Random Sequences and Linear Complexity

Abstract

Stream ciphers utilize deterministically generated “random” sequences to encipher the message stream. Since the running key generator is a finite state machine, the key stream necessarily is (ultimately) periodic. Thus the best one can hope for is to make the first period of a periodic key stream resemble the output of a binary symmetric source (BSS). A BSS is a device which puts out with equal probability a zero or a one independently of the previous output bits, or in other words, a BSS realizes a fair coin tossing experiment. (Note that we have tacitly assumed the sequences under investigation to be defined over GF(2)). The period of the key stream necessarily is a finite quantity. Thus we are confronted with the problem of characterizing the randomness of a finite sequence. But how can this be done in light of the fact that every finite output sequence of a BSS is equally likely? It seems difficult to define adequately the concept of randomness (in a mathematical sense) for finite sequences. Still, nearly everyone would agree that something like a “typical” output sequence of a BSS exists. A finite coin tossing sequence, for example, would “typically” exhibit a balanced distribution of single bits, pairs, triples, etc. of bits, and long runs of one symbol would be very rare. This in contrast to infinite coin tossing sequences, where local nonrandomness is sure to occur.