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The essential elements of any pseudorandom number generator (PRNG) are a seed value and a deterministic algorithm for generating a stream of pseudorandom bits.

**PSEUDORANDOM
NUMBER GENERATION USING HASH FUNCTIONS AND MACS**

The essential
elements of any pseudorandom number
generator (PRNG) are a seed value and a deterministic algorithm
for generating a stream of pseudorandom bits.
If the algorithm is used as a pseudorandom function (PRF) to produce a
required value, such as a session key, then
the seed should only be known to the user of
the PRF. If the algorithm
is used to produce a stream encryption function, then the seed
has the role of a secret key that must be known to the sender and the receiver.

We noted in Chapters 7 and 10 that,
because an encryption algorithm pro- duces an apparently random output, it can serve
as the basis of a (PRNG). Similarly, a hash function or MAC produces apparently random output and
can be used to build a PRNG. Both ISO standard 18031 (*Random Bit Generation*) and NIST SP 800-90 (*Recommendation** for Random Number Generation
Using Deterministic Random Bit Generators*) define
an approach for random number
generation using a cryptographic hash function. SP 800-90
also defines a random number generator based
on HMAC. We look at these two approaches in turn.

PRNG Based on Hash Function

Figure 12.12a shows the basic strategy
for a hash-based PRNG specified in SP 800-90 and ISO 18031. The algorithm
takes as input:

V = seed

seedlen
= bit length of V >= k + 64,
where k is a
desired security level expressed
in bits

n =
desired number of output bits

The algorithm
uses the cryptographic hash function H with an hash value
out- put of *outlen *bits. The basic
operation of the algorithm is

The SP 800-90 specification also provides for periodically updating *V *to enhance security. The specification
also indicates that there are no known or sus- pected weaknesses in the
hash-based approach for a strong cryptographic hash algorithm, such as SHA-2.

PRNG Based on MAC Function

Although there are no known or suspected
weaknesses in the use of a cryptographic hash function for a PRNG in the manner
of Figure 12.12a,
a higher degree
of confi- dence can
be achieved by using a MAC. Almost invariably, HMAC is used
for constructing a MAC-based
PRNG. This is because HMAC is a widely used stan- dardized MAC function
and is implemented in many protocols and applications. As SP 800-90 points out, the disadvantage of this approach
compared to the hash-based
approach is that the execution time is twice as long, because HMAC involves two executions of the underlying hash function for each output
block. The advantage of the
HMAC approach is that it provides
a greater degree
of confidence in its security,
compared to a pure hash-based approach.

For the MAC-based approach, there are two
inputs: a key *K *and a seed *V*. In effect, the combination of *K *and *V *form
the overall seed for the PRNG specified
in SP 800-90. Figure
12.12b shows the basic structure of the PRNG mechanism, and the
leftmost column of Figure 12.13
shows the logic.
Note that the key remains
the same for each block of output, and the data input for each block is equal to the tag output of the previous block. The SP
800-90 specification also provides for periodically
updating *K *and *V *to enhance security.

It is instructive to compare the SP 800-90
recommendation with the use of HMAC for a PRNG in some applications, and this is shown in Figure 12.13.
For the IEEE 802.11i wireless
LAN security standard
(Chapter 17), the data input consists
of the seed concatenated with a counter.
The counter is incremented for each block *wi *of output. This approach
would seem to offer enhanced
security compared to the
SP 800-90 approach. Consider that for SP 800-90, the data input
for output block
*wi *is just the output
*wi *- 1 of the previous
execution of HMAC. Thus, an opponent who is
able to observe the pseudorandom output knows both the input and output of

HMAC. Even so, with the assumption that HMAC is secure,
knowledge of the input
and output should not be sufficient to recover K and hence not sufficient to predict future
pseudorandom bits.

The approach taken by the Transport Layer
Security protocol (Chapter 16) and the Wireless
Transport Layer
Security Protocol (Chapter
17) involves invoking HMAC twice for each block
of output *wi*. As with IEEE 802.11, this is done in such a
way that the output does not yield direct information about the input.
The double use of HMAC doubles
the execution burden and would seem to be security
overkill.

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Cryptography and Network Security Principles and Practice : Cryptographic Data Integrity Algorithms : Message Authentication Codes : Pseudorandom Number Generation Using Hash Functions and MACS |

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