Abstract
In this work, we study functions that can be obtained by restricting a vectorial Boolean function F :mathbb {F}_{2}^n rightarrow mathbb {F}_{2}^n to an affine hyperplane of dimension n-1 and then projecting the output to an n-1-dimensional space. We show that a multiset of 2 cdot (2^n-1)^2 EA-equivalence classes of such restrictions defines an EA-invariant for vectorial Boolean functions on mathbb {F}_{2}^n. Further, for all of the known quadratic APN functions in dimension n < 10, we determine the restrictions that are also APN. Moreover, we construct 6368 new quadratic APN functions in dimension eight up to EA-equivalence by extending a quadratic APN function in dimension seven. A special focus of this work is on quadratic APN functions with maximum linearity. In particular, we characterize a quadratic APN function F :mathbb {F}_{2}^n rightarrow mathbb {F}_{2}^n with linearity of 2^{n-1} by a property of the ortho-derivative of its restriction to a linear hyperplane. Using the fact that all quadratic APN functions in dimension seven are classified, we are able to obtain a classification of all quadratic 8-bit APN functions with linearity 2^7 up to EA-equivalence.
Highlights
Let us be given two integers n, m ∈ N with m < n and two vectorial Boolean functions F : Fn2 → Fn2 and G : Fm2 → Fm2
A special focus of this work is on quadratic APN functions with maximum linearity
Compared to the case in which only a coordinate is discarded, we show that this operation is sound in the sense that (1), any function on Fn2−1 that is a restriction of F is EA-equivalent to a trim of F and, (2), EA-equivalent functions yield EA-equivalent trims when considering all 2 · (2n − 1)2 possibilities to choose the affine hyperplanes and the component functions
Summary
Let us be given two integers n, m ∈ N with m < n and two vectorial Boolean functions F : Fn2 → Fn2 and G : Fm2 → Fm2. 3, we define an operation called trimming, which restricts a vectorial Boolean function F on Fn2 to a linear or affine hyperplane of dimension n − 1 and projects the output to an n − 1-dimensional space by discarding one component function. By using a recursive tree search with backtracking, we are able to construct 6,368 new quadratic APN functions in dimension n = 8 up to EA-equivalence. We show that for the orthoderivative πG of G, we have πG (α), L(α) = 1 for all α ∈ Fn2 \ {0} with (α) = 0 This observation allows us in particular to classify all quadratic APN functions with maximum linearity in dimension eight, by using the recent classification of all quadratic 7-bit. We conclude by listing several open problems for future work
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