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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
##### Keywords:
elliptic curves; canonical lifts; Frobenius substitutions
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##### References:
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