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APN (almost perfect nonlinear) functions over finite fields of even characteristic are interesting and have many applications to the design of symmetric ciphers resistant to differential attacks. This notion was generalized to GAPN (generalized APN) for arbitrary characteristic p by Kuroda and Tsujie. In this paper, we completely classify GAPN monomial functions x d for the case when the exponent d has exactly two non-zero digits when represented in base p; these functions can be viewed as generalizations of the APN Gold functions. In particular, we characterise all the monomial GAPN functions over F p 2 . We also obtain a new characterization for certain GAPN functions over F p n of algebraic degree p using the multivariate algebraic normal form; this allows us to explicitly construct a family of GAPN functions of algebraic degree p for n = 3 and arbitrary prime p ≥ 3 . [ABSTRACT FROM AUTHOR]

We propose a method for finding an almost perfect nonlinear (APN) function. It is based on translation into SAT-problem and using SAT-solvers. We construct several formulas defining the conditions for finding an APN function and introduce two representations of the function, sparse and dense, which are used to describe the problem of finding one-to-one vectorial Boolean functions and APN functions. We also propose a new method for finding a vector APN function with additional properties. It is based on the idea of representing the unknown vectorial Boolean function as a sum of a known APN function and two unknown Boolean functions, , where is a known APN function. It is shown that this method is more efficient than the direct construction of APN function using SAT for dimensions 6 and 7. As a result, the method described in the paper can prove the nonexistence of cubic APN functions in dimension 7 representable in the form of the sum described above. [ABSTRACT FROM AUTHOR]

Linear codes generated by component functions of perfect nonlinear (PN for short) and almost perfect nonlinear (APN for short) functions and the first-order Reed-Muller codes have been an object of intensive study in coding theory. The objective of this paper is to investigate some binary shortened codes of two families of linear codes from APN functions and some $p$ -ary shortened codes associated with PN functions. The weight distributions of these shortened codes and the parameters of their duals are determined. The parameters of these binary codes and $p$ -ary codes are flexible. Many of the codes presented in this paper are optimal or almost optimal. The results of this paper show that the shortening technique is very promising for constructing good codes. [ABSTRACT FROM AUTHOR]

In this paper, we present two new families of APN functions. The first family is in bivariate form $\big (x^{3}+xy^{2}+ y^{3}+xy, x^{5}+x^{4}y+y^{5}+xy+x^{2}y^{2} \big)\,\,\vphantom {_{\int _{\int }}}$ over ${\mathbb F}_{2^{m}}^{2}$. It is obtained by adding certain terms of the form $\sum _{i}(a_{i}x^{2^{i}}y^{2^{i}},b_{i}x^{2^{i}}y^{2^{i}})$ to a family of APN functions recently proposed by Gölo&gcaron;lu. The $\vphantom {_{\int _{\int }}}$ second family has the form $L(z)^{2^{m}+1}+vz^{2^{m}+1}$ over ${\mathbb F}_{{2^{3m}}}$ , which generalizes a family of APN functions by Bracken et al. from 2011. By calculating the $\Gamma $ -rank of the constructed APN functions over ${\mathbb F}_{2^{8}}$ and ${\mathbb F}_{2^{9}}$ , we demonstrate that the two families are CCZ-inequivalent to all known families. In addition, the two new families cover two known sporadic APN instances over ${\mathbb F}_{2^{8}}$ and ${\mathbb F}_{2^{9}}$ , which were found by Edel and Pott in 2009 and by Beierle and Leander in 2021, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we propose three new classes of permutation trinomials over F q 3 . The first two classes are of the form x + L (x q 2 + q − 1) , where L (x) ∈ { x + A x q , x + A x q 2 } , q even, and the third class is x + A x q 2 − q + 1 + A 2 x q 2 , q odd. In fact we prove a conjecture proposed by Gong, Gao and Liu [8] about permutation trinomials of the form x q + 1 + L (x) ∈ F q 3 [ x ] as a particular case of our results. Moreover, we determine necessary and sufficient conditions for the permutation trinomials studied. We also show that the permutation trinomials proposed in this paper are new in the sense that they are not quasi-multiplicative equivalent to permutation trinomials over F q 3 known so far. [ABSTRACT FROM AUTHOR]

Almost perfect nonlinear functions possess optimal resistance to differential cryptanalysis and are widely studied. Most known APN functions are defined using their representation as a polynomial over a finite field and very little is known about combinatorial constructions of them on F 2 n . In this work we propose two approaches for obtaining quadratic APN functions on F 2 n . The first approach exploits a secondary construction idea, it considers how to obtain a quadratic APN function in n + 1 variables from a given quadratic APN function in n variables using special restrictions on the new terms. The second approach is searching for quadratic APN functions that have a matrix representation partially filled with the standard basis vectors in a cyclic manner. This approach allows us to find a new APN function in 7 variables. We prove that the updated list of quadratic APN functions in dimension 7 is complete up to CCZ-equivalence. Also, we observe that the quadratic parts of some APN functions have a low differential uniformity. This observation allows us to introduce a new subclass of APN functions, the so-called stacked APN functions. These are APN functions of algebraic degree d such that eliminating monomials of degrees k + 1,..., d for any k < d results in APN functions of algebraic degree k. We provide cubic examples of stacked APN functions for dimensions up to 6. [ABSTRACT FROM AUTHOR]

This article determines all the solutions in the finite field F 2 4 n of the equation x 2 3 n + 2 2 n + 2 n − 1 + (x + 1) 2 3 n + 2 2 n + 2 n − 1 = b. Specifically, we explicitly determine the set of b 's for which the equation has i solutions for any positive integer i. Such sets, which depend on the number of solutions i , are given explicitly and expressed nicely, employing the absolute trace function over F 2 n , the norm function over F 2 4 n relatively to F 2 n and the set of (2 n + 1) st roots of unity in F 2 4 n . The equation considered in this paper comes from an article by Budaghyan et al. ([2]) in which the authors have investigated novel approaches for obtaining alternative representations for functions from the known infinite APN families. In particular, they have been interested in determining the differential spectrum of some power functions among them is the one F (x) = x 2 3 n + 2 2 n + 2 n − 1 defined over F 2 4 n . The problem of the determination of such spectrum has led to a conjecture (Conjecture 27 in the preprint (2020) [2] for which an updated version will appear in 2022 at the IEEE Transactions Information Theory) stated by Budaghyan et al. As an immediate consequence of our results, we prove that the above equation has 2 2 n solutions for one value of b , 2 2 n − 2 n solutions for 2 n values of b in F 2 4 n and has at most two solutions for all remaining points b , leading to complete proof of the conjecture raised by Budaghyan et al. We highlight that the recent work of Li et al., in [9] gives the complete differential spectrum of F and also gives an affirmative answer to the conjecture of Budaghyan et al. However, we emphasize that our approach is interesting and promising by being different from Li et al. Indeed, on the opposite to their article, our technique allows to determine ultimately the set of b 's for which the considered equation has solutions as well as the solutions of the equation for any b in F 2 4 n . [ABSTRACT FROM AUTHOR]

A Walsh zero space (WZ space) for f : F 2 n → F 2 n is an n-dimensional vector subspace of F 2 n × F 2 n whose all nonzero elements are Walsh zeros of f. We provide several theoretical and computer-free constructions of WZ spaces for Gold APN functions f (x) = x 2 i + 1 on F 2 n where n is odd and gcd (i , n) = 1 . We also provide several constructions of trivially intersecting pairs of such spaces. We illustrate applications of our constructions that include constructing APN permutations that are CCZ equivalent to f but not extended affine equivalent to f or its compositional inverse. [ABSTRACT FROM AUTHOR]

In this paper, we establish a lower bound on the total number of inequivalent APN functions on the finite field with 2 2 m elements, where m is even. We obtain this result by proving that the APN functions introduced by Pott and the second author [22] , which depend on three parameters k , s and α , are pairwise inequivalent for distinct choices of the parameters k and s. Moreover, we determine the automorphism group of these APN functions. [ABSTRACT FROM AUTHOR]

The problem of finding APN permutations of F 2 n where n is even and n > 6 has been called the Big APN Problem. Li, Li, Helleseth and Qu recently characterized APN functions defined on F q 2 of the form f(x) = x3q + a1x2q+ 1 + a2xq+ 2 + a3x3, where q = 2m and m ≥ 4. We will call functions of this form Kim-type functions because they generalize the form of the Kim function that was used to construct an APN permutation of F 2 6 . We prove that Kim-type APN functions with m ≥ 4 (previously characterized by Li, Li, Helleseth, and Qu) are affine equivalent to one of two Gold functions G1(x) = x3 or G 2 (x) = x 2 m − 1 + 1 . Combined with the recent result of Göloğlu and Langevin who proved that, for even n, Gold APN functions are never CCZ equivalent to permutations, it follows that for m ≥ 4 Kim-type APN functions on F 2 2 m are never CCZ equivalent to permutations. [ABSTRACT FROM AUTHOR]