Welcome to the resource topic for 2021/332
Title:
An O(\log^2 p) Approach to Point-Counting on Elliptic Curves From a Prominent Family Over the Prime Field \mathbb{F}_p
Authors: Yuri Borissov, Miroslav Markov
Abstract:We elaborate an approach for determining the order of an elliptic curve from the family \mathcal{E}_p = \{E_a: y^2 = x^3 + a \pmod p, a \not = 0\} where p is a prime number > 3. The essence of this approach consists in combining the well-known Hasse bound with an explicit formula for that order reduced to modulo p. It allows to advance an efficient technique of complexity O(\log^2 p) for computing simultaneously the six orders associated with the family \mathcal{E}_p when p \equiv 1 \pmod 3, thus improving the best known algorithmic solution for that problem with an order of magnitude.
ePrint: https://eprint.iacr.org/2021/332
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