Random numbers play an important role in the use of encryption for various net- work security applications. In this section, we provide a brief overview of the use of random numbers in cryptography and network security and then focus on the prin- ciples of pseudorandom number generation.

**PRINCIPLES OF PSEUDORANDOM NUMBER GENERATION**

Random numbers play an important role in the use of encryption for
various net- work security applications. In this section,
we provide a brief overview
of the use of random numbers
in cryptography and network security
and then focus on the prin-
ciples of pseudorandom number generation.

The
Use of Random Numbers

A
number of network security algorithms and protocols based on cryptography make
use of random binary numbers. For example,

•
Key distribution and reciprocal authentication schemes, such
as those discussed in Chapters 14 and 15. In such schemes, two communicating parties
cooperate by exchanging messages to distribute keys and/or authenticate
each other. In many cases, nonces
are used for handshaking to prevent replay
attacks. The use of random numbers for the nonces
frustrates an opponent’s efforts to deter- mine or guess the nonce.

•
Session key generation. We will see a number of protocols
in this book where a secret
key for symmetric encryption is generated for use for a short
period of time. This
key is generally called a session key.

•
Generation of keys for the RSA public-key
encryption algorithm (described in Chapter 9).

•
Generation
of a bit stream for symmetric stream encryption (described in this chapter).

These
applications give rise to two distinct and not necessarily compatible
requirements for a sequence of random numbers:
randomness and unpredictability.

*RANDOMNESS*** **Traditionally, the concern
in the generation of a sequence of allegedly
random numbers has been that the sequence of numbers be random in some well- defined statistical sense. The following two criteria
are used to validate that a
sequence of numbers is random:

•
**Uniform distribution: **The distribution of bits in the sequence should be uniform; that is, the frequency
of occurrence of ones and zeros should be approximately equal.

•
**Independence: **No one subsequence in the sequence
can be inferred from the others.

Although
there are well-defined tests for determining that a sequence of bits matches a
particular distribution, such as the uniform distribution, there is no such test to “prove” independence.
Rather, a number of tests can be applied to demon- strate if a sequence
does not exhibit
independence. The general
strategy is to apply
a number of such tests until the confidence that independence exists is sufficiently strong.

In the context of our discussion, the use of a sequence
of numbers that appear
statistically random often
occurs in the
design of algorithms related to cryptography. For example, a fundamental requirement of the RSA public-key encryption scheme discussed in Chapter
9 is the ability to generate prime numbers. In general, it is dif- ficult
to determine if a given large number *N *is prime. A brute-force approach would be to divide *N *by every odd integer less than 1*N*. If *N *is on the order,
say, of 10150, which is a not uncommon
occurrence in public-key cryptography, such a brute-force approach is beyond the reach of human analysts
and their computers. However, a number
of effective algorithms exist that test the primality of a number by using a sequence of randomly chosen integers as input to relative__ly ____si__mple
computations. If the sequence is sufficiently long (but far, far less than 210150), the primality of a number
can be determined with near certainty. This
type of approach, known as randomization, crops
up frequently in the design
of algorithms. In essence,
if a problem is too hard or
time-consuming to solve exactly, a simpler,
shorter approach based on
randomization is used to provide an answer with any desired level of
confidence.

** UNPREDICTABILITY **In applications such
as reciprocal authentication, session key generation,
and stream ciphers, the requirement is not just that the sequence of numbers be
statistically random but that the successive members of the sequence are unpredictable.
With “true” random sequences, each
number is statistically independent of other numbers in the sequence and
therefore unpredictable. However, as is discussed shortly, true random
numbers are seldom used; rather,
sequences of numbers that appear to be random are generated by some algorithm.
In this latter case, care must be taken that an opponent
not be able to predict
future elements of the sequence on the basis of earlier
elements.

TRNGs, PRNGs, and PRFs

Cryptographic applications typically make use of
algorithmic techniques for ran- dom number generation. These algorithms are deterministic and therefore produce sequences of numbers that are not statistically random.
However, if the algorithm is good,
the resulting sequences
will pass many reasonable tests of randomness. Such numbers are referred to as **pseudorandom numbers**.

You may be somewhat uneasy about the concept of using numbers
generated by a deterministic algorithm as if they were random numbers.
Despite what might be called philosophical
objections to such a practice,
it generally works. As one expert
on probability theory
puts it [HAMM91]:

For practical purposes we are forced
to accept the
awkward concept of
“relatively random” meaning that with regard to the proposed use we can see no reason why they will not perform as if they were
random (as the theory usually requires). This is highly subjective and is not very palatable to purists,
but it is what statisticians regu- larly appeal to when they take “a random sample”
—they hope that any
results they use will have approximately the same properties as a complete counting
of the whole sample space that occurs in their theory.

Figure 7.1 contrasts a **true random number
generator **(TRNG) with two forms of
pseudorandom number generators. A TRNG takes
as input a source that is effec- tively random; the source
is often referred
to as an **entropy
source**. We discuss such sources in Section 7.6. In essence,
the entropy source is drawn from the physical environment of the computer and
could include things such as keystroke timing
patterns, disk electrical activity, mouse
movements, and instantaneous values of the system clock. The source, or combination of sources, serve as input to an algorithm
that produces random binary output.
The TRNG may simply
involve conversion of an analog source to a binary output.
The TRNG may involve
additional processing to overcome
any bias in the source;
this is discussed in Section 7.6.

In contrast, a PRNG takes as input a fixed value, called
the **seed**, and produces a sequence of output
bits using a deterministic algorithm. Typically, as shown, there is some feedback path by which
some of the results of the algorithm are fed back as
** **

input as additional output bits are produced. The important
thing to note is that the output bit stream is determined solely by the input value or values,
so that an adver- sary who knows the algorithm and the seed can reproduce the entire bit stream.

Figure 7.1 shows two different forms of PRNGs, based on
application.

•
**Pseudorandom number generator: **An algorithm
that is used to produce
an open-ended sequence of bits is referred to as a PRNG.
A common application for an open-ended sequence
of bits is as input
to a symmetric stream cipher,
as discussed in Section
7.4. Also, see Figure 3.1a.

•
**Pseudorandom function (PRF):
**A PRF is used to produced
a pseudorandom string of bits
of some fixed length. Examples are symmetric encryption keys and nonces.
Typically, the PRF takes as input a seed plus some context specific values, such as a user ID or an application ID. A number of examples of PRFs
will be seen throughout this book, notably
in Chapters 16 and 17.

Other than the number of bits produced, there is no
difference between a PRNG and a PRF. The same algorithms can be used in both
applications. Both require a seed and both must exhibit randomness and
unpredictability. Further, a PRNG application may also employ context-specific
input. In what follows, we make no distinction between these two applications.

PRNG Requirements

When
a PRNG or PRF is used for a cryptographic application, then the basic
requirement is that an adversary who does not know the seed is unable to deter-
mine the pseudorandom string. For example, if the pseudorandom bit stream is used in a stream cipher,
then knowledge of the pseudorandom bit stream would enable the adversary to
recover the plaintext from the
ciphertext. Similarly, we wish to protect the output value of a PRF. In
this latter case, consider the following scenario. A 128-bit seed, together
with some context-specific values, are used to generate a 128-bit secret key
that is subsequently used for symmetric encryption. Under normal circumstances,
a 128-bit key is safe from a brute-force attack. However, if the PRF does not
generate effectively random 128-bit output values, it may be possible for an
adversary to narrow the possibilities and successfully use a brute force attack.

This general requirement for secrecy of the output
of a PRNG or PRF leads to specific requirements in the areas of randomness, unpredictability, and the character- istics of the seed.
We now look at these in turn.

** RANDOMNESS **In terms of randomness, the requirement for a PRNG is that the generated
bit stream appear random even though it is deterministic. There is no single test that can determine if a PRNG generates numbers that have the characteristic of randomness. The best that can be done is to apply a
sequence of tests to the PRNG. If the PRNG exhibits
randomness on the basis of multiple tests, then
it can be assumed to satisfy the randomness requirement. NIST SP 800-22
(

•
**Uniformity: **At any point in the generation of a sequence of
random or pseudorandom bits, the
occurrence of a zero or one is equally likely,
i.e., the probability of each is exactly 1/2. The expected number of zeros (or ones) is *n*/2, where *n *= the sequence
length.

•
**Scalability:
**Any test applicable to a sequence can also be applied to
subse- quences extracted at random. If a sequence
is random, then any such extracted
subsequence should also be random. Hence,
any extracted subsequence should pass any test for randomness.

•
**Consistency:
**The behavior of a
generator must be consistent across starting values (seeds). It is inadequate to test a PRNG based on the output from a sin- gle
seed or an TRNG on the basis of an output produced
from a single physi- cal output

SP
800-22 lists 15 separate tests of randomness. An understanding of these tests requires
a basic knowledge of statistical analysis, so we don’t attempt
a techni- cal description
here. Instead, to give some flavor for the tests, we list three of the tests and the purpose
of each test,
as follows.

•
**Frequency
test: **This is the most basic test and must be included in any
test suite. The purpose of this test is to determine whether
the number of ones and zeros in a sequence
is approximately the same as would be expected for a truly random sequence.

**Runs test: **The focus of this test is the total number of
runs in the sequence, where a run is an uninterrupted sequence
of identical bits bounded before and after
with a bit of the opposite value. The purpose
of the runs test is to determine whether the number of runs of ones and zeros of various lengths
is as expected for a random
sequence.

•
**Maurer’s universal statistical test: **The focus of this test is the number
of bits between matching
patterns (a measure that is related to the length of a com- pressed sequence).
The purpose of the test is to detect whether or not the sequence can be significantly compressed without loss of information. A signifi-
cantly compressible sequence
is considered to be non-random.

** UNPREDICTABILITY
**A stream of pseudorandom numbers should exhibit two forms
of unpredictability:

•
**Forward unpredictability**: If the seed is unknown,
the next output
bit in the sequence should be unpredictable in spite of any knowledge of previous bits in
the sequence.

•
**Backward**** ****unpredictability**: It should also not be feasible
to determine the seed
from knowledge of any generated
values. No correlation between a seed and
any value generated from that seed should be evident; each element of the
sequence should appear to be the outcome of an independent random event whose probability is 1/2.

The same set of tests for randomness also provide a test of unpredictability. If the generated bit stream appears
random, then it is not possible to
predict some bit or bit sequence
from knowledge of any previous
bits. Similarly, if the bit sequence
appears random, then there is no feasible way to deduce the seed based on the
bit sequence. That is, a random sequence
will have no correlation with a fixed value (the seed).

** SEED REQUIREMENTS **For cryptographic applications, the seed that serves
as input to the PRNG must be secure. Because the
PRNG is a deterministic algorithm, if the adversary can deduce the seed, then
the output can also be determined. Therefore, the
seed must be unpredictable. In fact, the seed itself must be a random or
pseudorandom number.

Typically,
the seed is generated by a TRNG, as shown in
Figure 7.2. This is the scheme recommended by SP800-90. The reader may wonder, if a TRNG is available, why it is necessary
to use a PRNG. If the application is a stream cipher, then
a TRNG is not practical. The sender would need to generate a
keystream of bits as long as the plaintext and then transmit the
keystream and the ciphertext securely to the receiver. If
a PRNG is used, the sender need only find a
way to deliver the stream cipher key, which is typically
54 or 128 bits, to the receiver in a
secure fashion.

Even in the case of a PRF
application, in which only a limited number of bits is generated, it is
generally desirable to use a TRNG to provide the seed to the PRF and use the
PRF output rather then use the TRNG directly. As is explained in a Section 7.6,
a TRNG may produce a binary string with some bias. The PRF would have the effect
of “randomizing” the output of the TRNG so as to eliminate that bias.

Finally, the mechanism used to
generate true random numbers may not be able to generate bits at a rate
sufficient to keep up with the application requiring the random bits.

Algorithm Design

Cryptographic
PRNGs have been the subject of much research over the years, and a wide variety
of algorithms have been developed. These fall roughly into two categories.

•
**Purpose-built**** ****algorithms:**** **These are algorithms designed specifically and
solely for the purpose of generating pseudorandom bit streams.
Some of these algo- rithms
are used for a variety
of PRNG applications; several of these
are described in the next
section. Others are designed
specifically for use in a stream
cipher. The most important example of the latter is RC4, described
in Section 7.5.

•
**Algorithms based
on existing cryptographic algorithms: **Cryptographic algo- rithms have the effect of
randomizing input. Indeed, this is a requirement of such algorithms. For example, if a symmetric block cipher produced
ciphertext that had certain
regular patterns in it, it would aid in the process of cryptanalysis. Thus, cryptographic algorithms can serve as the core of PRNGs. Three broad categories of cryptographic algorithms are commonly used to create
PRNGs:

—**Symmetric block ciphers: **This approach is discussed in Section 7.3.

—**Asymmetric**** ****ciphers:**** **The number theoretic concepts used for an asymmetric cipher can also be adapted for a PRNG; this approach is examined
in Chapter 10.

—**Hash functions and
message authentication codes:
**This approach is examined in
Chapter 12.

Any of these
approaches can yield a cryptographically
strong PRNG. A purpose-built algorithm may be provided by an operating
system for general use. For applications that already use certain cryptographic algorithms for encryption or authentication, it makes sense to reuse the same code for
the PRNG. Thus, all of these approaches are in common
use.

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