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Counting points in medium characteristic using Kedlaya's algorithm

Pierrick Gaudry 1, 2 Nicolas Gürel 1, 2
1 TANC - Algorithmic number theory for cryptology
Inria Saclay - Ile de France, LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]
Abstract : Recently, many new results have been found concerning algorithms for counting points on curves over finite fields of characteristic p, mostly due to the use of p-adic liftings. The complexity of these new methods is exponential in p therefore they behave well when p is small, the ideal case being p=2. When applicable, these new methods are usually faster than those based on SEA algorithms, and are more easily extended to non-elliptic curves. We investigate more precisely this dependance on the characteristic, and in particular, we show that after a few modifications using fast algorithms for radix-conversion, Kedlaya's algorithm works in time almost linear in p. As a consequence, this algorithm can also be applied to medium values of p. We give an example of a cryptographic size genus 3 hyperelliptic curve over a finite field of characteristic 251.
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https://hal.inria.fr/inria-00514830
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Submitted on : Friday, September 3, 2010 - 1:09:20 PM
Last modification on : Friday, April 17, 2020 - 3:42:06 PM
Long-term archiving on: : Saturday, December 4, 2010 - 2:46:53 AM

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Pierrick Gaudry, Nicolas Gürel. Counting points in medium characteristic using Kedlaya's algorithm. Experimental Mathematics, Taylor & Francis, 2003, 12, pp.395-402. ⟨inria-00514830⟩

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