Data for the average degree density add_3 and add_4 for all the functions of degree 4 in 7 variables

The file contains the values that we computed for the average degree density add_3 and add_4 (as defined in the paper for which this a supplement) for all the functions of degree 3 in 7 variables. We used the representatives of all such functions as provided in the dataset of P. Langevin at https://langevin.univ-tln.fr/project/rm742/rm742.htmlArticle abstractIn cryptographic applications, Boolean functions are typically represented in algebraic normal form, i.e. as multivariate polynomial functions over the finite field F₂. For such a function f, we consider, for each degree k, the density of monomials of degree k in f, i.e. the number of monomials of degree k that appear in f, normalized by the total number of possible monomials of degree k. We then average this number over all functions which are affine equivalent to f; we call the resulting quantity, denoted by add_k (f), the average degree-k monomial density of f. This quantity was defined in previous work, and it was shown that it is closely related to a probabilistic test for deciding whether deg(f) < k. In this paper, we give lower and upper bounds for add_k (f) for functions of any degree d (only the particular case d = k having been dealt with in previous work). The lower bound is reached; while in general the upper bound is not reached, we show that, except for some border cases, is not far from the actual maximum. There are several consequences of these bounds. Firstly, it answers negatively the following question: Does there exist a function f which has no monomials of a particular degree k (with k k is equal to 0.5, the distribution of the values is somewhat surprising; when n ≥ 20, n − k ≥ 9 and d − k ≥ 6, low values of add_k (f) exist (reaching approximately 1/2d−k), but there are no values higher than around 0.5005. We also report experimental results for computing exact values of add_k (f) for functions in 7 variables and results for running the probabilistic test on functions describing the output of the ciphers Trivium, Grain-128a, and SNOW-V.© the authors