Motivated by the problem of constructing bijective maps with low differential uniformity, we introduce the notion of permutation resemblance of a function, which looks to measure the distance a given map is from being a permutation. We prove several results concerning permutation resemblance and show how it can be used to produce low differentially uniform bijections. We also study the permutation resemblance of planar functions, which over fields of odd characteristic are known not to be bijections and to have the optimal differential uniformity.

Abstract In this work we give several generalizations of the isotopic shift construction, introduced recently by Budaghyan et al. (IEEE Trans Inform Theory 66:5299–5309, 2020), when the initial function is a Gold function. In particular, we derive a general construction of APN functions which covers several unclassified APN functions for $$n=8$$ n = 8 and produces fifteen new APN functions for $$n=9$$ n = 9 .

Almost perfect nonlinear (APN) functions over fields of characteristic 2 play an important role in cryptography, coding theory and, more generally, mathematics and information theory. In this paper we deduce a new method for constructing APN functions by studying the isotopic equivalence, concept defined for quadratic planar functions in fields of odd characteristic. In particular, we construct a family of quadratic APN functions which provides a new example of an APN mapping over${\mathbb F}_{2^{9}}$and includes an example of another APN function$x^{9}+ \mathop {\mathrm {Tr}}\nolimits (x^{3})$over${\mathbb F}_{2^{8}}$, known since 2006 and not classified up to now. We conjecture that the conditions for this family are satisfied by infinitely many APN functions.

CCZ-equivalence is the most general currently known equivalence relation for functions over finite fields preserving planarity and APN properties. However, for the particular case of quadratic planar functions isotopic equivalence is more general than CCZ-equivalence. A recent construction method for APN functions over fields of even characteristic, so-called isotopic shift construction, was instigated by the notion of isotopic equivalence. In this paper we discuss possible applications of the idea of isotopic shift for the case of planar functions. We show that, surprisingly, some of the known planar functions are actually isotopic shifts of each other. This confirms practically the pertinence of the notion of isotopic shift not only for APN functions but also for planar maps.

Bent functions are maximally nonlinear Boolean functions. Introduced by Rothaus and first examined by Dillon, these important functions have subsequently been studied by many researchers over the last four decades. Since a complete classification of bent functions appears elusive, many researchers concentrate on methods for constructing bent functions. In this paper, we investigate constructions of bent functions from involutions over finite fields in even characteristic. We present a generic construction technique, study its equivalence issues and show that linear involutions (which are an important class of permutations) over finite fields give rise to bent functions in bivariate representations. In particular, we exhibit new constructions of bent functions involving binomial linear involutions, whose dual functions are directly obtained without computation. The existence of bent functions from involutions relies heavily on solving systems of equations over finite fields.