Alina Ostafe

Pseudorandomness and Dynamics of Fermat Quotients

We obtain some theoretical and experimental results concerning various properties (the number of fixed points, image distribution, cycle lengths) of the dynamical system naturally associated with Fermat quotients acting on the set

Structure of pseudorandom numbers derived from fermat quotients

\{0,\dots,p-1\}

On the degree growth in some polynomial dynamical systems and nonlinear pseudorandom number generators

. In particular, we improve the lower bound of Vandiver [Bull. Amer. Math. Soc., 22 (1915), pp. 61-67] on the image size of Fermat quotients on the above set (from

On the Length of Critical Orbits of Stable Quadratic Polynomials

p^{1/2}-1

Pseudorandom numbers and hash functions from iterations of multivariate polynomials

to

On pseudorandom numbers from multivariate polynomial systems

(1+o(1))p(\log p)^{-2}

On the generalized joint linear complexity profile of a class of nonlinear pseudorandom multisequences

). We also consider pseudorandom properties of Fermat quotients such as uniform distribution and linear complexity.

Algebraic entropy, automorphisms and sparsity of algebraic dynamical systems and pseudorandom number generators

We study the distribution of s-dimensional points of Fermat quotients modulo p with arbitrary lags. If no lags coincide modulo p the same technique as in [21] works. However, there are some interesting twists in the other case. We prove a discrepancy bound which is unconditional for s = 2 and needs restrictions on the lags for s > 2.We apply this bound to derive results on the pseudorandomness of the binary threshold sequence derived from Fermat quotients in terms of bounds on the well-distribution measure and the correlation measure of order 2, both introduced by Mauduit and Sarkozy. We also prove a lower bound on its linear complexity profile. The proofs are based on bounds on exponential sums and earlier relations between discrepancy and both measures above shown by Mauduit, Niederreiter and Sarkozy. Moreover, we analyze the lattice structure of Fermat quotients modulo p with arbitrary lags.

On stable quadratic polynomials

In this paper we study a class of dynamical systems generated by iterations of multivariate polynomials and estimate the degree growth of these iterations. We use these estimates to bound exponential sums along the orbits of these dynamical systems and show that they admit much stronger estimates than in the general case and thus can be of use for pseudorandom number generation.

On the Carlitz rank of permutations of Fq and pseudorandom sequences

We use the Weil bound of multiplicative character sums together with some recent results of N. Boston and R. Jones, to show that the critical orbit of quadratic polynomials over a finite field of q elements is of length O q 3/4 � , improving upon the trivial bound q.