Group

A GROUP is a set in which you can perform one operation (usually addition or multiplication mod n) that satisfies the four properties : closure, associativity, identity element,commutative and invertibility

• Closed — any operation performed with elements from the set returns an element contained in the original set.
• Associative — if you have (a· b) ·c, it’s the same as a ·(b ·c)
• Identity — there exists a neutral element (usually 1) such that a · 1 = a
• Inverse — within the set there’s another element such that a · (a)^-1= 1
• Commutative — the order of operations doesn’t matter: a · b = b · a A group is said to be “abelian” if x ∗ y = y ∗ x for every x, y ∈ G.

Ring

A RING is a set equipped with two operations, called addition and multiplication. A RING is a GROUP under addition and satisfies some of the properties of a group for multiplication.

Field

A FIELD is a GROUP under both addition and multiplication.

Finite field (Galois Field)

• A Finite field can be defined as a set of numbers that we can add, subtract, multiply and divide together and only ever end up with a result that exists in our set of numbers
• The number of elements of a finite field is called its order or, sometimes, its size. A finite field of order q exists if and only if q is a prime power p^k (where p is a prime number and k is a positive integer). q= p^q, example 256= 2^8. Here q is the order of the field and p the prime number is the characteristic of the field
• The most crucial property of a finite field is that it has p^m elements where p is a prime number and m is whatever you choose. A finite field with 11 elements can be defined as GF(11^1). A finite field with 256 elements would be written as GF(2^8). Where GF is Galois Field
  • If m = 1 then we get prime fields
  • If m > 1 then we get extension fields

Prime field (p) - finite field with integers (0,1,2,… p-1)

5 → {0,1,2,3,4}, 3+4 mod 5 = 2

Extension field

Unlike finite fields, whose elements are integers, ****extension fields’ elements are polynomials. Extension fields = GF(2^m) where m > 1