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

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 cu...

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Bibliographic Details
Published in:Mathematical and computer modelling Vol. 49; no. 1; pp. 80 - 87
Main Authors: Sohn, Gyoyong, Kim, Hoil
Format: Journal Article
Language:English
Published: Kidlington Elsevier Ltd 2009
Elsevier
Subjects:
Algebra
Algebraic geometry
Baby-step giant-step algorithm
Counting points
Cryptography
Exact sciences and technology
Mathematical analysis
Mathematics
Methods of scientific computing (including symbolic computation, algebraic computation)
Number theory
Numerical analysis. Scientific computation
Picard curves
Sciences and techniques of general use
Counting points
Picard curves
Cryptography
Baby-step giant-step algorithm
Chaos
Finite field
Characteristic polynomial
Algorithm
Scientific computation
Applied mathematics
Hyperelliptic curve
Mathematical model
Computer aided analysis
ISSN:0895-7177, 1872-9479
Online Access:Get full text
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Summary: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.
ISSN:0895-7177
1872-9479
DOI:10.1016/j.mcm.2008.03.012