Abstract MQBP V6.2 develops an exact prime-indexed computational architecture in which computational complexity is transferred from the cardinality of a nominal representative space to the structure and evaluability of its objective-preserving quotient. This is Q-Day, effectively, mathematically. Rather than treating a state carrier XnX_n or computational-basis space as the primitive object to be searched, the framework constructs a class map π:Xn→Cn class-indexed fold invariants, polynomial layers, Cantor complexity coordinates, determinantal degeneracy data, algebraic certificates, Thom encodings, and an invariant rank ρ:Cn→Rn. For every objective admitting the exact factorization F=Φ∘ρ∘π representative-level enumeration is mathematically unnecessary: the objective is constant on quotient classes and may be evaluated entirely from the reduced carrier. Thus the exponential cardinality of the representative space and the computational complexity of the objective become distinct mathematical quantities; the operative complexity is that of constructing and evaluating the objective-preserving class and rank maps. The architecture is quantified over arbitrary prime nn and arbitrary arity r≥2r\ge2. For an occupancy class m=(m0,…,mr−1), ∑dmd=n, the class-indexed fold hierarchy M_j([m])=\sum_{d=0}^{r-1}m_d d^j defines the polynomial layer λ([m])=M1([m]), while (M_1,…,M_r−1)forms an exact finite occupancy coordinate by Vandermonde reconstruction. The degree-k polynomial channel is c_{r,n,k}=[t^k](1+t+\cdots+t^{r-1})^n, with n(r−1)+1 layers and prime-index Frobenius residues. Recursive Cantor coordinates canonically couple occupancy class, polynomial layer, fold order, and fold value. The moduli-degeneracy carrier augments these class coordinates with exact determinantal rank-drop witnesses, canonical integer-polynomial representations, quadratic-module Positivstellensatz certificates, signed Thom derivative encodings, stabilizer data, and generalized isogenic fold anchors. The same discrete bit word indexes both the CPU carrier and its computational-basis label, permitting quotient, fold, Cantor, layer, and objective maps to execute directly in the CPU model. Accordingly, V6.2 does not formulate cryptographic computation as faster exploration of an enormous state space. It formulates a stronger possibility: when the cryptographic objective descends through an efficiently computable exact quotient, the nominal state space is no longer the computational problem. Hardness is thereby relocated from raw state-space cardinality to the construction and evaluation of the objective-preserving quotient itself. Runtime evaluations instantiate this universal theorem carrier across prime indices from 11 through 2,147,483,647 including exact class folds, Cantor coordinates, polynomial-layer counts, endogenous exponent inverses, power/inverse class memory, and multiprecision Hilbert/Frobenius calculations. The resulting architecture provides a formal basis for studying quantum cryptography as exact class computation over certified moduli-degeneracy structure rather than representative-space search.
Abstract MQBP V6.2 develops an exact prime-indexed computational architecture in which computational complexity is transferred from the cardinality of a nominal representative space to the structure and evaluability of its objective-preserving quotient. This is Q-Day, effectively, mathematically. Rather than treating a state carrier XnX_n or computational-basis space as the primitive object to be searched, the framework constructs a class map π:Xn→Cn class-indexed fold invariants, polynomial layers, Cantor complexity coordinates, determinantal degeneracy data, algebraic certificates, Thom encodings, and an invariant rank ρ:Cn→Rn. For every objective admitting the exact factorization F=Φ∘ρ∘π representative-level enumeration is mathematically unnecessary: the objective is constant on quotient classes and may be evaluated entirely from the reduced carrier. Thus the exponential cardinality of the representative space and the computational complexity of the objective become distinct mathematical quantities; the operative complexity is that of constructing and evaluating the objective-preserving class and rank maps. The architecture is quantified over arbitrary prime nn and arbitrary arity r≥2r\ge2. For an occupancy class m=(m0,…,mr−1), ∑dmd=n, the class-indexed fold hierarchy M_j([m])=\sum_{d=0}^{r-1}m_d d^j defines the polynomial layer λ([m])=M1([m]), while (M_1,…,M_r−1)forms an exact finite occupancy coordinate by Vandermonde reconstruction. The degree-k polynomial channel is c_{r,n,k}=t^k^n, with n(r−1)+1 layers and prime-index Frobenius residues. Recursive Cantor coordinates canonically couple occupancy class, polynomial layer, fold order, and fold value. The moduli-degeneracy carrier augments these class coordinates with exact determinantal rank-drop witnesses, canonical integer-polynomial representations, quadratic-module Positivstellensatz certificates, signed Thom derivative encodings, stabilizer data, and generalized isogenic fold anchors. The same discrete bit word indexes both the CPU carrier and its computational-basis label, permitting quotient, fold, Cantor, layer, and objective maps to execute directly in the CPU model. Accordingly, V6.2 does not formulate cryptographic computation as faster exploration of an enormous state space. It formulates a stronger possibility: when the cryptographic objective descends through an efficiently computable exact quotient, the nominal state space is no longer the computational problem. Hardness is thereby relocated from raw state-space cardinality to the construction and evaluation of the objective-preserving quotient itself. Runtime evaluations instantiate this universal theorem carrier across prime indices from 11 through 2,147,483,647 including exact class folds, Cantor coordinates, polynomial-layer counts, endogenous exponent inverses, power/inverse class memory, and multiprecision Hilbert/Frobenius calculations. The resulting architecture provides a formal basis for studying quantum cryptography as exact class computation over certified moduli-degeneracy structure rather than representative-space search.
La cryptographie post-quantique est une branche de la cryptographie qui vise à concevoir des systèmes cryptographiques non quantiques (c'est-à-dire classiques), qui sont protégés contre un adversaire possédant un ordinateur quantique. Dans cette thèse, nous nous concentrons sur l'étude de deux problèmes fondamentaux pour la cryptographie post-quantique : le problème du plus court vecteur (SVP) et le problème de la somme de sous-ensembles aléatoires. Le SVP demande de trouver le plus court vecteur non nul d'un réseaux euclidien donné. Il sert de jauge pour quantifier la sécurité de la cryptographie reposant sur les réseaux euclidiens, qui est considérée comme prometteuse pour l'ère post-quantique. Les principales approches pour résoudre le SVP sont les algorithmes de tamisage, qui utilisent un temps et un espace exponentiels, et les algorithmes d'énumération, qui utilisent un temps superexponentiel et un espace polynomial. Dans cette thèse, nous donnons tout d'abord un compromis temps-mémoire prouvable qui interpole approximativement entre les garanties de ressources des algorithmes de tamisage et celles des algorithmes d'énumération. Nous montrons ensuite l'algorithme quantique prouvable le plus rapide connu qui résout le SVP en temps 2^(0.9535n) et en mémoire classique 2^(0.5n+o(n)) avec seulement un nombre poly(n) de qubits. Nous montrons également le meilleur algorithme prouvable classique connu dont la complexité en mémoire est de 2^(0.5n+o(n)). Quant aux algorithmes d'énumération, on pensait auparavant qu'ils pouvaient bénéficier d'une accélération quadratique quantique grâce à l'algorithme de Grover. Nous montrons que cette accélération quadratique peut effectivement être obtenue pour l'algorithme d'énumération de base et ses variantes d'élagage cylindrique et discret, mais avec des algorithmes quantiques plus sophistiqués tels que la marche quantique sur les arbres. Notre accélération quantique s'applique également à l'élagage extrême où l'on répète le processus d'énumération sur plusieurs bases réduites. Le problème de la somme de sous-ensembles aléatoires sous-tend également certains schémas cryptographiques visant à la sécurité post-quantique, bien qu'il soit principalement de nature académique. Dans un contexte de cryptanalyse, il sert souvent d'outil car de nombreux problèmes sur lesquels se fonde la cryptographie moderne peuvent être exprimés comme des variantes (vectorielles) du problème de la somme de sous-ensembles. Plus récemment, il a également été démontré qu'il peut être utilisé comme élément de base dans certains algorithmes de décalage caché quantique, qui ont des applications dans la cryptanalyse quantique de schémas symétriques et de schémas reposant sur les isogénies. Dans cette thèse, nous présentons de nouveaux algorithmes classiques et quantiques pour la résolution du problème de la somme de sous-ensembles aléatoires. Nous obtenons d'abord l'algorithme classique le plus rapide connu en temps Õ(2^0.283n)). Ensuite, nous améliorons l'état de l'art des algorithmes quantiques pour ce problème dans plusieurs directions. Nous concevons un algorithme avec un temps asymptotique Õ(2^0.236n), avec l'avantage d'utiliser une mémoire classique avec accès aléatoire quantique. Les algorithmes connus précédemment utilisaient des marches quantique, et nécessitaient une mémoire quantique avec accès aléatoire quantique.Nous proposons également des marches quantiques plus rapides pour ce problème en temps Õ(2^0.216n). Ce temps dépend d'une heuristique concernant le temps de mise à jour dans les marches quantiques. Les algorithmes précédents pour ce problème dépendaient eux aussi de cette heuristique. Nous montrons comment surmonter partiellement cette heuristique, et nous obtenons un algorithme quantique de temps Õ(2^0.218n) ne nécessitant que les heuristiques standard du problème de la somme de sous-ensembles aléatoires.
Abstract. We develop a general exact architecture joining linear subshell capacity, unrestricted isotropic fold vectors, conjugacy/class memory under power and inverse power, common objective ideals, class-objective rank factorization, and a bit–qubit class equivalence on the computational-basisvcarrier. The state is any finite vector q ∈ NL ; its fold carrier is the unrestricted sequence Fq(j) = XLs=1qj+1s , j ∈ N, and the bridge family is arbitrary in arity, βr(q)s = min(qs, r − 1), r ≥ 2. The bridge is diadic at r = 2, triadic at r = 3, and stabilizes exactly to q whenever r > maxs qs. Power and inverse-power operations that respect a class relation descend to the quotient, so vector composites of powered class invariants remain class memories. For four objective ideals I1, . . . , I4, the common objective ideal Iobj = I1 ∩ I2 ∩ I3 ∩ I4 is the maximal ideal simultaneously contained in all four. If a physical or computational objective factors through a surjective class map, F = ρ ◦ π, then its minimizers are exactly the preimages of minimizing classes. A second central theorem identifies the isotropic pathway count with the sum of quotient-class multiplicities. For capacity vector c = (c1, . . . , cL) and total fill Z, let QZ (c) = {q ∈ NL:Xsqs = Z, 0 ≤ qs ≤ cs}. The permutation class of q has multiplicity µ(q) = Z!Qsqs!. Then the total number of capacity-admissible labeled pathways is Pc(Z) = Xq∈QZ (c)µ(q). Thus the isotropic pathway number is exactly the sum of the collapsed vector-class quotients. In the uncapped case this reduces to the multinomial identity P(Z) = LZ. The 118-element periodic-table instantiation uses the same architecture. Each element receives a canonical 19-subshell capacity-fill vector, fold moments, diadic-to-arbitrary bridge vectors, aVbaryonic-spatial Cantor rank from reference-isotope neutron count and covalent radius, class-memory power/inverse-power checks, and exact pathway multiplicities. The compiled C++ benchmark executed 166,218 checks with all passing; 150,507 acted directly on the 118-element carrier, 15,104 exhaustively compared direct and class-rank marking predicates over all 118 targets and all 128 seven-bit codes, and 359 checks verified the exact isotropic pathway partition recurrence. An independent Python implementation executed 52,166 checks with all passing, including 10,000 arbitrary generated fold/bridge vectors. Four chemistry-oriented structural benchmarks are integrated into the same computational module. Contents 1. Architecture and attribution 3 2. Linear subshell maximum fill 3 3. General isotropic fold vectors 4 3.1. Diadic through arbitrary arity 4 4. Isotropic pathway sum as quotient-class partition 5 5. Class memory: powers, inverse powers, and composites 7 6. Common objective ideal and class-objective rank 7 7. Tibedo Bit–Qubit Class Equivalence Theorem 8 MQBP V5: CLASS MEMORY AND ISOTROPIC PATHWAYS 3 8. 118-element periodic-table instantiation 9 8.1. Canonical fill 9 8.2. Baryonic-spatial coordinate 9 9. Integrated benchmark execution 10 9.1. FCI-like high-arity benchmark 11 9.2. 198-site triadic benchmark 11 9.3. Isotope benchmark 11 9.4. Prime-channel benchmark 11 10. Formalization and software architecture 11 10.1. Coq 12 10.2. C++ 12 10.3. Lean 4 12 11. Structural distinction 12 12. Conclusion 13 Appendix A. Complete 118-element dataset table 13 Appendix B. Dataset contents 17 References 17
Paper 3 of 11 — A General Theory of Answerability The eleven-paper TUS OS mathematical research programme Abstract This paper develops an exact finite local-to-global analysis of the six-dimensional Hamming-cube foundation Atlas underlying the TUS OS construction. Every admissible completed foundation carries the same exact local qubit prepare–measure representation, while the complete repeated-common-effect Atlas has exact minimum complex Hilbert dimension four. The paper classifies the full ambient family of completed four-state foundations in the six-dimensional Hamming cube. Every foundation is uniquely an affine coset associated with a perfect matching of the six coordinates. There are exactly 240 unordered foundations and 5,760 ordered codebooks. Every foundation induces the same exact 27-class tetrahedral-SIC qubit operational quotient, with carrier-fibre profile 1⁸, 2¹², 4⁶, 8¹ and the corresponding Bloch-radius structure of eight pure states, twelve states of radius √(2/3), six states of radius 1/√3, and one maximally mixed state. Each ambient chart reduces losslessly to a signed perfect matching. The chart-coloured union of two matchings obeys an exact component-parity collision law, yielding the complete seven-value image spectrum over all 28,680 ambient foundation pairs. Exactly 10,800 pairs jointly reconstruct the full 64-state carrier, giving an ambient mathematical chart-cover number of two. The frozen TUS selector defines a canonical subatlas of 36 distinct foundations and 64 anchor-indexed chart instances across nine matching classes. Its selected-foundation reconstruction graph is bipartite and eight-regular with 144 edges, while selector multiplicity yields exactly 400 reconstructing pairs among the 2,016 canonical chart-instance pairs. Only this canonical subatlas inherits the native LANDING and GUIDANCE structure. At the sequential level, all 384 active map/chart descent tests fail: no active LANDING map descends to a state-only transformation on a fixed 27-class local chart quotient. Under the complete principal future language, every canonical chart refines to all 64 deterministic predictive states, with maximum separating depth two. The subordinate GUIDANCE witness is also classified: its endpoint map has image size 34, eight two-cycle attractors and maximum transient depth five, while the additional GUIDANCE-selected chart raises principal path reconstruction from 138/192 contexts to 169/192. The paper then establishes the exact global static common-effect Hilbert dimension. Although every individual foundation admits an exact qubit representation, no unrestricted qutrit state/common-effect family can reproduce all 4,096 Atlas probabilities while simultaneously satisfying all 240 overlapping foundation POVM identities. Together with an explicit chart-independent dimension-four construction, this proves d_q^atlas = 4 for the complete static Atlas. The attaining construction uses two declared qubit sectors; unrestricted uniqueness or rigidity of minimum representations is not claimed. The result therefore exhibits a precise local-to-global separation: exact qubit prepare–measure form exists on every individual foundation, but an exact chart-independent repeated-common-effect representation of the complete Atlas requires complex Hilbert dimension four, while the complete deterministic future language preserves all 64 predictive identities. Local operational state, global relational state and deterministic predictive state are consequently distinct mathematical structures. The paper explicitly keeps these distinct from the still-unresolved whole-position process state. The scope boundary is explicit. The paper does not establish a physical quantum carrier, operational access to all 240 ambient foundations, LANDING as a quantum channel, GUIDANCE as a quantum observable, endpoint channel or instrument, contextuality, nonlocality, quantum advantage, unrestricted rigidity of minimum representations, or whole-position process synthesis. Its conclusions are exact at the static Atlas, canonical-chart and deterministic predictive scopes defined in the manuscript. Supporting General Theory Programme This paper is Paper 3 in the TUS OS quantum-operational and Answerability research programme. It extends the results of Papers 1–2 from the canonical local geometry and exact qubit prepare–measure behaviour to the complete finite relational Atlas. It classifies the full family of admissible foundations, proves that every local chart carries the same exact 27-class qubit operational quotient, identifies the canonical TUS subatlas, and shows that the complete native future language refines those local quotients back to all 64 deterministic predictive states. Its principal global result is that the complete common-effect Atlas has exact minimum quantum dimension d=4, while remaining distinct from the larger 64-state predictive structure and from later process-level dynamics. The paper also classifies the subordinate GUIDANCE witness and its finite reconstruction contribution without promoting it into a channel or instrument. The wider programme then develops complete-future and process closure in Paper 4, the Anchored Analog Quantum System framework in Paper 5, contract-relative resource and operational-observability theory in Papers 6–7, generic Answerability theory in Papers 8–9, complete TUS OS formal and executable realisation in Paper 10, and culminates in Paper 11 with A General Theory of Answerability. Paper 1 — Canonical Tetrahedral Qubit-State Geometry in the TUS OS TriadXOR Completion, Receipt Verification and the Qubit-Geometric Semantic WitnessEstablishes the canonical tetrahedral geometry, unique XOR completion, exact qubit-SIC representation and centred-face trine geometry. Paper 2 — The TUS OS Operational CalculusClassical Construction of Exact Qubit Prepare–Measure Behaviour from a Receipt-Custodied Engine ArchitectureEstablishes exact full-carrier qubit prepare–measure behaviour from the deterministic classical engine under the declared Coordinate-Incidence Operational Harness. For each lawful foundation, the 64 six-bit carriers map to exactly 27 anchor-relative qubit preparations under one foundation-defined tetrahedral-SIC Harness, while the separately declared face Harness gives the exact trine table. Paper 3 — From Finite Relational Structure to Qubit Operational FormAmbient Foundations, Canonical Subatlas, Exact Global Quantum Dimension, GUIDANCE Witness, Sequential Closure and Global Representation in TUS OSExtends the local prepare–measure construction to the complete ambient family of 240 completed foundations and the canonical TUS OS subatlas. Every completed foundation has the same 27-class qubit operational quotient; pair relations between charts provide exact carrier-reconstruction structure; the complete repeated-common-effect static Atlas has exact minimum complex Hilbert dimension 4; and the frozen complete principal future language has deterministic predictive-state minimum 64. The dimension-4 result is a static representation theorem and does not by itself supply common quantum process dynamics. Paper 4 — Technical Volume I — Formal Foundations and the Complete-Future CorpusNative Contract, Statistical Discrimination and the Invariant Probability ModuleFixes the formal native domain, complete principal terminal future-experiment language and one-common-word experiment class; establishes complete statistical separation of distinct carrier pairs, exact parity-resolved discrimination results and complement reachability; and proves that the rational invariant probability module has dimension 33. Paper 4 — Technical Volume II — Orthogonality, Process Closure and Native Triadic SynthesisEstablishes universal complement-support orthogonality and its propagation under exact common-CPTP dynamics, yielding the certified same-parity support graph and the bound 32 ≤ d_CPTP^min ≤ 64, with cross-parity forcing and the exact state-only common-CPTP minimum remaining unresolved. It further establishes recovery-complete LANDING and GUIDANCE results, exact 64-dimensional selected-line instrument/readout minima at their declared contracts, an exact classical–quantum hybrid construction, and the native structural/predictive quotient correspondence. Paper 5 — TUS OS as an Anchored Analog Quantum SystemNative Witnesses, Predictive Synthesis, and Access-Contract Resource SeparationDefines Analog Quantum Views and Anchored Analog Quantum Systems, proves that TUS OS is a proper Anchored Analog Quantum System on its 64-input abstract canonical corpus, elevates the inherited structural/predictive correspondence into the Predictive Synthesis Identity Theorem, organises the inherited exact quantum-operational results through a typed access-contract resource law, and composes the recognition, predictive, resource and downstream-conservation results into the TUS OS Anchored Analog Quantum System Theorem. Paper 6 — Contract-Relative Quantum Realisations of Classical ProcessesPredictive Compression, Vanishing Worst-Case Causal-State Transparency, and Finite-Dimensional BreakdownExtends the programme beyond the TUS-specific discovery corpus by formalising the contract-relative achievable-resource object (S,C,𝔐,ε) ⟼ Ach(S,C;𝔐,ε), and establishes three exact contract-relative regimes: strict finite separation, unbounded finite separation and finite-capacity breakdown. These are instantiated respectively by the TUS OS 4→64 separation, the periodically modulated decay 2→N exact prediction-to-transparency separation, and dual-Poisson finite-capacity breakdown for every positive uniform worst-case transparency threshold. Paper 7 — Operational Observability and Contract-Induced Resource Transitions in Quantum RealisabilityStructural Classification, Family Scaling, and Finite-Resource BreakdownDevelops a structural classification
Avec la possibilité de l’existence d’un ordinateur quantique, les primitives cryptographiques basées sur la théorie des nombres risquent de devenir caduques. Il devient donc important de concevoir des schémas résistants à ce nouveau type de menaces. Les réseaux euclidiens et les codes correcteurs d’erreurs sont deux outils mathématiques permettant de construire des problèmes d’algèbre linéaire, pour lesquels il n’existe aujourd’hui pas d’algorithme quantique permettant d’accélérer significativement leur résolution. Dans cette thèse, nous proposons quatre primitives cryptographiques de ce type : deux schémas de signatures (dont une signature traçable) basés sur les réseaux, un protocole de délégation de signature utilisant du chiffrement complètement homomorphe, et une nouvelle approche permettant de construire des cryptosystèmes très efficaces en pratique basés sur les codes. Ces contributions sont accompagnées de paramètres concrets permettant de jauger les coûts calculatoires des primitives cryptographique dans un monde post-quantique.
The Substrate Language Compiler is the executable computation branch of the SAM Research Project. The current frozen revision, SLCV1.2-H14F-EA18WA-D9, preserves the N100 native basis and H14F exact variable-elimination fallback of the earlier SLC line while installing the complete Exact Algebra of a Write, the directional T18/Z4^9 Weil adapter with reconstructible W9 reciprocal history, and an exact reversible dyadic block interface at depths 1 through 9. The source grammar remains the object of computation: N100 is not replaced by a smaller physical problem, and the current machine still returns exact global tensors whose coefficient sums close to the complete formal state count. The canonical N100 frustrated-spin computation retains all 2^100 formal binary configurations of the unchanged 100-spin, 260-edge Hamiltonian. The structure-first route detects eight exact components of sizes 15,15,15,15,15,10,10,5, evaluates 165,920 local assignments, and reconvolves them into the same coefficientwise-identical tensor returned by the full H14F path. The current SLC therefore treats graph structure, separator width, quotient rank, reversible history, and readout type as computational certificates rather than as after-the-fact annotations. SLCV1.2 adds a native write engine whose exact finite semantics include the complete 256-state W8 shell, B-gated X1 activation, W9 record emission and W9-to-W8 return, signed Z4^9 action, endpoint reversal, the 161-coordinate nonconstant history quotient, completed-mode projection, analytic Bt/Et/J/P+/- operators, and multiplicative-ledger readback. A 4,096-write program applied at every one of 262,144 addresses represents 1,073,741,824 logical transitions; the compiled group-law path produces bit-identical output and exact inverse closure with a recorded same-task speedup of 1784.531387x. The directional Weil fusion replaces the former NumPy write seam with this native exact write while leaving the prime-spacing and completed-Weil channels unchanged. The machine now joins exact dense state computation, typed record semantics, reciprocal direction, and read-only Weil custody in one frozen architecture. The result is a compiler whose central principle is unchanged: preserve the complete mathematical object, identify the information that must remain live, contract the rest exactly, and retain enough custody that the result can be independently reconstructed
The development of quantum computing has been rapid, and an existential threat to computational hardness assumptions that underlie today's public-key cryptography has been encountered. This manuscript is a complete and thoroughly organized review of Post-Quantum Cryptography (PQC), discussing the mathematical concepts, complexity, and mechanisms of the algorithms that underpin quantum-safe security infrastructures. Rather than the principles of quantum mechanics, the analysis mathematically defines the basic principles of qubits, superposition, and entanglement, and clarifies common misunderstandings of the theory of parallelism in quantum mechanics. The research places cryptographic problems into the Bounded-Error Quantum Polynomial-Time (BQP) complexity class, thus highlighting the exact defects of classical algorithms. The quantum gate complexity of Shor's algorithm is compared to classical bounds as in Pollard's rho algorithm for the Elliptic Curve Discrete Logarithm Problem (ECDLP), for a detailed comparative cryptanalysis. The comparison highlights a very important “quantum security inversion” in that Elliptic Curve Cryptography (ECC) is much more susceptible to the early-stage fault-tolerant quantum computer as compared to the approach used today in RSA. It unfolds the architectural dependency of quantum algorithms, one by one, from the Quantum Fourier Transform (QFT) to Quantum Phase Estimation (QPE) and Shor's algorithm to Grover's algorithm. Last but not least, the manuscript examines emerging attack primitives such as lattice-based, code-based, and hash-based cryptography, their security models, side channel vulnerabilities, and the recently finalised Federal Information Processing Standards (FIPS 205, 203, and 204) that have been published by the National Institute of Standards and Technology (NIST).