Use of elliptic curves in cryptography
Lecture notes in computer sciences; 218 on Advances in cryptology---CRYPTO 85
Mathematics of Computation
Efficient Arithmetic on Koblitz Curves
Designs, Codes and Cryptography - Special issue on towards a quarter-century of public key cryptography
Elliptic curves in cryptography
Elliptic curves in cryptography
CM-Curves with Good Cryptographic Properties
CRYPTO '91 Proceedings of the 11th Annual International Cryptology Conference on Advances in Cryptology
An Elliptic Curve Implementation of the Finite Field Digital Signature Algorithm
CRYPTO '98 Proceedings of the 18th Annual International Cryptology Conference on Advances in Cryptology
Elliptic Curves: Number Theory and Cryptography
Elliptic Curves: Number Theory and Cryptography
Guide to Elliptic Curve Cryptography
Guide to Elliptic Curve Cryptography
Advances in Elliptic Curve Cryptography (London Mathematical Society Lecture Note Series)
Advances in Elliptic Curve Cryptography (London Mathematical Society Lecture Note Series)
New Frobenius expansions for elliptic curves with efficient endomorphisms
ICISC'02 Proceedings of the 5th international conference on Information security and cryptology
SAC'05 Proceedings of the 12th international conference on Selected Areas in Cryptography
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In elliptic curve cryptosystems, scalar multiplications performed on the curves have much effect on the efficiency of the schemes, and many efficient methods have been proposed. In particular, recoding methods of the scalars play an important role in the performance of the algorithm used. For integer radices, the non-adjacent form (NAF) [21] and its generalizations (e.g., the generalized non-adjacent form (GNAF) [6] and the radix-r non-adjacent form (rNAF) [28]) have been proposed for minimizing the non-zero densities in the representations of the scalars. On the other hand, for subfield elliptic curves, the Frobenius expansions of the scalars can be used for improving efficiency [25]. Unfortunately, there are only a few methods apply the techniques of NAF or its analogue to the Frobenius expansion, namely t -adic NAF techniques on Koblitz curves [16, 27, 3] and hyperelliptic Koblitz curves [10]. In this paper, we try to combine these techniques, namely recoding methods for reducing non-zero density and the Frobenius expansion, and propose two new efficient recoding methods of scalars on more general family of subfield elliptic curves in odd characteristic. We also prove that the non-zero densities for the new methods are same as those for the original GNAF and rNAF. As a result, the speed of the proposed methods improve between 8% and 50% over that for the Frobenius expansion method.