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Introduction to Cryptography

A brief historical overview of the development of cryptography: Atbashi, Stsitala, plaque Aeneas, Polybius square ciphers; Cæsar cipher; Alberti encryption disk; Vigenère, Richelieu, Napoleon ciphers; Vernam code; "Enigma" disk encryptor.
Block and stream ciphers. A mathematical model of substitution cipher. Attacks on ciphers, perfect ciphers, Shannon's theorem. Modes of block ciphers. GOST 28147-89 and DES encryption standards.
General methods of cryptanalysis. "Brute force" attack. Analytical methods. "Meet in the middle" attack. "Divide and conquer" principle. Correlation attacks on stream ciphers. Cryptanalysis of Vigenère cipher.
Statistical methods for cryptanalysis. Linear and differential cryptanalysis of block ciphers. Examples of attacks on block ciphers.
Cryptographic protocols. Authentication protocols using digital signature (EDS).
A formal model of protocol-based EDS-sided function. The concept of a hash function. Methods of constructing hash functions. Basic properties of hash functions. Key and keyless hash functions.
Key management system. Preliminary distribution of secret keys. Forward keys.
Methods of distribution of public keys. Public keys infrastructure. Public keys certification center.
Cryptographic tools and methods of data protection software. Basic requirements for data security in software systems. Means and methods of protection.
Integrity monitoring software.

Lecturer: associate professor Primenko E. A.

Mathematical Theory of Information

Information and entropy of a finite probabilistic scheme. Khinchin and Faddeev axioms. Uniqueness theorem for the type of the function of entropy. A theorem on the connection between entropy of probabilistic scheme with entropy AB and schemes A and B. Mutual information, the average mutual information, proprietary information, average own information. Relationships additivity of information. A theorem on monotonicity of mutual information with surjective mappings schemes.
A theorem on the convexity of the average mutual information.
Message sources. A mathematical model of a discrete message source. Entropy of the source. Discrete source without memory. First Shannon's theorem for the sources of this type. Second Shannon's theorem for the sources without memory and the corollary of this theorem. Markov sources. Stepping entropy and entropy of the sign for the Markov source. First Shannon's theorem for Markov sources.
Ergodic source. Birkhoff's theorem. Examples of ergodic and nonergodic sources.
Entropy on the sign and the steps entropy for an ergodic source. Macmillan's theorem.
Optimal Coding. The concepts of the theory of "compressive codes". Kraft's inequality. R. Fano's and K. Shannon's coding algorithms. An optimal codes and its properties. Huffman's algorithm. Examples of optimal codes. Huffman's Algorithm for d-alphabets. The theorem on the upper and lower limits for the average length of codeword.

Lecturer: associate professor Dukhin A. A.

Mathematical Bases of Cryptology

Number theory models in cryptology. Euclidean algorithm. Representation Theorem for GCD. The fundamental theorem of arithmetic. Properties of prime divisors.
Comparisons and their properties. Algorithm for computing the inverse element. Chinese remainder theorem. Euler's totient function and its properties. Möbius function and its properties. RSA cryptosystem. Fast exponentiation. Quadratic residues, Legendre symbol and its properties. Algorithms for the solution of comparisons of the second degree. Jacobi symbol and its properties. Blum's numbers and their properties. Quadratic residues and the factorization problem.
Algebraic models in cryptology. Groups, Lagrange's theorem. Cyclic group, order of the element, the number of generators, the criterion for a generator. The fundamental theorem of finite abelian group. Theorems on the multiplicative cyclic group modulo a prime power. Permutation group on a finite set. Multiple transitivity and primitivity. Systems of generators of symmetric and alternating groups. Rings, ideals, principal ideals. The ring of polynomials over a field, Bézout's theorem, Euclidean algorithm. Finite fields and their structure.
Theorem of the primitive element. Primitive polynomial and its properties, the criterion of primitivity of a polynomial. Algorithms for computing discrete logarithms in a finite field. ElGamal cryptosystem, Diffie-Hellman public key exchange protocol.
Elliptic curves over finite fields. An elliptic curve over a prime field.
Addition operation of elliptic curve's points. f Postnikov's theorem.Theorems on the order of elliptic curve for certain types of primes. Hasse's theorem
Elliptic curves over fields of characteristic two. Supersingular and nonsupersingular curves and their properties.
Mathematical models for the generation of pseudorandom sequences. Linear recurrent sequences over a finite field. The characteristic polynomial of LRP, period, condition of periodicity. Annihilating polynomial of LRS. The period of the sum of two LRS with characteristic relatively prime polynomials.
Relationship between LRP and the period of its characteristic polynomial.Construction of an LRS of maximal period, linear feedback shift registers.

Lecturer: associate professor Primenko E. A.

Theoretical Bases of Computer Security

The technology of building secure computer systems. Business processes and information support. Opponents, damage, threat, vulnerability. Policy. Risks.
Axiom of security as a firewall.
Classification and categorization of information. The proof of consistency and completeness. Automatic categorization.
Discretionary security policy. Insecurity to the attacks with a Trojan horse. The coplexity of control distribution rights in the discretionary policy problem. "Take-Grant" model.
Role-permission model for security policy. A theorem on the relationship of role model and discretionary policy.
The simplest information flows. Multi-Level Security policy (MLS). MLS resistance to attacks by Trojan horse. Terms of maintaining safe conditions for the functioning of the system with multilevel security policy. Bell-La Padulla Model. BST theorem. "LOW WATER MARK" model.
Complex information flows. Hidden channels. Examples of security violations through a hidden channels. Invisibility of violation by protection system.
The model of an isolated software environment.
Distributed Systems. Critical systems. Rationale for a new core set of security requirements for large distributed systems. Threats in distributed systems. The simplest model of secure distributed systems.
Mechanisms of protection.
The architectural implementation of multilevel security policy. Proof of security for streams in general.

Lecturer: professor Grusho A. A.

Coding Theory

Error-correcting codes. The notion of a linear code, generating and check matrices. Minimum distance. Error detection and correction. Systematic codes. Boundaries for the parameters of codes. Perfect and equidistant codes. Decoding of linear codes using a standard location, the modified method using syndromes.
The method of sequential decoding. Hamming code and its properties. The generating matrix of the Hamming code. Methods of constructing new codes from the specified. MacWilliams identities. Hadamard codes and Hadamard matrices, Reed-Muller codes. Irreducible polynomials over finite fields. Cyclic codes, generators and the check matrix generating polynomials and verification.
Algebraic structure of cyclic codes. Bose-Chaudhuri-Hoquenghem codes. A theorem on the distance of BCH-code. Building codes correcting a given number of errors.
Discrete channels without memory and associated coding theorems. Discrete channels without memory and its capacity. Calculating the capacity of communication channels, symmetric with respect to entry, symmetric. Theorem on the probability distribution that maximizes mutual information between input and output channels. An iterative algorithm for finding the capacity of a discrete channel without memory. Finding the channel capacity in the case of a nondegenerate matrix of transition probabilities. Direct and converse Shannon's theorems for the binary symmetric channel. Theorems for a discrete source without memory. Converse Wolfowitz theorem.

Lecturer: associate professor Dukhin A. A.

The Protection of Information Processes in Computer Systems

The technology of building secure computer systems. Business processes and information support. Opponents, damage, threat, vulnerability. Policy. Risks. Protecting access as an axiom of security. Attack as a basic concept of computer security. Some characteristics of the attacks.

Hierarchical decomposition of computer systems, wide telecommunications, databases, distributed systems.
Undeclared processors' capabilities. On the possible existence of "invisible" agents in the processors.
Invisibility to protect the agents on the OS level (execution environment nevliyaniya).
Theorems on the possibility of creating a network of agents that are invisible to the remedies.
Hostile multi-agent systems. The possibility of invisible attack of the application. Attacks on LDAP.
The problem of agent interaction with the environment. Firewalls. IPsec.
Attacks on VPN protected segments using hidden channels.
Construction of steganographic channels using IP-protocol.
Methods of embedding hostile software into a computer system.
Anonymous calculations as the protection of computing in a hostile environment.
Architectural methods of protection.

Lecturer: professor Grusho A. A.

Cryptographic Protocols

The notion of cryptographic protocol. Secret sharing protocols. Voting protocols.
Remote coin flip protocols. Playing poker on the phone.
Authentication and digital signature protocols. EDS standards.
Hashing algorithms.
Zero-knowledge proofs.
Distribution of secret keys protocols. Open distribution of keys. Protocols of the prior distribution of secret keys.
Hidden transmission.
Models of cryptographic protocols on elliptic curves.

Lecturer: associate professor Primenko E. A.