Let p be a small prime and q = pn. Let E be an elliptic curve over double-struck F signq. We propose an algorithm which computes without any preprocessing the j-invariant of the canonical lift of E with the cost of O(log n) times the cost needed to compute a power of the lift of the Frobenius. Let μ be a constant so that the product of two n-bit length integers can be carried out in O(nμ) bit operations, this yields an algorithm to compute the number of points on elliptic curves which reaches, at the expense of a O(n5/2) space complexity, a theoretical time complexity bound equal to O(nmax(1.19,μ)+μ+1/2 log n). When the field has got a Gaussian Normal Basis of small type, we obtain furthermore an algorithm with O(log(n)n2μ) time and O(n2) space complexities. From a practical viewpoint, the corresponding algorithm is particularly well suited for implementations. We outline this by a 100002-bit computation. © International Association for Cryptologic Research 2003.
CITATION STYLE
Lercier, R., & Lubicz, D. (2003). Counting points on elliptic curves over finite fields of small characteristic in quasi quadratic time. Lecture Notes in Computer Science (Including Subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 2656, 360–373. https://doi.org/10.1007/3-540-39200-9_22
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