Advances in Applied Mathematics
Use of elliptic curves in cryptography
Lecture notes in computer sciences; 218 on Advances in cryptology---CRYPTO 85
A course in computational algebraic number theory
A course in computational algebraic number theory
Elliptic curves and their applications to cryptography: an introduction
Elliptic curves and their applications to cryptography: an introduction
Elliptic curves in cryptography
Elliptic curves in cryptography
A method for obtaining digital signatures and public-key cryptosystems
Communications of the ACM
Elliptic Curve Public Key Cryptosystems
Elliptic Curve Public Key Cryptosystems
Software Implementation of the NIST Elliptic Curves Over Prime Fields
CT-RSA 2001 Proceedings of the 2001 Conference on Topics in Cryptology: The Cryptographer's Track at RSA
Algorithms for Multi-exponentiation
SAC '01 Revised Papers from the 8th Annual International Workshop on Selected Areas in Cryptography
Fast Simultaneous Scalar Multiplication on Elliptic Curve with Montgomery Form
SAC '01 Revised Papers from the 8th Annual International Workshop on Selected Areas in Cryptography
Faster Point Multiplication on Elliptic Curves with Efficient Endomorphisms
CRYPTO '01 Proceedings of the 21st Annual International Cryptology Conference on Advances in Cryptology
More Flexible Exponentiation with Precomputation
CRYPTO '94 Proceedings of the 14th Annual International Cryptology Conference on Advances in Cryptology
Optimal Extension Fields for Fast Arithmetic in Public-Key Algorithms
CRYPTO '98 Proceedings of the 18th Annual International Cryptology Conference on Advances in Cryptology
Efficient Elliptic Curve Exponentiation Using Mixed Coordinates
ASIACRYPT '98 Proceedings of the International Conference on the Theory and Applications of Cryptology and Information Security: Advances in Cryptology
Fast Implementation of Elliptic Curve Arithmetic in GF(pn)
PKC '00 Proceedings of the Third International Workshop on Practice and Theory in Public Key Cryptography: Public Key Cryptography
Software Implementation of Elliptic Curve Cryptography over Binary Fields
CHES '00 Proceedings of the Second International Workshop on Cryptographic Hardware and Embedded Systems
High-speed hardware implementations of Elliptic Curve Cryptography: A survey
Journal of Systems Architecture: the EUROMICRO Journal
CT-RSA'03 Proceedings of the 2003 RSA conference on The cryptographers' track
Fast elliptic curve arithmetic and improved weil pairing evaluation
CT-RSA'03 Proceedings of the 2003 RSA conference on The cryptographers' track
Fast multi-computations with integer similarity strategy
PKC'05 Proceedings of the 8th international conference on Theory and Practice in Public Key Cryptography
Improved precomputation scheme for scalar multiplication on elliptic curves
IMACC'11 Proceedings of the 13th IMA international conference on Cryptography and Coding
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Our development of efficient methods for the precomputation of multi-scalar multiplication for elliptic curve cryptosystems (ECCs) is presented. Multi-scalar multiplication is required in many forms of ECC, including schemes for the verification of ECDSA signatures. The simultaneous method is one known method for fast multi-scalar multiplication. The method has two stages: a precomputation stage and an evaluation stage. Points for use in the evaluation stage are computed in the precomputation stage. The actual multiscalar multiplication is carried out on the basis of the precomputed points in the evaluation stage. In the evaluation stage of the simultaneous method, we are able to quickly compute the points of the multiscalar multiple because few additions are required. On the other hand, if we use a large window width, we have to compute an enormous number of points in the precomputation stage. Hence, we have to compute an abundance of inversions, which carries a high computational cost. The result is that a large amount of time is required by the precomputation stage. This is the well-known draw-back of the simultaneous method. In our proposed method, we apply the Montgomery trick to reduce the number of inversions required with a width window w from O(22w) to O(w). In addition, our proposed method computes uP and vQ for any u, v, then compute uP + vQ, where P,Q are elliptic points. This procedure enables us to remove points that will not be used later from the process of precomputation. Without our proposed method, an algorithm to compute precomputation table would have to be changed dependently on unused points. Compared with the method without Montgomery trick, our proposed method is 3.6 times faster than the conventional simultaneous method, i.e., than in the absence of the Montgomery trick. Moreover, the optimal window width for our proposed method is 3, whereas the corresponding width for conventional simultaneous methods is 2.