Finding Optimal Chudnovsky-Chudnovsky Multiplication Algorithms

Abstract

AbstractThe Chudnovsky-Chudnovsky method provides today’s best known upper bounds on the bilinear complexity of multiplication in large extension of finite fields. It is grounded on interpolation on algebraic curves: we give a theoretical lower threshold for the smallest bounds that one can expect from this method (with exceptions). This threshold appears often reachable: we moreover provide an explicit method for this purpose.We also provide new bounds for the multiplication in small- algebras over \(\mathbf {F}_2\). Building on these ingredients, we: explain how far elliptic curves can provide upper bounds for the multiplication over \(\mathbf {F}_2\); using these curves, improve the bounds for the multiplication in the NIST-size extensions of \(\mathbf {F}_2\); thus, turning to curves of higher genus, further improve these bounds with the well known family of classical modular curves. Although illustrated only over \(\mathbf {F}_2\), the techniques introduced apply to all characteristics.KeywordsElliptic modular curvesFinite field arithmeticChudnovsky-Chudnovsky interpolationTensor rankOptimal algorithms