The Nonhomomorphicity of Boolean Functions

Abstract

Abstract. We introduce the notion of nonhomomorphicity as an alternative criterion that forecasts nonlinear characteristics of a Boolean function. Although both nonhomomorphicity and nonlinearity reflect a "difference" between a Boolean function and all the affine functions, they are measured from different perspectives. We are interested in nonhomomorphicity due to several reasons that include (1) unlike other criteria, we have not only established tight lower and upper bounds on the nonhomomorphicity of a function, but also precisely identified the mean of nonhomomorphicity over all the Boolean functions on the same vector space, (2) the nonhomomorphicity of a function can be estimated efficiently, and in fact, we demonstrate a fast statistical method that works both on large and small dimensional vector spaces.

An Improved Correlation Attack Against Irregular Clocked and Filtered Keystream Generators

“…We start by analyzing the boolean function f (x) for a correlation property that we will use in the attack. A similar property is analyzed in [18] where they look at the nonhomomorphicity of functions. In this paper we identify the probability…”

Improving bounds on probabilistic affine tests to estimate the nonlinearity of Boolean functions

“…A generalization of the tests above was proposed by Zhang and Zheng in [14], where the authors defined the notion of (k + 1)-st order nonhomomorphicity of a function f as the probability P k (f ) of failing the test…”

“…It was shown that for k odd, f is affine if and only if P k (f ) = 0; for k even, f is linear if and only if P k (f ) = 0; also, still for k even, f is affine if and only if P k (f ) ∈ {0, 1}. Furthermore, some bounds on P k (f ) with respect to d A (f ), for k odd, were given in [14].…”

“…In this paper, we firstly improve both the upper and lower bounds presented by Zhang and Zheng in [14] for P k (f ) with k odd (see Sections 3 and 4). Our lower bound holds for arbitrary k and generalizes the lower bound proven in [1] for k = 2, 3.…”

Further results on semi-bent functions in polynomial form

“…The so-called plateaued functions in n variables (or r-plateaued functions) have been introduced in 1999 by Zheng and Zhang in [16,17] for 0 < r < n. They were intensively studied as good candidates for designing cryptographic functions in symmetric cryptography. For r ∈ {0, 1, 2}, r-plateaued functions have been actively studied and have attracted much attention due to their cryptographic, algebraic and combinatorial properties.…”