An Analysis of Exponentiation Based on Formal Languages

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 signe...

• Type: Conference
• Year: 1999
• Where: EUROCRYPT
• Authors: Luke O'Connor
• Keywords: Cryptology, EUROCRYPT 1999, Exponentiation Ae, Recoding, Sliding Window