Remarks on the k-error linear complexity of p(n)-periodic sequences

Recently the first author presented exact formulas for the number of 2ⁿn-periodic binary sequences with given 1-error linear complexity, and an exact formula for the expected 1-error linear complexity and upper and lower bounds for the expected k-error linear complexity, k >2, of a random 2ⁿn-periodic binary sequence. A crucial role for the analysis played the Chan-Games algorithm. We use a more sophisticated generalization of the Chan-Games algorithm by Ding et al. to obtain exact formulas for the counting function and the expected value for the 1-error linear complexity for pⁿn-periodic sequences over Fp, p prime. Additionally we discuss the calculation of lower and upper bounds on the k-error linear complexity of pⁿn-periodic sequences over Fp

Rationality and meromorphy of zeta functions

This article is all about two theorems on equations over finite fields which have been proved in the past decade. First, the finiteness of the rigid cohomology of a variety over a finite field. Second, the p-adic meromorphy of the unit root zeta function of a family of varieties over a finite field of characteristic p. The purpose of the article is to explain what these theorems mean, and also to give an outline of the proof of the first one. The intended audience is mathematicians with an interest in finite field, but no especial expertise on the vast literature which surrounds the topic of equations over finite filelds. © 2005 Elsevier Inc. All rights reserved

Computing zeta functions of Kummer curves via multiplicative characters

We present a practical polynomial-time algorithm for computing the zeta function of a Kummer curve over a finite field of small characteristic. Such algorithms have recently been obtained using a method of Kedlaya based upon Monsky-Washnitzer cohomology, and are of interest in cryptography. We take a different approach. The problem is reduced to that of computing the L-function of a multiplicative character sum. This latter task is achieved via a cohomological formula based upon the work of Dwork and Reich. We show, however, that our method and that of Kedlaya are very closely related. © 2002 SFoCM

Counting solutions to equations in many variables over finite fields

We present a polynomial-time algorithm for computing the zeta function of a smooth projective hypersurface of degree d over a finite field of characteristic p, under the assumption that p is a suitably small odd prime and does not divide d. This improves significantly upon an earlier algorithm of the author and Wan which is only polynomial-time when the dimension is fixed

Homotopy methods for equations over finite fields

This paper describes an application of some ideas from homotopy theory to the problem of computing the number of solutions to a multivariate polynomial equation over a finite field. The benefit of the homotopy approach over more direct methods is that the runningtime is far less dependent on the number of variables. The method was introduced by the author in another paper, where specific complexity estimates were obtained for certain special cases. Some consequences of these estimates are stated in the present paper. © Springer-Verlag Berlin Heidelberg 2003

Decomposition of polytopes and polynomials

Motivated by a connection with the factorization of multivariate polynomials, we study integral convex polytopes and their integral decompositions in the sense of the Minkowski sum. We first show that deciding decomposability of integral polygons is NP-complete then present a pseudo-polynomial-time algorithm for decomposing polygons. For higher-dimensional polytopes, we give a heuristic algorithm which is based upon projections and uses randomization. Applications of our algorithms include absolute irreducibility testing and factorization of polynomials via their Newton polytopes

Computing the error linear complexity spectrum of a binary sequence of period 2n

Binary sequences with high linear complexity are of interest in cryptography. The linear complexity should remain high even when a small number of changes are made to the sequence. The error linear complexity spectrum of a sequence reveals how the linear complexity of the sequence varies as an increasing number of the bits of the sequence are changed. We present an algorithm which computes the error linear complexity for binary sequences of period ℓ = 2n using O(l(logl)2) bit operations. The algorithm generalizes both the Games-Chan and Stamp-Martin algorithms, which compute the linear complexity and the k-error linear complexity of a binary sequence of period ℓ = 2n, respectively. We also discuss an application of an extension of our algorithm to decoding a class of linear subcodes of Reed-Muller codes

Factoring polynomials via polytopes

We introduce a new approach to multivariate polynomial factorisation which incorporates ideas from polyhedral geometry, and generalises Hensel lifting. Our main contribution is to present an algorithm for factoring bivariate polynomials which is able to exploit to some extent the sparsity of polynomials. We give details of an implementation which we used to factor randomly chosen sparse and composite polynomials of high degree over the binary field. Copyright 2004 ACM

Computing zeta functions of Artin-Schreier curves over finite fields II

We describe a method which may be used to compute the zeta function of an arbitrary Artin-Schreier cover of the projective line over a finite field. Specifically, for covers defined by equations of the form Zp - Z = f (X) we present, and give the complexity analysis of, an algorithm for the case in which f (X) is a rational function whose poles all have order 1. However, we only prove the correctness of this algorithm when the field characteristic is at least 5. The algorithm is based upon a cohomological formula for the L-function of an additive character sum. One consequence is a practical method of finding the order of the group of rational points on the Jacobian of a hyperelliptic curve in characteristic 2. © 2003 Elsevier Inc. All rights reserved