Use of elliptic curves in cryptography
Lecture notes in computer sciences; 218 on Advances in cryptology---CRYPTO 85
A survey of fast exponentiation methods
Journal of Algorithms
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
Efficient elliptic curve exponentiation
ICICS '97 Proceedings of the First International Conference on Information and Communication Security
Algorithms for Multi-exponentiation
SAC '01 Revised Papers from the 8th Annual International Workshop on Selected Areas in Cryptography
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
Analysis of linear combination algorithms in cryptography
ACM Transactions on Algorithms (TALG)
Theoretical Computer Science
Improved techniques for fast exponentiation
ICISC'02 Proceedings of the 5th international conference on Information security and cryptology
A note on the signed sliding window integer recoding and a left-to-right analogue
SAC'04 Proceedings of the 11th international conference on Selected Areas in Cryptography
ISPEC'10 Proceedings of the 6th international conference on Information Security Practice and Experience
Fast elliptic curve cryptography using minimal weight conversion of d integers
AISC '12 Proceedings of the Tenth Australasian Information Security Conference - Volume 125
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One of the most frequent operations in modern cryptosystems is a multi-scalar multiplication with two scalars. Common methods to compute it are the Shamir method and the Interleave method whereas their speed mainly depends on the (joint) Hamming weight of the scalars. To increase the speed, the scalars are usually deployed using some general representation which provides a lower (joint) Hamming weight than the binary representation. However, by using such general representations the precomputation and storing of some points becomes necessary and therefore more memory is required. Probably the most famous method to speed up the Shamir method is the joint sparse form (JSF). The resulting representation has an average joint Hamming weight of 1/2 and it uses the digits 0,± 1. To compute a multi-scalar multiplication with the JSF, the precomputation of two points is required. While for two precomputed points both the Shamir and the Interleave method provide the same efficiency, until now the Interleave method is faster in any case where more points are precomputed. This paper extends the used digits of the JSF in a natural way, namely we use the digits 0, ±1, ±3 which results in the necessity to precompute ten points. We will prove that using the proposed scheme, the average joint Hamming density is reduced to 239/661 ≈ 0.3615. Hence, a multi-scalar multiplication can be computed more than 10% faster, compared to the JSF. Further, our scheme is superior to all known methods using ten precomputed points and is therefore the first method to improve the Shamir method such that it is faster than the Interleave method. Another advantage of the new representation is, that it is generated starting at the most significant bit. More specific, we need to store only up to 5 joint bits of the new representation at a time. Compared to representations which are generated starting at the least significant bit, where we have to store the whole representation, this yields a significant saving of memory.