INTERNET-DRAFT Randomness Requirements for Security
01 October 1993
Expires 31 March 1994
Randomness Requirements for Security
---------- ------------ --- --------
Donald E. Eastlake 3rd, Stephen D. Crocker, & Jeffrey I. Schiller
Status of This Document
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Abstract
Security systems today are built on increasingly strong cryptographic
algorithms that foil pattern analysis attempts. However, the security
of these systems is dependent on generating secret quantities for
passwords, cryptographic keys, and similar quantities. The use of
pseudo-random processes to generate secret quantities can result in
pseudo-security. The sophisticated attacker of these security
systems will often find it easier to reproduce the environment that
produced the secret quantities, searching the resulting small set of
possibilities, than to locate the quantities in the whole of the
number space.
Choosing random quantities to foil a resourceful and motivated
attacker is surprisingly difficult. This paper points out many
pitfalls in using traditional pseudo-random number generation
techniques for choosing such quantities, 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.
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Acknowledgements
Useful comments on this document that have been incorporated were
received from (in alphabetic order) the following:
David M. Balenson (TIS)
Carl Ellison (Stratus)
Marc Horowitz (MIT)
Charlie Kaufman (DEC)
Steve Kent (BBN)
Hal Murray (DEC)
Neil Haller (Bellcore)
Richard Pitkin (DEC)
Tim Redmond (TIS)
Doug Tygar (CMU)
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Table of Contents
Status of This Document....................................1
Abstract...................................................1
Acknowledgements...........................................2
Table of Contents..........................................3
1. Introduction............................................4
2. Requirements............................................5
3. Traditional Pseudo-Random Sequences.....................7
4. Unpredictability........................................9
4.1 Problems with Clocks and Serial Numbers................9
4.2 Timing and Content of External Events.................10
4.3 The Fallacy of Complex Manipulation...................10
4.4 The Fallacy of Selection from a Large Database........11
5. Hardware for Randomness................................12
5.1 Volume Required.......................................12
5.2 Sensitivity to Skew...................................12
5.2.1 Using Stream Parity to De-Skew......................13
5.2.2 Using Transition Mappings to De-Skew................14
5.2.3 Using Compression to De-Skew........................15
5.3 Using Sound/Video Input...............................15
6. Recommended Non-Hardware Strategy......................17
6.1 Mixing Functions......................................17
6.1.1 A Trivial Mixing Function...........................17
6.1.2 Stronger Mixing Functions...........................18
6.1.3 Using a Mixing Function to Stretch Random Bits......19
6.1.4 Other Factors in Choosing a Mixing Function.........20
6.2 Non-Hardware Sources of Randomness....................20
6.3 Cryptographically Strong Sequences....................21
7. US DoD Recommendations for Password Generation.........23
8. Examples of Randomness Required........................24
8.1 Password Generation..................................24
8.2 A Very High Security Cryptographic Key................24
8.2.1 Effort per Key Trial................................25
8.2.2 Meet in the Middle Attacks..........................25
8.2.3 Other Considerations................................26
9. Security Considerations................................27
References................................................27
Authors Addresses.........................................29
Expiration and File Name..................................29
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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 so far has been 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 sample
applications.
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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 very hard to guess.)
Many other requirements come from the cryptographic arena.
Cryptographic techniques can be used to provide a variety of services
including confidentiality and authentication. Such services are
based on quantities, traditionally called "keys", that are unknown to
and unguessable by an adversary.
In some cases, such as the use of symmetric encryption with the one
time pads [CRYPTO*] or the US Data Encryption Standard [DES], 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 (NIST) requires good random
numbers for each signature. And encrypting with a one time pad, in
principle the strongest possible encryption technique, requires a
volume of randomness equal to all the messages to be processed.
In all of these cases, an adversary may try to determine the "secret"
key by trial and error. (This is possible as long as the key is
enough smaller than the message that the correct key can be uniquely
identified.) The probability of an adversary succeeding at this must
be made acceptably low, depending on the particular application. The
size of the space the adversary must search is related to the amount
of key "information" present in the information theoretic sense
[SHANNON]. This depends on the number of different secret values
possible and the probability of each value as follows:
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-----
\
Bits-of-info = \ - p * log ( p )
/ i 2 i
/
-----
where i varies from 1 to the number of possible secret values and p
sub i is the probability of the value numbered i. (Since p sub i is
less than one, the log will be negative so each term in the sum will
be non-negative.)
If there are 2^n different values of equal probability, then n bits
of information are present and an adversary would, on the average,
have to try half of the values, or 2^(n-1) , before guessing the
secret quantity. If the probability of different values is unequal,
then there is less information present and fewer guesses will, on
average, be required by an adversary. In particular, any values that
the adversary can know are impossible, or are of low probability, can
be 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.
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3. Traditional Pseudo-Random Sequences
Most traditional sources of random numbers use deterministic sources
of "pseudo-random" numbers. These typically start with a "seed"
quantity and use numeric or logical operations to produce a sequence
of values.
[KNUTH] has a general exposition on pseudo-random numbers.
Applications he mentions are simulation of natural phenomena,
sampling, numerical analysis, testing computer programs, decision
making, and games. None of these have the same characteristics as
the sort of security uses we are talking about. Only in the last two
could there be an adversary trying to find the random quantity.
However, in these cases, the adversary normally has only a single
chance to use a guessed value. In guessing passwords or attempting
to break an encryption scheme, the adversary normally has many,
perhaps unlimited, chances at guessing the correct value and should
be assumed to be aided by a computer.
For testing the "randomness" of numbers, Knuth suggests a variety of
measures including statistical and spectral. These tests check
things like autocorrelation between different parts of a "random"
sequence or distribution of its values. They could be met by a
constant stored random sequence, such as the "random" sequence
printed in the CRC Standard Mathematical Tables [CRC].
A typical pseudo-random number generation technique, known as a
linear congruence pseudo-random number generator, is modular
arithmetic where the N+1th value is calculated from the Nth value by
V = ( V * a + b )(Mod c)
N+1 N
The above technique has a strong relationship to linear shift
register pseudo-random number generators, which are well understood
cryptographically [SHIFT*]. In such generators bits are introduced
at one end of a shift register as the Exclusive Or (binary sum
without carry) of bits from selected fixed taps into the register.
For example:
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+----+ +----+ +----+ +----+
| B | <-- | B | <-- | B | <-- . . . . . . <-- | B | <-+
| 0 | | 1 | | 2 | | n | |
+----+ +----+ +----+ +----+ |
| | | |
| | V +-----+
| V +----------------> | |
V +-----------------------------> | XOR |
+---------------------------------------------------> | |
+-----+
V = ( ( V * 2 ) + B .xor. B ...B )(Mod 2^n)
N+1 N 0 1 n
The goodness of traditional pseudo-random number generator algorithm
is measured by statistical tests on such sequences. Carefully chosen
values of the initial V and a, b, and c or the placement of shift
register tap in the above simple processes can produce excellent
statistics.
These sequences may be adequate in simulations (Monte Carlo
experiments) as long as the sequence is orthogonal to the structure
of the space being explored. Even there, subtle patterns may cause
problems. [ref to come - Marsaglia] However, such sequences are
clearly bad for use in security applications. They are fully
predictable if the initial state is known. Depending on the form of
the pseudo-random number generator, the sequence may be determinable
from observation of a short portion of the sequence. For example,
with the generators above, one can determine V(n+1) given knowledge
of V(n). In fact, it has been shown that with them even if only one
bit of the pseudo-random values are released, the seed can be
determined from short sequences. [ref to come - Frieze, Hastad,
Kannan, Lagaris, & Shamir]
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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,
provide significantly fewer real bits of unpredictability than might
appear from their specifications.
Tests have been done on clocks on numerous systems and it was found
that their behavior can vary widely and in unexpected ways. One
version of an operating system running on one set of hardware may
actually provide, say, microsecond resolution in a clock while a
different configuration of the "same" system may always provide the
same lower bits and only count in the upper bits at much lower
resolution. This means that successive reads on the clock may
produce identical values even if enough time has passed that the
value "should" change based on the nominal clock resolution. There
are also cases where frequently reading a clock can produce
artificial sequential values because of extra code that checks for
the clock being unchanged between two reads and increases it by one!
Designing portable application code to generate unpredictable numbers
based on such system clocks is particularly challenging because the
system designer does not always know the properties of the system
clocks that the code will execute on.
Use of a hardware serial number such as an Ethernet address may also
provide fewer bits of uniqueness than one would guess. Such
quantities are usually heavily structured and subfields may have only
a limited range of possible values or values easily guessable based
on approximate date of manufacture or other data. For example, it is
likely that most of the Ethernet cards installed on Digital Equipment
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.
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4.2 Timing and Content of External Events
It is possible to measure the timing of mouse movement, key strokes,
and the like. This is 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 hard to build
standard software intended for distribution to a large range of
machines based on this technique.
The amount of mouse movement or keys actually hit are usually easier
to access than timings but may yield less unpredictability as the
user may provide highly repetative input.
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 seed value.
Another serious strategy error is to assume that a very complex
pseudo-random number generation algorithm will produce strong random
numbers when there has been no theory behind or analysis of the
algorithm. There is a excellent example of this fallacy right near
the beginning of [KNUTH] where the author describes a complex
algorithm. It was intended that the machine language program
corresponding to the algorithm would be so complicated that a person
trying to read the code without comments wouldn't know what the
program was doing. Unfortunately, actual use of this algorithm
showed that it almost immediately converged to a single repeated
value in one case and a small cycle of values in another case.
Not only does complex manipulation not help you if you have a limited
range of seeds but blindly chosen complex manipulation can destroy
the randomness in a good seed!
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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
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.
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5. Hardware for Randomness
Is there any hope for strong portable randomness in the future?
There might be. All that's needed is a physical source of
unpredictable numbers.
A thermal noise or radioactive decay source and a fast, free-running
oscillator would do the trick. 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 in Section 8, 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 few hundred random bits generated once a
day would be enough using such techniques. Even if the random bits
are generated as slowly as one per second and it is not possible to
overlap the generation process, it should be tolerable in high
security applications to wait 200 seconds occasionally.
These numbers are trivial to achieve. It could be done by a person
repeatedly tossing a coin. Almost any hardware process is likely to
be much faster.
5.2 Sensitivity to Skew
Is there any specific requirement on the shape of the distribution of
the random numbers? The good news is the distribution need not be
uniform. All that is needed is a conservative estimate of how non-
uniform it is to bound performance. Two simple techniques to de-skew
the bit stream are given below and stronger techniques are mentioned
in Section 6.1.2 below.
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5.2.1 Using Stream Parity to De-Skew
Consider taking a sufficiently long string of bits and map the string
to "zero" or "one". The mapping will not yield a perfectly uniform
distribution, but it can be as close as desired. One mapping that
serves the purpose is to take the parity of the string. This has the
advantages that it is robust across all degrees of skew up to the
estimated maximum skew and is absolutely trivial to implement in
hardware.
The following analysis gives the number of bits that must be sampled:
Suppose the ratio of ones to zeros is 0.5 + e : 0.5 - e, where e is
between 0 and 0.5 and is a measure of the "eccentricity" of the
distribution. Consider the distribution of the parity function of N
bit samples. The probabilities that the parity will be one or zero
will be the sum of the odd or even terms in the binomial expansion of
(p + q)^N, where p = 0.5 + e, the probability of a one, and q = 0.5 -
e, the probability of a zero.
These sums can be computed easily as
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].
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.
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+---------+--------+-------+
| 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 |
+------+-----------------------------------------+
This technique will completely eliminate any bias but at the expense
of taking an indeterminate number of input bits for any particular
desired number of output bits. The probability of any particular
pair being discarded is 0.5 + 2e^2 so the expected number of input
bits to produce X output bits is X/(0.25 - e^2).
This technique assumes that the bits are from a stream where each bit
has the same probability of being a 0 or 1 as any other bit in the
stream and that bits are not correlated, i.e., that the bits are
identical independent distributions. If alternate bits were from two
different sources, for example, the above analysis breaks down.
The above technique also provides another illustration of how a
simple statistical analysis can mislead if one is not always on the
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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.
5.2.3 Using Compression to De-Skew
Reversible compression techniques also provide a crude method of de-
skewing a skewed bit stream. This follows directly from the
definition of reversible compression and the formula in Section 2
above for the amount of information in a sequence. Since the
compression is reversible, the same amount of information must be
present in the shorter output than was present in the longer input.
By the Shannon information equation, this is only possible if, on
average, the probabilities of the different shorter sequences are
more uniformly distributed than were the probabilities of the longer
sequences. Thus the shorter sequences are de-skewed relative to the
input.
However, many compression techniques add a somewhat predicatable
preface to their output stream and may insert such a sequence again
periodically in their output or otherwise introduce subtle patterns
of their own. They should be considered only a rough technique
compared with those described above or in Section 6.1.2. At a
minimum, the beginning of the compressed sequence should be skipped
and only later bits used for applications requiring random bits.
5.3 Using Sound/Video Input
Increasingly computers are being built with inputs that digitize some
real world analog source, such as sound from a microphone or video
input from a camera. Under appropriate circumstances, such input can
provide reasonably high quality random bits. The "input" from a
sound digitizer with no source plugged in or a camera with the lens
cap on, if the system is high enough gain to detect anything, is
essentially thermal noise.
For example, on a Sparkstation, one can read from the /dev/audio
device with nothing plugged into the microphone jack. Such data is
essentially random noise although it should not be trusted without
some checking in case of hardware failure. It will, in any case,
need to be de-skewed as described elsewhere.
Thus, combining this with compression to de-skew, one can in UNIXese
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generate a hugh amount of relatively random data by doing
cat /dev/audio | compress - >random-bits-file
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6. Recommended Non-Hardware Strategy
What is the best overall strategy for meeting the requirement for
unguessable random numbers in the absence of a reliable hardware
source? It is to obtain random input from a 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 different complex non-
linear function of all the input bits. On average, changing any
input bit will change about half the output bits. But because the
relationship is complex and non-linear, no particular output bit is
guaranteed to change when any particular input bit is changed.
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
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
Exclusive Or'ing them which is examined as a trivial mixing function
immediately below. Use of stronger mixing functions to extract more
of the randomness in a stream of skewed bits is examined in Section
6.1.2.
6.1.1 A Trivial Mixing Function
A trivial example for single bit inputs is the Exclusive Or function,
which is equivalent to addition without carry, as show in the table
below. This is a degenerate case in which the one output bit always
changes for a change in either input bit but it will still provide a
useful illustration.
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+-----------+-----------+----------+
| input 1 | input 2 | output |
+-----------+-----------+----------+
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
+-----------+-----------+----------+
If inputs 1 and 2 are uncorrelated and combined in this fashion then
the output will be an even better (less skewed) random bit than the
inputs. If we assume an "eccentricity" e as defined in Section 5.2
above, then the output eccentricity relates to the input eccentricity
as follows:
e = 2 * e * e
output input 1 input 2
Since e is never greater than 1/2, the eccentricity is always
improved except in the case where at least one input is a totally
skewed constant. This is illustrated in the following table where
the top and left side values are the two input eccentricities and the
entries are the output eccentricity:
+--------+--------+--------+--------+--------+--------+--------+
| e | 0.00 | 0.10 | 0.20 | 0.30 | 0.40 | 0.50 |
+--------+--------+--------+--------+--------+--------+--------+
| 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 0.10 | 0.00 | 0.02 | 0.04 | 0.06 | 0.08 | 0.10 |
| 0.20 | 0.00 | 0.04 | 0.08 | 0.12 | 0.16 | 0.20 |
| 0.30 | 0.00 | 0.06 | 0.12 | 0.18 | 0.24 | 0.30 |
| 0.40 | 0.00 | 0.08 | 0.16 | 0.24 | 0.32 | 0.40 |
| 0.50 | 0.00 | 0.10 | 0.20 | 0.30 | 0.40 | 0.50 |
+--------+--------+--------+--------+--------+--------+--------+
However, keep in mind that the above calculations assume that the
inputs are not correlated. If the inputs were, say, the parity of
the number of minutes from midnight on two clocks accurate to a few
seconds, then each might appear random if sampled at random intervals
much longer than a minute. Yet if they were both sampled and
combined with xor, the result would usually 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
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non-linear 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 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, ref to come - Odlyzko].
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
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fewer bits than are input will tend to strengthen the randomness of
the output the way using Exclusive Or to produce one bit from two did
above.
The last table in Section 6.1.1 shows that mixing a random bit with a
constant bit with Exclusive Or will produce a random bit. While this
is true, it does not provide a way to "stretch" one random bit into
more than one. If, for example, a random bit is mixed with a 0 and
then with a 1, this produces a two bit sequence but it will always be
either 01 or 10. Since there are only two possible values, there is
still only the one bit of original randomness.
6.1.4 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 have 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.
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 noise 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.
Unfortunately, any of these sources can produce limited or
predicatable values under some circumstances.
Some of the sources listed above would be quite strong on multi-user
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systems where, in essence, each user of the system is a source of
randomness. However, on a small single user system, such as a
typical IBM PC or Apple Macintosh, it might be possible for an
adversary to assemble a similar configuration. This could give the
adversary inputs to the mixing process that were sufficiently
correlated to those used originally as to make exhaustive search
practical.
The use of multiple random inputs with a strong mixing function is
recommended and can overcome weakness in any particular input. For
example, the timing and content of requested "random" user keystrokes
can yield hundreds of random bits but conservative assumptions need
to be made. For example, assuming a few bits of randomness if the
inter-keystroke interval is unique in the sequence up to that point
and a similar assumption if the key hit is unique but assuming that
no bits of randomness are present in the initial key value or if the
timing or key value duplicate previous values. The results of mixing
these timings and characters typed could be further combined with
clock values and other inputs.
This strategy may make practical portable code to produce good random
numbers for security even if some of the inputs are very weak on some
of the target systems. However, it may fail against a high grade
attack on small single user systems, especially if the adversary has
even been able to observe the generation process in the past. A
hardware random source is still preferable.
6.3 Cryptographically Strong Sequences
In cases where a series of random quantities must be generated, an
adversary may learn some values in the sequence. In general, they
should not be able to predict other values from the ones that they
know.
The correct technique is to start with a strong random seed, take
cryptographically strong steps from that seed [CRYPTO2], and do not
reveal the complete state of the generator in the sequence elements.
If each value in the sequence can be calculated in a fixed way from
the previous value, then when any value is compromised, all future
values can be determined. This would be the case, for example, if
each value were a constant function of the previous values, even if
the function were a very strong, non-invertible message digest
function.
A good way to achieve a strong sequence is to have the values be
produced by hashing the quantities produced by concatenating the seed
with successive integers or the like and then mask the values
obtained so as to limit the amount of generator state available to
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the adversary. It may also be possible to use an encryption
algorithm with a random key and seed value to encrypt and feedback
some of the output encrypted value into the value to be encrypted for
the next iteration. Appropriate feedback techniques will usually be
recommended with the encryption algorithm. An example is shown below
where shifting and masking are used to combine the cypher output
feedback. This type of feedback is recommended in connection with
DES [DES MODES].
+---------------+
| V |
| | n |
+--+------------+
| | +---------+
| +---------> | | +-----+
+--+ | Encrypt | <--- | Key |
| +-------- | | +-----+
| | +---------+
V V
+------------+--+
| V | |
| n+1 |
+---------------+
Note that if a shift of one is used, this is the same as the shift
register technique described in Section 3 above but with the all
important difference that the feedback is determined by a complex
non-linear function of all bits rather than a simple linear
combination of output from a few bit position taps.
To predict values of a sequence from others when the sequence was
generated by these techniques is equivalent to breaking the
cryptosystem or inverting the "non-invertible" hashing involved with
only partial information available. The less information revealed
each iteration, the harder it will be for an adversary to predict the
sequence. Thus it is best to use only one bit from each value. It
has been shown that some cases this makes it impossible to break a
system even when the cryptographic system is invertible and can be
broken if all of the generated values were revealed.
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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.
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8. Examples of Randomness Required
Below are two examples showing rough calculations of needed
randomness for security.
8.1 Password Generation
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
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
D. Eastlake, S. Crocker, & J. Schiller [Page 24]
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key values. Assume further that there is no systematic weakness in
the cryptographic system so that brute force trial of keys is the
best the adversary can do.
8.2.1 Effort per Key Trial
How much effort will it take to try each key? For very high security
applications it is best to assume a low value of effort. Even if it
would clearly take tens of thousands of computer cycles or more to
try a single key, there may be some pattern that enables huge blocks
of key values to be tested with much less effort per key. Thus it is
probably best to assume no more than a 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
and resourceful adversary could break the key in 2 weeks (on average
they need try only half the keys).
8.2.2 Meet in the Middle Attacks
If chosen or known plain text and the resulting encrypted text are
available, a "meet in the middle" attack is possible if the structure
of the encryption algorithm allows it. (In a known plain text
attack, the adversary knows all or part of the messages being
encrypted, possibly some standard header or trailer fields. In a
chosen plain text attack, the adversary can force some known plain
text to be encrypted, possibly by "leaking" an exciting text that
would then be sent by the adversary over an encrypted channel.)
An oversimplified explanation of the meet in the middle attack attack
is as follows: the adversary can half-encrypt the know or chosen
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
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is found, the full key can be assembled from the halves and used to
decrypt other parts of the message or other messages. At its best,
this type of attack can halve the exponent of the work required by
the adversary while adding a moderate constant factor. To be assured
of safety against this, a doubling of the amount of randomness in the
key to a minimum of 96 bits is required.
The meet in the middle attack assumes that the cryptographic
algorithm can be decomposed in this way but we can not rule that out
without a deep knowledge of the algorithm. Even if a basic algorithm
is not subject to a meet in the middle attack, an attempt to produce
a stronger algorithm by applying the basic algorithm twice with
different keys may gain much less than would be expected. Such a
composite algorithm would be subject to this type of attack.
Enormous resources may be required to mount a meet in the middle
attack but they are probably within the range of the national
security services of a major nation. 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.
8.2.3 Other Considerations
Since we have not even considered the possibilities of special
purpose code breaking hardware or just how much of a safety margin we
want beyond our assumptions above, probably a good minimum for a very
high security cryptographic key is 128 bits of randomness which
implies a minimum key length of 128 bits. If the two parties agree
on a key by Diffie-Hellman exchange [D-H], then in principle only
half of this randomness would have to be supplied by each party.
However, there is probably some correlation between their random
inputs so it is probably best to assume that each party needs to
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.
D. Eastlake, S. Crocker, & J. Schiller [Page 26]
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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.
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-STD-002-85
as one of its appendicies.)
[KNUTH] - The Art of Computer Programming, Volume 2: Seminumerical
Algorithms, Chapter 3: Random Numbers. Addison Wesley Publishing
Company, Second Edition 1982, Donald E. Knuth.
D. Eastlake, S. Crocker, & J. Schiller [Page 27]
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[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
[SHANNON] - The Mathematical Theory of Communication, University of
Illinois Press, 1963, Claude E. Shannon. (originally from: Bell
System Technical Journal, July and October 1948)
[SHIFT1] - Shift Register Sequences, Aegean Park Press, Revised
Edition 1982, Solomon W. Golomb.
[SHIFT2] - Cryptanalysis of Shift-Register Generated Stream Cypher
Systems, Aegean Park Press, 1984, Wayne G. Barker.
D. Eastlake, S. Crocker, & J. Schiller [Page 28]
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Authors Addresses
Donald E. Eastlake 3rd
Digital Equipment Corporation
550 King Street, LKG2-1/BB3
Littleton, MA 01460
Telephone: +1 508 486 6577(w) +1 508 287 4877(h)
EMail: dee@skidrow.lkg.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 and File Name
This draft expires 31 March 1994.
Its file name is draft-ietf-security-randomness-01.txt.
D. Eastlake, S. Crocker, & J. Schiller [Page 29]
%%% overflow headers %%%
Cc: David M. Balenson <balenson@tis.com>, Stephen D. Crocker <crocker@tis.com>,
Beast (Donald E. Eastlake,III) <dee@skidrow.lkg.dec.com>,
Carl Ellison <cme@ellisun.sw.stratus.com>,
Neil Haller <nmh@thumper.bellcore.com>, Marc Horowitz <marc@mit.edu>,
Charlie Kaufman <kaufman@abyss.enet.dec.com>,
Steve Kent <kent@bbn.com>, Hal Murray <murray@decsrc.enet.dec.com>,
Richard Pitkin <pitkin@ranger.enet.dec.com>,
Tim Redmond <redmond@tis.com>, Jeffrey I. Schiller <jis@mit.edu>,
Doug Tygar <doug.tygar@cs.cmu.edu>,
Eirikur Hallgrimsson <eirikur@ranger.enet.dec.com>,
Al Kent <arkent@world.std.com>,
Jim Burrows (Brons) <burrows@brons.enet.dec.com>
%%% end overflow headers %%%