Functions Of Many-valued Logic Research Articles

Synthesis method for S-boxes satisfying the criterion of correlation immunity of Boolean and 4-functions

The improvement of cryptanalysis methods, in particular, using the mathematical apparatus of many-valued logic functions, determines the need to create methods for the synthesis of cryptographic primitives, first of all, S-boxes that satisfy the criteria of cryptographic quality both in the sense of component Boolean functions, and in the sense of their possible representations by functions of many-valued logic. The most important criterion characterizing the ability of the S-box, and hence, of the entire cryptographic transformation in which it is used, to resist attacks of correlation cryptoanalysis, is the criterion of the correlation immunity of component functions. At the moment, there are no methods for the synthesis of S-boxes that satisfy the criterion of correlation immunity both in the sense of component Boolean functions and in the sense of component functions of many-valued logic, which complicates the further development of symmetric cryptographic algorithms, considering their possible representation using many-valued logic functions. In this paper, we propose a method for the synthesis of a full class of S-boxes of length N=16 corresponding to both the criterion of the correlation immunity of component Boolean functions and the criterion of the correlation immunity of component 4-functions. With the help of the developed method, it was possible to construct a set of 18 304 S-boxes that satisfies the criterion of correlation immunity both in the sense of component Boolean functions and in the sense of component 4-functions. The synthesized class of S-boxes can be used to increase the diffusion and confusion of modern symmetric cryptographic algorithms, as well as to increase their protection against promising cryptanalysis attacks based on many-valued logic functions.

A method for synthesis of S-boxes with good avalanche characteristics of component Boolean and quaternary functions

The widespread use of block symmetric ciphers in modern information security systems makes the task of increasing their efficiency urgent, as well as the task of synthesizing high-quality cryptographic primitives used in modern block symmetric ciphers, primarily S-boxes. In particular, the S-boxes must have good avalanche characteristics not only of their component Boolean functions, but also of the component functions of many-valued logic. In this paper, we propose a recursive method for the synthesis of large lengths S-boxes that satisfy the strict avalanche criterion of component 4-functions. As a raw material for the construction of such an S-boxes, small-length S-boxes are used that satisfy the strict avalanche criterion of component 4-functions. In the constructed set of S-boxes that satisfy the strict avalanche criterion of component 4-functions, we found those that simultaneously satisfy the criterion of the maximum avalanche effect of component Boolean functions. The avalanche characteristics of these S-boxes are of high quality both from the point of view of their representation with help of Boolean functions, and from the point of view of their representation using 4-functions. These S-boxes can be recommended for practical use to modernize existing symmetric cryptographic algorithms, as well as to build new perspective ciphers.

Nonlinearity of Many-Valued Logic Component Functions of Modern Cryptographic Algorithms S-boxes

Block symmetric cryptographic algorithms are the essential and inalienable element of any information security system. The main component of block symmetric cryptographic algorithms is the S-box, which largely determines the speed and cryptographic quality of the entire cryptographic transformation. The rapid development of cryptanalysis methods, including the development of quantum computers, requires a more careful research of all possible representations of nonlinear elements of block symmetric cryptographic algorithms. In view of the fact that in addition to the mathematical apparatus of Boolean functions used today to estimate the cryptographic quality of S-boxes, a cryptanalyst can apply the mathematical apparatus of functions of many-valued logic, which makes relevant the task of researching and comparing the cryptographic quality of S-boxes of modern cryptographic algorithms, represented by functions of many-valued logic. This paper presents a research and comparative analysis of nonlinear transformations of the cryptographic algorithms AES, Kalyna, BelT, and Kuznechik when represented by functions of many-valued logic. It was found that only the nonlinear transformation of the BelT cryptographic algorithm is characterized by growth when represented by many-valued logic functions, while the Kalyna cryptographic algorithm demonstrates the greatest decrease in nonlinearity when represented by 16-functions among the cryptographic algorithms researched. The results obtained in the paper indicate a significant unused reserve of cryptographic quality, which can be used in the design of new cryptographic algorithms and their structural elements.

On classes of functions of many-valued logic with minimal logarithmic growth rate

AbstractWe obtain a criterion for the minimal logarithmic growth rate for an arbitrary set with a given set of operations defined on it, i.e., we describe all finite sets A with operations on them such that the growth rate differs by at most a constant from the logarithmic growth rate to base ∣A∣.

AVALANCHE CHARACTERISTICS OF CRYPTOGRAPHIC FUNCTIONS OF TERNARY LOGIC

Context. The development and aplication of cryptographic algorithms based on many-valued logic functions makes it important to research their cryptographic properties and develop effective criteria for the cryptographic quality of their components. The development of efficient methods for the synthesis of high-quality cryptographic primitives based on the functions of many-valued logic is also an important task. The object of this research is the process of improving the efficiency of cryptographic algorithms based on many-valued logic functions. Objective. The purpose of this paper is to generalize the error propagation criterion and the strict avalanche criterion for the case of functions of three-valued logic. Method. The emergence of cryptography based on many-valued logic functions led to the understanding that today’s dominant cryptographic algorithms based on binary algebraic constructions are only a special case of more general trends. Numerous researches show that the use of cryptographic constructions based on many-valued logic functions leads to the creation of cryptoalgorithms that more fully implement the principles of diffusion and confusion. One of the most important cases of many-valued logic functions are 3-functions, which are also used in quantum cryptography. This article is another step towards developing cryptographic constructions based on many-valued logic functions. Results. The definition of the propagation criterion was extended to the case of functions of three-valued logic. On the basis of the propagation criterion for the functions of three-valued logic, the definition of a strict avalanche criterion was introduced, which describes the stability of cryptographic constructions against differential cryptanalysis attacks. We experimentally determined the number of 3-functions of length N=9, satisfying the strict avalanche criterion. A method based on three constructive rules is proposed, which allows to synthesize a complete set of 864 S-boxes of length N=9 satisfying strict avalanche criterion. This set of Sboxes is basic for the application of Kim’s construction, which allows to recurrently increase the length of the S-box to the required value. The paper shows that using Kim’s construction to increase the length preserves the S-box satisfying to a strict avalanche criterion, while allowing to obtain S-boxes with satisfactory non-linearity value as well as small output and input vectors correlation. Conclusions. The most important criterion of cryptographic quality, which shows the stability of the cryptographic algorithm to attacks of differential cryptanalysis is the propagation criterion that was generalized to the case of 3-functions. The existence of 3- functions of length N=9 satisfying the strict avalanche criterion is shown, and their full set is found. On the basis of the proposed constructive method, a complete set of S-boxes of length N=9 that satisfy the strict avalanche criterion was synthesized. It is shown that the Kim scheme can be applied to recurrently increase the length of S-boxes based on many-valued logic functions. As an actual direction for the continuation of the research, the development of regular and constructive methods for th

On bases of closed classes of vector functions of many-valued logic

AbstractWe consider the functional system of vector functions of many-valued logic with the naturally defined operation of superposition and construct examples of closed classes of special type without a basis and with a countable basis.

On bases of closed classes of vector functions of many-valued logic

Рассматривается функциональная система вектор-функций многозначной логики с естественным образом определенной операцией суперпозиции. Построены примеры замкнутых классов специального вида без базиса и со счeтным базисом.

Certain sufficient conditions of uniformity for systems of functions of many-valued logic

For any finite system A of functions of the k-valued logic taking values in the set E s = {0,1,…, s − 1}, k ≥ s ≥ 2, such that the closed class generated by restriction of functions from A on the set E s contains a near-unanimity function, it is proved that there exist constants c and d such that for an arbitrary function f ∈ [A] the depth D A (f) and the complexity L A (f) of f in the class of formulas over A satisfy the relation D A (f) ≤ clog2 L A (f) + d.

Uniformity of a certain systems of functions of many-valued logic

For any finite system A of functions of many-valued logic taking values in the set {0,1} such that a projection of A generates the class of all monotone boolean functions, it is proved that there exists constants c and d such that for an arbitrary function f e [A] the depth D(f) and the complexity L(f) of f in the class of formulas over A satisfy the relation D(f) ≤ clog2L(f) + d.

FE classification of functions of many-valued logic

On the set of functions of many-valued logic, we consider the closure operator defined on the basis of systems of functional equations (the operator of FE closure). This operator generates an FE classification of the functions of many-valued logic, whose kernel consists of classes of SG-type defined by groups of permutations of G. A number of results are obtained to guarantee FE precompleteness of classes of only this type in the classes of SG type. The results obtained are illustrated by examples of functions of 2-, 3-, and 4-valued logics.

The uniform id-decomposition of functions of many-valued logic over homogeneous functions

We consider the uniform id -decomposition of functions of the class P k of functions of k -valued logic over the class of structural homogeneous functions and the class D k of homogeneous functions generated by the dual discriminator d . We find the degrees of the uniform id -decomposition of the class P k over the classes and D k and give the methods of construction of homogeneous functions over which the uniform id -decomposition of the class P k is realised.

A-classification of idempotent functions of many-valued logic

A-closure on the set of functions of many-valued logic is defined as the closure under the operations of superposition and of taking the dual function with respect to a permutation in the alternating group. A-classification of functions of many-valued logic is defined with the use of the A-closure operator. The class I k of idempotent functions is one of two (for k⩾5) and one of four (for k=4) A-precomplete classes in P k . Twelve types of standard relations, which are called main, are defined on the set E k . It is proved that each A-closed class of functions from I k can be defined by a suitable choice of the main relations.

A-closed classes of idempotent functions of many-valued logic definable by binary relations

A-closure on a set of functions of many-valued logic is defined as the closure with respect to the operations of superposition and transition to dual functions with respect to permutations of the alternating group. The class I k of idempotent functions for k⩾5 is one of two, and for k=4 is one of four A-precomplete classes in P k . We define 12 types of standard relations over the set E k which are called basic. We prove that any A-closed class of functions in I k which is defined by arbitrary binary relations can be also specified by means of a suitable set of basic relations.

On expressibility of functions of many-valued logic in some logical-functional languages

Article On expressibility of functions of many-valued logic in some logical-functional languages was published on January 1, 1999 in the journal Discrete Mathematics and Applications (volume 9, issue 6).

The S-classification of functions of many-valued logic

On the set of functions of many-valued logic, we consider the closure operator defined on the basis of systems of functional equations (the operator of FE closure). This operator generates an FE classification of the functions of many-valued logic, whose kernel consists of classes of SG-type defined by groups of permutations of G. A number of results are obtained to guarantee FE precompleteness of classes of only this type in the classes of SG type. The results obtained are illustrated by examples of functions of 2-, 3-, and 4-valued logics.

Characteristic of some algebras of functions of many-valued logic

Characteristic of some algebras of functions of many-valued logic

Extension of not everywhere defined functions of many-valued logic

Extension of not everywhere defined functions of many-valued logic

Monotonic functions of many-valued logic and supermatroids

The accuracy of the gradient method of solving the convex integer-valued programming problem is studied in terms of monotonic functions of many-valued logic. To this end, special monotonic function-supermatroids are investigated.

The realization of functions of the Pk by formulae (k ? 3)

In this note we study the realization of the functions of many-valued logic by formulae in a special basis subject to the condition that the functions of the basis are of arbitrary weight. An asymptotic bound is obtained for Shannon's function in this case.

Minimization methods for functions of many-valued logic

Minimization methods for functions of many-valued logic