---
title: Cryptography Math
description: This topic covers the mathematical foundations of secure communication, including number theory, abstract algebra, and probability. Learners will understand how algorithms like RSA and elliptic curve cryptography secure data transmission and digital signatures.
category: mathematics
subcategory: applied-mathematics
difficulty: beginner, intermediate, advanced
url: /subject/cryptography-math
---

# Cryptography Math

This topic covers the mathematical foundations of secure communication, including number theory, abstract algebra, and probability. Learners will understand how algorithms like RSA and elliptic curve cryptography secure data transmission and digital signatures.

## Available Resources

4 Books

## Books

### 1. Elliptic Curves: Number Theory and Cryptography (2nd Edition)

**Author:** Lawrence C. Washington

Reference on elliptic curves over finite fields: the group law, torsion points, endomorphisms, Weil and Tate-Lichtenbaum pairings, point counting, and the discrete logarithm attacks that determine ECC key sizes. Covers projective, Jacobian, and Edwards coordinates.

**Difficulty:** Advanced | **Language:** English | **Price:** Paid

**Link:** https://www.amazon.com/dp/1420071467?tag=edmonddante07-20

**Tags:** elliptic-curves, elliptic-curve-cryptography, number-theory, pairings, discrete-logarithm

### 2. A Graduate Course in Applied Cryptography

**Author:** Dan Boneh, Victor Shoup

Graduate treatment of provable security: definitions, reductions, and proofs for ciphers, message authentication, public-key encryption, elliptic curves, lattices, zero-knowledge proofs, and multiparty computation. Appendices develop the number theory and probability used throughout. Complete PDF free from the authors.

**Difficulty:** Advanced | **Language:** English | **Price:** Free

**Link:** https://toc.cryptobook.us/

**Tags:** provable-security, public-key-cryptography, symmetric-cryptography, zero-knowledge-proofs, lattice-cryptography

### 3. A Computational Introduction to Number Theory and Algebra (2nd Edition)

**Author:** Victor Shoup

Covers congruences, distribution of primes, quadratic reciprocity, groups, rings, fields, and discrete probability, together with the algorithms and complexity analysis that public-key cryptography depends on. Published by Cambridge; the author hosts the complete PDF at no cost.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://shoup.net/ntb/

**Tags:** number-theory, abstract-algebra, computational-number-theory, algorithms, discrete-probability

### 4. An Introduction to Mathematical Cryptography (2nd Edition)

**Author:** Jeffrey Hoffstein, Jill Pipher, Joseph H. Silverman

Undergraduate textbook developing the number theory, abstract algebra, and probability behind RSA, Diffie-Hellman, elliptic curve cryptography, lattice schemes, and digital signatures. Each chapter builds the required mathematics first, then the cryptosystem, with worked examples and exercises.

**Difficulty:** Intermediate | **Language:** English | **Price:** Paid

**Link:** https://www.amazon.com/dp/1493917102?tag=edmonddante07-20

**Tags:** cryptography, number-theory, rsa, elliptic-curves, lattice-cryptography

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