Deterministic equation solving over finite fields

It is shown how to solve diagonal forms in many variables over finite fields by means of a deterministic efficient algorithm. Applications to norm equations, quadratic forms, and elliptic curves are given.Thomas Stieltjes Institute for MathematicsUBL - phd migration 201

Improving the Berlekamp Algorithm for Binomials x n − a

In this paper, we describe an improvement of the Berlekamp algorithm, a method for factoring univariate polynomials over finite fields, for binomials xn −a over finite fields Fq. More precisely, we give a deterministic algorithm for solving the equation h(x)q≡h(x) (mod xn−a) directly without applying the sweeping-out method to the corresponding coefficient matrix. We show that the factorization of binomials using the proposed method is performed in O˜, (n log q) operations in Fq if we apply a probabilistic version of the Berlekamp algorithm after the first step in which we propose an improvement. Our method is asymptotically faster than known methods in certain areas of q, n and as fast as them in other areas

Finite element methods for deterministic simulation of polymeric fluids

In this work we consider a finite element method for solving the coupled Navier-Stokes (NS) and Fokker-Planck (FP) multiscale model that describes the dynamics of dilute polymeric fluids. Deterministic approaches such as ours have not received much attention in the literature because they present a formidable computational challenge, due to the fact that the analytical solution to the Fokker-Planck equation may be a function of a large number of independent variables. For instance, to simulate a non-homogeneous flow one must solve the coupled NS-FP system in which (for a 3-dimensional flow, using the dumbbell model for polymers) the Fokker-Planck equation is posed in a 6-dimensional domain. In this work we seek to demonstrate the feasibility of our deterministic approach. We begin by discussing the physical and mathematical foundations of the NS-FP model. We then present a literature review of relevant developments in computational rheology and develop our deterministic finite element based method in detail. Numerical results demonstrating the efficiency of our approach are then given, including some novel results for the simulation of a fully 3-dimensional flow. We utilise parallel computation to perform the large-scale numerical simulations

On Taking Square Roots without Quadratic Nonresidues over Finite Fields

We present a novel idea to compute square roots over finite fields, without being given any quadratic nonresidue, and without assuming any unproven hypothesis. The algorithm is deterministic and the proof is elementary. In some cases, the square root algorithm runs in $\tilde{O}(\log^2 q)$ bit operations over finite fields with $q$ elements. As an application, we construct a deterministic primality proving algorithm, which runs in $\tilde{O}(\log^3 N)$ for some integers $N$.Comment: 14 page

Counting Value Sets: Algorithm and Complexity

Let $p$ be a prime. Given a polynomial in \F_{p^m}[x] of degree $d$ over the finite field \F_{p^m}, one can view it as a map from \F_{p^m} to \F_{p^m}, and examine the image of this map, also known as the value set. In this paper, we present the first non-trivial algorithm and the first complexity result on computing the cardinality of this value set. We show an elementary connection between this cardinality and the number of points on a family of varieties in affine space. We then apply Lauder and Wan's $p$-adic point-counting algorithm to count these points, resulting in a non-trivial algorithm for calculating the cardinality of the value set. The running time of our algorithm is $(pmd)^{O(d)}$. In particular, this is a polynomial time algorithm for fixed $d$ if $p$ is reasonably small. We also show that the problem is #P-hard when the polynomial is given in a sparse representation, $p=2$, and $m$ is allowed to vary, or when the polynomial is given as a straight-line program, $m=1$ and $p$ is allowed to vary. Additionally, we prove that it is NP-hard to decide whether a polynomial represented by a straight-line program has a root in a prime-order finite field, thus resolving an open problem proposed by Kaltofen and Koiran in \cite{Kaltofen03,KaltofenKo05}