Signed bits and fast exponentiation

Abstract

An exact analysis is given of the benefits of using the non-adjacent form representation for integers (rather than the binary representation), when computing powers of elements in a group in which inverting is easy. By counting the number of multiplications for a random exponent requiring a given number of Its in its binary representation, we arrive at a precise version of the known asymptotic result that on average one in three signed bits in the non-adjacent form is non-zero. This shows that the use of signed bits (instead of bits for ordinary repeated squaring and multiplication) reduces the cost of exponentiation by one ninth.