In this talk, we present an ergodicity criterion of a certain class of 1-Lipschitz functions on $mathbb{Z}_p$ for arbitrary primes $p,$ known as $mathcal{B}$-functions. These functions are locally analytic functions of order 1 (and therefore contain polynomials). For arbitrary primes $pgeq 5,$ this erodicity criterion leads to an efficient and practical method of constructing ergodic polynomials on $Z_p$ that realize a given unicyclic permutation modulo $p.$ In particular, for polynomials over $mathbb{Z}_3$, we provide a complete ergodicity criterion in terms of its coefficients. This method can be applied to a $mathbb{Z}_p$ for general primes $p.$