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


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.

Table of Contents

 About This Book

 Geometric Introduction to Elliptic Curves

 Algebraic Introduction to Elliptic Curves

 Abelian Group and Elliptic Curves

 Discrete Logarithm Problem (DLP)

Finite Fields

 Generators and Cyclic Subgroups

 Reduced Elliptic Curve Groups

 Elliptic Curve Subgroups

 tinyec - Python Library for ECC

 EC (Elliptic Curve) Key Pair

 ECDH (Elliptic Curve Diffie-Hellman) Key Exchange

 ECDSA (Elliptic Curve Digital Signature Algorithm)

 ECES (Elliptic Curve Encryption Scheme)

 EC Cryptography in Java

 Standard Elliptic Curves



 Full Version in PDF/EPUB