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On p-adic Point Counting Algorithms for Elliptic Curves over Finite Fields

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Algorithmic Number Theory (ANTS 2002)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2369))

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Abstract

Let p be a prime and let q:= p N. Let E be an elliptic curve over F q. We are interested in efficient algorithms to compute the order of the group E(F q) of F q-rational points of E. An l-adic algorithm, known as the SEA algorithm, computes #E(F q) with O((logq)4+ɛ) bit operations (with fast arithmetic) and O((logq)2) memory. In this article, we survey recent advances in p-adic algorithms. For a fixed small p, the computational complexity of the known fastest p-adic point counting algorithm is O(N 3+ɛ) in time and O(N 2) in space. If we accept some precomputation depending only on p and N or a certain restriction on N, the time complexity is reduced to O(N 2.5+ɛ) still with O(N 2) space requirement.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Saitama University, Urawa, Saitama, 338-8570, Japan

    Takakazu Satoh

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  1. Takakazu Satoh
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  1. School of Mathematics and Statistics, F07, University of Sydney, Sydney, NSW, 2006, Australia

    Claus Fieker  & David R. Kohel  & 

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Satoh, T. (2002). On p-adic Point Counting Algorithms for Elliptic Curves over Finite Fields. In: Fieker, C., Kohel, D.R. (eds) Algorithmic Number Theory. ANTS 2002. Lecture Notes in Computer Science, vol 2369. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45455-1_5

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  • DOI: https://doi.org/10.1007/3-540-45455-1_5

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