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Orthogonal Arrays Theory And Applications

In this work, we employ the concept of {\em composite representation} of Boolean functions, which represents an arbitrary Boolean function as a composition of one Boolean function and one vectorial function, for the purpose of specifying new secondary constructions of bent/plateaued functions. This representation gives a better understanding of the existing secondary constructions and it also allows us to provide a general construction framework of these objects. This framework essentially gives rise to an {\em infinite number} of possibilities to specify such secondary construction methods (with some induced sufficient conditions imposed on initial functions) and in particular we solve several open problems in this context. We provide several explicit methods for specifying new classes of bent/plateaued functions and demonstrate through examples that the imposed initial conditions can be easily satisfied. Our approach is especially efficient when defining new bent/plateaued functions on larger variable spaces than initial functions. For instance, it is shown that the indirect sum methods and Rothaus' construction are just special cases of this general framework and some explicit extensions of these methods are given. In particular, similarly to the basic indirect sum method of Carlet, we show that it is possible to derive (many) secondary constructions of bent functions without any additional condition on initial functions apart from the requirement that these are bent functions. In another direction, a few construction methods that generalize the secondary constructions which do not extend the variable space of the employed initial functions are also proposed.

A general framework for secondary constructions of bent and plateaued functions

In this work, we employ the concept of composite representation of Boolean functions, which represents an arbitrary Boolean function as a composition of one Boolean function and one vectorial function, for the purpose of specifying new secondary constructions of bent/plateaued functions. This representation gives a better understanding of the existing secondary constructions and it also allows us to provide a general construction framework of these objects. This framework essentially gives rise to an infinite number of possibilities to specify such secondary construction methods (with some induced sufficient conditions imposed on initial functions) and in particular we solve several open problems in this context. We provide several explicit methods for specifying new classes of bent/plateaued functions and demonstrate through examples that the imposed initial conditions can be easily satisfied. Our approach is especially efficient when defining new bent/plateaued functions on larger variable spaces than initial functions. For instance, it is shown that the indirect sum methods and Rothaus’ construction are just special cases of this general framework and some explicit extensions of these methods are given. In particular, similarly to the basic indirect sum method of Carlet, we show that it is possible to derive (many) secondary constructions of bent functions without any additional condition on initial functions apart from the requirement that these are bent functions. In another direction, a few construction methods that generalize the secondary constructions which do not extend the variable space of the employed initial functions are also proposed.

A notion of the graph of minimal distances of bent functions is introduced. It is an undirected graph (V, E) where V is the set of all bent functions in 2k variables and $$(f, g) \in E$$
 if the Hamming distance between f and g is equal to $$2^k$$
 . It is shown that the maximum degree of the graph is equal to $$2^k (2^1 + 1) (2^2 + 1) \cdots (2^k + 1)$$
 and all its vertices of maximum degree are quadratic bent functions. It is obtained that the degree of a vertex from Maiorana—McFarland class is not less than $$2^{2k + 1} - 2^k$$
 . It is proven that the graph is connected for $$2k = 2, 4, 6$$
 , disconnected for $$2k \ge 10$$
 and its subgraph induced by all functions EA-equivalent to Maiorana—McFarland bent functions is connected.

The graph of minimal distances of bent functions and its properties

We investigate sets of mutually orthogonal latin squares (MOLS) generated by cellular automata (CA) over finite fields. After introducing how a CA defined by a bipermutive local rule of diameter d over an alphabet of q elements generates a Latin square of order $$q^{d-1}$$, we study the conditions under which two CA generate a pair of orthogonal Latin squares. In particular, we prove that the Latin squares induced by two Linear Bipermutive CA (LBCA) over the finite field $$\mathbb {F}_q$$ are orthogonal if and only if the polynomials associated to their local rules are relatively prime. Next, we enumerate all such pairs of orthogonal Latin squares by counting the pairs of coprime monic polynomials with nonzero constant term and degree n over $$\mathbb {F}_q$$. Finally, we present a construction for families of MOLS based on LBCA, and prove that their cardinality corresponds to the maximum number of pairwise coprime polynomials with nonzero constant term. Although our construction does not yield all such families of MOLS, we show that the resulting lower bound is asymptotically close to their actual number.

Mutually orthogonal latin squares based on cellular automata

A detailed overview of the problems, solutions and experience of the first international student’s Olympiad in cryptography, NSUCRYPTO’2014, is given. We start with rules of participation and description of rounds. All 15 problems of the Olympiad and their solutions are considered in detail. There are discussed solutions of the mathematical problems related to cipher constructing such as studying of differential characteristics of S-boxes, S-box masking, determining of relations between cyclic rotation and additions modulo 2 and 2n, constructing of special linear subspaces in F2 ; problems about the number of solutions of the equation F (x)+F (x+a) = b over the finite field F2n and APN functions. Some unsolved problems in symmetric cryptography are also considered.

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Problems, solutions and experience of the first international student''s Olympiad in cryptography

In this paper1 we show that the minimal Hamming distance in class of bent functions in n variables is equal to 2n/2 We prove that two bent functions are at the minimal distance if and only if they differ on an affine subspace and these functions are affine on it We describe a simple algorithm for constructing bent functions at the minimal distance from the given one We give distribution of Hamming distances for bent functions of small dimension

Bent functions on the minimal distance

A detailed overview of the mathematical problems and their solutions for the Second International Students’ Olympiad in Cryptography (NSUCRYPTO’2015) is given. The authors consider mathemat...

Mathematical problems of the Second International Students’ Olympiad in Cryptography

Problems and their solutions of the Fifth International Students’ Olympiad in cryptography NSUCRYPTO’2018 are presented. We consider problems related to attacks on ciphers and hash functions, Boole...

Nikolay Kolomeec

Papers

The graph of minimal distances of bent functions and its properties

Problems, solutions and experience of the first international student''s Olympiad in cryptography

Bent functions on the minimal distance

Mathematical problems of the Second International Students’ Olympiad in Cryptography

The Fifth International Students’ Olympiad in cryptography—NSUCRYPTO: Problems and their solutions