| Reciprocity laws and their residue symbols have applications not only in number theory but also in other fields like cryptography. In their effort to develop cryptosystems with security equivalent to the difficulty of integer factorization, mathematicians utilized these objects to develop such schemes in cyclotomic fields of degree lambda - 1, where lambda = 2, 3, and 5. A crucial part of such schemes is an efficient and fast residue symbol algorithm. Such algorithms were devised for lambda = 2, 3, 5 but not for 7. Here we develop a fast and efficient residue symbol algorithm for lambda = 7. We accomplish this by giving explicit conditions on integers in Q (zeta) (with zeta a primitive 7th root of unity) to be primary, by formulating explicit forms of the complementaries to Kummer's 7th degree reciprocity law, and by using the norm-Euclidean algorithm in Q (zeta). We also reformulate the complementaries we obtain using Dickson's system of quadratic Diophantine equations. |
