Alev Topuzoğlu

Factorization of a class of polynomials over finite fields

Abstract We study the factorization of polynomials of the form F r ( x ) = b x q r + 1 − a x q r + d x − c over the finite field F q . We show that these polynomials are closely related to a natural action of the projective linear group PGL ( 2 , q ) on non-linear irreducible polynomials over F q . Namely, irreducible factors of F r ( x ) are exactly those polynomials that are invariant under the action of some non-trivial element [ A ] ∈ PGL ( 2 , q ) . This connection enables us to enumerate irreducibles which are invariant under [ A ] . Since the class of polynomials F r ( x ) includes some interesting polynomials like x q r − x or x q r + 1 − 1 , our work generalizes well-known asymptotic results about the number of irreducible polynomials and the number of self-reciprocal irreducible polynomials over F q . At the same time, we generalize recent results about certain invariant polynomials over the binary field F 2 .

On the linear complexity profile of nonlinear congruential pseudorandom number generators of higher orders

Nonlinear congruential methods are attractive alternatives to the classical linear congruential method for pseudorandom number generation. Generators of higher orders are of interest since they admit longer periods. We obtain lower bounds on the linear complexity profile of nonlinear pseudorandom number generators of higher orders. The results have applications in cryptography and in quasi-Monte Carlo methods.

On the Carlitz rank of permutations of Fq and pseudorandom sequences

L. Carlitz proved that any permutation polynomial f over a finite field Fq is a composition of linear polynomials and inversions. Accordingly, the minimum number of inversions needed to obtain f is defined to be the Carlitz rank of f by Aksoy et al. The relation of the Carlitz rank of f to other invariants of the polynomial is of interest. Here we give a new lower bound for the Carlitz rank of f in terms of the number of nonzero coefficients of f which holds over any finite field. We also show that this complexity measure can be used to study classes of permutations with uniformly distributed orbits, which, for simplicity, we consider only over prime fields. This new approach enables us to analyze the properties of sequences generated by a large class of permutations of Fp, with the advantage that our bounds for the discrepancy and linear complexity depend on the Carlitz rank, not on the degree. Hence, the problem of the degree growth under iterations, which is the main drawback in all previous approaches, can be avoided.

Enumeration of quadratic functions with prescribed Walsh spectrum

The Walsh transform f̂ of a quadratic function f: F(pⁿ) → F_p satisfies |f̂| ∈ {0,p^n+s/2} for an integer 0 ≤ s ≤ n-1, depending on f. In this paper, quadratic functions of the form F_p,n(x) = Tr_n(Σ_i=0^ka_ix^pi+1) are studied, with the restriction that a_i ∈ F_p, 0 ≤ i ≤ k. Three methods for enumeration of such functions are presented when the value for s is prescribed. This paper extends earlier enumeration results significantly, for instance, the generating function for the counting function is obtained, when n is odd and relatively prime to p, or when n = 2 m, for odd m and p = 2. The number of bent and semibent functions for various classes of n is also obtained.

Enumeration of a class of sequences generated by inversions

Any permutation of a finite field F-q can be represented by a polynomial P-n(x) = (. . . ((a(0)x a(1))(q-2) + a(2))(q-2) ... a(n))(q-2) + a(n+1), for some n >= 0. In this note we present the number of distinct permutations of the types P-2(x) and P-3(x) with full cycle. These results extend earlier work on the inversive pseudorandom number generator and on P-1.

Complete mappings and Carlitz rank

The well-known Chowla and Zassenhaus conjecture, proven by Cohen in 1990, states that for any

Spectra of a class of quadratic functions: Average behaviour and counting functions

d\ge 2

On the Carlitz Rank of Permutation Polynomials Over Finite Fields: Recent Developments

d≥2 and any prime