A New Implementation of Miura-Arita Algorithm for Miura Curves

Authors: A. Basiri, S. Rahmany, D. Khatibi

Abstract:

The aim of this paper is to review some of standard fact on Miura curves. We give some easy theorem in number theory to define Miura curves, then we present a new implementation of Arita algorithm for Miura curves.

Keywords: Miura curve, discrete logarithm problem, algebraic curve cryptography, Jacobian group.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1335528

References:

[1] F. K. Abu Salem, K. Khuri Makdisi, Fast Jacobian group operations for C3,4 curves over a large finite field, LMS Journal of Computation and Mathematics 10 (2007), 307-328.
[2] S. Arita. Algorithms for computations in Jacobian group of Cab curve and their application to discrete-log based public key cryptosystems. IEICE Transactions, J82-A(8):1291-1299, 1999. In Japanese. English translation in the proceedings of the Conference on The Mathematics of Public Key Cryptography, Toronto 1999.
[3] S. Arita, S. Miura, and T. Sekiguchi. An addition algorithm on the jacobian varieties of curves. Journal of the Ramanujan Mathematical Society, 19(4):235-251, December 2004.
[4] A. Basiri, A. Enge, J.-C. Faug`ere, and N. G¨urel. Implementing the arithmetic of c3,4 curves. In Lecture Notes in Computer Science, Proceedings of ANTS, pages 87-101. Springer-Verlag, June 2004.
[5] A. Basiri, A. Enge, J.-C. Faug`ere, and N. G¨urel. The arithmetic of jacobian groups of superelliptic cubics. Math. Comp., 74:389-410, 2005.
[6] S.-D. Galbraith, S. Paulus, and N.-P. Smart. Arithmetic on superelliptic curves. Mathematics of Computation, 71(237):393-405, 2002.
[7] R. Harasawa and J. Suzuki. Fast Jacobian group arithmetic on Cab curves. In W. Bosma, editor, Algorithmic Number Theory ÔÇö ANTS-IV, volume 1838 of Lecture Notes in Computer Science, pages 359-376, Berlin, 2000. Springer-Verlag.