Cryptography And Mathematical Security Systems

Abstract

The impending arrival of cryptographically relevant quantum computing threatens classical public‑key infrastructures. This paper reviews the latest developments (2025–2026) in post‑quantum cryptography (PQC), fully homomorphic encryption (FHE), and zero‑knowledge proofs (ZKP). NIST has advanced nine signature candidates to its third evaluation round and selected HQC as a backup encryption standard. Novel primitives include bio‑inspired RNA‑based cryptography, algebraic hash signatures, and topology‑mined lattice schemes. FHE has reached its fifth generation with the GL scheme and the MadPanthera virtual processor, while lightweight ZKPs such as Microsoft’s Vega enable mobile‑friendly verification. These advances demonstrate rapid maturation toward deployable quantum‑safe systems.