Cyclotomy and some applications in cryptology

Cyclotomic numbers are quite useful algebraic arguments in cryptology. However calculation of these numbers in terms of the definition is getting harder while parameters are getting larger. Some of the properties of cyclotomic numbers provide to express them in terms of Diophant equations. In this way the problem is linked to number theory and in some cases the calculations become easier. Primary works on the problem are usually on Z_p and later generalizations to F_{p^n} are sought. On Z_p there are several works on order 2-24 [12,16], on F_{p^n} there are works on order 2-6 and 8 [4,5,20]. In this work, order 7 cyclotomic numbers on F_{p^n} are examined and the results are compiled. Additionally some applications of cyclotomy on cryptology are mentioned and Sidel'nikov sequences which are the current and important applications are explained by examples.Keywords: Cyclotomy, cyclotomic numbers, uniform cyclotomy, Sidel'nikov sequences, autocorrelation, autocorrelation distribution.