Pi and the AGM: a study in the analytic number theory and computational complexity
Pi and the AGM: a study in the analytic number theory and computational complexity
Satoh's algorithm in characteristic 2
Mathematics of Computation
A Memory Efficient Version of Satoh's Algorithm
EUROCRYPT '01 Proceedings of the International Conference on the Theory and Application of Cryptographic Techniques: Advances in Cryptology
ASIACRYPT '02 Proceedings of the 8th International Conference on the Theory and Application of Cryptology and Information Security: Advances in Cryptology
On p-adic Point Counting Algorithms for Elliptic Curves over Finite Fields
ANTS-V Proceedings of the 5th International Symposium on Algorithmic Number Theory
Fast computation of canonical lifts of elliptic curves and its application to point counting
Finite Fields and Their Applications
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In 2000 T. Satoh gave the first p-adic point counting algorithm for elliptic curves over finite fields. Satoh's algorithm was followed by the Satoh-Skjernaa-Taguchi algorithm and furthermore by the arithmetic-geometric mean and modified SST algorithms for characteristic two only. All four algorithms are important to Elliptic Curve Cryptography. In this paper, we present the general framework for p-adic point counting and we apply it to elliptic curves on Legendre form. We show how the @l-modular polynomial can be used for lifting the curve and Frobenius isogeny to characteristic zero and we show how the associated multiplier gives the action of the lifted Frobenius isogeny on the invariant differential. The result is a point counting algorithm for elliptic curves on Legendre form. The algorithm runs in a time complexity of O(n^2^@m^+^1^/^(^@m^+^1^)) for fixed p and a space complexity of O(n^2) where p^n is the field size. We include results from experimeriments in characteristic p=3,5,...,19.