Some cryptographic properties of Boolean functions

We investigate some cryptographic properties of Boolean functions. Some of the properties we are going to consider include weight, balancedness, nonlinearity and resiliency. Mainly, we study how the properties of a Boolean function can be related to the properties of some other functions in a lower dimension. We utilize these relations to construct balanced and resilient functions. Another aspect which we consider is the set of linear structures of Boolean functions. Our interest is in construction of balanced functions which have a trivial set of linear structures. It is well-known that block ciphers may suffer from two main attacks, namely, differential attacks and linear attacks. APN functions are known to provide the best resistance against differential attacks. We look at some properties of APN functions in even dimension. We study the linear structures of their components. We show that there must be at least a component whose set of linear structures is trivial. In particular, we determine the possible size of the set of linear structures for any component of an APN permutation. Based on the sizes of the sets of linear structures for the components, we establish a simple characterization of quadratic APN functions, and this knowledge is useful in proving some results on a general form for the number of bent components. We further consider counting bent components in any quadratic power functions. Based on the behaviour of second order derivatives, we derive some quantities which are used for characterization of quadratic and cubic APN functions. We show that these quantities can also be used to characterize quadratic and cubic Bent functions. Furthermore, we show that these derived quantities can be linked to the size of the set of linear structures for any quadratic and cubic partially-bent functions.