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 is...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.展开更多
In this paper,we present the generalized Huff curves that contain Huff＇s model as a special case.First,it is proved that every elliptic curve with three points of order 2 is isomorphic to a generalized Huff curve.The...In this paper,we present the generalized Huff curves that contain Huff＇s model as a special case.First,it is proved that every elliptic curve with three points of order 2 is isomorphic to a generalized Huff curve.Then,the fast and explicit formulae are derived for generalized Huff curves in projective coordinates.This paper also enumerates the number of isomorphism classes of generalized Huff curves over finite fields.Finally,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.展开更多
Let F be an algebracially closed field of characteristic p】2, and L be the p<sup>n</sup>-dimensional Zassenhaus algebra with the maximal invariant subalgebra L<sub>0</sub> and the standard fil...Let F be an algebracially closed field of characteristic p】2, and L be the p<sup>n</sup>-dimensional Zassenhaus algebra with the maximal invariant subalgebra L<sub>0</sub> and the standard filtration {L<sub>i</sub>}|<sub>i=-1</sub><sup>p<sup>n</sup>-2</sup>. Then the number of isomorphism classes of simple L-modules is equal to that of simple L<sub>0</sub>-modules, corresponding to an arbitrary character of L except when its height is biggest. As to the number corresponding to the exception there was an earlier result saying that it is not bigger than p<sup>n</sup>.展开更多
Let p be an odd prime, and D2p = （a,b I aP = b2 = l,bab= a 1） the dihedral group of order 2p. In this paper, we completely classify the cubic Cayley graphs on D2p up to isomorphism by means of spectral method. By th...Let p be an odd prime, and D2p = （a,b I aP = b2 = l,bab= a 1） the dihedral group of order 2p. In this paper, we completely classify the cubic Cayley graphs on D2p up to isomorphism by means of spectral method. By the way, we show that two cubic Cayley graphs on D2p are isomorphic if and only if they are cospectral. Moreover, we obtain the number of isomorphic classes of cubic Cayley graphs on D2 by using Gauss＇ celebrated law of quadratic reciprocity.展开更多
基金supported by National Natural Science Foundation of China (Grant Nos. 10501049,60821002)973 Project (Grant No. 2004CB318000)
文摘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.
基金Supported by the National Natural Science Foundation of China (11101002,10990011)
文摘In this paper,we present the generalized Huff curves that contain Huff＇s model as a special case.First,it is proved that every elliptic curve with three points of order 2 is isomorphic to a generalized Huff curve.Then,the fast and explicit formulae are derived for generalized Huff curves in projective coordinates.This paper also enumerates the number of isomorphism classes of generalized Huff curves over finite fields.Finally,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.
基金Supported in part by the National Natural Science Foundation of China Grant 19801022the Scientifictechnological Major Project of Educational Ministry of China, Grant 99036.
文摘Let F be an algebracially closed field of characteristic p】2, and L be the p<sup>n</sup>-dimensional Zassenhaus algebra with the maximal invariant subalgebra L<sub>0</sub> and the standard filtration {L<sub>i</sub>}|<sub>i=-1</sub><sup>p<sup>n</sup>-2</sup>. Then the number of isomorphism classes of simple L-modules is equal to that of simple L<sub>0</sub>-modules, corresponding to an arbitrary character of L except when its height is biggest. As to the number corresponding to the exception there was an earlier result saying that it is not bigger than p<sup>n</sup>.
基金Supported by National Natural Science Foundation of China(Grant Nos.11671344 and 11531011)
文摘Let p be an odd prime, and D2p = （a,b I aP = b2 = l,bab= a 1） the dihedral group of order 2p. In this paper, we completely classify the cubic Cayley graphs on D2p up to isomorphism by means of spectral method. By the way, we show that two cubic Cayley graphs on D2p are isomorphic if and only if they are cospectral. Moreover, we obtain the number of isomorphic classes of cubic Cayley graphs on D2 by using Gauss＇ celebrated law of quadratic reciprocity.
