Fast elliptic curve point counting using Gaussian normal basis

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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 less than or equal to 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-pN to Z(pN) in a natural way. From the specific properties of GNBs, efficient multiplication and the Frobenius substitution axe available. Thus a fast norm computation algorithm is derived, which runs in O(N-2mu log N) with O(N-2) space, where the time complexity of multiplying two n-bit objects is O(n(mu)). As a result, for all small characteristic p, we reduced the time complexity of the SST-algorithm from O(N2mu+0.5) to O(N2mu+ 1/mu+1) and the space complexity 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].
Publisher
SPRINGER-VERLAG BERLIN
Issue Date
2002
Language
English
Article Type
Article; Proceedings Paper
Keywords

FINITE-FIELDS

Citation

ALGORITHMIC NUMBER THEORY BOOK SERIES: LECTURE NOTES IN COMPUTER SCIENCE, v.2369, pp.292 - 307

ISSN
0302-9743
URI
http://hdl.handle.net/10203/83656
Appears in Collection
MA-Journal Papers(저널논문)
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