Classification of all the minimal bilinear algorithms for computing the coefficients of the product of two polynomials modulo a polynomial, part I: The algebra G[u]<Q(u)l>, l>1

AbstractIn this paper we will classify all the minimal bilinear algorithms for computing the coefficients of(∑i=0n-1xiui)(∑i=0n-1yiui) mod Q(u)l where deg Q(u)=j,jl=n and Q(u) is irreducible.The case where l=1 was studied in [1]. For l>1 the main results are that we have to distinguish between two cases: j>1 and j=1. The first case is discussed here while the second is classified in [4]. For j>1 it is shown that up to equivalence every minimal (2n-1 multiplications) bilinear algorithm for computing the coefficients of (∑i=0n-1xiui)(∑i=0n-1yiui) mod Q(u)l is done by first computing the coefficients of (∑i=0n-1xiui)(∑i=0n-1yiui) and then reducing it modulo Q(u)l (similar to the case l = 1, [1])

Solving the "Isomorphism of Polynomials with Two Secrets" Problem for all Pairs of Quadratic Forms

We study the Isomorphism of Polynomial (IP2S) problem with m=2 homogeneous quadratic polynomials of n variables over a finite field of odd characteristic: given two quadratic polynomials (a, b) on n variables, we find two bijective linear maps (s,t) such that b=t . a . s. We give an algorithm computing s and t in time complexity O~(n^4) for all instances, and O~(n^3) in a dominant set of instances. The IP2S problem was introduced in cryptography by Patarin back in 1996. The special case of this problem when t is the identity is called the isomorphism with one secret (IP1S) problem. Generic algebraic equation solvers (for example using Gr\"obner bases) solve quite well random instances of the IP1S problem. For the particular cyclic instances of IP1S, a cubic-time algorithm was later given and explained in terms of pencils of quadratic forms over all finite fields; in particular, the cyclic IP1S problem in odd characteristic reduces to the computation of the square root of a matrix. We give here an algorithm solving all cases of the IP1S problem in odd characteristic using two new tools, the Kronecker form for a singular quadratic pencil, and the reduction of bilinear forms over a non-commutative algebra. Finally, we show that the second secret in the IP2S problem may be recovered in cubic time

Improved method for finding optimal formulae for bilinear maps in a finite field

In 2012, Barbulescu, Detrey, Estibals and Zimmermann proposed a new framework to exhaustively search for optimal formulae for evaluating bilinear maps, such as Strassen or Karatsuba formulae. The main contribution of this work is a new criterion to aggressively prune useless branches in the exhaustive search, thus leading to the computation of new optimal formulae, in particular for the short product modulo X 5 and the circulant product modulo (X 5 -- 1). Moreover , we are able to prove that there is essentially only one optimal decomposition of the product of 3 x 2 by 2 x 3 matrices up to the action of some group of automorphisms

Symmetric tensor decomposition

We present an algorithm for decomposing a symmetric tensor, of dimension n and order d as a sum of rank-1 symmetric tensors, extending the algorithm of Sylvester devised in 1886 for binary forms. We recall the correspondence between the decomposition of a homogeneous polynomial in n variables of total degree d as a sum of powers of linear forms (Waring's problem), incidence properties on secant varieties of the Veronese Variety and the representation of linear forms as a linear combination of evaluations at distinct points. Then we reformulate Sylvester's approach from the dual point of view. Exploiting this duality, we propose necessary and sufficient conditions for the existence of such a decomposition of a given rank, using the properties of Hankel (and quasi-Hankel) matrices, derived from multivariate polynomials and normal form computations. This leads to the resolution of polynomial equations of small degree in non-generic cases. We propose a new algorithm for symmetric tensor decomposition, based on this characterization and on linear algebra computations with these Hankel matrices. The impact of this contribution is two-fold. First it permits an efficient computation of the decomposition of any tensor of sub-generic rank, as opposed to widely used iterative algorithms with unproved global convergence (e.g. Alternate Least Squares or gradient descents). Second, it gives tools for understanding uniqueness conditions, and for detecting the rank

Algorithms in algebraic number theory

In this paper we discuss the basic problems of algorithmic algebraic number theory. The emphasis is on aspects that are of interest from a purely mathematical point of view, and practical issues are largely disregarded. We describe what has been done and, more importantly, what remains to be done in the area. We hope to show that the study of algorithms not only increases our understanding of algebraic number fields but also stimulates our curiosity about them. The discussion is concentrated of three topics: the determination of Galois groups, the determination of the ring of integers of an algebraic number field, and the computation of the group of units and the class group of that ring of integers.Comment: 34 page