Mathematics for Cryptography & Cyber Security

What it is

The mathematical toolkit cryptography and security are built on: number theory and abstract algebra (the structure behind RSA, Diffie–Hellman, elliptic curves, and lattice schemes), probability and statistics (security definitions, randomness, and attacks), linear algebra (lattices, coding theory, LWE), and information theory (entropy, perfect secrecy, channel coding).

Why it matters — security & society

Every security guarantee ultimately rests on these foundations — you can’t reason about what a scheme protects, or how it breaks, without them. Strengthening the math literacy behind cryptography is what lets people build systems that genuinely protect privacy, payments, and critical infrastructure rather than ones that merely look secure.

Start here — spans several areas

• A Computational Introduction to Number Theory and Algebra — Victor Shoup. Free; builds the number theory, abstract algebra, and discrete probability used across cryptography from scratch. The single best crypto-oriented anchor. (book)

Algorithms and Complexity

• Introduction to Algorithms - Cormen, Leiserson, Rivest and Stein

• Algorithm Design - Kleinberg and Tardos

• Computational Complexity - Oded Goldreich

Number theory

• An Introduction to Mathematical Cryptography — Hoffstein, Pipher & Silverman. The standard math-for-crypto text, tying number theory directly to schemes. Link also here.

• A Course in Computational Algebraic Number Theory - Henri Cohen. An advanced book on computational number-theoretic algorithms.

• See also Shoup, above.

Probability & statistics

• Introduction to Probability — Joseph Blitzstein & Jessica Hwang

• A FIRST COURSE IN PROBABILITY - Sheldon Ross

Linear algebra

• Linear Algebra Done Right — Sheldon Axler. Free (open access, 4th ed.); a clean, proof-based treatment of vector spaces and operators.

• Linear Algebra - Rao and Bhimasankaram

Abstract algebra

• Abstract Algebra - Dummit and Foote. Also, here.

• Abstract Algebra: Theory and Applications — Thomas Judson. (groups → rings → fields, with built-in coding-theory and cryptography applications.)

Information theory

• Information Theory, Inference, and Learning Algorithms — David MacKay. Free from the author; Shannon’s theory alongside inference and coding. Also, here.

• Elements of Information Theory - Cover and Thomas. Also, here.

Discrete Mathematics

• Disceret and Combinatorial Mathematics - Grimaldi.