Multiplicative group of a prime field
Let be a prime. The multiplicative group of the prime field for the prime , is defined in the following equivalent ways:
- Literally, the multiplicative group of the prime field
- The group which as a set is nonzero congruence classes mod , with multiplication coming from integer multiplication
The multiplicative group of a prime field, as an abstract group, is a cyclic group of order . 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 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 is termed the discrete logarithm problem.