STATISTICAL PROPERTIES OF PSEUDORANDOM SEQUENCES

Random numbers (in one sense or another) have applications in computer simulation, Monte Carlo integration, cryptography, randomized computation, radar ranging, and other areas. It is impractical to generate random numbers in real life, instead sequences of numbers (or of bits) that appear to be ``random yet repeatable are used in real life applications. These sequences are called pseudorandom sequences. To determine the suitability of pseudorandom sequences for applications, we need to study their properties, in particular, their statistical properties. The simplest property is the minimal period of the sequence. That is, the shortest number of steps until the sequence repeats. One important type of pseudorandom sequences is the sequences generated by feedback with carry shift registers (FCSRs). In this dissertation, we study statistical properties of N-ary FCSR sequences with odd prime connection integer q and least period (q-1)/2. These are called half-ℓ-sequences. More precisely, our work includes: The number of occurrences of one symbol within one period of a half-ℓ-sequence; The number of pairs of symbols with a fixed distance between them within one period of a half-ℓ-sequence; The number of triples of consecutive symbols within one period of a half-ℓ-sequence. In particular we give a bound on the number of occurrences of one symbol within one period of a binary half-ℓ-sequence and also the autocorrelation value in binary case. The results show that the distributions of half-ℓ-sequences are fairly flat. However, these sequences in the binary case also have some undesirable features as high autocorrelation values. We give bounds on the number of occurrences of two symbols with a fixed distance between them in an ℓ-sequence, whose period reaches the maximum and obtain conditions on the connection integer that guarantee the distribution is highly uniform. In another study of a cryptographically important statistical property, we study a generalization of correlation immunity (CI). CI is a measure of resistance to Siegenthaler\u27s divide and conquer attack on nonlinear combiners. In this dissertation, we present results on correlation immune functions with regard to the q-transform, a generalization of the Walsh-Hadamard transform, to measure the proximity of two functions. We give two definitions of q-correlation immune functions and the relationship between them. Certain properties and constructions for q-correlation immune functions are discussed. We examine the connection between correlation immune functions and q-correlation immune functions

Distributional properties of d-FCSR sequences

AbstractIn this paper we study the distribution properties of d-FCSR sequences. These sequences have efficient generators and have several good statistical properties. We show that for d=2 the number of occurrences of an fixed size subsequence differs from the average number of occurrences by at most a small constant times the square root of the average

Maximum-order complexity and 22-adic complexity

The $2$-adic complexity has been well-analyzed in the periodic case. However, we are not aware of any theoretical results on the $N$th $2$-adic complexity of any promising candidate for a pseudorandom sequence of finite length $N$ or results on a part of the period of length $N$ of a periodic sequence, respectively. Here we introduce the first method for this aperiodic case. More precisely, we study the relation between $N$th maximum-order complexity and $N$th $2$-adic complexity of binary sequences and prove a lower bound on the $N$th $2$-adic complexity in terms of the $N$th maximum-order complexity. Then any known lower bound on the $N$th maximum-order complexity implies a lower bound on the $N$th $2$-adic complexity of the same order of magnitude. In the periodic case, one can prove a slightly better result. The latter bound is sharp which is illustrated by the maximum-order complexity of $\ell$-sequences. The idea of the proof helps us to characterize the maximum-order complexity of periodic sequences in terms of the unique rational number defined by the sequence. We also show that a periodic sequence of maximal maximum-order complexity must be also of maximal $2$-adic complexity