produce good random
numbers for security even when some of the inputs are very weak on
some of the target systems. However, even this may fail against a
high grade attack on small single user systems if a hardware random
source is not available.
6.3 Cryptographically Strong Sequences
In cases where a series of random quantities must be generated, an
adversary may learn some values in the sequence. In general, they
should not be able to predict other values from the ones that they
know. The correct technique is to start with a strong random seed
and take cryptographically strong steps from that seed [CRYPTO2]. If
each value in the sequence can be calculated in a fixed way from the
previous value, then when any value is compromised, all future values
can be determined. This would be the case, for example, if each
value were a constant function of the previous values, even if the
function were a very strong, non-invertible message digest function.
The best way to achieve a strong sequence is to have the values be
produced by successive multiple "encryption" of a random seed under a
random key or by hashing the quantities produced by concatenating the
seed with successive integers or the like. To predict values of a
sequence from others when the sequence was generated by these
techniques is equivalent to breaking the cryptosystem or inverting
the "non-invertible" hashing involved.
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7. US DoD Recommendations for Password Generation
The United States Department of Defense has specific recommendations
for password generation [DoD]. They suggest using the US Data
Encryption Standard [DES] in Output Feedback Mode [DES MODES] as
follows:
use an initialization vector determined from
the system clock,
system ID,
user ID, and
date and time;
use a key determined from
system interrupt registers,
system status registers, and
system counters; and,
as plain text, use an external randomly generated 64 bit quantity
such as 8 characters typed in by a system administrator.
The password can then be calculated from the 64 bit "cipher text"
generated in 64-bit Output Feedback Mode. As many bits as are needed
can be taken from these 64 bits and expanded into a pronounceable
word, phrase, or other format.
8. Examples of Randomness Required
Below are two examples showing rough calculations of needed
randomness for security.
8.1 A Low Security Password
Assume that user passwords change once a year and a probability of
less than one in a thousand that an adversary could guess the
password for a particular account is desired. The key question is
how often they can try possibilities. Assume that delays have been
introduced into a system so that, at most, an adversary can make one
password try every six seconds. That's 600 per hour or about 15,000
per day or about 5,000,000 tries in a year. Assuming any sort of
monitoring, it is unlikely someone could actually try continuously
for a year. In fact, even if log files are only checked monthly,
500,000 tries is more plausible before the attack is noticed and
steps taken to change passwords and make it harder to try more
passwords. (All this assumes that sending a password to the system
is the only way to try a password.)
To have a one in a thousand chance of guessing the password in
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500,000 tries implies a universe of at least 500,000,000 passwords or
about 2^29. Thus 29 bits of randomness are needed. This can probably
be achieved using the US DoD recommended inputs for password
generation as it has 8 inputs which probably average over 5 bits of
randomness each. Using a list of 1000 words, the password could be
expressed as a three word phrase (1,000,000,000 possibilities) or,
using case insensitive letters and digits, six would suffice
((26+10)^6 = 2,176,782,336 possibilities).
For a higher security password, the number of bits required goes up.
To decrease the probability by 1,000 requires increasing the universe
of passwords by the same factor which adds about 12 bits. Thus to
have only a one in a million chance of a password being guessed under
the above scenario would require 31 bits of randomness and a password
that was a four word phrase from a 1000 word list or eight
letters/digits. To go to a one in 10^9 chance, 43 bits of randomness
are needed implying a five word phrase or ten letter/digit password.
8.2 A Very High Security Cryptographic Key
Assume that a very high security key is needed for symmetric
encryption/decryption between two parties. Assume an adversary can
observe communications and knows the algorithm being used. Within
the field of random possibilities, the adversary can exhaustively try
key values.
How much effort will it take to try each key? For very high security
applications it is best to assume a low value of effort. Even if it
would clearly take tens of thousands of computer cycles or more to
try a single key, there may be some pattern that enables huge blocks
of key values to be tested with much less effort per key. Thus it is
probably best to assume no more than a hundred cycles per key.
(There is no clear lower bound on this as computers operate in
parallel on a number of bits and a poor encryption algorithm could
allow many keys or even groups of keys to be tested in parallel.
However, we need to assume some value and can hope that a reasonably
strong algorithm has been chosen for our hypothetical high security
task.)
If the adversary can command a highly parallel processor or a large
network of work stations, 10^10 cycles per second is probably a
minimum assumption for availability today. Looking forward just a
few years, there should be at least an order of magnitude
improvement. Thus assuming 10^9 keys could be checked per second or
3.6*10^11 per hour or 6*10^13 per week or 2.4*10^14 per month is
reasonable. This implies a need for a minimum of 48 bits of
randomness in keys to be sure they cannot be found in a week. Even
then it is possible that, a few years from now, a highly determined
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and resourceful adversary could break the key in 2 weeks (on average
they need try only half the keys).
Assuming a known plain text attack, where the adversary can force
some known plain text to be encrypted or knows some standard part of
messages, the structure of the encryption algorithm may allow a "meet
in the middle" attack. An oversimplified explanation of this type of
attack is as follows: the adversary can half-encrypt the know plain
text with all possible first half-keys, sort these, then half-decrypt
the encoded text with all the second half-keys. If a match is found,
the full key can be assembled from the halves and used to decrypt
other parts of the message or other messages.
At its best, this type of attack can halve the exponent of the work
required by the adversary requiring a doubling of the amount of
randomness in the key to a minimum of 96 bits. This assumes that the
cryptographic algorithm can be decomposed in this way but we can not
rule that out without a deep knowledge of the algorithm. Enormous
resources may be required for this sort of attack but they are
probably within the range of the national security services of a
major nation. Almost all nations spy on other nations government
traffic and Some nations are known to spy on commercial traffic and
give the information to their domestic companies to assist them
against foreign competition.
Since we have not even considered the possibilities of special
purpose code breaking hardware or just how much of a safety margin we
want beyond our assumptions above, probably a good minimum for a very
high security cryptographic key is 128 bits of randomness which
implies a minimum key length of 128 bits. If the two parties agree
on a key by Diffie-Hellman exchange [D-H], then in principle only
half of this randomness would have to be supplied by each party.
However, there is probably some correlation between their random
inputs so it is probably best to assume that each party needs to
provide at least 96 bits worth of randomness for very high security.
This amount of randomness is probably beyond the limit of that in the
inputs recommended by the US DoD for password generation and could
require user typing timing, hardware random number generation, or
other sources.
It should be noted that key length calculations such at those above
are controversial and depend on various assumptions about the
cryptographic algorithms in use. In some cases, a professional with
a deep knowledge of code breaking techniques and of the strength of
the algorithm in use could be satisfied with less than half of the
key size derived above.
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9. Security Considerations
The entirety of this draft concerns techniques and recommendations
for generating "random" quantities for use as passwords,
cryptographic keys, and similar security uses.
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References
[ASYMMETRIC] - Secure Communications and Asymmetric Cryptosystems,
edited by Gustavus J. Simmons, AAAS Selected Symposium 69, Westview
Press, Inc.
[CRC] - C.R.C. Standard Mathematical Tables, Chemical Rubber
Publishing Company.
[CRYPTO1] - Cryptography: A Primer, by Alan G. Konheim, A Wiley-
Interscience Publication, John Wiley & Sons, 1981, Alan G. Konheim.
[CRYPTO2] - Cryptography: A New Dimension in Computer Data Security,
A Wiley-Interscience Publication, John Wiley & Sons, 1982, Carl H.
Meyer & Stephen M. Matyas.
[DES] - Data Encryption Standard, United States of America,
Department of Commerce, National Institute of Standards and
Technology, Federal Information Processing Standard (FIPS) 46-1.
- Data Encryption Algorithm, American National Standards Institute,
ANSI X3.92-1981.
(See also FIPS 112, Password Usage, which includes FORTRAN code for
performing DES.)
[DES MODES] - DES Modes of Operation, United States of America,
Department of Commerce, National Institute of Standards and
Technology, Federal Information Processing Standard (FIPS) 81.
- Data Encryption Algorithm - Modes of Operation, American National
Standards Institute, ANSI X3.106-1983.
[D-H] - New Directions in Cryptography, IEEE Transactions on
Information Technology, November, 1976, Whitfield Diffie and Martin
E. Hellman.
[DoD] - Password Management Guideline, United States of America,
Department of Defense, Computer Security Center, CSC-STD-002-85.
(See also FIPS 112, Password Usage, which incorporates CSC-S002-85 as
one of its appendicies.)
[KNUTH] - The Art of Computer Programming, Volume 2: Seminumerical
Algorithms, Chapter 3: Random Numbers. Addison Wesley Publishing
Company, 1971, Donald E. Knuth.
[MD2] - The MD2 Message-Digest Algorithm, RFC1319, April 1992, B.
Kaliski
[MD4] - The MD4 Message-Digest Algorithm, RFC1320, April 1992, R.
Rivest
[MD5] - The MD5 Message-Digest Algorithm, RFC1321, April 1992, R.
Rivest
D. Eastlake, S. Crocker, & J. Schiller [Page 20]
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[SHANNON] - The Mathematical Theory of Communication, University of
Illinois Press, 1963, Claude E. Shannon. (originally from: Bell
System Technical Journal, July and October 1948)
Authors Addresses
Donald E. Eastlake 3rd
Digital Equipment Corporation
30 Porter Road, MS: LJO2/I4
Littleton, MA 01460
Telephone: +1 508 486 2358(w) +1 617 244 2679(h)
EMail: dee@ranger.enet.dec.com
NIC Handle: [DEE]
Stephen D. Crocker
Trusted Information Systems, Inc.
3060 Washington Road
Glenwood, MD 21738
Telephone: +1 301 854 6889
EMail: crocker@tis.com
NIC Handle: [SDC1]
Jeffrey I. Schiller
Massachusetts Institute of Technology
77 Massachusetts Avenue
Cambridge, MA 02139
Telephone: +1 617 253 0161
EMail: jis@mit.edu
NIC Handle: [JIS]
Expiration
This draft expires 24 September 1993.
D. Eastlake, S. Crocker, & J. Schiller [Page 21]
