Algebraic complexities and algebraic curves over finite fields

Abstract

Algebraic schemes of computation of bilinear forms over various rings of scalars are examined. The problem of minimal complexity of these schemes is considered for computation of polynomial multiplication and multiplication in commutative algebras, and finite extensions of fields. While for infinite fields minimal complexities are known (Winograd, Fiduccia, Strassen), for finite fields precise minimal complexities are not yet determined. We prove lower and upper bounds on minimal complexities. Both are linear in the number of inputs. These results are obtained using the relationship with linear coding theory and the theory of algebraic curves over finite fields.