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by Hae Young Kim, Jung Youl Park, Jung Hee Cheon, Je Hong Park, Jae Heon Kim, Sang Geun Hahn
Proc. of ANTS V, Lecture Notes in Comput. Sci. 2369
http://crypt.kaist.ac.kr/pre_papers/ANTS_final.pdf
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Abstract:
Abstract. In this paper we present an improved algorithm for counting points on elliptic curves over finite fields. It is mainly based on Satoh-Skjernaa-Taguchi algorithm [SST01], and uses a Gaussian Normal Basis (GNB) of small type t ≤ 4. In practice, about 42 % (36 % for prime N) of fields in cryptographic context (i.e., for p = 2 and 160 < N < 600) have such bases. They can be lifted from F p N to Z p N in a natural way. From the specific properties of GNBs, efficient multiplication and the Frobenius substitution are available. Thus a fast norm computation algorithm is derived, which runs in O(N 2µ log N) with O(N 2) space, where the time complexity of multiplying two n-bit objects is O(n µ). As a result, for all small characteristic p, we reduced the time complexity of the SSTalgorithm from O(N 2µ+0.5) to O(N still fits in O(N 2). Our approach is expected to be applicable to the AGM since the exhibited improvement is not restricted to only [SST01].
Citations
|
301
|
Use of elliptic curves in cryptography
– Miller
- 1985
|
|
242
|
Elliptic Curve Public Key Cryptosystems
– Menezes
- 1993
|
|
112
|
Elliptic curves over finite fields and the computation of square roots mod p
– Schoof
- 1985
|
|
90
|
Algebraic Number Theory
– Lang
- 1994
|
|
46
|
Applications of Finite Fields
– Menezes
- 1993
|
|
45
|
The number of points on an elliptic curve modulo a prime. Email on the Number Theory mailing list
– Atkin
- 1991
|
|
35
|
Counting Points on Hyperelliptic Curves using Monsky-Washnitzer Cohomology
– Kedlaya
- 2001
|
|
32
|
On Artin’s conjecture
– Hooley
- 1967
|
|
31
|
The canonical lift of an ordinary elliptic curve over a finite field and its point counting
– Satoh
- 2000
|
|
28
|
Die Typen der Multiplikatorenringe elliptischer Funktionenkörper
– Deuring
- 1941
|
|
25
|
Elliptic and modular curves over finite fields and related computational issues
– Elkies
- 1998
|
|
12
|
On Satoh's algorithm and its implementation
– Fouquet, Gaudry, et al.
- 2000
|
|
11
|
Artin’s conjecture for primitive roots
– Murty
- 1988
|
|
8
|
Elliptic curves and formal groups. In Lecture notes prepared in connection with the seminars held at the Summer Institute on Algebraic Geometry
– Lubin, Serre, et al.
- 1964
|
|
7
|
Fast Multiplication in Finite Fields GF(2 N
– Silverman
- 1999
|
|
7
|
Fast Computation of Canonical Lifts of Elliptic curves and its Application to Point Counting
– Satoh, Skjernaa, et al.
- 2001
|
|
5
|
A Memory Efficient Version of Satoh’s Algorithm
– Vercauteren, Preneel, et al.
- 2001
|
|
4
|
Counting points with the arithmetic-geometric mean(joint work with
– Harley
- 2001
|
|
3
|
Satoh Point Counting in characteristic 2
– Skjernaa
- 2000
|
|
1
|
Couveignes, Computing l-isogenies using the p-Torsion, Algorithmic number theory
– M
- 1996
|
|
1
|
Elliptic curve cyptosystem
– Koblitz
- 1998
|
|
1
|
On the number of nonscalar multiplications necessary to evaluate polynomials
– Parterson, Stockmeyer
- 1973
|
