Post-quantum Cryptography: Foundations, Algorithms, and Implementation

Large-scale quantum computers will break RSA, Diffie-Hellman, and elliptic-curve cryptography completely - not slowly, not partially, but with a polynomial-time algorithm that leaves no safe margin to retreat to. This is the only cryptographic transition in history where the endpoint isn't in question, only the timeline.Post-Quantum Cryptography is a technical guide for readers who want to understand this transition at the level of the mathematics and the algorithms, not the marketing. Starting from groups, rings, lattices, and quantum gates, it builds up to a rigorous treatment of Shor's and Grover's algorithms, then surveys every major post-quantum family - lattice-based, code-based, hash-based, multivariate, and isogeny-based - including the 2022 Rainbow and SIKE breaks, still among the most instructive cryptanalytic events in the field's history.The book covers NIST's finalized standards - ML-KEM, ML-DSA, and SLH-DSA - in full engineering depth: key generation, encapsulation, the Number Theoretic Transform, Fiat-Shamir with aborts, and the side-channel and implementation risks that determine whether a mathematically sound scheme survives contact with real hardware. A worked hand-computation of Shor's algorithm and a fully worked Learning With Errors example ground the abstractions in arithmetic you can check yourself. Case studies cover TLS at internet scale, Apple's iMessage PQ3, Signal's PQXDH, and national security migration deadlines.Written for readers comfortable with linear algebra, modular arithmetic, and asymptotic complexity - engineers, researchers, and students who need to understand not just that these schemes are secure, but why, and where the open questions still are.