As we have seen in Theorem 9.16, for a prime is a cyclic group of order p - 1. This means that there exists a generator, such that for all, α can be written uniquely as α = γx, where x is an integer with 0 ≤ x < p - 1; the integer x is called the discrete logarithm of α to the base γ, and is denoted logγ α.

This chapter discusses some computational problems in this setting; namely, how to efficiently find a generator γ, and given γ and α, how to compute logγ α.

More generally, if γ generates a subgroup G of of order q, where q | (p - 1), and α ∈ G, then logγ α is defined to be the unique integer x with 0 ≤ x < q and α = γx. In some situations it is more convenient to view logγ α as an element of ℤq. Also for x ∈ ℤq, with x = [a]q, one may write γx to denote γa. There can be no confusion, since if x = [a′]q, then γa′ = γa. However, in this chapter, we shall view logγ α as an integer.

Although we work in the group, all of the algorithms discussed in this chapter trivially generalize to any finite cyclic group that has a suitably compact representation of group elements and an efficient algorithm for performing the group operation on these representations.