Monte Carlo methods for index computation (𝑚𝑜𝑑𝑝)

Abstract

We describe some novel methods to compute the index of any integer relative to a given primitive root of a prime p. Our first method avoids the use of stored tables and apparently requires O ( p 1 / 2 ) O(p^{1/2}) operations. Our second algorithm, which may be regarded as a method of catching kangaroos, is applicable when the index is known to lie in a certain interval; it requires O ( w 1 / 2 ) O(w^{1/2}) operations for an interval of width w, but does not have complete certainty of success. It has several possible areas of application, including the factorization of integers.