Counting points on genus-3 hyperelliptic curves with explicit real multiplication
| Autor: | Abelard, Simon, Gaudry, Pierrick, Spaenlehauer, Pierre-Jean |
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| Rok vydání: | 2018 |
| Předmět: | |
| Zdroj: | Open Book Series 2 (2019) 1-19 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.2140/obs.2019.2.1 |
| Popis: | We propose a Las Vegas probabilistic algorithm to compute the zeta function of a genus-3 hyperelliptic curve defined over a finite field $\mathbb F_q$, with explicit real multiplication by an order $\mathbb Z[\eta]$ in a totally real cubic field. Our main result states that this algorithm requires an expected number of $\widetilde O((\log q)^6)$ bit-operations, where the constant in the $\widetilde O()$ depends on the ring $\mathbb Z[\eta]$ and on the degrees of polynomials representing the endomorphism $\eta$. As a proof-of-concept, we compute the zeta function of a curve defined over a 64-bit prime field, with explicit real multiplication by $\mathbb Z[2\cos(2\pi/7)]$. Comment: Proceedings of the ANTS-XIII conference (Thirteenth Algorithmic Number Theory Symposium) |
| Databáze: | arXiv |
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