Algorithms for Constructing Balanced Plateaued Functions With Maximal Algebraic Degrees

Abstract

It is important to study constructions of plateaued functions with balancedness and high algebraic degrees for preventing cryptographic attacks. Our goal of this paper is to find practical construction methods for producing infinite families of balanced r-plateaued functions with maximal algebraic degrees for every positive integer r. We first present a theoretical framework for secondary constructions of plateaued functions. From this framework, we derive three practical algorithms by controlling initial input vectors. These algorithms produce (s+1)-plateaued functions from a given bent function and s-plateaued functions in a recursive way for any nonnegative integer s ; therefore, we obtain r-plateaued functions for every r > s. Then we obtain three concrete construction methods of balanced r-plateaued functions with maximal algebraic degrees from the algorithms. For implementation, in the tables, we list up some initial bent (0-plateaued) functions, which guarantee the maximality of algebraic degrees of plateaued functions. Furthermore, we discuss the complexities of the three algorithms, which shows the feasibility of our methods. We emphasize that this is the first time to give constructions of balanced r-plateaued functions with maximal algebraic degrees for every positive integer r as far as we know. © 1963-2012 IEEE.