Abstract
In this paper we want to estimate the nonlinearity of Boolean functions, by probabilistic methods, when it is computationally very expensive, or perhaps not feasible to compute the full Walsh transform (which is the case for almost all functions in a larger number of variables, say more than 30). Firstly, we significantly improve upon the bounds of Zhang and Zheng (1999) on the probabilities of failure of affinity tests based on nonhomomorphicity, in particular, we prove a new lower bound that we have previously conjectured. This new lower bound generalizes the one of Bellare et al. (IEEE Trans. Inf. Theory 42(6), 1781–1795 1996) to nonhomomorphicity tests of arbitrary order. Secondly, we prove bounds on the probability of failure of a proposed affinity test that uses the BLR linearity test. All these bounds are expressed in terms of the function’s nonlinearity, and we exploit that to provide probabilistic methods for estimating the nonlinearity based upon these affinity tests. We analyze our estimates and conclude that they have reasonably good accuracy, particularly so when the nonlinearity is low.
Highlights
Introduction and motivationBoolean functions are defined on a vector space over the binary finite field F2 with output in F2
Computing the Walsh transform is not feasible in practice when the number of variables is large and the function is given as a “black box”
Using the bounds on failing the BLR linearity test from [1], which depend on the distance to the closest linear function, we show that similar bounds hold for min(P2(f ), P2(f + 1)), but this time the bounds depend on the distance to the closest affine function
Summary
Boolean functions are defined on a vector space over the binary finite field F2 with output in F2. G is treated as a “black box” function, and f can be evaluated at any given input using several calls to g Another test used in the literature for deciding whether a Boolean function is affine is to check whether the equation f (u + v + w) + f (u) + f (v) + f (w) = 0 holds, for some u, v, w ∈ Fn2 chosen uniformly at random. There are functions f, g such that f has higher probability of failing the test than g, even though f has lower nonlinearity than g This was shown in [2] for the BLR test and in [14] for the tests based on the (k + 1)-st order nonhomomorphicity with k odd. We intend to push further the connection between the probability of failing these tests and the nonlinearity, as well as look at estimating the nonlinearity of functions of cryptographic interest
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