Cryptographic Protocols Notes for Lecture 1 Scribe :

A group is a mathematical structure 〈G; ∗〉 consisting of a non-empty set G and a binary operation ∗ : G×G→ G and satisfying the following axioms: (A1) The operation ∗ is associative, i.e., for any x, y, z ∈ G, x ∗ (y ∗ z) = (x ∗ y) ∗ z. (A2) There exists a neutral element e for ∗, i.e., x ∗ e = e ∗ x = x for all x ∈ G. (A3) Every element x ∈ G has an inverse x̂, i.e., x ∗ x̂ = x̂ ∗ x = e. If ∗ is commutative, then G is called an abelian or commutative group. Often, ∗ is denoted by +, inverses by −, and the neutral element by 0, in which case one speaks of an additive group. Similarly, if the above are denoted by ·, −1, and 1, respectively, one speaks of a multiplicative group.