CSE-1215:Discrete Mathematics Lecture - 9 Syeda Tamanna Alam Monisha Lecturer, CSE, Leading University E-mail: monisha_cse@lus.ac.bd NUMBER THEORY AND CRYPTOGRAPHY 4.3 PRIMES AND GREATEST COMMON DIVISORS Discrete Mathematics and Its Applications by Rosen 7th Edition Primes Greatest Common Divisors Greatest Common Divisors Least common multiple The Euclidean Algorithm The Euclidean Algorithm NUMBER THEORY AND CRYPTOGRAPHY 4.6 CRYPTOGRAPHY Discrete Mathematics and Its Applications by Rosen 7th Edition Cryptography • Cryptography is the practice and study of techniques for secure communication in the presence of third parties called adversaries. • Number theory plays a key role in cryptography, the subject of transforming information so that it cannot be easily recovered without special knowledge. • Number theory is the basis of many classical ciphers, first used thousands of years ago, and used extensively until the 20th century. • These ciphers encrypt messages by changing each letter to a different letter, or each block of letters to a different block of letters. Terminologies • Cryptology: • The scientific study of cryptography and cryptanalysis. • Cryptography: • Cryptography is the practice and study of techniques for secure communication in the presence of third parties called adversaries. • Cryptanalysis: • Cryptanalysis is used to breach cryptographic security systems and gain access to the contents of encrypted messages, even if the cryptographic key is unknown. Types Of Cryptography • In general there are two types of cryptography: • Symmetric Key Cryptography: • It is an encryption system where the sender and receiver of message use a single common key to encrypt and decrypt messages. • Symmetric Key Systems are faster and simpler but the problem is that sender and receiver have to somehow exchange key in a secure manner. • Asymmetric Key Cryptography: • • • • • • This is also known as Public-key cryptography Under this system a pair of keys is used to encrypt and decrypt information Each person in the communication has a pair of keys: a public key and a private key A public key is used for encryption and a private key is used for decryption Public key and private Key are different Even if the public key is known by everyone the intended receiver can only decode it because he alone knows the private key. Caesar cipher • One of the earliest known uses of cryptography was by Julius Caesar • He made messages secret by shifting each letter three letters forward in the alphabet • Caesar’s encryption method can be represented by the function f f (p) = (p + 3) mod 26 In the encrypted version of the message, the letter represented by p is replaced with the letter represented by (p + 3) mod 26. • To recover the original message from a secret message decryption can be done using f−1(p) = (p − 3) mod 26 In other words, to find the original message, each letter is shifted back three letters in the alphabet. Caesar cipher Encryption Decryption Encryption Caesar cipher Shift cipher Here the integer k is called the key where 0 ≤ 𝑘 ≤ 25 .