Let \(n\geq 2\) be a positive integer and \(D_0\) a multiplicative subgroup of \(\mathbb{Z}^*_n\) (integer \(\text{mod }n\), coprime to \(n\)) of index \(d\). Let \(D_j= g_jD_0\), \(j= 1,2,\dots, d-1\). We call \(D_j\) the generalized cyclotomic classes of order \(d\) when \(n\) is composite and the classical cyclotomic classes of order \(d\) when \(n\) is prime. The generalized cyclotomic numbers \((i,j)\) of order \(d\) are defined by \[ (i,j)= |(D_i+ 1)\cap D_j|,\quad i,j= 0,1,\dots,d-1. \] For different multiplicative subgroups \(D_0\), we get different cyclotomies and cyclotomic numbers of order \(d\).
Classical cyclotomy was developed by Gauss (1801), later followed by L. E. Dickson in his beautiful paper “Cyclotomy, higher congruences and Waring’s problem” [Am. J. Math. 57, 391-424 (1935; Zbl 0012.01203)]. Other names associated with classical and generalized cyclotomy are Whiteman, Storer, Williams, Lehmer, Berndt, Evans, to name a few.
In the present paper, the authors introduce a new generalized cyclotomy with respect to \(p^{e_1}_1\cdots p^{e_r}_r\), calculate cyclotomic numbers of order 2 and look into some applications in cryptography and coding theory.