Torsion Limits and Riemann-Roch Systems for Function Fields and Applications

The Ihara limit (or -constant) $A(q)$ has been a central problem of study in the asymptotic theory of global function fields (or equivalently, algebraic curves over finite fields). It addresses global function fields with many rational points and, so far, most applications of this theory do not require additional properties. Motivated by recent applications, we require global function fields with the additional property that their zero class divisor groups contain at most a small number of $d$-torsion points. We capture this by the torsion limit, a new asymptotic quantity for global function fields. It seems that it is even harder to determine values of this new quantity than the Ihara constant. Nevertheless, some non-trivial lower- and upper bounds are derived. Apart from this new asymptotic quantity and bounds on it, we also introduce Riemann-Roch systems of equations. It turns out that this type of equation system plays an important role in the study of several other problems in areas such as coding theory, arithmetic secret sharing and multiplication complexity of finite fields etc. Finally, we show how our new asymptotic quantity, our bounds on it and Riemann-Roch systems can be used to improve results in these areas.Comment: Accepted for publication in IEEE Transactions on Information Theory. This is an extended version of our paper in Proceedings of 31st Annual IACR CRYPTO, Santa Barbara, Ca., USA, 2011. The results in Sections 5 and 6 did not appear in that paper. A first version of this paper has been widely circulated since November 200

New uniform and asymptotic upper bounds on the tensor rank of multiplication in extensions of finite fields

International audienceWe obtain new uniform upper bounds for the tensor rank of the multiplication in the extensions of the finite fields $\mathbb{F}_q$ for any prime power $q$; moreover these uniform bounds lead to new asymptotic bounds as well. In addition, we also give purely asymptotic bounds which are substantially better by using a family of Shimura curves defined over $\mathbb{F}_q$, with an optimal ratio of $\mathbb{F}_{q^t}$-rational places to their genus, where $q^t$ is a square