We apply the Information-Theoretic Unification (ITU) framework (Terada 2026, concept DOI 10.5281/zenodo.20109209; current version v2.0.0 at 10.5281/zenodo.20133709) to cryptography. The ITU axiom δS = δ⟨K⟩ simultaneously governs noise correction (quantum error correction, QECC) and adversary protection (cryptography); these are dual applications of the same information-protection principle. This is Tier 1 paper #3, completing the ITU engineering triangle of Quantum Computing (Tier 1 #1, DOI 10.5281/zenodo.20139391), Machine Consciousness / ASI (Tier 1 #2, DOI 10.5281/zenodo.20150501), and Cryptography. Phase 51: Establishes the ITU foundation. Shannon's perfect-secrecy theorem (one-time pad), BB84 information-theoretic security, and post-quantum computational security all follow from the ITU axiom applied to adversary-resilient code design. We numerically verify Shannon's bound (I(M;C) ~ 0 when H(K) ≥ H(M)) and reproduce the 11% BB84 security threshold. Phase 52: BB84 detailed analysis with photon loss and decoy-state protocol. Standard BB84 supports ~200 km fibre links at practical rates. We propose a naive ITU enhancement (embed the [[5,1,3]] code from Tier 1 #1 in transmitted photons) and report a HONEST NEGATIVE FINDING: the naive 5-to-1 embedding underperforms standard BB84 because overhead dominates the loss-recovery gain. This refines the ITU prescription — protective code embedding must respect channel-specific overhead trade-offs. Phase 53: Lattice-based post-quantum cryptography (CRYSTALS-Kyber, CRYSTALS-Dilithium, NIST 2024 standards). A toy LWE KEM is implemented and verified; exponential brute-force scaling is observed. Kyber-1024 provides ~256/240-bit (classical/quantum) security, future-proof through the 2050s. We propose an ITU 3-TIER cryptographic framework: Tier 1 classical (AES-256), Tier 2 quantum-safe (PQC + QKD), Tier 3 ASI-safe (Φ_ITU-embedded protocols). Serial application yields combined 584-768-bit security against adversaries up to ASI. Phase 54: Synthesis into a 2024-2040 cryptography migration roadmap with: PQC adoption timeline (mainstream by 2029) QKD network deployment (~500 cities by 2040) Cost estimates by organisation type (individual to military, spanning $0 to ~$500B) Quantum-computer capability vs crypto-threshold projection 10 falsifiable predictions (Kyber-1024 secure to 2050; ASI cryptanalysis ∝ Φ_ITU; etc.) Policy recommendations at national, corporate, and individual levels Central thesis: under ITU, cryptography is the DUAL of quantum error correction, both governed by δS = δ⟨K⟩; a 3-tier defence-in-depth is the optimal response to the ASI-era threat landscape (ITU central prediction: ASI by 2030). Includes 4 theory documents, 4 Python numerical experiments, 4 figures, 4 JSON summaries, paper metadata. Total runtime ~2 minutes on a modern laptop.
Background: The imminent threat posed by cryptographically relevant quantum computers to classical public-key infrastructure specifically RSA and Elliptic-Curve Cryptography (ECC), whose security rests on the conjectured computational intractability of integer factorisation and the discrete logarithm problem respectively has propelled lattice-based cryptography to the centre of global post-quantum standardisation efforts. On 13 August 2024, the United States National Institute of Standards and Technology (NIST) published three landmark Federal Information Processing Standards (FIPS): FIPS 203 (ML-KEM, derived from CRYSTALS-Kyber), FIPS 204 (ML-DSA, derived from CRYSTALS-Dilithium), and FIPS 205 (SLH-DSA, derived from SPHINCS+), with FIPS 206 (FN-DSA, derived from FALCON) in advanced draft. The first three of these standards rest on algebraic structures intimately connected to classical number theory: polynomial rings over integer quotients, cyclotomic fields, and the algebraic geometry of ideal lattices in rings of integers of number fields. Objectives: This article provides: (1) a rigorous mathematical exposition of the lattice problems underpinning NIST’s post-quantum standards, specifically the Learning With Errors (LWE), Ring-LWE (RLWE), Module-LWE (MLWE), Short Integer Solution (SIS), and NTRU problems, and their connections to algebraic number theory; (2) a comparative performance analysis of NIST-standardised lattice-based algorithms against classical RSA and ECC benchmarks; (3) a security analysis examining known classical and quantum attacks on each hard problem; (4) a comparative evaluation of the four NIST-selected algorithms across key size, signature size, and computational benchmarks; and (5) a strategic assessment of the migration challenges facing deployed cryptographic infrastructure. Methods: The study synthesises peer-reviewed literature, NIST official standards documentation (FIPS 203, 204, 205, and draft FIPS 206), and independently published performance benchmarks published between June 2024 and June 2026. Mathematical security analysis is grounded in the worst-case to average-case reductions established by Regev (2005) for LWE and by Lyubashevsky, Peikert, and Regev (2013) [11, 18] for Ring-LWE. Performance data are drawn from Garani (2024) [8], the ARM Cortex-M0+ benchmarking study of Ratnayaka and Fischer (2026) [4], and the cross-platform evaluation by Alzahrani et al. (2025) [2]. Results: NIST’s three August 2024 standards establish a comprehensive lattice-based post-quantum portfolio. ML-KEM-768 produces a 1,184-byte public key and 1,088-byte ciphertext, achieving IND-CCA2 security under the MLWE assumption at NIST Level 3 (AES-192 equivalent). ML-DSA-65 produces 1,952-byte public keys and 3,309-byte signatures under the Module-LWE and Module-SIS assumptions at Level 3. FN-DSA-512 (FALCON, draft FIPS 206) produces 897-byte public keys and 666-byte signatures under the NTRU assumption, offering the most compact lattice-based signatures. Performance benchmarks on an Intel Core i7-12700K (3.6 GHz) confirm that ML-KEM key generation is orders of magnitude faster than RSA-3072 key generation. The algebraic structure of cyclotomic polynomial rings ∑₂ⁿ = Z[x]/(x^n + 1) is central to the efficiency of all three lattice-based NIST standards. Conclusion: Lattice-based cryptography, grounded in the algebraic number theory of ideal lattices over cyclotomic fields, provides the most comprehensive and mathematically well-founded response to the quantum computational threat. The August 2024 NIST standardisation of ML-KEM, ML-DSA, and SLH-DSA represents a landmark in cryptographic history, and the imminent finalisation of FN-DSA will complete the post-quantum toolbox. Organisations should prioritise immediate migration planning, leveraging hybrid classical-quantum schemes during transition, with full transition to NIST post-quantum standards recommended by 2030 per NSA and CISA guidance.
Post-quantum cryptography (PQC) is moving from evaluation to deployment as NIST finalizes standards for ML-KEM, ML-DSA, and SLH-DSA. This survey maps the space from foundations to practice. We first develop a taxonomy across lattice-, code-, hash-, multivariate-, isogeny-, and MPC-in-the-Head families, summarizing security assumptions, cryptanalysis, and standardization status. We then compare performance and communication costs using representative, implementation-grounded measurements, and review hardware acceleration (AVX2, FPGA/ASIC) and implementation security with a focus on side-channel resistance. Building upward, we examine protocol integration (TLS, DNSSEC), PKI and certificate hygiene, and deployment in constrained and high-assurance environments (IoT, cloud, finance, blockchain). We also discuss complementarity with quantum technologies (QKD, QRNGs) and the limits of near-term quantum computing. Throughout, we emphasize crypto-agility, hybrid migration, and evidence-based guidance for operators. We conclude with open problems spanning parameter agility, leakage-resilient implementations, and domain-specific rollout playbooks. This survey aims to be a practical reference for researchers and practitioners planning quantum-safe systems, bridging standards, engineering, and operations.