Finite Fields

This chapter provides an introduction to Finite Fields. Topics covered include definition of finite fields; examples of finite fields: prime fields GF(p), binary fields GF(2n) and polynomial fields GF(pn); field order as the number of elements; field characteristic p is the least positive integer that the scalar multiplication of p and 1 is 0.

These sections are omitted from this Web preview version. To view the full content, see information on how to obtain the full version this book.

What Is Finite Field

Prime Fields GF(p) - Finite Fields with Modular Arithmetic

Prime Field Example - GF(11)

Binary Fields GF(2n) on Binary Polynomials

Binary Field Example - GF(22) with Modulo x2+x+1

Binary Field Example - GF(23) with Modulo x3+x+1

Binary Format of Binary Fields GF(2n)

Binary Format for GF(23) with Modulo x3+x+1

Polynomial Fields GF(pn) on Prime Polynomials

Polynomial Field Example - GF(32) with Modulo x2+1

Field Order and Field Characteristic

Field Characteristic Is a Prime Number

Field Order Is Prime Power

Takeaways:

• An Finite Field is a set of elements that is both an Additive Abelian Group and a Multiplicative Abelian Group. Their operations satisfy the distributivity condition.
• A prime field, GF(p), is a finite field defined on integers of 0, 1, 2, ..., p-1 for a given prime number p.
• A binary field, GF(2n), is a finite field defined on binary polynomials of n-1 degree, cn-1xn-1 + ... + c2x2 + c1x1 + c0x0, where coefficients are 0 or 1.
• A polynomial field, GF(pn), is a finite field defined on prime polynomials of n-1 degree, cn-1xn-1 + ... + c2x2 + c1x1 + c0x0, where coefficients are 0, 1, 2, ..., or p-1 for a given prime number p.
• The field order is defined as the total number of elements in a finite field. And it is a power of a prime number.
• The field characteristic is defined as the least positive integer, p, such that the scalar multiplication of p and 1 is 0. And it is a prime number.

Note that finite field is not closely related to elliptic curve Abelian groups. Skipping this chapter will have little impact on understanding elliptic curve cryptography.