Executive Summary : | Permutation polynomials over finite fields are of great importance due to their applications in cryptography, coding theory, and combinatorial designs, etc. From the implementation point of view, permutation polynomials, which have only a few terms, are quite significant. It may be noted that the classification of permutation polynomials derived from power maps (monomials) is complete and well-known. However, the classification of permutation polynomials over finite fields that are binomials, trinomials, quadrinomials, and pentanomials is not yet understood completely and appears to be a challenging problem. Our first objective, thus, shall remain the construction of permutation polynomials over finite fields, particularly those with fewer terms. The substitution box (S-box) is one of the most crucial constituents of block ciphers. In the language of mathematics, an S-box is nothing but a function from a finite field to itself. For a robust (which is immune to differential and boomerang attacks) S-box , this function must have some important attributes such as low boomerang and differential uniformities, among others. Our second goal will be to compute the (generalized) boomerang and differential uniformities of some known classes of functions, as well as to construct some new classes of functions with low boomerang and differential uniformities |