Semi-bent Functions from Oval Polynomials

摘要

Although there are strong links between finite geometry and coding theory (it has been proved since the 1960's that all these connections between the two areas are important from a theoretical point of view and for applications), the connections between finite geometry and cryptography remain little studied. In 2011, Carlet and Mesnager have showed that projective finite geometry can also be useful in constructing significant cryptographic primitives such as plateaued Boolean functions. Two important classes of plateaued Boolean functions are those of bent functions and of semi-bent functions, due to their algebraic and combinatorial properties. In this paper, we show that oval polynomials (which are closely related to the hyperovals of the projective plane) give rise to several new constructions of infinite classes of semi-bent Boolean functions in even dimension. The following diagram gives an indication of the main topics and interconnections arising in this paper.