An Efficient Seventh Power Residue Symbol Algorithm

Power residue symbols and their reciprocity laws have applications not only in number theory, but also in other fields like cryptography. A crucial ingredient in certain public key cryptosystems is a fast algorithm for computing power residue symbols. Such algorithms have only been devised for the Jacobi symbol as well as for cubic and quintic power residue symbols, but for no higher powers. In this paper, we provide an efficient procedure for computing 7-th power residue symbols. The method employs arithmetic in the field Q(ζ), with ζ a primitive 7-th root of unity, and its ring of integers Z[ζ]. We give an explicit characterization for an element in Z[ζ] to be primary, and provide an algorithm for finding primary associates of integers in Z[ζ]. Moreover, we formulate explicit forms of the complementary laws to Kummer’s 7-th degree reciprocity law, and use Lenstra’s norm-Euclidean algorithm in the cyclotomic field.