Random Digit Representation of Integers

Abstract

Modular exponentiation, or scalar multiplication, is core to today's main stream public key cryptographic systems. In this article we generalize the classical fractional wNAF method for modular exponentiation - the classical method uses a digit set of the form {1, 3, . . . , m} which is extended here to any set of odd integers of the form {1, d2, . . . , dn}. We propose a general modular exponentiation algorithm based on a generalization of the frac-wNAF recoding and a new precomputation scheme. We also give general formula for the average density of non-zero therms in these representations, prove that there are infinitely many optimal sets for a given number of digits and show that the asymptotic behavior, when those digits are randomly chosen, is very close to the optimal case.