Restriction, Terms and Nonlinearity of Boolean Functions

Nonlinear characteristics of (Boolean) functions is one of the important issues both in the design and cryptanalysis of (private key) ciphers or encryption algorithms. This paper studies nonlinear properties of functions from three different but closely related perspectives: maximal odd weighting subspaces, restrictions to cosets, and hypergraphs, all associated with a function. Main contributions of this work include (1) by using a duality property of a function, we have obtained several results that are related to lower bounds on nonlinearity as well as on the number of terms, of the function, (2) we show that the restriction of a function on a coset has a significant impact on cryptographic properties of the function, (3) we identify relationships between the nonlinearity of a function and the distribution of terms in the algebraic normal form of the function, (4) we prove that cycles of odd length in the terms, as well as quadratic terms, in the algebraic normal form of a function play an important role in determining the nonlinearity of the function. We hope that these results contribute to the study of new cryptanalytic attacks on ciphers, and more importantly, of counter-measures against such attacks.