A New Implementation of Miura-Arita Algorithm for Miura Curves

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.





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.