ADLP: A Post-Quantum Cryptographic Framework from Apollonian Sphere Packing and Hyperbolic Group Theory

A complete post-quantum cryptographic framework based on the Apollonian Discrete Logarithm Problem (ADLP). The mathematical foundation is an infinite discrete subgroup Γ ⊂ SO(3,1) generated by 36 Lorentz transformations derived from the FCC lattice via Apollonian sphere packing in R^{3,1}. Security rests on two independent hard problems — ADLP and the conjugacy search problem CSP_Γ — neither admitting a known efficient classical or quantum algorithm. Three constructions are presented: ADLP-KEM (key encapsulation, reduces to CSP_Γ in the ROM), ADLP-PKE (OW-CPA, proof of concept due to numerical instability at security word lengths), and ADLP-Sign (Fiat-Shamir signatures, 0/100 forgeries). Public keys and ciphertexts measure 128 bytes at all security levels. Five open-source simulation scripts reproduce all results.
This version corrects the generator count from 24 to 36 (two trace classes at {0.3211, 3.1738}), corrects the coefficient field from Q(√2,√3) to Q(√11,√19), demotes the direct PKE construction to proof-of-concept status, and revises the security claim to OW-CPA with IND-CPA achieved via KEM + AES-GCM hybrid. Fourth paper in a series with Papers 1–2 (DOI 10.5281/zenodo.18793518, 10.5281/zenodo.18828073).