Introduction to Mathematical Cryptography

Chapter 12 presents applications and initiates a study on cryptography. In the modern busy digital world, the word “cryptography” is well known. Every day, knowingly or unknowingly, in many places different techniques of cryptography are used. Starting from the log-in into a PC, sending e-mails, withdrawal of money from an ATM using a PIN code, operating the locker at a bank with the help of a designated person from the bank, sending message using a mobile phone, buying things through the Internet using a credit card, transferring money digitally from one account to another over Internet, every where cryptography is applied. Every such case requires to hide some information or it is necessary to transfer information secretly. So cryptography has something to do with security. Naturally, the questions that come are: What is cryptography? How important is it in daily life? In this chapter, cryptography is introduced and a brief overview of the subject with its basic goal is presented both intuitively and mathematically. More precisely, various cryptographic notions starting from the historical ciphers to modern cryptographic notions like public-key encryption schemes, signature schemes, secret sharing schemes, oblivious transfer etc. by using mathematical tools mainly based on modern algebra are explained. Finally, the implementation issues of three public key cryptographic schemes, namely RSA, ElGamal and Rabin, using the open source software SAGE are discussed.