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Counting isomorphism classes of Pointed Hyperelliptic Curves of Genus 4 over Finite Fields with Even Characteristic

Acta Mathematica Sinica,English Series 2010年第6期26卷 1019-1054页

作者： Huah CHU Ying Pu DENG Tse-Chung YANG Department of Mathematics National Taiwan University Taipei Taiwan Key Laboratory of Mathematics Mechanization Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100190 P. R. China and State Key Laboratory of Information Security Graduate University of Chinese Academy of Sciences Beijing 100049 P. R. China

This paper is devoted to counting the number of isomorphism classes of pointed hyperelliptic curves over finite fields. We deal with the genus 4 case and the finite fields are of even characteristics. The number of isomorphism classes is computed and the explicit formulae are given. This number can be represented as a polynomial in q of degree 7, where q is the order of the finite field. The result can be used in the classification problems and it is useful for further studies of hyperelliptic curve cryptosystems, e.g. it is of interest for research on implementing the arithmetics of curves of low genus for cryptographic purposes. It could also be of interest for point counting problems; both on moduli spaces of curves, and on finding the maximal number of points that a pointed hyperelliptic curve over a given finite field may have.

关键词： hypereUiptic curves hyperelliptic curve cryptosystems Jacobians isomorphism classes stabilizer

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On the isomorphism classes of Legendre elliptic curves over finite fields

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Science China Mathematics 2011年第9期54卷 1885-1890页

作者： WU HongFeng FENG RongQuan College of Science North China University of technology Beijing 100144 China LMAM School of Mathematical Sciences Peking University Beijing 100871 China

In this paper,the number of isomorphism classes of Legendre elliptic curves over finite field is enumerated.

关键词： elliptic curve Legendre curve isomorphism classes cryptography

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Elliptic Curves in Huff's Model

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Wuhan University Journal of Natural Sciences 2012年第6期17卷 473-480页

作者： WU Hongfeng FENG Rongquan College of Sciences North China University of TechnologyBeijing 00China School of Mathematical Sciences/Laboratory of Mathematics and Applied Mathematics(LMAM) Peking UniversityBeijing 100871China

In this paper,we present the generalized Huff curves that contain Huff＇s model as a special ***,it is proved that every elliptic curve with three points of order 2 is isomorphic to a generalized Huff ***,the fast and explicit formulae are derived for generalized Huff curves in projective *** paper also enumerates the number of isomorphism classes of generalized Huff curves over finite ***,the explicit formulae are presented for the doubling step and addition step in Miller＇s algorithm to compute the Tate pairing on generalized Huff elliptic curves.

关键词： elliptic curve Huff curve cryptography scalar multiplication isomorphism classes

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