Arne Winterhof

Lattice Structure of Nonlinear Pseudorandom Number Generators in Parts of the Period

Recently, we showed that an extension of Marsaglia’s lattice test for segments of sequences over arbitrary fields and the linear complexity profile provide essentially equivalent quality measures for the intrinsic structure of pseudorandom number sequences. More precisely, the knowledge of the linear complexity profile yields a value S such that the largest dimension for passing the above lattice test is either S or S — 1. In the present paper for periodic sequences over finite fields and sufficiently long parts of the period we determine the exact value S or S —1. As an application we deduce from recently obtained lower bounds on the linear complexity profile of certain nonlinear pseudorandom number generators new results on their lattice structure.

On the Autocorrelation of Cyclotomic Generators

We extend a result of Ding and Helleseth on the autocorrelation of a cyclotomic generator in several ways. We define and analyze cyclotomic generators of arbitrary orders and over arbitrary finite fields, and we consider two, in general, different definitions of autocorrelation. Cyclotomic generators are closely related to the discrete logarithm. Hence, the results of this paper do not only describe interesting cryptographic properties of cyclotomic generators and their generalizations but also desirable features of the discrete logarithm.

Counting functions and expected values for the lattice profile at n

Recently, Dorfer and Winterhof introduced and analyzed a lattice test for sequences of length n over a finite field. We determine the number of sequences @h of length n with given largest dimension Sn(@h)=S for passing this test. From this result we derive an exact formula for the expected value of Sn(@h). For the binary case we characterize the (infinite) sequences @h with maximal possible Sn(@h) for all n.

A Note on the Linear Complexity Profile of the Discrete Logarithm in Finite Fields

We essentially improve lower bounds on the linear complexity of a sequence representing the residues of the discrete logarithm in a finite field modulo a divisor of the order of the multiplicative group. More generally, we present the result as a bound on the linear complexity profile. The proof is based on character sum bounds.

On the linear complexity profile of some new explicit inversive pseudorandom numbers

Linear complexity and linear complexity profile are interesting characteristics of a sequence for applications in cryptography and Monte-Carlo methods. We introduce some new explicit inversive pseudorandom number generators and prove lower bounds on their linear complexity profile which are close to the best possible.

Interpolation of the elliptic curve Diffie-Hellman mapping

We prove lower bounds on the degree of polynomials interpolating the Diffie-Hellman mapping for elliptic curves over finite fields and some related mappings including the discrete logarithm. Our results support the assumption that the elliptic curve Diffie-Hellman key exchange and related cryptosystems are secure.

On the distribution of points in orbits of PGL(2,q) acting on GF(qn)

We prove results on the distribution of points in an orbit of PGL(2,q) acting on an element of GF(q^n). These results support a conjecture of Klapper. More precisely, we show that the points in an orbit are uniformly distributed if n is small with respect to q.

Algebraic Curves and Finite Fields: Cryptography and Other Applications

This book collects the results of the workshops on Applications of Algebraic Curves and Applications of Finite Fieldsat the RICAMin 2013. These workshops brought together the most prominent researchers in the area of finite fields and their applications around the world, addressing old and new problems on curves and other aspects of finite fields, with emphasis on their diverse applications to many areas of pure and applied mathematics.

Uniform distribution and Quasi-Monte Carlo methods : discrepancy, integration and applications

The survey articles in this book focus on number theoretic point constructions, uniform distribution theory, and quasi-Monte Carlo methods. As deterministic versions of the Monte Carlo method, quasi-Monte Carlo rules enjoy increasing popularity, with many fruitful applications in mathematical practice, as for example in finance, computer graphics, and biology.

A Review of Number Theory and Algebra

Elementary number theory may be regarded as a prerequisite for this book, but since we, the authors, want to be nice to you, the readers, we provide a brief review of this theory for those who already have some background on number theory and a crash course on elementary number theory for those who have not. Apart from trying to be friendly, we also follow good practice when we prepare the ground for the coming attractions by collecting some basic notation, terminology, and facts in an introductory chapter, like a playwright who presents the main characters of the play in the first few scenes. Basically, we cover only those results from elementary number theory that are actually needed in this book. For more information, there is an extensive expository literature on number theory, and if you want to read the modern classics, then we recommend the books of Hardy and Wright [61] and of Niven, Zuckerman, and Montgomery [151].