Mathematics for Engineers

We present here the major techniques aiming at guaranteing the confidentiality and authenticity of messages.
We distinguish between ciphering and signature algorithms. Ciphering is the basic operation in cryptology. It consists of transforming a message ( plaintext) into a hidden, encrypted message ( ciphertext), only readable by its legitimate addressee. This is called a cryptogram. The transformation is performed by using a key (the ciphering key). The technique associated with ciphering systems is referred to as cryptography. Only the receiver possessing the key should be able to decipher the message. This system is reliable only if a third party cannot retrieve the message, i.e. cannot retrieve the key. Deciphering a message without possessing the key is called cryptanalysis. The term cryptology encompasses all the techniques for cryptography and cryptanalysis.
This discipline has a long history (examples are found in antiquity, with substitution ciphers), and initially found its justification in military and diplomatic areas. Today, the development of communications and commercial transactions on the Internet ("e-business") makes it an essential operation, in order to guarantee confidentiality of the exchanged data. New functions are now employed, such as signatures, the goal of which is to authenticate messages (both their content and their sender).
We distinguish between symmetric-key and public-key encryption. Symmetric-key encryption relies on the sharing of a secret-key between the sender and the receiver. Public-key algorithms do not have this weakness, but being slower they do not...