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(2^n) and polynomial fields GF(p^n); 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(2^n) on Binary Polynomials

Binary Field Example - GF(2^2) with Modulo x^2+x+1

Binary Field Example - GF(2^3) with Modulo x^3+x+1

Binary Format of Binary Fields GF(2^n)

Binary Format for GF(2^3) with Modulo x^3+x+1

Polynomial Fields GF(p^n) on Prime Polynomials

Polynomial Field Example - GF(3^2) with Modulo x^2+1

Field Order and Field Characteristic

Field Characteristic Is a Prime Number

Field Order Is Prime Power


Conclusion:

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)

 Terminology

 References

 Full Version in PDF/EPUB