IKBFU's Vestnik

2019 Issue №04

Back to the list Download an article

Analysis of Algo­rithms for Computing in Jacobian of a Picard Curve



In this article a representation of the elements of the Jacobian of a Picard curve is considered, which allows us to construct an algorithm for the reduc­tion of divisors with complexity O(deg(D)). Addition of divisors can be per­formed using the reduction algorithm.


1.  Barreirol E. R., Sarlabous J. E., Cherdieu J.-P. Efficient Reduction on the Jacobian Variety of Picard Curves // Coding Theory, Cryptography and Related Areas. 1998. P. 13—28.

2.  Sarlabous J. E., Barreirol E. R., Barceló J. A. P. On the Jacobian Varieties of Picard Cur­ves: Explicit Addition Law and Algebraic Structure // Mathematische Na­chri­ch­ten. 1999. № 208. P. 149—166.

3.  Flon S., Oyono R. Fast Arithmetic on Jacobians of Picard Curves // Public Key Cryp­tography — PKC 2004. 2004. P. 55—68.

4.  Oyono R. Arithmetik nicht-hyperelliptischer Kurven des Geschlechts 3 und ih­re Anwendung in der Kryptographie : PhD Diss. Univ. Duisburg-Essen, 2005.

5.  Handbook of Elliptic and Hyperelliptic Curve Cryptography / ed. H. Cohen, G. Frey. Chapman & Hall, 2006

6.  Sutherland A. V. Fast Jacobian Arithmetic for Hyperelliptic Curves of Genus 3 // ANTS XIII. 2019. P. 425—442.

7.  Thakur S. Abelian varieties in pairing-based cryptography. 2019. aXiv:1812. 11479v2 [math.NT].