Multiplication in a Galois ring

In this paper, we focus on the efficient multiplication in a Galois ring of the size 4n, where n is a positive integer. We consider to adapt the finite field multiplication methods to the Galois ring multiplication. We give the polynomial multiplication in the Galois ring as a Toeplitz matrix-vector multiplication design with a modification used in finite fields of characteristic two. By this method, we reduce the multiplication complexity. Note that the proposed approach can be easily generalized to Galois rings of arbitrary characteristic. To the best of our knowledge, this is the first study to have a subquadratic space complexity to multiply two elements in the Galois rings.