Concrete mathematics: a foundation for computer science
Concrete mathematics: a foundation for computer science
Exponentiating faster with addition chains
EUROCRYPT '90 Proceedings of the workshop on the theory and application of cryptographic techniques on Advances in cryptology
An introduction to the analysis of algorithms
An introduction to the analysis of algorithms
Redundant Integer Representations and Fast Exponentiation
Designs, Codes and Cryptography - Special issue dedicated to Gustavus J. Simmons
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
A method for obtaining digital signatures and public-key cryptosystems
Communications of the ACM
Handbook of Applied Cryptography
Handbook of Applied Cryptography
Introduction To Automata Theory, Languages, And Computation
Introduction To Automata Theory, Languages, And Computation
Signed Digit Representations of Minimal Hamming Weight
IEEE Transactions on Computers
CM-Curves with Good Cryptographic Properties
CRYPTO '91 Proceedings of the 11th Annual International Cryptology Conference on Advances in Cryptology
Speeding up Elliptic Cryptosystems by Using a Signed Binary Window Method
CRYPTO '92 Proceedings of the 12th Annual International Cryptology Conference on Advances in Cryptology
An Improved Algorithm for Arithmetic on a Family of Elliptic Curves
CRYPTO '97 Proceedings of the 17th Annual International Cryptology Conference on Advances in Cryptology
CRYPTO '89 Proceedings of the 9th Annual International Cryptology Conference on Advances in Cryptology
On String Replacement Exponentiation
Designs, Codes and Cryptography
Minimal Weight Digit Set Conversions
IEEE Transactions on Computers
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A recoding rule for exponentiation is a method for reducing the cost of the exponentiation ae by reducing the number of required multiplications. If w(e) is the (hamming) weight of e, and e the result of applying the recoding rule A to e, then the purpose is to reduce wA(e) as compared to w(e). A well-known example of a recoding rule is to convert a binary exponent into a signed-digit representation in terms of the digits { 1;1; 0 } where 1 = -1, by recoding runs of 1's. In this paper we show how three recoding rules can be modelled via regular languages to obtain precise information about the resulting weight distributions. In particular we analyse the recoding rules employed by the 2k-ary, sliding window and optimal signed-digit exponentiation algorithms. We prove that the sliding window method has an expected recoded weight of approximately n/(k +1) for relevant k-bit windows and n-bit exponents, and also that the variance is small. We also prove for the optimal signed digit method that the expected weight is approximately n/3 with a variance of 2n/27. In general the sliding window method provides the best performance, and performs less than 85% of the multiplications required for the other methods for a majority of exponents.