Cryptographic systems rely fundamentally on the algebraic structure of groups. Informally, a group is a set GG equipped with a binary operation (often denoted “⋅⋅” or “+”) satisfying four axioms. First, closure: for all a,b∈Ga,b∈G, the product a⋅ba⋅b is also in GG. Second, associativity: (a⋅b)⋅c=a⋅(b⋅c)(a⋅b)⋅c=a⋅(b⋅c) for all a,b,c∈Ga,b,c∈G. Third, an identity element ee exists in GG such that e⋅a=a⋅e=ae⋅a=a⋅e=a for every a∈Ga∈G. Fourth, every element has a unique inverse: for each a∈Ga∈G the...
