Multiplicative group of a prime field

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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.