Bent functions of maximal degree

In this article a technique for constructing p-ary bent functions from plateaued functions is presented. This generalizes earlier techniques of constructing bent from near-bent functions. The Fourier spectrum of quadratic monomials is analysed, examples of quadratic functions with highest possible absolute values in their Fourier spectrum are given. Applying the construction of bent functions to the latter class of functions yields bent functions attaining upper bounds for the algebraic degree when $p=3,5$. Until now no construction of bent functions attaining these bounds was known

A general approach to construction and determination of the linear complexity of sequences based on cosets

We give a general approach to $N$-periodic sequences over a finite field \F_q constructed via a subgroup $D$ of the group of invertible elements modulo $N$. Well known examples are Legendre sequences or the two-prime generator. For some generalizations of sequences considered in the literature and for some new examples of sequence constructions we determine the linear complexity

On the cycle structure of permutation polynomials

L. Carlitz observed in 1953 that for any a € F*q, the transposition (0 a) can be represented by the polynomial Pa(x) = -a[2](((x - a)[q-2] + a-[1])[q-2] - a)[q-2] which shows that the group of permutation polynomials over Fq is generated by the linear polynomials ax + b, a, b € Fq, a≠0, and x[q-2]. Therefore any permutation polynomial over Fq can be represented as Pn = (...((a[0]x + a[1])[q-2] +a[2]) [q-2] ... + a[n])[q-2] + a[n+1], for some n ≥ 0. In this thesis we study the cycle structure of permutation polynomials Pn, and we count the permutations Pn, n ≤ 3, with a full cycle. We present some constructions of permutations of the form Pn with a full cycle for arbitrary n. These constructions are based on the so called binary symplectic matrices. The use of generalized Fibonacci sequences over Fq enables us to investigate a particular subgroup of Sq, the group of permutations on Fq. In the last chapter we present results on this special group of permutations

A construction of bent functions from plateaued functions

In this presentation, a technique for constructing bent functions from plateaued functions is introduced and analysed. This generalizes earlier techniques for constructing bent from near-bent functions. Using this construction, we obtain a big variety of inequivalent bent functions, some weakly regular and some non-weakly regular. Classes of bent function with some additional properties that enable the construction of strongly regular graphs are constructed, and explicit expressions for bent functions with maximal degree are presented

Recent advances in the theory of nonlinear pseudorandom number generators

The classical linear congruential method for generating uniform pseudorandom numbers has some deficiencies that can render them useless for some simulation problems. This fact motivated the design and the analysis of nonlinear congruential methods for the generation of pseudorandom numbers. In this thesis, we aim to review the recent developments in the study of nonlinear congruential pseudorandom generators. Our exposition concentrates on inversive generators. We also describe the so-called power generator and the quadratic exponential generator which are particularly interesting for cryptographic applications. We give results on the period length and theoretical analysis of these generators. The emphasis is on the lattice structure, discrepancy and linear complexity of the generated sequences