Number theory is crucial to proving that RSA, Diffie-Hellman, and other cryptographic algorithms and protocols work properly. Suppose that p is an odd prime number. Then the set Z p={1,2,3,. ..,( p1)} is the set of integers mod p relatively prime to p. A Group in Abstract Algebra is a set together with a binary operation ab on the set which has three properties: the operation is associative, it has a neutral element, denoted as 1, and each element a in the set has an inverse, denoted a 1 . The operation may or may not be also commutative. If the operation is commutative then the Group is called an Abelian Group.

a) Prove that Z pwith the operation multiplication mod p is an Abelian Group.

b) Consider the Group Z23 . What is the inverse of 18?