A survey of fast exponentiation methods
Journal of Algorithms
On convergence rates in the central limit theorems for combinatorial structures
European Journal of Combinatorics
Modern computer algebra
The State of Elliptic Curve Cryptography
Designs, Codes and Cryptography - Special issue on towards a quarter-century of public key cryptography
Optimal Left-to-Right Binary Signed-Digit Recoding
IEEE Transactions on Computers - Special issue on computer arithmetic
Concrete Mathematics: A Foundation for Computer Science
Concrete Mathematics: A Foundation for Computer Science
Distribution results for low-weight binary representations for pairs of integers
Theoretical Computer Science - Combinatorics of the discrete plane and tilings
Analysis of linear combination algorithms in cryptography
ACM Transactions on Algorithms (TALG)
The Complexity of Certain Multi-Exponentiation Techniques in Cryptography
Journal of Cryptology
Analysis of linear combination algorithms in cryptography
ACM Transactions on Algorithms (TALG)
Theoretical Computer Science
A New Upper Bound for the Minimal Density of Joint Representations in Elliptic Curve Cryptosystems
IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
Unbalanced digit sets and the closest choice strategy for minimal weight integer representations
Designs, Codes and Cryptography
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
An advanced method for joint scalar multiplications on memory constraint devices
ESAS'05 Proceedings of the Second European conference on Security and Privacy in Ad-Hoc and Sensor Networks
New minimal weight representations for left-to-right window methods
CT-RSA'05 Proceedings of the 2005 international conference on Topics in Cryptology
Fractional windows revisited: improved signed-digit representations for efficient exponentiation
ICISC'04 Proceedings of the 7th international conference on Information Security and Cryptology
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Several cryptosystems rely on fast calculations of linear combinations in groups. One way to achieve this is to use joint signed binary digit expansions of small “weight.” We study two algorithms, one based on nonadjacent forms of the coefficients of the linear combination, the other based on a certain joint sparse form specifically adapted to this problem. Both methods are sped up using the sliding windows approach combined with precomputed lookup tables. We give explicit and asymptotic results for the number of group operations needed, assuming uniform distribution of the coefficients. Expected values, variances and a central limit theorem are proved using generating functions.Furthermore, we provide a new algorithm that calculates the digits of an optimal expansion of pairs of integers from left to right. This avoids storing the whole expansion, which is needed with the previously known right-to-left methods, and allows an online computation.