Scalar multiplication on Weierstraß elliptic curves from Co-Z arithmetic

被引：40
作者：
Goundar R.R. [1 ]
Joye M. [2 ]
Miyaji A. [3 ]
Rivain M. [4 ]
Venelli A. [5 ]
机构：
[1] not available
[2] Technicolor, Security and Content Protection Labs, 35576 Cesson-Sévigné Cedex
[3] Japan Advanced Institute of Science and Technology, Nomi, Ishikawa, 923-1292
[4] CryptoExperts, 75002 Paris
[5] Inside Secure, 13790 Rousset, Avenue Victoire
来源：
Journal of Cryptographic Engineering | 2011年 / 1卷 / 02期
关键词：
Elliptic curves; Embedded systems; Implementation attacks; Jacobian coordinates; Meloni's technique; Regular ladders;
D O I：
10.1007/s13389-011-0012-0
中图分类号：
学科分类号：
摘要：
In 2007, Meloni introduced a new type of arithmetic on elliptic curves when adding projective points sharing the same Z-coordinate. This paper presents further co-Z addition formulæ (and register allocations) for various point additions on Weierstraß elliptic curves. It explains how the use of conjugate point addition and other implementation tricks allow one to develop efficient scalar multiplication algorithms making use of co-Z arithmetic. Specifically, this paper describes efficient co-Z based versions of Montgomery ladder, Joye's double-add algorithm, and certain signed-digit algorithms, as well as faster (X, Y)-only variants for left-to-right versions. Further, the proposed implementations are regular, thereby offering a natural protection against a variety of implementation attacks. © 2011 Springer-Verlag.
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收藏
页码：161 / 176
页数：15
相关论文
共 35 条
  • [1] Koblitz N., Elliptic curve cryptosystems, Math. Comput., 48, 177, pp. 203-209, (1987)
  • [2] Miller V.S., Use of elliptic curves in cryptography, Advances in Cryptology - CRYPTO'85, 218, pp. 417-426, (1985)
  • [3] Avanzi R., Cohen H., Doche C., Frey G., Lange T., Nguyen K., Vercauteren F., Handbook of Elliptic and Hyperelliptic Curve Cryptography, (2005)
  • [4] Blake I.F., Seroussi G., Smart N.P., Advances in Elliptic Curve Cryptography, 317, (2005)
  • [5] Meloni N., New point addition formulæ for ECC applications, Arithmetic of Finite Fields (WAIFI 2007), 4547, pp. 189-201, (2007)
  • [6] Goundar R.R., Joye M., Miyaji A., Co-Z addition formulæ and binary ladders on elliptic curves, Cryptographic Hardware and Embedded Systems - CHES 2010, 6225, pp. 65-79, (2010)
  • [7] Venelli A., Dassance F., Faster side-channel resistant elliptic curve scalar multiplication, Contemp. Math., 521, pp. 29-40, (2010)
  • [8] Rivain M., Fast and regular algorithms for scalar multiplication over elliptic curves, Cryptology ePrint Archive, (2011)
  • [9] Kocher P.C., Jaffe J., Jun B., Differential power analysis, Advances in Cryptology - CRYPTO '99, 1666, pp. 388-397, (1999)
  • [10] Yen S.M., Joye M., Checking before output may not be enough against fault-based cryptanalysis, IEEE Trans. Comput., 49, 9, pp. 967-970, (2000)
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