Fractional jumps: complete characterisation and an explicit infinite family

In this paper we provide a complete characterisation of transitive fractional jumps by showing that they can only arise from transitive projective automorphisms. Furthermore, we prove that such construction is feasible for arbitrarily large dimension by exhibiting an infinite class of projectively primitive polynomials whose companion matrix can be used to define a full orbit sequence over an affine space

Full Orbit Sequences in Affine Spaces via Fractional Jumps and Pseudorandom Number Generation

Let $n$ be a positive integer. In this paper we provide a general theory to produce full orbit sequences in the affine $n$-dimensional space over a finite field. For $n=1$ our construction covers the case of the Inversive Congruential Generators (ICG). In addition, for $n>1$ we show that the sequences produced using our construction are easier to compute than ICG sequences. Furthermore, we prove that they have the same discrepancy bounds as the ones constructed using the ICG.Comment: To appear in Mathematics of Computatio

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