Discrete mathematicsTopics

The main concepts and elements of combinatorics, principles of counting in finite sets and methods of solving combinatorial problems are studied within the subject. the concept of the cartesian product, the construction of the cartesian product of sets and its main properties are reviewed. also, the concept of boolean sets, their structure and application in discrete mathematics is investigated.

In the framework of the topic, the concept of relationship and the mathematical expression of relationships between the elements of sets are studied. the concept of binary relationship, its methods of presentation and main properties are considered. the equivalence relation, its properties of reflexivity, symmetry and transitivity, as well as the principles of dividing the set into equivalence classes are also investigated.

Within the scope of the topic, the concept of a function, the domain of a function and the set of values ​​are studied. the main types of functions and methods of their assignment, as well as the composition of functions and their main properties are considered. also, the concept of the power of a set, the principles of determining and comparing the power of finite and infinite sets are examined.

The main concepts of mathematical logic and logical operations are studied within the subject. the essence, truth tables and basic properties of negation, conjunction, disjunction, implication and equivalence operations are reviewed. also, the construction, analysis and application of logical expressions in solving various practical issues are investigated.

Within the subject, the concept of logical formula, the structure and main features of formulas are studied. the concept of binary function and its relationship with logical functions are considered. the principle of duality, the construction of dual forms of logical expressions and the application of this principle in boolean algebra are also investigated.

Detection and evaluation of engineering conditions. determining the boundaries of spill and fire zones. problem solving.

In the framework of the topic, methods of separation of bul functions according to variables and their main features are studied. the principles of expression of bul functions in the form of perfect disjunctive normal form (pmf) and perfect conjunctive normal form (mknf) are considered. also, the concept of jegalkin polynomial, presentation of boolean functions in algebraic normal form and methods of constructing jegalkin polynomial are investigated.

The basic concepts of algebraic structures are studied within the subject. the concept of group, the main properties of group operation, neutral and inverse elements are reviewed. the concepts of ring and square, their main properties and differences between them are explained. the role of group, ring, and field structures in discrete mathematics and other applied fields is also explored.

Within the subject, the basic principles of calculations and the concept of comparison are studied by module. addition, subtraction, multiplication and exponentiation operations on integers are considered. the concept of the class of residues (z_m), the structure of the classes of residues and the principles of performing algebraic operations on them are investigated.

The main numerical functions used in number theory and their properties are studied within the subject. basic concepts related to the euler function and modulus calculations are reviewed. mathematical basis of RSA algorithm, selection of prime numbers, generation of public and private keys, encryption and decryption processes are explained. also, the security of the RSA algorithm based on the problem of dividing large integers into primes is investigated.

In the framework of the topic, finite periodic groups, their structure and main properties are studied. group composition, element composition, and generator concepts are reviewed. the nature of primitive elements and their role in finite periodic groups are explained. also, the mathematical nature of the discrete logarithmization problem, the computational complexity of its solution, and its role in the security of cryptographic systems are investigated.

The concept of finite field, its structure and main properties are studied within the subject. the principles of constructing simple finite squares and performing basic algebraic operations on their elements are considered. the structure of extended finite squares, their relationship with simple finite squares and their main properties are explained. also, the application of finite fields in coding, cryptography and other computational fields is investigated.

Within the framework of the topic, the principles of constructing finite squares and their application in cryptography are studied. the structure of simple and extended finite squares, the creation of extended squares based on irreducible polynomials, and the performance of algebraic operations on square elements are considered. the structure of the finite square (gf(2^8)) used in the AES cipher, the polynomial expression of bytes, and the addition and multiplication operations on the finite square are explained. also, the role of finite field operations in the basic transformations of the AES algorithm is investigated.

Within the subject, the concept of primitive polynomial, its relation with finite fields and its main properties are studied. the differences between irreducible and primitive polynomials, the principles of defining primitive polynomials, and their role in the construction of finite fields are reviewed. also, the application of primitive polynomials in generating pseudorandom sequences, coding and cryptographic systems is investigated.

The main concepts and elements of graph theory are studied within the subject. concepts of graph, vertex, thread, directed and undirected, weighted and unweighted graph, as well as path and connectivity in graphs are reviewed. the working principle of dijkstra's algorithm and its application in finding the shortest paths from a given starting vertex to other vertices in weighted graphs are explained. also, the main stages of the algorithm and application conditions are investigated.