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Discrete Logarithm in Galois Rings

Since Galois rings are the generalization of Galois fields, the question we tried to answer is: How to move from the discrete logarithm in Galois fields to the one in Galois rings? That concept of the discrete logarithm in Galois rings is a little bit different from the one in Galois fields. Here, the discrete logarithm of an element is the tuple, which is not the case in Galois fields. However, thanks to the multiplicative representation of elements in Galois rings, each element can be uniquely represented in the form: ; where k is a nonnegative integer, is a generator of the Galois ring (the definition of a generator in a Galois ring will be given later on). Then the tuple will be called: the discrete logarithm of . The notion of generators in Galois rings comes from the one in the group theory. The Knowledge of the generators in multiplicative groups allows, as well to determine the generators in Galois rings ; p is a prime number and m is a nonnegative integer greater than or equal to two. These new concepts of discrete logarithm and generators in Galois rings will help to securely share common information and to perform ElGamal encryption in Galois rings.