Efficient algorithms for Koblitz curves over fields of characteristic three

Abstract The nonadjacent form method of Koblitz [Advances in Cryptology (CRYPTO'98), in: Lecture Notes in Comput. Sci., vol. 1462, 1998, pp. 327–337] is an efficient algorithm for point multiplication on a family of supersingular curves over a finite field of characteristic 3. In this paper, a further discussion of the method is given. A window nonadjacent form method is proposed and its validity is proved. Efficient reduction and pre-computations are given. Analysis shows that more than 30% of saving can be achieved.

[1]  Victor S. Miller,et al.  Use of Elliptic Curves in Cryptography , 1985, CRYPTO.

[2]  Willi Meier,et al.  Efficient Multiplication on Certain Nonsupersingular Elliptic Curves , 1992, CRYPTO.

[3]  Jerome A. Solinas,et al.  Efficient Arithmetic on Koblitz Curves , 2000, Des. Codes Cryptogr..

[4]  G. Frey,et al.  A remark concerning m -divisibility and the discrete logarithm in the divisor class group of curves , 1994 .

[5]  Alfred Menezes,et al.  Reducing elliptic curve logarithms to logarithms in a finite field , 1993, IEEE Trans. Inf. Theory.

[6]  Matthew K. Franklin,et al.  Identity-Based Encryption from the Weil Pairing , 2001, CRYPTO.

[7]  N. Koblitz.  Elliptic curve cryptosystems , 1987 .

[8]  Neal Koblitz,et al.  CM-Curves with Good Cryptographic Properties , 1991, CRYPTO.

[9]  Feng Dengguo,et al.  Fast Multiplication on a Family of Koblitz Elliptic Curves , 2003 .

[10]  Antoine Joux,et al.  A One Round Protocol for Tripartite Diffie–Hellman , 2000, Journal of Cryptology.

[11]  Steven D. Galbraith,et al.  Implementing the Tate Pairing , 2002, ANTS.

[12]  Paulo S. L. M. Barreto,et al.  Efficient Algorithms for Pairing-Based Cryptosystems , 2002, CRYPTO.

[13]  Alfred Menezes,et al.  Reducing elliptic curve logarithms to logarithms in a finite field , 1991, STOC '91.

[14]  Ian F. Blake,et al.  Elliptic curves in cryptography , 1999 .

[15]  Neal Koblitz,et al.  An Elliptic Curve Implementation of the Finite Field Digital Signature Algorithm , 1998, CRYPTO.