1st Class Discrete Structures الهياكل المتقطعة أستاذة المادة: م. م

References : 1- Theory and problems of Discrete mathematics, by Seymour Lipschutz & Marc Lars Lipson, Schaum’s Outline Series, third edition 2007 2- DISCRETE STRUCTURES, AMIN WITNO, Revision Notes and Problems 2006, www.witno.com 3- Discrete mathematics for New technology, Rowan Garnier & John Taylor (Second Edition 2002) 4- Discrete mathematical structures for computer science by Bernard Kolman & Robert C. Busby 5- http://www.math.uvic.ca/faculty/gmacgill/guide 6- http://rutherglen.ics.mq.edu.au/wchen/lndmfolder/lndm.html - 7- http://en.wikibooks.org/wiki/Discrete_mathematics/Set_theory

Syllabus 1- Fundamental of Computer Security • Computer Security Definition • Objective of Computer Security • Kinds of Security Breaches 2- Cryptography • Cryptographic Terminology • Substitution Cipher - Monoalphabetic - Homophonic - Polyalphabetic - Polygram • Transposition Cipher - Fixed Period - Columnar

Syllabus 3-Cryptanalysis • Symmetric Cryptography ( classical cipher system ) • Asymmetric Cryptography 4- Information Hiding 5- User Authentication 6- Malicious Code - Virus - Trojan Hours - Worm 7- Network Security

COMPUTER AND NETWORK SECURITY INTRODUCTION: Cryptograph: Is the science and study of secret writing. Cipher: Is a secret method of writing. Where by plaintext (or clear text) is transformed in to ciphertext (cryptogram). Encipherment (encryption): The process of transforming plaintext in to ciphertext. Decipherment(Decryption): Is the reverse process of encipherment,of transforming ciphertext into plaintext. Cryptanalysis: The art and science of breaking ciphertext.

Types of Key-based algorithems : 1- Symmetric algorithms (privet key):- are algorithms where the encryption key can be calculated from the decryption key and vice versa. In most symmetric algorithms, the encryption key and the decryption key are the same. Symmetric algorithms can be divided in to two categories: Stream cipher: which operator on the plaintext a single bit at a time. Block cipher: which operator on the plaintext in a group of bits called blocks at a time. 2- Asymmetric algorithms (public key):- Are designed so that the key used for encryption is different from the key used from decryption and cant'nt easily calculated from each others.

Encryption and Decryption symmetric Process Encipher plaintext Key ciphertext Decipher Figure -1 Encryption and Decryption symmetric Process K1 Encipher plaintext ciphertext Decipher K2 Figure -2 Encryption and Decryption Asymmetric Process

Cryptography classification Cryptology Figure-3 Cryptography classification Cryptography Cryptanalysis Asymmetric Symmetric Block cipher Stream cipher Block cipher Cipher system Modern Conventional Figure-4 Cipher system classification Asymmetric Symmetric Block cipher Stream cipher Block cipher

CLASSICAL CIPHER SYSTEMS Classical ciphers have been used since ancient Egypt to exchange classified messages between authorized persons. Since then, different methods and techniques are used in order to increase security level of such information. Most of theses methods and techniques are based on the idea that each natural language has its own distribution characteristics. The encryption process aims to uniformly flatten the statistical characteristics of language and obscure any correlation and dependencies between plaintext and ciphertext by diffusion and confusion. Cryptographic systems have been classified into two systems: Symmetric (one-key) cryptosystems: In symmetric or one-key systems, the enciphering and deciphering key are the same. Asymmetric ( two-key) cryptosystem: In asymmetric or two-key systems, enciphering and deciphering keys are different in such away that at least one key is computationally infeasible to determine from the other.

1- TRNSPOSITION CIPHERS: Transposition ciphers rearrange characters according to some scheme. This is done classically by some geometric figure, the figure is 2-dimentional array, and often called "permutation". Example: Encrypt the word “RENAISSANCE" using 3x4 figure, using [2 4 1 3] scheme. 1 2 3 4 Ciphertext: ESCAARINNSE. Many transposition ciphers permute characters of the plaintext with fixed period d. let Zd be the integers 1….d, and f : Zd → Zd be a permutation over Zd, then successive blocks of d characters are encrypted by permuting the characters according to f. M = m1 …md,md+1,…...m2d,…..is encrypted as Ek(M) = mf(1)….mf(d), md+f(1) … md+f(d) R E N A I S C

1- TRNSPOSITION CIPHERS: Example: suppose d = 4 and fE =[ 2 4 1 3] and FD = [ 3 1 4 2] ,thus: I 1 2 3 4 F(i) M = RENA ISSA NCE Ex(M) = EARN SAIS CNE Dx(C) = RENA ISSA NCE

SIMPLE SUBSTITUTION CIPHERS: In simple substitution (or mono alphabetic) ciphers, each character of the plaintext replaced with a corresponding character of ciphertext. A single one-to-one mapping function (f) from plaintext to ciphertext character is used to encrypt the entire message using the same key (k); such that: Ek(M) = F(m1) F(m2) … ..F(mN) =C Where: N : is the length of the message. M : is plaintext message given by M = ( m1, m2, … ..,mN). C : is ciphertext message given by C = (c1,c2,… .., cN). There are many types of simple substitution ciphers according to its equations used to encryption, they: Shifted alphabet (Caesar cipher): F(a) = (a + k) mod n Where k : is the number of positions to be shifted. a : is a single character of the alphabet> n : is the size of the alphabet.

SIMPLE SUBSTITUTION CIPHERS: Example: If k =3 then we can encrypt the following message as: M = R E N A I S S A N C E Ek(M) = U H Q D L V V D Q F H