Some cryptographic properties of Boolean functions

Venue: Room A219 – Polo Ferrari 1 – Via Sommarive, 5  Povo - Trento
Time: 11.00 a.m.

• Augustine Musukwa - PhD in Mathematics

Abstract:
Boolean functions with good cryptographic properties play a crucial role in the design of block ciphers and stream ciphers. This talk presents some contributions to the study of cryptographic properties of Boolean functions. We consider how the properties of a Boolean function, such as weight, balancedness, nonlinearity and resiliency, can be related to the properties of some other functions in a lower dimension. We employ these relations to construct balanced and resilient functions. Another aspect in consideration is the construction of balanced Boolean functions with trivial linear spaces.
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. In this talk a derivation of some quantities used for characterization of quadratic and cubic APN functions based on the behaviour of second order derivatives is presented. These quantities can also be used to characterize quadratic and cubic bent functions.
We also consider APN functions in even dimension and show that there must be at least a component whose linear space is trivial. In particular, we show that the size of the linear space for any component of an APN permutation is at most 1. Based on the sizes of the linear spaces 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.

Supervisor: Massimiliano Sala