We bring concepts from all chapters of the course together in the presentation of public key crypto systems. These concepts are
Chapter 1 Integers and Algorithm:.
The operation \(\fmod\text{,}\) the Euclidean algorithm, and Bézout's identity for finding inverse in the group \((\Z_p^\otimes,\otimes)\text{.}\)
Chapter 2 Sets and Functions:.
The encoding function \(C\) for converting text into a sequence of numbers.
Chapter 3 Numbers and Counting:.
Prime numbers, binary numbers needed for fast exponentiation, and the representation of text by numbers.
Chapter 4 Groups:.
The groups \((\Z_p^\otimes,\otimes)\text{,}\) (fast) exponentiation, and the discrete logarithm needed to attack the crypto systems.
In particular we present the Diffie-Hellman key exchange and the ElGamal crypto system, which are both widely used in practice. The Diffie-Hellman key exchange is used to initiate secure connections such as the secure communication between web browser and web server. ElGamal is applied in the encryption of email and other forms a secure communication.
