Interest in normal bases over finite fields stems both from mathematical theory and practical applications. There has been a lot of literature dealing with various properties of normal bases (for finite fields and for Galois extension of arbitrary fields). The advantage of using normal bases to represent finite fields was noted by Hensel in 1888. With the introduction of optimal normal bases, large finite fields, that can be used in secure and e#cient implementation of several cryptosystems, have recently been realized in hardware. The present thesis studies various theoretical and practical aspects of normal bases in finite fields. We first give some characterizations of normal bases. Then by using linear algebra, we prove that F q n has a basis over F q such that any element in F q represented in this basis generates a normal basis if and only if some groups of coordinates are not simultaneously zero. We show how to construct an irreducible polynomial of degree 2 n with linearly i...

PRNGlib provides several pseudo-random number generators through a common interface on any Shared or Distributed Memory Parallel architecture. Common routines are specified to initialize the generators with appropriate seeds on each processor and to generate uniform or (normal, Poisson, exponential) distributed random vectors. We concentrate on those generators which assure high quality (i.e., passing most of the empirical and theoretical tests), have a long period, and can be calculated quickly, also in parallel, i.e., it must be possible to generate the same random sequence independent of the number of processors. This splitting facility implies a method to skip over n pseudo-random numbers without calculating all intermediate values, i.e., an O(log n) algorithm is required. Taking into account these criteria Lagged Fibonacci, Generalized Shift Register, and Multiplicative Linear Congruential generators are implemented with (almost) arbitrary specifications for lags, multipliers, m...

this report we shall present the fundamentals of random number generation on parallel processors. We shall exhibit how the practical task of carrying out stochastic simulation on a parallel machine leads deeply into number theory and algebra. We shall see that some classical algorithms which have proved to be excellent for single-processor machines, are either useless or require greatest care in the case of parallel processors. Stochastic simulation is one of the important tasks for single- as well as multiprocessor machines. Computer simulations of real-life processes based on stochastic models have become one of the most interesting -- and demanding -- applications of mathematics. Due to the computational complexity of the problems, parallelization of the underlying algorithms is receiving increasing attention. As a basic condition to any research, we should be able to reproduce and to verify a scientific experiment. These two requirements and, further, considerations of storage and computational effectiveness rule out physical sources for random numbers, such as radioactive decay or electronic noise. The efficient generation of random numbers of high statistical quality is an absolute necessity for stochastic simulation. In his well-known monograph, Ripley [19, p.2] writes: "The first thing needed for a stochastic simulation is a source of randomness. This is often taken for granted but is of fundamental importance. Regrettably many of the so-called random functions supplied with the most widespread computers are far from random, and many simulation studies have been invalidated as a consequence." D5H-1/Rel 1.0/April 27, 1994 Random number generators for parallel processors PACT The following statement from Ripley[19, p.14] does not exaggerate the actual situation:...

: 2 1 Introduction 2 2 Some measures of the quality of uniform distribution of sequences and parallel Weyl--sequences 3 3 Parallel Generation and Independence of Weyl Sequences 9 R5Z-4/Rel 1.0/Oktober 31 1994 Introduction PACT 0 Abstract: The paper ist part of the NEWTON project. New Technology of Numerics intends to be the amalgam of number theoretic multivariate numerics and the advanced technology of parallel computers. We propose in the present paper parallel Weyl generators for pseudo random sequences of points in the s--dimensional unit cube. The methods proposed are useful for multivariate numerical and simulation problems. 1 Introduction Many questions of contemporary numerical analysis such as multivariate integration, approximation and interpolation and simulation as well are connected with the theory of uniform distributed sequences. The most efficient methods of multivariate analysis and simulation (i. e. pseudo--random generators) are based on deep methods of number th...

This thesis describes various efficient architectures for computation in Galois fields of the type GF(2^k).

Abslracl-A construction of perfect maps, i.e., periodic r X u binary arrays in which each n X m binary matrix appears exactly once, is given. A similar constrnction leads to arrays in which only the zero n X m matrix does not appear and to a construction in which only a few n X m binary matrices do not appear. A generalization to the nonbinary case is also given. The constructions involve an interesting problem in shift register theory. We give the solution for almost all the cases of this problem. P I.