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Fast computation of canonical lifts of elliptic curves and its application to point counting. (English) Zbl 1106.14302
Summary: Let \(p\) be a fixed small prime. We give an algorithm with preprocessing to compute the \(j\)-invariant of the canonical lift of a given ordinary elliptic curve \(E/\mathbb F_q\) (\(q=p^N\), \(j(E)\notin F_{p^2}\)) modulo \(p^{N/2+O(1)}\) in \(O(N^{2\mu+1/\mu+1})\) bit operations (assuming the time complexity of multiplying two \(n\)-bit objects is \(O(n^\mu)\)) using \(O(N^2)\) memory, not including preprocessing. This is faster than the algorithm of F. Vercauteren et al. [Lect. Notes Comput. Sci. 2045, 1–13 (2001; Zbl 1009.11052 )] by a factor of \(N^{\mu/\mu+1}\). Let \(K\) be the unramified extension field of degree \(N\) over \(\mathbb Q_p\). We also develop an algorithm to compute \(N_{K/\mathbb Q_p}(x) \pmod p^{N/2+O(1)}\) with \(O(N^{2\mu+0.5})\) bit operations and \(O(N^2)\) memory when \(x\in K\) satisfies certain conditions, which are always satisfied when applied to our point counting algorithm. As a result, we get an \(O(N^{2\mu+0.5})\) time, \(O(N^2)\) memory algorithm for counting the \(\mathbb F_q\)-rational points on \(E/\mathbb F_q\), which turns out to be very fast in practice for cryptographic size elliptic curves.

MSC:
14G50 Applications to coding theory and cryptography of arithmetic geometry
11Y16 Number-theoretic algorithms; complexity
11G20 Curves over finite and local fields
14G05 Rational points
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