Curves with many points and multiplication complexity in any extension of F-q

From the existence of algebraic function fields having some good properties , we obtain some new upper bounds on the bilinear complexity of multiplicat ion in all extensions of the finite held F-q, where q is an arbitrary prime power. So we prove that the bilinear complexity of multiplication in the f inite fields F-q(n) is linear uniformly in q with respect to the degree n. (C) 1999 Academic Press.