Finite Fields and Their Applications

On the Distribution and Lattice Structure of Nonlinear Congruential Pseudorandom Numbers

“…For parts of the period, analogous results have been obtained only in the case where the numbers u L can be represented modulo p as the values of a polynomial over % N whose degree is much smaller than p. It is clear, however, that in most cases the degree of this polynomial will be close to p, and so the known results cover only a minority of cases. In particular, in the very important case where f is a polynomial of small degree which generates a sequence of the largest possible period t"p, it is shown in the proof of Theorem 5 that the degree of the polynomial representing the numbers u L is of order p. We refer to [3,4,7,8, 11}13] for more details and precise references to original papers. The case of periods t(p is of interest as well.…”

Irreducible polynomials from a cubic transformation

“…Linear feedback shift registers are used to generate sequences over a finite field in a range of applications, from pseudo-random number generation to the design of stream ciphers, see [11,Chapter 8, Section 1]. Transformation shift registers (TSRs) can be considered a generalization of linear feedback shift registers.…”

Algorithms for computing parameters of graph-based extensions of BCH codes

“…We use standard concepts concerning algorithms, graphs, and codes, following [7,12,14,17,18,[21][22][23][24]27,[31][32][33][34]36]. Throughout the word "graph" means a directed graph without multiple edges but possibly with loops.…”