Finding Optimal Formulae for Bilinear Maps

Abstract

Abstract. We describe a unified framework to search for optimal formulae evaluating bilinear -or quadratic -maps. This framework applies to polynomial multiplication and squaring, finite field arithmetic, matrix multiplication, etc. We then propose a new algorithm to solve problems in this unified framework. With an implementation of this algorithm, we prove the optimality of various published upper bounds, and find improved upper bounds.

Some new results on binary polynomial multiplication

“…Because of applications in a variety of areas, such as cryptography and coding theory, new techniques for improving polynomial multiplication have been presented in numerous papers, e.g., [1,4,5,7,8,[13][14][15][16][17][18]20,[23][24][25]27,28]. For cryptographic applications, arithmetic in the binary extension field F 2 n is often used and, of the basic operations in F 2 n , multiplication contributes most to the total number of bit operations.…”

“…6] + a[0] t143 = a[10] + t86 t201 = t182 + a[8] t259 = t250 + t256 t317 = t217 + t315 t28 = a[6] * b[7] t86 = a[7] + a[1] t144 = a[11] + t87 t202 = t183 + b[6] t260 = t251 + t257 c0 = t1 t29 = a[7] * b[6] …”

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

“…Specifically, we compute all the decompositions for the short product of polynomials P and Q modulo X 5 and the product of 3 × 2 by 2 × 3 matrices. The latter problem was out of reach with the method used in [1]. We prove, in particular, that the set of possible decompositions for this matrix product is essentially unique, up to the action of the automorphism group.…”

“…In [20], Oseledets proposes a heuristic approach to solve the bilinear rank problem for the polynomial product over F 2 . Later, Barbulescu et al proposed in [1] a unified framework, extending the idea proposed by Oseledets. This allows the authors to compute the bilinear rank of different applications, such as the short product or the middle product over a finite field.…”

“…Contributions. The work presented is an improvement to the algorithm introduced in [1], allowing one to increase the family of bilinear maps over a finite field for which we are able to compute all the optimal formulae. Our algorithm relies on the automorphism group stabilizing a bilinear map, and on the notion of "stem" of a vector space associated to such a bilinear map.…”