Notes on two constructions of zero-difference balanced (ZDB) functions are made in this letter. Then ZDB functions over Ze×∏ki=0 Fqi are obtained. And it shows that all the known ZDB functions using cyclotomic cosets over Zn are special cases of a generic construction. Moreover, applications of these ZDB functions are presented.

Permutation polynomials over Zpn are useful in the design of cryptographic algorithms. In this paper, we obtain an equivalent condition for polynomial functions over Zpn to be permutations, and this equivalent condition can help us to analysis the randomness of such functions. Our results provide a method to distinguish permutation polynomials from random functions. We also introduce how to improve the randomness of permutation polynomials over Zpn.

For any positive integer n, define an iterated function $f(n)=left{ egin{array}{ll} n/2, & mbox{ $n$ even, } 3n+1, & mbox{ $n$ odd. } end{array} ight.$ Suppose k (if it exists) is the lowest number such that fk(n)