899 research outputs found
On Equivalence of Known Families of APN Functions in Small Dimensions
In this extended abstract, we computationally check and list the
CCZ-inequivalent APN functions from infinite families on for n
from 6 to 11. These functions are selected with simplest coefficients from
CCZ-inequivalent classes. This work can simplify checking CCZ-equivalence
between any APN function and infinite APN families.Comment: This paper is already in "PROCEEDING OF THE 20TH CONFERENCE OF FRUCT
ASSOCIATION
On the Equivalence of Quadratic APN Functions
Establishing the CCZ-equivalence of a pair of APN functions is generally
quite difficult. In some cases, when seeking to show that a putative new
infinite family of APN functions is CCZ inequivalent to an already known
family, we rely on computer calculation for small values of n. In this paper we
present a method to prove the inequivalence of quadratic APN functions with the
Gold functions. Our main result is that a quadratic function is CCZ-equivalent
to an APN Gold function if and only if it is EA-equivalent to that Gold
function. As an application of this result, we prove that a trinomial family of
APN functions that exist on finite fields of order 2^n where n = 2 mod 4 are
CCZ inequivalent to the Gold functions. The proof relies on some knowledge of
the automorphism group of a code associated with such a function.Comment: 13 p
On Some Properties of Quadratic APN Functions of a Special Form
In a recent paper, it is shown that functions of the form
, where and are linear, are a good source for
construction of new infinite families of APN functions. In the present work we
study necessary and sufficient conditions for such functions to be APN
On relations between CCZ- and EA-equivalences
In the present paper we introduce some sufficient conditions and a procedure for checking whether, for a given function, CCZ-equivalence is more general than EA-equivalence together with taking inverses of permutations. It is known from Budaghyan et al. (IEEE Trans. Inf. Theory 52.3, 1141–1152 2006; Finite Fields Appl. 15(2), 150–159 2009) that for quadratic APN functions (both monomial and polynomial cases) CCZ-equivalence is more general. We prove hereby that for non-quadratic APN functions CCZ-equivalence can be more general (by studying the only known APN function which is CCZ-inequivalent to both power functions and quadratics). On the contrary, we prove that for power non-Gold APN functions, CCZ equivalence coincides with EA-equivalence and inverse transformation for n ≤ 8. We conjecture that this is true for any n.acceptedVersio
Relation between o-equivalence and EA-equivalence for Niho bent functions
Boolean functions, and bent functions in particular, are considered up to so-called EA-equivalence, which is the most general known equivalence relation preserving bentness of functions. However, for a special type of bent functions, so-called Niho bent functions there is a more general equivalence relation called o-equivalence which is induced from the equivalence of o-polynomials. In the present work we study, for a given o-polynomial, a general construction which provides all possible o-equivalent Niho bent functions, and we considerably simplify it to a form which excludes EA-equivalent cases. That is, we identify all cases which can potentially lead to pairwise EA-inequivalent Niho bent functions derived from o-equivalence of any given Niho bent function. Furthermore, we determine all pairwise EA-inequivalent Niho bent functions arising from all known o-polynomials via o-equivalence.publishedVersio
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