Alan G. B. Lauder

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.

Counting Solutions to Equations in Many Variables over Finite Fields

Abstract 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.

Random Krylov Spaces over Finite Fields

Motivated by a connection with block iterative methods for solving linear systems over finite fields, we consider the probability that the Krylov space generated by a fixed linear mapping and a random set of elements in a vector space over a finite field equals the space itself. We obtain an exact formula for this probability and from it we derive good lower bounds that approach 1 exponentially fast as the size of the set increases.

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.

Computing Zeta Functions of Artin–schreier Curves over Finite Fields

The authors present a practical polynomial-time algorithm for computing the zeta function of certain Artin–Schreier curves over finite fields. This yields a method for computing the order of the Jacobian of an elliptic curve in characteristic 2, and more generally, any hyperelliptic curve in characteristic 2 whose affine equation is of a particular form. The algorithm is based upon an efficient reduction method for the Dwork cohomology of one-variable exponential sums.

Hensel lifting and bivariate polynomial factorisation over finite fields

This paper presents an average time analysis of a Hensel lifting based factorisation algorithm for bivariate polynomials over finite fields. It is shown that the average running time is almost linear in the input size. This explains why the Hensel lifting technique is fast in practice for most polynomials.

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.

Computations with classical and p -adic modular forms

We present p-adic algorithms for computing Hecke polynomials and Hecke eigenforms associated to spaces of classical modular forms using the theory of overconvergent modular forms. The algorithms have a running time which grows linearly with the logarithm of the weight and are well suited to investigating the dimension variation of certain p-adically defined spaces of classical modular forms.

Efficient Computation of Rankin p-Adic L-Functions

We present an efficient algorithm for computing certain special values of Rankin triple product p-adic L-functions and give an application of this to the explicit construction of rational points on elliptic curves.

Computing Zeta Functions of Kummer Curves via Multiplicative Characters

AbstractWe 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.