New classes of zero-difference balanced (ZDB) functions derived from Fermat quotients are proposed in this letter. Based on the new ZDB functions, some applications, such as the construction of optimal frequency hopping sequences set and perfect difference systems of sets, are introduced.

Recently, trace representation of a class of balanced quaternary sequences of period p from the classical cyclotomic classes was given by Yang et al. (Cryptogr. Commun.,15 (2023): 921-940). In this letter, based on the generalized cyclotomic classes, we define a class of balanced quaternary sequences of period pn, where p = ef + 1 is an odd prime number and satisfies e ≡ 0 (mod 4). Furthermore, we calculate the defining polynomial of these sequences and obtain the formula for determining their trace representations over ℤ4, by which the linear complexity of these sequences over ℤ4 can be determined.

In this letter, we give a new construction of a family of sequences of period pk-1 with low correlation value by using additive and multiplicative characters over Galois rings. The new constructed sequence family has family size (M-1)(pk-1)rpkr(e-1) and alphabet size Mpe. Based on the characters sum over Galois rings, an upper bound on the correlation of this sequence family is presented.

For an odd prime p and a positive integer r, new classes of binary sequences with period pr+1 are proposed from Euler quotients in this letter, which include several known classes of binary sequences derived from Fermat quotients and Euler quotients as special cases. The advantage of the new constructions is that they allow one to choose their support sets freely. Furthermore, with some constrains on the support set, the new sequences are proved to possess large linear complexities under the assumption of 2p-1 ≢ 1 mod p2.