book

Elliptic Curve Cryptography for Developers

Name: Elliptic Curve Cryptography for Developers
Author: Michael G. Rosing
ISBN: 9781633437944

by Michael G. Rosing

March 2025

Intermediate to advanced

352 pages

10h 58m

English

Manning Publications

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Elliptic Curve Cryptography for Developers
copyright
contents
dedication
preface
acknowledgments
about this book
about the author
about the cover
1 Pairings over elliptic curves in cryptography
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

Part 1: Basics
2 Description of finite field mathematics
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 Explaining the core of elliptic curve mathematics
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 Key exchange using elliptic curves
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 Prime field elliptic curve digital signatures explained
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 Finding good cryptographic elliptic curves
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
Part 2: Interlude
7 Description of finite field polynomial math
7.1 Field extension7.2 Polynomial setup7.3 Polynomial addition7.4 Polynomial utilities
8 Multiplication of polynomials explained
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 Computing powers of polynomials
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 Description of polynomial division using Euclid’s algorithm
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 Creating irreducible polynomials
11.1 Basic theory of irreducible polynomials11.2 Code for finding irreducible polynomials
12 Taking square roots of 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
Part 3: Pairings
13 Finite field extension curves described
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 Finding low embedding degree elliptic curves
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 General rules of elliptic curve pairing explained
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 Weil pairing defined
16.1 Weil pairing formula16.2 Pairing subroutines16.3 Example code with tiny curves
17 Tate pairing defined
17.1 Tate pairing mathematics17.2 The Tate pairing subroutine described17.3 Testing the Tate pairing using a tiny example
18 Exploring BLS multi-signatures
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 Proving knowledge and keeping secrets: Zero knowledge using pairings
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
Appendix A. Code and tools
A.1 GNU Multiple Precision Arithmetic LibraryA.2 PARI/gpA.3 Building KangarooTwelve library
Appendix B. Hilbert class polynomials
B.1 Generation codeB.2 Hilbert class polynomial file
Appendix C. Variables list
selected bibliography

Content preview from Elliptic Curve Cryptography for Developers

14 Finding low embedding degree elliptic curves

This chapter covers

Algorithms for finding low embedding degree field extensions
The j-invariant and Hilbert class polynomials
The method of complex multiplication, used to find elliptic curves
Routines to find secure curves with low embedding degree

In this chapter, I’ll explain the reasons for choosing a specific size of extension field to create a secure pairing-friendly curve. The optimal size depends on the security level desired—too small and it might be broken; too large and it may not be efficient for practical use. I then discuss algorithms that can find curves of a low embedding degree. This involves searching for primes using mathematical functions. The parameters found during ...

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Elliptic Curve Cryptography for Developers

by Michael G. Rosing