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

TL;DR

This paper introduces the torsion limit for global function fields, explores bounds on this new quantity, and demonstrates its applications in coding theory, secret sharing, and finite field multiplication complexity.

Contribution

It defines the torsion limit, derives bounds for it, and applies Riemann-Roch systems to improve results in multiple areas such as coding and cryptography.

Findings

Derived non-trivial bounds on the torsion limit

Introduced Riemann-Roch systems of equations

Applied these concepts to enhance results in coding theory and cryptography

Abstract

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.