Explicit bounds of polynomial coefficients and counting points on Picard curves over finite fields

Authors:
Gyoyong Sohn;Hoil Kim
Affiliations:
Department of Industrial and Applied mathematics, Kyungpook National University, 1370 Sankyuk-dong, Buk-gu, Daegu 702-701, Republic of Korea;Department of Industrial and Applied mathematics, Kyungpook National University, 1370 Sankyuk-dong, Buk-gu, Daegu 702-701, Republic of Korea
Venue:
Mathematical and Computer Modelling: An International Journal
Year:
2009

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Abstract

In this paper, we describe an algorithm for computing the order of the Jacobian varieties of Picard curves over finite fields. This is an extension of the algorithm of Matsuo, Chao and Tsujii (MCT) [K. Matsuo, J. Chao, S. Tsujii, An improved baby step algorithm for point counting of hyperelliptic curves over finite fields, in: LNCS vol. 2369, Springer-Verlag, 2005, pp. 461-474] for hyperelliptic curves. We study the characteristic polynomials and the Jacobian varieties of algebraic curves of genus three over finite fields. Based on this, we investigate the explicit computable bounds for coefficients of the characteristic polynomial and improve a part of the baby-step giant-step of the counting points algorithm. Usefulness of the proposed method is illustrated and verified by the simple examples.