7.2 Groups
Groups are the most basic mathematical structure in which public-key cryptography can take place. So, let’s plunge right into the math and explain the properties of a so-called abelian group.
Let M be a nonempty set, and let ⋆ be an operation on M, which means ⋆ is a function M × M → M, which maps pairs of elements of M to elements of M. The pair (M,⋆) is called an abelian group 𝔾 if the following properties hold:
(G1) ⋆ is an associative operation that means for all cases of a,b, and c ∈ M, we have
(G2) In M exists a neutral element e with the property that for all cases of a ∈ M,
(G3) For all cases of a ∈ M exists an inverse element a−1 ∈ M with the property a−1 ⋆ a = a ⋆ a−1 = e.
(G4) ⋆ is a commutative operation, which means for all cases of a,b ∈ M, we have
The number of elements of M is called the order of the group 𝔾...
