# Multiplicative group of a prime field

## Definition

Let $p$ be a prime. The multiplicative group of the prime field for the prime $p$, is defined in the following equivalent ways:

• Literally, the multiplicative group of the prime field $F_p$
• The group which as a set is nonzero congruence classes mod $p$, with multiplication coming from integer multiplication

The multiplicative group of a prime field, as an abstract group, is a cyclic group of order $p-1$. However, there is no direct procedure to find a generator for this multiplicative group; even given a generator, constructing a bijection between this multiplicative group and the additive group modulo $p - 1$ is a hard task.

The computational way of viewing this is that the multiplicative group of a prime field is a black-box cyclic group, with the multiplicative structure being the encoding. The problem of finding a generator is termed the primitive root-finding problem and the problem of constructing an explicit bijection with an additive group of order $p-1$ is termed the discrete logarithm problem.