Boolean functions with restricted input and their robustness; application to the FLIP cipher

Abstract

We study the main cryptographic features of Boolean functions (balancedness, nonlinearity, algebraic immunity) when, for a given number n of variables, the input to these functions is restricted to some subset E of

A new construction of weightwise perfectly balanced Boolean functions

“…Furthermore, it is proved that n-variable WPB Boolean functions exist only if n is a power of 2 (see [1]). In this paper, we always consider the Boolean functions on 2 m variables, where m is a positive integer.…”

“…An early version of FLIP faces an attack given by Duval et al [3], which leads the design of the filter functions to become more complicated to reach better criteria on a subset of F n 2 . In paper [1], it is shown that, for Boolean functions with restricted input, balancedness and nonlinearity parameters continue to play an important role with respect to the corresponding attacks on the framework of FLIP ciphers.…”

“…The parameters, balancedness, and nonlinearity are strongly related to the resistance against the distinguishing attack and affine approximation attack, respectively. In 2017, Carlet, Méaux, and Rotella provided a security analysis on the FLIP cipher and gave the first study on cryptographic criteria of Boolean functions with restricted input [1]. In paper [9], the authors studied the Boolean functions' nonlinearity with restricted input.…”

A family of weightwise (almost) perfectly balanced boolean functions with optimal algebraic immunity

“…As primitives in stream ciphers and block ciphers, Boolean functions are classically studied with input defined on the whole vector space F n 2 . Very recently, Carlet et al [3] studied the main cryptographic features of Boolean functions with input ranging over a subset of vector space F n 2 . Boolean functions with restricted input play an important role in a new family of stream ciphers, called FLIP, which are proposed by Méaux et al [13] at Eurocrypt 2016.…”

“…More explicitly, the input of the filter function will only consist of those vectors in F n 2 with constant Hamming weight. It has been shown that on the framework of FLIP ciphers, the important cryptographic criteria of Boolean functions include the balancedness, the nonlinearity, and the algebraic immunity, when the input of functions is restricted to the vectors with constant Hamming weight (see [3]). Since an early version of FLIP faces an attack given by Duval et al [6], the design of the filter function in FLIP becomes more complicated in order to reach better cryptographic criteria on the subsets of F n 2 .…”