Groups, Rings, Fields

Modern cryptography relies, for its ability to convert plaintext into ciphertext that appears to be random sequences of symbols, on the basic notions of abstract algebra. We introduce in this chapter the basics of groups, rings, and fields, including subgroups, cyclic groups, the order of elements, and Lagrange’s Theorem. A group is a set that is closed under an operation that is usually referred to (and often is) either as addition or as multiplication, with additional properties. A ring has both an addition and a multiplication, but which may not have an operation that resembles division. A field has all the characteristics we normally associate with doing arithmetic. All three are in some sense merely descriptions in the abstract of ordinary arithmetic. Proofs will to a large extent be left to later, or not done at all. And since we are interested more in using groups, rings, and fields than in proving theorems about them as algebraic objects, this chapter can be viewed largely as simply providing definitions for and formal statements of the truth of what we observe when doing the operations for encrypting and decrypting.