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The binomial B(x)=x3+βx36 (where β is primitive in F24) over F210 is the first known example of an Almost Perfect Nonlinear (APN) function that is not CCZ-equivalent to a power function, and has remai...
On equivalence between known families of quadratic APN functions
CCZ-equivalence EA-equivalence APN
2019/7/15
We study a question whether the currently known families of quadratic APN polynomials are pairwise different up to CCZ-equivalence. We reduce the list of these families to those CCZ-inequivalent to ea...
On the EA-classes of known APN functions in small dimensions
EA-equivalence CCZ-equivalence Boolean functions
2019/4/11
Recently Budaghyan, Calderini and Villa (2018) introduced a procedure for investigating if CCZ-equivalence can be more general than EA-equivalence together with inverse transformation (when applicable...
We investigate the differential properties of a construction in which a given function F:F2n→F2nF:F2n→F2n is modified at K∈NK∈N points in order to obtain a new function GG. This is motivated by the qu...
Whether there exist Almost Perfect Non-linear permutations (APN) operating on an even number of bit is the so-called Big APN Problem. It has been solved in the 6-bit case by Dillon et al. in 2009 but,...
Constructing APN functions through isotopic shifts
Boolean function APN isotopic equivalence
2018/8/28
Almost perfect nonlinear (APN) functions over fields of characteristic 2 play an important role in cryptography, coding theory and, more generally, information theory as well as mathematics. Building ...
On the exponents of APN power functions and Sidon sets, sum-free sets, and Dickson polynomials
sum-free sets Dickson polynomials
2017/12/11
On the exponents of APN power functions and Sidon sets, sum-free sets, and Dickson polynomials.
On differential equivalence of APN functions
Boolean function Almost perfect nonlinear function Almost bent function
2017/9/25
For a given vectorial Boolean function FF from Fn2F2n to itself it was defined an associated Boolean function γF(a,b)γF(a,b) in 2n2n variables by C.~Carlet, P.~Charpin, V.~Zinoviev in 1998 that takes ...
Componentwise APNness, Walsh uniformity of APN functions and cyclic-additive difference sets
secret-key cryptography cyclic-additive difference sets
2017/6/8
In the preprint [Characterizations of the differential uniformity of vectorial functions by the Walsh transform, IACR ePrint Archive 2017/516], the author has, for each even positive δδ, characterized...
Some Results on the Known Classes of Quadratic APN Functions
APN function quadratic function Walsh spectrum
2017/1/3
In this paper, we determine the Walsh spectra of three classes of quadratic APN functions and we prove that the class of quadratic trinomial APN functions constructed by G\"olo\u glu is affine equival...
A generalisation of Dillon's APN permutation with the best known differential and linear properties for all fields of size $2^{4k+2}$
Boolean function Sbox APN
2016/12/9
The existence of Almost Perfect Nonlinear (APN) permutations operating on an even number of variables was a long-standing open problem, until an example with six variables was exhibited by Dillon et a...
Cryptanalysis of a Theorem: Decomposing the Only Known Solution to the Big APN Problem (Full Version)
Boolean functions APN Butterfly structure
2016/6/2
The existence of Almost Perfect Non-linear (APN) permutations
operating on an even number of bits has been a long standing open
question until Dillon et al., who work for the NSA, provided an exampl...
On a remarkable property of APN Gold functions
Boolean function Almost perfect nonlinear function Almost bent function
2016/3/16
In [13] for a given vectorial Boolean function F from \mathbb{F}_2^n to itself it was defined an associated Boolean function \gamma_F(a,b) in 2n variables that takes value~1 iff a\neq{\bf 0} and equat...
On the (non-)existence of APN $(n,n)$-functions of algebraic degree $n$
almost perfect nonlinear almost bent Boolean function
2016/2/23
In this paper, we study the problem of existence of almost perfect nonlinear (APN) functions of algebraic degree n over \ftwon. We characterize such functions by means of derivatives and power moments...
On values of vectorial Boolean functions and related problems in APN functions
Vectorial Boolean function APN function differentially δ-uniform function
2016/1/26
In this paper we prove that there are only differential 4-uniform functions which are on distance 1 from an APN function. Also we prove that there are no APN functions of distance 1 from another APN f...