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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]

Combinatorial t‐designs have nice applications in coding theory, finite geometries, and several engineering areas. A classical method for constructing t‐designs is by the action of a permutation group that is t‐transitive or t‐homogeneous on a point set. This approach produces t‐designs, but may not yield (t+1)‐designs. The objective of this paper is to study how to obtain 3‐designs with 2‐transitive permutation groups. The incidence structure formed by the orbits of a base block under the action of the general affine groups, which are 2‐transitive, is considered. A characterization of such incidence structure to be a 3‐design is presented, and a sufficient condition for the stabilizer of a base block to be trivial is given. With these general results, infinite families of 3‐designs are constructed by employing almost perfect nonlinear functions. Some 3‐designs presented in this paper give rise to self‐dual binary codes or linear codes with optimal or best parameters known. Several conjectures on 3‐designs and binary codes are also presented. [ABSTRACT FROM AUTHOR]

Differentially 4-uniform permutations onwith high nonlinearity are often chosen as substitution boxes in block ciphers. Recently, Quet al.used the powerful switching method to construct permutations with low differential uniformity from the inverse function [Quet al., Constructing differentially 4-uniform permutations over via the switching method, IEEE Trans. Inform. Theory 59(7) (2013), pp. 4675–4686, Quet al., More constructions of differentially 4-uniform permutations on, Des. Codes Cryptogr. DOI 10.1007/s10623-014-0006-x] and proposed a sufficient but not necessary condition for these permutations to be differentially 4-uniform. In this paper, a sufficient and necessary condition is presented. We also give a compact estimation for the number of constructed differentially 4-uniform permutations. Comparing with those constructions in [11,12], the number of functions constructed here is much bigger. As an application, a new class of differentially 4-uniform permutations is constructed. The obtained functions in this paper may provide more choices for the design of substitution boxes. [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]

Almost perfect nonlinear (APN) functions play an important role in the design of block ciphers as they offer the strongest resistance against differential cryptanalysis. Despite more than 25 years of research, only a limited number of APN functions are known. In this paper, we show that a recent construction by Taniguchi provides at least φ (m) 2 2 m + 1 3 m inequivalent APN functions on the finite field with 2 2 m elements, where φ denotes Euler’s totient function. This is a great improvement of previous results: for even m, the best known lower bound has been φ (m) 2 ⌊ m 4 ⌋ + 1 ; for odd m, there has been no such lower bound at all. Moreover, we determine the automorphism group of Taniguchi’s APN functions. [ABSTRACT FROM AUTHOR]

Almost perfect nonlinear (APN) functions are of great interest to many researchers since they have the optimal resistance to the differential attack. The existence of bijective APN functions in even number of variables is an important open problem, and there is only one known example of such a function at present. In this paper we consider a special subclass of 2-to-1 vectorial Boolean functions that can allow us to search and construct APN permutations. We proved that each 2-to-1 function is potentially EA-equivalent to a permutation and proposed an algorithm that generates special symbol sequences for constructing 2-to-1 APN functions. Also, we described two methods for searching APN permutations, that are based on sequences generated by this algorithm. [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]

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]

The switching method is a very powerful method to construct new APN functions and differentially 4-uniform permutations over finite fields of even characteristic. In this paper, using this method, we present several new constructions of infinite classes of nonpower APN functions and two new classes of bent functions in finite fields of odd characteristic. [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]

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]

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]

A method for constructing piecewise differentially 4-uniform permutations has been recently introduced by Zha, Hu and Sun. By using this method, we provide two new infinite families of differentially 4-uniform permutations from the known APN functions. The CCZ inequivalence between these constructed functions and the known differentially 4-uniform permutations are also investigated by computation. [ABSTRACT FROM AUTHOR]