Finite Fields

EC Cryptography Tutorials - Herong's Tutorial Examples

∟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.

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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, c_n-1x^n-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, c_n-1x^n-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.