Substitution boxes (aka S-boxes) are the only nonlinear part of a substitution-permutation network as a cryptosystem. Without them, adversaries would compromise the system with ease. Bent functions are a special kind of Boolean functions that achieve maximum nonlinearity. Therefore, it is important to study bent functions since S-Boxes are composed of highly nonlinear Boolean functions. Conventionally, researchers study and analyze Boolean functions in their Algebraic Normal Form. In this work we use cyclotomic cosets to construct nonlinear Boolean functions in their Univariate Polynomial Form. We have three conjectures as our research results and we have found one order 4 bent function with 8 variables. Finally, we analyze the new functions in terms of other design criteria for S-boxes such as strict avalanche and bit independence.We have found a highly nonlinear and balanced Boolean function with 6 variables that fulfills the design criteria and therefore would be a good candidate for constructing an S-box.
