Joachim von zur Gathen

A bound on solutions of linear integer equalities and inequalities

Consider a system of linear equalities and inequalities with integer coefficients. We describe the set of rational solutions by a finite generating set of solution vectors. The entries of these vectors can be bounded by the absolute value of a certain subdeterminant. The smallest integer solution of the system has coefficients not larger than this subde- terminant times the number of indeterminates. Up to the latter factor, the bound is sharp. Let A, B, C, D be m x zz-, m x \-,p x n-,p x 1-matrices respectively with integer entries. The rank of A is r, and s is the rank of the (m + p) X n- matrix (c). Let M be an upper bound on the absolute values of those (s — 1) X (s — 1)- or s X i-subdeterminants of the (m + p) X (n + l)-matrix (c d)> which are formed with at least r rows from (A, B). Theorem. If Ax = B and Cx > D have a common integer solution, then they have one with coefficients bounded by (n + \)M. Let Mx, M2, and M3 be upper bounds on the absolute values of the r X r-subdeterminants, the subdeterminants, and the entries of (A, B) respectively. Taking the zz X zz-identity matrix for C and D = 0, we have the following

Algorithms for Exponentiation in Finite Fields

Gauss periods yield (self-dual) normal bases in finite fields, and these normal bases can be used to implement arithmetic efficiently. It is shown that for a small prime power q and infinitely many integersn , multiplication in a normal basis of Fqn over Fq can be computed with O(n logn loglog n), division with O(n log2n loglog n) operations in Fq, and exponentiation of an arbitrary element in Fqn withO (n2loglog n) operations in Fq. We also prove that using a polynomial basis exponentiation in F 2 n can be done with the same number of operations in F 2 for all n. The previous best estimates were O(n2) for multiplication in a normal basis, andO (n2log n loglog n) for exponentiation in a polynomial basis.

Irreducibility of multivariate polynomials

Abstract This paper deals with the problem of computing the degrees and multiplicities of the irreducible factors of a given multivariate polynomial. This includes the important question of testing for irreducibility. A probabilistic reduction from multivariate to bivariate polynomials is given, over an arbitrary (effectively computable) field. It uses an expected number of field operations (and certain random choices) that is polynomial in the length of a computation by which the input polynomial is presented, and the degree of the polynomial. Over algebraic number fields and over finite fields, we obtain polynomial-time probabilistic algorithms. They are based on an effective version of Hilberts irreducibility theorem.

Factoring sparse multivariate polynomials

Abstract This paper presents a probabilistic reduction for factoring polynomials from multivariate to the bivariate case, over an arbitrary (effectively computable) field. It uses an expected number of field operations (and certain random choices) that is polynomial in the size of sparse representations of input plus output, provided the number of irreducible factors is bounded. We thus obtain probabilistic polynomial-time factoring procedures over algebraic number fields and over finite fields. The reduction is based on an effective version of Hilberts irreducibility theorem.

Fast algorithms for Taylor shifts and certain difference equations

Joachim von zur Gathen Jurgen Gerhard Fachbereich 17 Mathematik-Informatik Universitat-GH Paderborn D-33095 Paderborn, Germany {gathen, jngerhar}@uni-paderborn. de http://www.uni-paderborn .de/cs/gathen .html

Feasible arithmetic computations: Valiant's hypothesis

An account of Valiants theory of p-computable versus p-definable polynomials, an arithmetic analogue of the Boolean theory of P versus NP, is presented, with detailed proofs of Valiants central results.

Factoring polynomials over finite fields: a survey

This survey reviews several algorithms for the factorization of univariate polynomials over finite fields. We emphasize the main ideas of the methods and provide an up-to-date bibliography of the problem.

Polynomials with two values

This paper investigates the minimal degree of polynomialsf∈R[x] that take exactly two values on a given range of integers {0,...n}. We show that thegap, defined asn-deg(f), isO(n548). The maximal gap forn≤128 is 3. As an application, we obtain a bound on the Fourier degree of symmetric Boolean functions.

Constructing normal bases in finite fields

An efficient probabilistic algorithm to find a normal basis in a finite field is presented. It can, in fact, find an element of arbitrary prescribed additive order. It is based on a density estimate for normal elements. A similar estimate yields a probabilistic polynomial-time reduction from finding primitive normal elements to finding primitive elements.

Efficient FPGA-based karatsuba multipliers for polynomials over F 2

We study different possibilities of implementing the Karatsuba multiplier for polynomials over F 2 on FPGAs. This is a core task for implementing finite fields of characteristic 2. Algorithmic and platform dependent optimizations yield efficient hardware designs. The resulting structure is hybrid in two different aspects. On the one hand, a combination of the classical and the Karatsuba methods decreases the number of bit operations. On the other hand, a mixture of sequential and combinational circuit design techniques includes pipelining and can be adapted flexibly to time-area constraints. The approach-both theory and implementation _ can be viewed as a further step towards taming the machinery of fast algorithmics for hardware applications.