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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]

Little theoretical work has been done on (n, m)-functions when n 2 < m < n , even though these functions can be used in Feistel ciphers, and actually play an important role in several block ciphers. Nyberg has shown that the differential uniformity of such functions is bounded below by 2 n - m + 2 if n is odd or if m > n 2 . In this paper, we first characterize the differential uniformity of those (n, m)-functions of the form F (x , z) = ϕ (z) I (x) , where I(x) is the (m, m)-inverse function and ϕ (z) is an (n - m , m) -function. Using this characterization, we construct an infinite family of differentially Δ -uniform (2 m - 1 , m) -functions with m ≥ 3 achieving Nyberg's bound with equality, which also have high nonlinearity and not too low algebraic degree. We then discuss an infinite family of differentially 4-uniform (m + 1 , m) -functions in this form, which leads to many differentially 4-uniform permutations. We also present a method to construct infinite families of (m + k , m) -functions with low differential uniformity and construct an infinite family of (2 m - 2 , m) -functions with Δ ≤ 2 m - 1 - 2 m - 6 + 2 for any m ≥ 8 . The constructed functions in this paper may provide more choices for the design of Feistel ciphers. [ABSTRACT FROM AUTHOR]

Using recent results on solving the equation X 2 k + 1 + X + a = 0 over a finite field F 2 n provided by the second and the third authors, we address an open question raised by the first author in WAIFI 2014 concerning the APN-ness of the Kasami functions x ↦ x 2 2 k - 2 k + 1 with gcd (k , n) = 1 , x ∈ F 2 n . [ABSTRACT FROM AUTHOR]

For every positive integers $n$ and $m$ and every even positive integer $\delta $ , we derive inequalities satisfied by the Walsh transforms of all vectorial $(n,m)$ -functions and prove that the case of equality characterizes differential $\delta $ -uniformity. This provides a generalization to all differentially $\delta $ -uniform functions of the characterization of Almost Perfect Nonlinear (APN) functions due to Chabaud and Vaudenay, by means of the fourth moment of the Walsh transform. Such generalization has been missing since the introduction of the notion of differential uniformity by Nyberg in 1994 and since Chabaud-Vaudenay’s result in the same year. Moreover, for each even $\delta \geq 2$ , we find several (in fact, an infinity of) such characterizations. In particular, when $\delta =2$ and $\delta =4$ , we have that, for any $(n,n)$ -function [resp. any $(n,n-1)$ -function)], the arithmetic mean of $W_{F}^{2}(u_{1},v_{1})W_{F}^{2}(u_{2},v_{2})W_{F}^{2}(u_{1}+u_{2},v_{1}+v_{2})$ when $u_{1},u_{2}$ range independently over ${\mathbb F}_{2}^{n}$ and $v_{1},v_{2}$ are nonzero and distinct and range independently over ${\mathbb F}_{2}^{m}$ is at least $2^{3n}$ , and that $F$ is APN (resp. is differentially 4-uniform) if and only if this arithmetic mean equals $2^{3n}$ (which is the value we would get with a bent function if such function could exist). These inequalities give more knowledge on the Walsh spectrum of $(n,m)$ -functions. We deduce in particular a property of the Walsh support of highly nonlinear functions. We also consider the completely open question of knowing if the nonlinearity of APN functions is necessarily non-weak (as it is the case for known APN functions); we prove new lower bounds which cover all power APN functions (and hence a large part of known APN functions), which explain why their nonlinearities are not bad, and we discuss the question of the nonlinearity of APN quadratic functions (since almost all other known APN functions are quadratic). [ABSTRACT FROM AUTHOR]

Almost perfect nonlinear (APN) functions on finite fields of characteristic two have been studied by many researchers. Such functions have useful properties and applications in cryptography, finite geometries and so on. However, APN functions on finite fields of odd characteristic do not satisfy desired properties. In this paper, we modify the definition of APN function in the case of odd characteristic, and study its properties. [ABSTRACT FROM AUTHOR]

NSUCRYPTO is the unique cryptographic Olympiad containing scientific mathematical problems for professionals, school and university students from any country. Its aim is to involve young researchers in solving curious and tough scientific problems of modern cryptography. From the very beginning, the concept of the Olympiad was not to focus on solving olympic tasks but on including unsolved research problems at the intersection of mathematics and cryptography. The Olympiad history starts in 2014. In 2019, it was held for the sixth time. We present the problems and their solutions of the Sixth International Olympiad in cryptography NSUCRYPTO 2019. Under consideration are the problems related to attacks on ciphers and hash functions, protocols, Boolean functions, Dickson polynomials, prime numbers, rotor machines, etc. We discuss several open problems on mathematical countermeasures to side-channel attacks, APN involutions, S-boxes, etc. The problem of finding a collision for the hash function Curl27 was partially solved during the Olympiad. [ABSTRACT FROM AUTHOR]

We consider binomial functions over a finite field of order 2n. Some necessary condition is found for such a binomial function to be a permutation. It is proved that there are no permutation binomial functions in the case that 2n − 1 is prime. Permutation binomial functions are constructed in the case when n is composite and found for n ≥ 8. [ABSTRACT FROM AUTHOR]