Abstract
We define a family of efficiently computable invariants for (n,m)-functions under EA-equivalence, and observe that, unlike the known invariants such as the differential spectrum, algebraic degree, and extended Walsh spectrum, in the case of quadratic APN functions over mathbb {F}_{2^n} with n even, these invariants take on many different values for functions belonging to distinct equivalence classes. We show how the values of these invariants can be used constructively to implement a test for EA-equivalence of functions from mathbb {F}_{2}^{n} to mathbb {F}_{2}^{m}; to the best of our knowledge, this is the first algorithm for deciding EA-equivalence without resorting to testing the equivalence of associated linear codes.
Highlights
Let F2n denote the finite field with 2n elements for some positive integer n, and let F∗2n denote its multiplicative group
We typically assume that m = n, i.e. we concentrate on functions from a finite field of characteristic two to itself; the approach given in the present paper can be applied to an arbitrary pair of dimensions (n, m)
The partitions Fm2 = K1⊕K2⊕· · ·⊕Ks can be precomputed for representatives from e.g. all known EA-classes of Almost perfect nonlinear (APN) functions; in particular, we refer to our computational results described in Section 5 where we describe how we provide such pre-computed results for all currently known APN functions over F2n up to dimension n = 10
Summary
Let F2n denote the finite field with 2n elements for some positive integer n, and let F∗2n denote its multiplicative group. In the case of cryptographically optimal vectorial Boolean functions, the most general equivalence relation preserving both the differential uniformity and the nonlinearity is the so-called Carlet-Charpin-Zinoviev-equivalence, or CCZ-equivalence [9]. One such property is the algebraic degree, which is preserved by EA-equivalence, but not by CCZ-equivalence This is not terribly useful for classifying APN function either since, as mentioned above, most known instances of APN functions are quadratic. Each of the individual steps comprising the algorithm has a concrete and meaningful input and output that can be monitored and verified This precludes the possibility of false positives or negatives as in the case of the current CCZ-equivalence test
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