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Counting points on hyperelliptic curves in large characteristic : algorithms and complexity

Simon Abelard 1
1 CARAMBA - Cryptology, arithmetic : algebraic methods for better algorithms
LORIA - ALGO - Department of Algorithms, Computation, Image and Geometry, Inria Nancy - Grand Est
Abstract : Counting points on algebraic curves has drawn a lot of attention due to its many applications from number theory and arithmetic geometry to cryptography and coding theory. In this thesis, we focus on counting points on hyperelliptic curves over finite fields of large characteristic $p$. In this setting, the most suitable algorithms are currently those of Schoof and Pila, because their complexities are polynomial in $\log q$. However, their dependency in the genus $g$ of the curve is exponential, and this is already painful even in genus 3. Our contributions mainly consist of establishing new complexity bounds with a smaller dependency in $g$ of the exponent of $\log p$. For hyperelliptic curves, previous work showed that it was quasi-quadratic, and we reduced it to a linear dependency. Restricting to more special families of hyperelliptic curves with explicit real multiplication (RM), we obtained a constant bound for this exponent.In genus 3, we proposed an algorithm based on those of Schoof and Gaudry-Harley-Schost whose complexity is prohibitive in general, but turns out to be reasonable when the input curves have explicit RM. In this more favorable case, we were able to count points on a hyperelliptic curve defined over a 64-bit prime field
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Submitted on : Tuesday, September 18, 2018 - 12:49:05 PM
Last modification on : Tuesday, December 18, 2018 - 4:18:26 PM
Long-term archiving on: : Wednesday, December 19, 2018 - 2:41:35 PM


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  • HAL Id : tel-01876314, version 1


Simon Abelard. Counting points on hyperelliptic curves in large characteristic : algorithms and complexity. Number Theory [math.NT]. Université de Lorraine, 2018. English. ⟨NNT : 2018LORR0104⟩. ⟨tel-01876314⟩



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