book

Elliptic Curve Cryptography for Developers

by Michael G. Rosing

March 2025

Intermediate to advanced

352 pages

10h 58m

English

Manning Publications

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1.1 What is elliptic curve cryptography?1.2 Why use elliptic curve cryptography?1.3 Elliptic curves come to public key cryptography1.3.1 General description of key exchange1.3.2 Digital signature algorithms explained1.3.3 How multiple people can sign the same document1.3.4 Zero knowledge, or how to keep a secret and prove you have one1.4 Who this book is for

2.1 Basic mathematics of finite fields2.2 Elliptic curves form groups of points over a finite field2.3 Basic subroutines for finite field arithmetic2.4 Computing quadratic residues over a prime field2.5 Computing the square root mod p
3.1 Elliptic curve algebra3.1.1 Point representation3.1.2 Elliptic curves over finite fields3.1.3 Point addition3.1.4 Point multiplication3.1.5 Embedding data on a curve3.2 Elliptic curve subroutines3.2.1 Code to represent curves and points3.2.2 Code for point addition3.2.3 Code for point multiplication3.3 Code for embedding data on a curve3.4 Miscellaneous routines
4.1 Diffie-Hellman algorithm description4.1.1 Elliptic curve math4.1.2 Hash function4.1.3 Key generation4.1.4 Computing shared keys4.2 MQV algorithm4.2.1 Elliptic curve math for the MQV algorithm4.2.2 MQV code4.3 Example test code4.3.1 Test curves4.3.2 Diffie-Hellman test routines4.3.3 MQV test routine
5.1 Schnorr digital signature5.1.1 Schnorr elliptic curve math5.1.2 Schnorr sign subroutine5.1.3 Schnorr verify subroutine5.1.4 Schnorr test example5.2 NIST elliptic curve digital signature algorithm5.2.1 ECDSA sign subroutine5.2.2 ECDSA verify subroutine5.2.3 ECDSA example code
6.1 PARI/gp for elliptic curves6.1.1 Starting PARI/gp6.1.2 PARI/gp elliptic curves over finite fields6.1.3 Libpari with elliptic curves6.2 General ordinary curves6.2.1 Variables and initialization6.2.2 Main loop6.3 Bad curves
7.1 Field extension7.2 Polynomial setup7.3 Polynomial addition7.4 Polynomial utilities
8.1 Defining irreducible polynomials8.2 Irreducible polynomial as modulus8.3 Building the matrix8.4 Multiplication code8.4.1 Creating the multiplication table8.4.2 Polynomial multiply8.5 Miscellaneous multiply routines
9.1 Using square and multiply to rapidly compute powers9.2 Polynomial powers code for general exponents9.3 Explicit polynomial example9.4 Powers of field prime
10.1 Euclid’s algorithm and gcd10.2 Inversion and division of polynomials10.3 Euclid’s algorithm code10.4 Gcd code10.5 Inversion modulo a prime polynomial10.6 Division modulo a prime polynomial
11.1 Basic theory of irreducible polynomials11.2 Code for finding irreducible polynomials
12.1 Mathematics for square root modulo a prime polynomial12.2 Code for square roots modulo a prime polynomial12.2.1 Content routine12.2.2 Pseudo-division routine12.2.3 Resultant subroutine12.2.4 Quadratic residue12.2.5 Polynomial square root routine
13.1 Field extension properties13.2 Elliptic curve routines13.2.1 Polynomial curve setup13.2.2 Polynomial curve utilities13.2.3 Polynomial curve point embedding13.2.4 Polynomial curve random point13.2.5 Polynomial elliptic curve addition13.2.6 Polynomial elliptic curve point multiplication13.3 Tiny example13.3.1 Tiny example variables13.3.2 Tiny example base curve13.3.3 Tiny example field extension curve
14.1 Security of field extensions for elliptic curve pairing14.2 Low embedding degree14.3 Complex multiplication14.4 Factoring a Hilbert class polynomial14.5 Code for finding pairing-friendly curves14.5.1 Pairing sweep14.5.2 Finding the curve
15.1 Mathematical rules of elliptic curve pairings15.1.1 Elliptic curve point pairing rule of bilinearity15.1.2 Nondegeneracy rule with the point at infinity15.2 Algorithms for pairing15.2.1 Function \(h_{P,Q}(R)\)15.2.2 Miller’s algorithm15.3 Subroutine \(h_{P,Q}\)15.4 Miller’s algorithm code
16.1 Weil pairing formula16.2 Pairing subroutines16.3 Example code with tiny curves
17.1 Tate pairing mathematics17.2 The Tate pairing subroutine described17.3 Testing the Tate pairing using a tiny example
18.1 Introduction to multi-signatures18.2 Multi-signatures with key aggregation18.2.1 The hash functions used in key aggregation algorithms18.2.2 Key aggregation algorithm for multi-signatures18.2.3 Digital multi-signature mathematics and verification algorithm18.3 Multi-signatures code description18.3.1 Key generation code explained18.3.2 Hash functions code description18.3.3 Computing the public key hash \(a_j\)18.3.4 Multi-signature and verification subroutines18.4 Accountable subgroup multi-signatures18.5 Description of the subgroup multi-signatures code18.6 Aggregate BLS signatures example program18.6.1 Initial test parameters18.6.2 Multi-signatures simulated key generation test program18.6.3 Multi-signatures signing and verifying simulation18.6.4 Creating subgroup multi-signatures simulation program
19.1 SNARK defined19.2 Concept of the quadratic arithmetic program explained19.3 Lagrange interpolant functions described19.4 The common reference string19.5 zk-SNARK example code19.5.1 Common subroutines in snarkbase.c19.5.2 Program to compute QAP parameters19.5.3 Common reference string creation19.5.4 Creating the proof for a record19.5.5 Zero-knowledge verification of a record
A.1 GNU Multiple Precision Arithmetic LibraryA.2 PARI/gpA.3 Building KangarooTwelve library
B.1 Generation codeB.2 Hilbert class polynomial file

Overview

Make your public key protocols smaller and more secure with this accessible guide to Elliptic Curve Cryptography.

Elliptic Curve Cryptography for Developers introduces the mathematics of elliptic curves—a powerful alternative to the prime number-based RSA encryption standard. You’ll learn to deliver zero-knowledge proofs and aggregated multi-signatures that are not even possible with RSA mathematics. All you need is the basics of calculus you learned in high school.

Elliptic Curve Cryptography for Developers includes:

Clear, well-illustrated introductions to key ECC concepts
Implementing efficient digital signature algorithms
State of the art zero-knowledge proofs
Blockchain applications with ECC-backed security

The book gradually introduces the concepts and subroutines you’ll need to master with diagrams, flow charts, and accessible language. Each chapter builds on what you’ve already learned, with step-by-step guidance until you’re ready to write embedded systems code with advanced mathematical algorithms.

About the Technology
The Elliptic Curve Cryptography (ECC) protocol secures everything from credit card transactions to the blockchain. With a little C code, high school calculus, and the techniques in this book, you can implement ECC cryptographic protocols that are smaller and more secure than the RSA-based systems in common use today.

About the Book
Elliptic Curve Cryptography for Developers teaches you how ECC protocols work and how to implement them seamlessly in C code. Unlike academic cryptography books, this practical guide sticks to the minimum math and theory you need to get the job done. Author Mike Rosing illustrates each concept with clear graphics, detailed code, and hands-on exercises. As you go, you’ll practice what you learn by building two encryption systems for a blockchain application.

What's Inside

Efficient digital signature algorithms
Zero-knowledge proofs
ECC security for blockchain applications

About the Reader
Readers need to understand basic calculus. Examples in C.

About the Author
Michael Rosing’s career as a scientist, hardware engineer, and software developer includes high-energy physics, telephone switch engineering, and developing vision devices for the blind.

The technical editor on this book was Mark Bissen.

Quotes
Brilliantly demystifies elliptic curve pairings, sidestepping convoluted math notations in favor of commented source code and concrete examples.
- Jean-Philippe Aumasson, Co-Founder & Chief Security Officer, Taurus

The companion you’ve been missing. It will take you from what you’ve learned in theory books and academic papers to your applications.
- Kobi Gurkan, Head of Research, Geometry Research

There’s a lot of background and practical information necessary to use elliptic curves correctly. This book lays them out in a friendly and complete manner. If you want to know about elliptic curves, this book is an excellent place to start.
- Victor Miller, Principal Computer Scientist, SRI