Cryptology ePrint Archive: Report 2014/298
Torsion Limits and Riemann-Roch Systems for Function Fields and Applications
Ignacio Cascudo and Ronald Cramer and Chaoping Xing
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 with the notion of 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 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 each of these areas: arithmetic secret sharing, symmetric bilinear complexity of multiplication in finite fields, frameproof codes and the theory of error correcting codes.
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.
Category / Keywords: Algebraic curves, Jacobian, torsion limit, Ihara limit, secret sharing, complexity of multiplication, frameproof codes
Original Publication (in the same form): IEEE Transactions on Information Theory (to appear)
Date: received 28 Apr 2014
Contact author: ignacio at cs au dk
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Version: 20140430:120958 (All versions of this report)
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