Use of elliptic curves in cryptography
Lecture notes in computer sciences; 218 on Advances in cryptology---CRYPTO 85
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
On p-adic Point Counting Algorithms for Elliptic Curves over Finite Fields
ANTS-V Proceedings of the 5th International Symposium on Algorithmic Number Theory
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 Elliptic Curve Point Counting Using Gaussian Normal Basis
ANTS-V Proceedings of the 5th International Symposium on Algorithmic Number Theory
A Timing-Resistant Elliptic Curve Backdoor in RSA
Information Security and Cryptology
Quasi-quadratic elliptic curve point counting using rigid cohomology
Journal of Symbolic Computation
Fast arithmetic in unramified p-adic fields
Finite Fields and Their Applications
A p-adic point counting algorithm for elliptic curves on legendre form
Finite Fields and Their Applications
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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/F"q (q=p^N, j(E)@?F"p"^"2) modulo p^N^/^2^+^O^(^1^) in O(N^2^@m^+^1^/^@m^+^1) bit operations (assuming the time complexity of multiplying two n-bit objects is O(n^@m)) using O(N^2) memory, not including preprocessing. This is faster than the algorithm of Vercauteren et al. [14] by a factor of N^@m^/^@m^+^1. Let K be the unramified extension field of degree N over Q"p. We also develop an algorithm to compute N"K"/"Q"""p(x)modp^N^/^2^+^O^(^1^) with O(N^2^@m^+^0^.^5) bit operations and O(N^2) memory when x@?K satisfies certain conditions, which are always satisfied when applied to our point counting algorithm. As a result, we get an O(N^2^@m^+^0^.^5) time, O(N^2) memory algorithm for counting the F"q-rational points on E/F"q, which turns out to be very fast in practice for cryptographic size elliptic curves.