On the Propagation Criterion of Boolean Functions

Abstract

The propagation criterion is one of the main cryptographic criteria on Boolean functions used in block ciphers. Quadratic Boolean functions satisfying the propagation criterion of high degree were given by Preneel et al., but their algebraic degree is too small for a cryptograhic use. Then designing Boolean functions of high algebraic degree and high degree of propagation has been the goal of several papers. In this paper, we investigate the work of Kurosawa and Satoh in order to optimize the algebraic degree and the degree of propagation, and the work of Honda, Satoh, Iwata, and Kurosawa, by giving in particular a construction of Boolean functions satisfying PC(3) and having a very large algebraic degree. We also show that among symmetric functions, only the quadratic ones satisfy the propagation criterion of degree greater than 1. A particular case of this result is that symmetric bent functions must be quadratic – a result that needed a whole paper to be proved before.KeywordsBoolean functionsBlock-CipherPropagation criterionSymmetric functions