On the bounds of the bilinear complexity of multiplication in some finite fields (Q1762551)

Let \(\mu_q(n)\) denotes the bilinear complexity of multiplication in \(\mathbb F_{q^n}\). Let \(p\geq 5\) be a prime number. It is known that \(\mu_p(n)\leq 6n(1+p/(p-3))\) and \(\mu_p^2(n)\leq 2n(1+p/(p-3))\); cf. [Finite Fields Appl. 5, No. 4, 364--377 (1999; Zbl 0953.11022)]. In this paper the authors improve on both upper bounds above (and in particular on an asymptotic upper bound regarding this complexity for the prime finite fields of characteristic \(p>5\)). The new results are: \(\mu_p(n)\leq 3n(1+4/(p-3))\) and \(\mu_{p^2}(n)\leq 2n(1+2/(p-3))\).