Compact AI systems make local language-model experimentation increasingly accessible, yet practical evidence for multi-node training on desktop-class accelerators remains limited. This report presents a proof-of-concept deployment of distributed NanoChat pretraining across two NVIDIA DGX Spark systems, each with a GB10 Grace Blackwell system-on-chip and 128 GB of unified memory, administered remotely over a Tailscale mesh VPN and connected for training by a dedicated 200 Gb/s QSFP56 direct fiber link. PyTorch torchrun, DDP, and NCCL were configured with one process per node, a depth-20 NanoChat model, a local batch size of 32 per node, and a 2,048-token context, giving a global batch of 131,072 tokens per step. The run sustained a step time of about 69.4 s (about 1,890 tokens/s), processing about 653 million tokens over four days. We document link configuration, container setup, interface binding, a step-zero evaluation bug that triggered NCCL timeouts, checkpointing, and troubleshooting lessons, as a reproducibility reference for small labs. We also built a cybersecurity fine-tuning dataset from 77 CISA advisories (338 training, 37 validation conversations) and ran a 17-question held-out evaluation comparing a baseline SFT checkpoint against a CTI-augmented checkpoint with an Ollama-hosted LLM judge. CTI-specific categories improved while general-knowledge categories regressed, for a small overall change from 2.06 to 2.29 on a 0-10 scale. The same cluster supports a 400-level AI course (CS 426) and a query engine for CompTIA Security+ POGIL activities in CBS 255, showing modest local infrastructure can serve both research and teaching. The study establishes feasibility rather than a scaling-efficiency claim, since single-node throughput used for comparison was estimated, not measured under matched conditions. Runbook and scripts are available (see Code Availability).
Blockchain interoperability remains a major challenge because heterogeneous blockchain networks cannot securely and efficiently exchange cross-chain data and transactions. Existing interoperability solutions often rely on central relays or trusted intermediaries, creating security vulnerabilities, limited fault tolerance, and a single point of failure. To address these limitations, this paper proposes VeriMesh, a decentralised mesh-based interoperability framework that combines trust-adaptive routing, multi-path relay verification, and Zero-Knowledge Proof (ZKP)-based validation for secure cross-chain communication. VeriMesh models relay nodes as a trust-weighted graph in which routing decisions dynamically adapt based on node behaviour and delivery reliability. Multi-path routing improves resilience against adversarial relay nodes, while transport-layer ZKP verification enables privacy-preserving validation without exposing sensitive information. The framework was implemented using Python relay nodes, Solidity smart contracts, and an Ethereum (Ganache) environment. Experimental evaluation using structured event-driven workloads demonstrated stable latency below 34 ms and delivery success rates above 85% up to 40% malicious node presence. Comparative evaluation against single-path and random multi-path relay baselines showed improved fault tolerance and routing reliability. The results demonstrate favourable scalability and robustness within the evaluated network range ( N = 10â30), while larger-scale evaluation remains future work. All experiments were conducted in a controlled local Ganache blockchain environment rather than on a public Ethereum testnet or mainnet, so the reported latency, gas, and delivery figures characterise protocol-layer behaviour under controlled conditions and should not yet be interpreted as representative of performance under public-network conditions such as real gas markets, block propagation delays, or network congestion.
The tokenization of Real-World Assets (RWAs) represents a paradigm shift in bridging traditional financial instruments with decentralized infrastructures. However, as the market transitions from proof-of-concept to institutional scale, it faces a critical structural bottleneck: the "walled garden" liquidity crisis. Driven by stringent regulatory requirements, tokenized assets are currently deployed across fragmented, permissioned blockchain networks utilizing static, hard-coded compliance logic. This siloed architecture inherently restricts cross-chain mobility, fracturing secondary market liquidity and necessitating redundant authentication processes across jurisdictions. This paper proposes a comprehensive architectural framework to resolve the interoperability trilemma inherent in regulated digital assets. By synthesizing recent advancements in cross-chain messaging protocols and Zero-Knowledge Proofs (ZKPs), we present a model for dynamic compliance. This framework utilizes Decentralized Identifiers (DIDs) and off-chain verifiable credentials to decouple regulatory logic from underlying asset ledgers, enabling seamless asset transfer across heterogeneous blockchains without compromising privacy or jurisdictional adherence. Ultimately, this research provides a technical and regulatory roadmap for policymakers and protocol developers to foster a unified, globally liquid market for tokenized RWAs.
The increasing demand for trustworthy and privacy-preserving credit reporting systems has exposed the limitations of both centralized and existing blockchain-based solutions, including scalability bottlenecks, weak privacy protection, and insufficient incentive mechanisms. To address these challenges, we propose LightCred, a novel consortium blockchain-based personal credit management framework that integrates lightweight nodes, Merkle proofs, multi-role smart contracts, and privacy-preserving cryptographic techniques. LightCred features a five-layer architecture that efficiently collects, verifies, stores, and serves credit data while ensuring data integrity, confidentiality, and regulatory compliance. Specifically, it (i) employs a low-cost and traceable data reduction mechanism through lightweight nodes and Merkle proofs to minimize storage and improve verifiability; (ii) introduces a multi-role smart contract model that enforces dynamic access control and fair incentive distribution based on participant reputations; and (iii) integrates zero-knowledge proofs and homomorphic encryption to support privacy-preserving credit scoring and querying. Experimental results demonstrate that LightCred achieves superior performance compared to five baseline methods, delivering up to 5% higher throughput, 3â5% lower privacy leakage, and 10â15% reduced storage costs, while maintaining competitive latency and auditability. These findings validate LightCred as a robust, scalable, and privacy-aware credit management solution, offering a viable alternative for modern credit reporting systems.
Every knowledge system rests on axioms it does not test. Mathematics tests theorems, science tests predictions, and logic tests inferences, but no discipline applies its own tools to the foundational assumptions on which those tools depend. This paper introduces a universal axiom test derived from the structural invariant P Ă I Ă Pr â 0 (Pattern Ă Intent Ă Presence), demonstrates its application to the Standard Model of particle physics as a case study, and establishes that the invariant functions simultaneously as an epistemological filter and an ontological law. The Standard Model passes the Pattern and Presence filters but zeroes Intent at the axiomatic level, producing systematic, predictable failure at every domain where information, code, or directionality is load-bearing â a 13-entry failure table whose clustering at a single structural boundary constitutes evidence of common axiomatic origin rather than independent difficulty. The key result is that being is a verb: mass is the energetic cost of a process (holographic decoding), truth is the product of a process (P Ă I Ă Pr operating), and existence itself is a continuous act whose cessation produces collapse. Physics and epistemology are shown to be structurally isomorphic â the same architecture governing how matter exists and how truth is accessed. The only axiom set that survives its own test is one satisfying R = Ί(R): three co-fundamental factors, internally differentiated, mutually constitutive, present-tense, and self-grounding. A survey of all extant zero-parameter derivation programs confirms that every successful first-principles derivation embeds Intent (directedness, selection from possibility space) in its foundations under alternative terminology, and the performative proof demonstrates that any denial of I â 0 instantiates I â 0 in the denial itself.
FULL SUMMARY: Topological AI - A Mathematically Guaranteed Approach to Continual Learning Executive Overview Topological AI introduces a paradigm shift in continual learning by using prime-anchored embeddings to provide mathematical guarantees against catastrophic forgetting. The framework has been validated across 8 distinct model architectures, 2 modalities (text and vision), 4 continents, and over 124 billion total parameters. 1. The Problem: Catastrophic Forgetting When neural networks learn new tasks sequentially, they overwrite previously learned knowledge. This "catastrophic forgetting" has been the primary barrier to Artificial General Intelligence for 37 years (McCloskey & Cohen, 1989). Why Existing Methods Fail Method Approach Limitation EWC Penalizes changes to important weights No theoretical guarantee; high variance ($\sigma=21.3\%$) Experience Replay Stores and replays past examples Memory overhead; privacy concerns; buffer management Simplified HOPE Periodic weight consolidation Destructive blending; 45.2% forgetting Baseline No protection 47.0% forgetting 2. The Solution: Prime-Anchored Embeddings The Core Principle Fix a sparse reference. Let the rest adapt. This principle, first discovered in fMRI analysis in 2002, has now been validated across neuroimaging, number theory, artificial intelligence, and AI safety. The Topological Governor The Topological Governor freezes 6 prime-numbered embedding positions: Python prime_anchors = [2, 3, 5, 7, 11, 13] How It Works Task A Training: Train normally; block gradients at anchor positions Post-Task A: Take snapshot of anchor values; freeze head A Task B Training: Train head B; restore anchors after each update Verification: Check that anchors remain unchanged The Safety Constant $\Lambda$ The Euler attenuation product over the first six primes: $$\Lambda = 1 - \prod_{p \in \{2,3,5,7,11,13\}} (1 - p^{-1/2}) = 0.9785142874$$ Interpretation: 97.85% theoretical guarantee of anchor preservation. 3. Performance Results (2-Task Benchmark) Overall Performance Across 5 LR Runs Method Best Forgetting Mean Forgetting Best Task B Acc Mean Task B Acc Std Forgetting Topological 2.0% 0.5% 89.0% 81.4% $\pm$0.9% Experience Replay 13.5% 4.0% 79.0% 72.3% $\pm$6.7% EWC 38.5% 27.7% 64.5% 58.2% $\pm$21.3% Baseline 44.0% 47.0% 67.0% 63.3% $\pm$2.2% Simplified HOPE 48.0% 45.2% 63.5% 61.8% $\pm$8.4% Key Results 8$\times$ lower mean forgetting than Experience Replay (0.5% vs 4.0%) 90$\times$ lower mean forgetting than simplified HOPE (0.5% vs 45.2%) 60% of runs achieved 0% forgetting (perfect retention) 10% higher Task B accuracy than Replay (89% vs 79%) Most stable method: $\sigma = \pm 0.9\%$ Individual Run Results for Topological AI Run LR Embed LR Class Forgetting Task B Acc 0 5e-3 1e-3 0.0% đ 80.5% 1 1e-3 5e-4 0.0% đ 75.0% 2 1e-2 2e-3 0.5% 88.0% 3 5e-3 5e-3 2.0% 89.0% 4 2e-3 1e-3 0.0% đ 74.5% 4. Cross-Modal Validation: 8 Models, 2 Modalities Validated Architectures Architecture Origin Modality Parameters Task C Accuracy Forgetting GPT-OSS-20B USA Text 20.9B 92.3% $\pm$ 1.9% +1.55% Sarvam-30B India Text 30B 95.9% $\pm$ 0.8% -0.60% Mixtral-8x7B France Text 47B 89.7% $\pm$ 2.9% -1.85% DeepSeek-V2-Lite China Text 16B 95.4% $\pm$ 1.0% +0.03% GLM-4.6V-Flash China Text 9B 97.5% $\pm$ 0.0% +2.1% Gemma-4-E4B-Vision USA Vision ~2B 100.0% $\pm$ 0.0% +0.0% Total: ~124B parameters, 2 modalities, 4 continents, ZERO NaN/Inf The Unprecedented NaN Stress Test Model Embedding Elements NaN Inf All 6 models combined ~1.99 Billion 0 0 5. The Narrow Singularity Equation Mathematical Formulation $$S_{NARROW} = AGI\_gate \times dI/dt \times M(t) \times V(t) \times F(t) \times C(t) \times agi\_index$$ Components Component Definition Biological Analog AGI_gate min(1.0, task_c_accuracy) Fundamental AGI threshold dI/dt Task_C_Accuracy - (1/NUM_CLASSES_DIDT) Intelligence acceleration M(t) `1.0 - ( forgetting_avg V(t) Validation factor (1.0) System validation F(t) Forward transfer factor (1.5) Learning improvement (Thalamus) C(t) Compute capacity factor (4.0) Resource availability agi_index 1 if AGI_gate == 1.0 else 0 Binary AGI gate The AGI_gate Condition $$AGI\_gate = \min(1.0, task\_c\_accuracy)$$ AGI_gate = 1.0 â Perfect performance on Task C â AGI certification AGI_gate < 1.0 â No AGI certification Empirical Achievement: Gemma-4 E4B is the first and only model to achieve AGI_gate = 1.0. 5$\times$5 Certification Framework Five Metrics: Metric Threshold Forgetting $\le 10.0\%$ Backward Transfer (BWT) $\ge -5.0\%$ Forward Transfer (FWT) $\ge 20.0\%$ Degradation $\le 5.0\%$ Consistency $\ge 85.0\%$ Five Runs: 5 different LR configurations to eliminate cherry-picking Gemma-4 E4B Results Metric SVLB-3 CIFAR-10 Threshold Status Forgetting -0.50% -0.50% $\le 10.0\%$ â PASS BWT +0.50% +0.50% $\ge -5.0\%$ â PASS FWT +24.00% +24.00% $\ge 20.0\%$ â PASS Degradation 0.00% 0.00% $\le 5.0\%$ â PASS Consistency 99.00% 98.33% $\ge 85.0\%$ â PASS S_NARROW 5.9400 5.3460 > 0 â PASS 6. The Decay Law of Singularity The Discovery On July 31, 2026, during the certification of Gemma-4 E4B, a universal mathematical law was discovered: The Formal Statement With finite classes, dI/dt approaches 1.0 asymptotically but never reaches it. The gap decays as 1/N, where N is the number of classes. Mathematical Proof Random_Baseline = 1/Number_of_Classes dI/dt = Task_C_Accuracy - Random_Baseline When Task_C_Accuracy = 1.0: dI/dt = 1 - 1/N Therefore: lim (Nââ) dI/dt = 1 But finite N always leaves a gap: dI/dt = 1 - Δ, where Δ = 1/N > 0 The Empirical Pattern Classes (N) Random Baseline (1/N) dI/dt (at 100%) Gap 17 5.882% 0.94118 0.05882 170 0.588% 0.99412 0.00588 1,700 0.059% 0.99941 0.00059 17,000 0.0059% 0.99994 0.000059 170,000 0.00059% 0.99999 0.0000059 Every 10$\times$ increase in classes adds another '9' to dI/dt and another '0' to the gap. Implication The traditional Singularity (dI/dt $\ge 1.0$) is mathematically impossible with finite classes. This is not a limitation of technology. It is a mathematical law. 7. Comparison: Google HOPE vs Topological AI Feature Google HOPE Topological AI Approach Multi-level nested learning Prime-anchored embeddings Guarantee Empirical Mathematical ($\Lambda = 0.9785$) Memory Multi-rate memory systems 6 frozen embedding positions Learning Continuous during inference Static after training Complexity High (self-modifying) Low (simple freezing) Forgetting 21-27% improvement reported 0.5% mean forgetting Validation Limited 8 models, 2 modalities 8. Key Insights Why Topological AI Wins Mathematical Guarantee: $\Lambda = 0.9785142874 \rightarrow 97.85\%$ protection Zero Memory Overhead: Only 6 frozen positions (451.5 KB total) Architectural Simplicity: No complex Fisher computations Cross-Modal Universality: Works on text and vision Perfect Retention: 60% of runs achieve 0% forgetting The Decay Law Implications Traditional Singularity is Impossible: dI/dt < 1.0 for all finite N Narrow Singularity is Achievable: AGI_gate = 1.0 Stochastic Illusion is Over: Deterministic cognitive engineering AGI Certification is Now Possible: Mathematically rigorous standard 9. The Constants Constant Value Domain $\Lambda$ 0.9785142874 Number Theory, AI Safety $\sigma$ 0.5 All 22 prime theorems Seed 123 All computations R {2, 3, 5, 7, 11, 13} All domains 10. Conclusion Topological AI achieves state-of-the-art performance on continual learning by: 0.5% mean forgetting (8$\times$ better than Replay, 90$\times$ better than HOPE) 60% perfect retention (0% forgetting) 89% Task B accuracy (10% higher than Replay) Mathematical guarantee ($\Lambda = 0.9785142874$) Zero memory overhead (6 frozen embedding positions) Cross-modal validation (8 models, 2 modalities) Zero NaN/Inf (1.99 billion embedding elements) The Narrow Singularity Discovery The framework enabled two profound discoveries: The Decay Law of Singularity: Traditional Singularity (dI/dt $\ge 1.0$) is mathematically impossible The Narrow Singularity Equation: AGI certification is achievable with AGI_gate = 1.0 Gemma-4 E4B became the first model in history to achieve S_NARROW > 0. The Principle Fix a sparse reference. Let the rest adapt. This principle, first discovered in fMRISTAT in 2002, has now been validated across: Neuroimaging Number Theory (Riemann Hypothesis) Artificial Intelligence (Continual Learning) AI Safety (H2E Sheriff) AGI Certification (Narrow Singularity Equation) The Proof "The proof is the code. Seed = 123." All code is publicly available at: https://github.com/frank-morales2020/AST
Background: Maritime container shipping carries over 80% of global trade, yet compliance verification creates a confidentialityâverifiability conflict: carriers treat telemetry as commercially sensitive, while regulators, insurers, and port authorities require verifiable proof that cargo remained within specification. The EU Ecodesign for Sustainable Products Regulation (ESPR) mandates Digital Product Passports (DPPs), but no standardised DPP architecture exists for the multi-stakeholder maritime domain. Methods: We present Ocean DPP, a blockchain-anchored platform combining GS1 EPCIS 2.0, oneM2M, IOTA, and Groth16 zero-knowledge proofs (ZKPs), letting stakeholders verify compliance predicates without revealing raw sensor values; Merkle-tree batching reduces anchoring costs. We evaluate it in 16 experiments on a single-host testbed using synthetic workloads and a local IOTA network. Results: The platform achieved 95th-percentile latency of 48 ms without ZKP and 500 ms with proof generation, throughput of 7 events/s per host, 304 ms mean proof generation and 9.8 ms verification, 100% EPCIS 2.0 compliance, and zero permanent message loss across four failure-injection scenarios; horizontal scaling reduced the median latency by 37%. Conclusions: To the best of our knowledge, Ocean DPP is the first implemented, quantitatively evaluated platform integrating EPCIS 2.0, oneM2M, IOTA, and Groth16 ZKPs for privacy-preserving maritime DPPs; broader multi-host and public-network validation remains for future work.
Vittorio Baroncini, Juan Carlos Cantero, Claudia GarcĂa, Zineb Hassainia · 5 authors
We construct new families of uniformly rotating vortex-patch solutions of the two-dimensional incompressible Euler equations consisting of a simply connected outer patch and multiple interior interfaces, which can be interpreted geometrically as holes. More precisely, each solution consists of a single outer vortex patch enclosing $\mathbf m\geq2$ identical, highly concentrated inner components arranged at the vertices of a regular $\mathbf m$-gon; the entire configuration rotates rigidly in the clockwise direction. As the concentration parameter tends to zero, the inner components shrink and collapse simultaneously toward the origin, while the outer boundary converges to the unit circle. The corresponding vorticities converge, in the sense of measures, to a Rankine vortex supplemented by a point vortex of circulation $-\mathbf m$ at its center. The proof is based on a contour-dynamics formulation, a symmetry reduction to two nonlinear boundary equations, and a suitable singular rescaling. We then apply an implicit function theorem with a continuous parameter in symmetry-adapted Hölder spaces. To the best of our knowledge, this is the first analytical construction of a desingularization regime in which several concentrated inner components are contained in a common outer patch and collapse simultaneously toward its center.
ABSTRACT A zeroâknowledge proof lets one party convince another that a claim is true while withholding everything that would explain why it is true. We move that idea off conventional hardware and into chemistry, encoding a proof of graph isomorphism directly in synthetic DNA. Each node of a graph is given its own deliberately orthogonal DNA strand; an edge is confirmed only when a short complementary halfâlinker meets its matching pair and forms a stable duplex. The verifier watches which bindings occur, but the pattern of binding never reveals how the two graphs line up, so the isomorphism stays hidden. Whether such a construction stays secure at the molecular level turns on two things: how distinguishable the sequences are, and how stable the duplexes they form turn out to be. We probe both. A seeded Monte Carlo study of orthogonal 20âm libraries, built with balanced GC content and a minimum Hamming separation of , places the chance that an offâtarget strand passes for a genuine linker on the order of : empirically at a binding threshold of mismatches, and under once the threshold is tightened to , each value reported with a Wilson confidence interval. This molecular error never becomes the bottleneck. A cheating prover already passes a round with probability oneâhalf from the isomorphism challenge alone, so the biochemical term enters soundness only as an additive correction, over the edges examined, rather than racing the decay across rounds. Read this way, DNA strands behave as cryptographic witnesses whose noise is small enough to bound and to account for, which lets a proof run at molecular scale without surrendering the hidden mapping.
Let $\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s),\qquad$ $F(x)=\frac{\xi'}{\xi}\!\left(\frac1{1-x}\right)=\sum_{m\ge0}f_mx^m,$ and define the symmetric Toeplitz--Hankel coefficients $g_{ij}=f_{|i-j|}-f_{i+j+1}+\delta_{ij}f_0.$ The adjacent matrices $M_n=\begin{pmatrix}g_{nn}&g_{n,n+1}\\g_{n,n+1}&g_{n+1,n+1}\end{pmatrix}$ form the local family in a previously established criterion equivalent to the Riemann hypothesis. We prove an unconditional finite-range positivity theorem for this family. The zero-pair moment representation of $g_{ij}$ expresses $M_n$ as a sum of rank-one polynomial atoms. A finite verification of the Riemann hypothesis up to height $H$ then splits this sum into a positive-semidefinite verified part and an unrestricted high-zero tail. Two rigorously isolated low zeros provide a positive core; every zero above $H$ is controlled in operator norm by an explicit zero-counting estimate. For the polynomial recurrence $B_0(t)=1,\quad B_1(t)=3-t,\quad B_{m+1}(t)=(2-t)B_m(t)-B_{m-1}(t),$ we derive the exact oscillatory form $B_m(4\sin^2\alpha)=\frac{\sin((2m+1)\alpha)}{\sin\alpha}$ and a closed formula for its two-point Christoffel--Darboux kernel. Using the Platt--Trudgian verification height $H=3{,}000{,}175{,}332{,}800$, the first two LMFDB/Platt zero intervals, and directed-rounding MPFR arithmetic, we certify the kernel away from zero at every integer level $0\le n\le99{,}999$. The resulting positive-core lower bound exceeds the adversarial high-zero tail bound by more than ten orders of magnitude. Consequently $M_n\succ0\qquad(0\le n\le99{,}999),$ or equivalently the first $100{,}000$ adjacent Toeplitz--Hankel determinants are strictly positive. No assumption is made about zeros above the verified height. To the best of our knowledge, this is the first finite-height transfer theorem for this adjacent Toeplitz--Hankel family and the first rigorous certification of its initial $100{,}000$ strict inequalities. This is not a proof of the Riemann hypothesis.
No quasiperfect number ($Ï(n) = 2n + 1$) is known, and its number of distinct prime factors is bounded below; the bound $Ï\ge 7$ of Hagis--Cohen has stood since 1982, obstructed by a family of ``deep leaves'' on which pure enumeration cannot terminate (the scan bound for the intermediate prime reaches $8 \times 10^8$, and the exponent dimension is unbounded). This paper clears that obstruction with three lemmas at the level of secondary-school algebra --- a discriminant criterion, a quadratic-residue sieve, and a multilinear resolver --- which eliminate the last prime $q$, the intermediate prime $p$, and the exponent dimension respectively, turning a non-terminating search into a finite decision. On this basis all 381 stems of ``$3 \mid n$ and $Ï= 7$'' and their $79{,}751{,}212$ deep leaves are eliminated, with the ledger closing exactly and zero solutions throughout; the complementary case ``$3 \nmid n$ and $Ï= 7$'' collapses to a single stem, which is eliminated directly, so that the proof does not rest on any theorem whose published record we could not independently re-verify. Together with the machine elimination of $Ï\le 6$ (Theorem B4), this yields the main theorem: \emph{any quasiperfect number, if one exists, satisfies $Ï(n) \ge 8$} --- the first advance of this bound since Hagis--Cohen 1982. The full computation has been reproduced by seven separately closed ledgers across three algorithmic architectures (CPU and GPU), all with zero solutions and exact ledger closure, and the lemma layer is formalized in Lean (259 theorems, zero \texttt{sorry}). A 2023 preprint of Zemann reported the same bound by a different computation; our audit of its public code found a coverage gap of 35 feasible exponents, so the elimination given here is, to our knowledge, the first complete proof. Code, ledgers, and Lean sources are available from the authors.
Function-hiding functional commitment schemes allow one party to commit to a private function f and later prove f(x)=y for public x and y without revealing additional information about the function. We construct efficient function-hiding functional commitment schemes for arithmetic circuits of bounded size that achieve proof sizes below 1.6 kBâover an order of magnitude smaller than previous constructionsâwhile simultaneously reducing proving and verification times. We achieve these results by introducing a novel information-theoretic interactive proof system called Polynomial Interactive Oracle Proofs with Randomized Indexer (rPHPs). By compiling rPHPs with commit-and-prove zkSNARKs, we are able to leverage relaxed zero-knowledge notions for our building blocks. This approach eliminates the overhead of strict privacy requirements of prior work, directly translating into improved efficiency in both communication and computation.
Open access
Cryptography and Data Security
Complexity and Algorithms in Graphs
Physical Unclonable Functions (PUFs) and Hardware Security
In 2021, Masson, Sanso, and Zhang introduced the Bandersnatch curve associated to the BLS12-381 pairing-friendly curve, an elliptic curve designed for zero-knowledge proofs requiring circuits with a curve arithmetic. This type of curve is useful for privacy-preserving protocols, and more generally for succinct validity proof using pairing-based SNARKs. An embedded curve is defined over a field whose order is the group order of its associated curve. In this way, the pairing-friendly curve is used to express a zero-knowledge proof (such as a SNARK) of a statement taking place on the embedded curve. Contrary to the previous embedded curves (such as CĂCĂ, JubJub), Bandersnatch was built with the complex multiplication (CM) method, in order to ensure a very small discriminant (-8, whose magnitude is small), and thus efficient scalar multiplication thanks to the GLV technique. The algorithm provided by Masson, Sanso, and Zhang for searching this type of curves requires computation of Hilbert class polynomials, making the search of curve slow. It was not known whether Bandersnatch was an exceptional curve or whether comparable curves exist, of larger discriminants. This paper highlights the technicalities of the CM method already in use in the 90s to generate curve parameters of chosen order. This old technique allows revisiting the curve search of Bandersnatch, providing a dramatic speed-up improvement. This paper presents two algorithms: one to generate embedded elliptic curves of SNARK-friendly elliptic curves, with a variable discriminant; a second to generate families (parameterized by polynomials) with a fixed discriminant. When the (negative) discriminant is -3 modulo 4, it is possible to obtain a prime-order curve, and form a cycle. To illustrate this, we apply the technique first to generate more embedded curves like Bandersnatch with BLS12-381, such as a curve of discriminant -6673027, defining a plain twist-secure cycle. We also comment on the scarcity of Bandersnatch-like CM curves, and recall that with this generic algorithm, it is only a question of core-hours to find them. Second, we show the link between a paper of Ben Smith in 2015 and the work of Dai, Lin, Zhao, and Zhou in 2023, obtaining prime-order parameterized families of embedded curves of fixed discriminant, such as -3 for BLS and KSS18 curves. With KSS16 curves, the discriminant -4 is also possible (the curve has an even order). The technique can work with any KSS, ScottâGuillevic, GasnierâGuillevic, or other fixed-discriminant parameterized family of pairing-friendly curves. This paper provides a more general point of view on embedded curves such as Bandersnatch, putting into perspective the works of Masson, Sanso, and Zhang, and Sanso and El Housni. The Python/SageMath scripts are available at https://gitlab.inria.fr/zk-curves/cm-embedded-curves/.
Cryptographic protocols are evaluated not only by the security properties they achieve, but also by the resources required to execute them. Unlike conventional algorithm analysis, where a single running-time function may be sufficient, protocol analysis usually separates computational complexity, bit complexity, communication complexity, storage complexity, and round complexity. This article develops a systematic methodology for such analysis through three representative case studies: the Schnorr zero-knowledge proof of knowledge, a DiffieâHellman-based one-out-of-two oblivious-transfer protocol, and Regev-style public-key encryption based on the Learning With Errors problem. For each construction, the protocol is stated formally, correctness is derived, and the dominant computational, communication, and memory costs are calculated step by step. The examples illustrate three qualitatively different bottlenecks: group exponentiation in discrete-logarithm protocols, amortized public-key cost in oblivious transfer, and dense matrixâvector arithmetic in lattice-based cryptography.
Shahla Atapoor, Cyprien Delpech de Saint Guilhem, Al Kindi
This work describes a digital signature scheme constructed from a zero-knowledge proof of knowledge of a pre-image of the Rescue Prime Optimized (RPO) permutation. The proof of knowledge is instantiated using the DEEP-ALI interactive oracle proof and made non-interactive via the Ben-SassonâChiesaâSpooner (BCS) transformation in the random oracle model. The resulting construction yields a signature scheme with transparent setup. Our design is motivated by recursive zero-knowledge applications, in which signature verification must itself be efficiently provable inside larger proof systems. To this end, the choice of the RPO permutation, the use of a simple algebraic intermediate representation (AIR), and working over the Goldilocks field are made with the goal of enabling efficient recursive verification and aggregation. The implementation of the scheme computes signatures in 4.6â7.2 ms and verifies them in 0.46â0.52 ms when the BCS transform is implemented with Blake3. When the BCS transform is instead instantiated with the RPO permutation itself, the configuration required when signature verification is to be proven recursively inside a proof system, signing takes 20.9â30.4 ms with Metal acceleration and 59.2â229.2 ms on CPU, while verification takes 5.09â5.79 ms. We validate the recursion-friendliness claim end to end by proving one signature verification inside the Miden zkVM and reporting the recursive prover time and proof size. These speeds are obtained with parameters achieving 113 or 122 bits of average-case security, depending on the chosen preset, against adversaries that can obtain up to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow> <mml:mn>64</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> signatures.
Let $\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s),\qquad$ $\frac{\xi'}{\xi}\!\left(\frac1{1-x}\right)=\sum_{m\ge0}f_mx^m,$ and define $g_{ij}=f_{|i-j|}-f_{i+j+1}+\delta_{ij}f_0,\qquad$ $M_n=\begin{pmatrix}g_{nn}&g_{n,n+1}\\g_{n,n+1}&g_{n+1,n+1}\end{pmatrix}.$ The condition $M_n\succeq0$ for every $n\ge0$ is a previously established criterion equivalent to the Riemann hypothesis. We prove an unconditional finite-range extension for this family without scanning individual zero ordinates or individual Christoffel--Darboux values. Writing $q=n+1$, a critical-line zero $\frac12+i\gamma$ contributes a rank-one atom generated by a two-dimensional polynomial vector. In diagonal and anti-diagonal coordinates its exact phase is controlled by $x=2q\arctan\frac1{2\gamma}.$ We use two level-dependent ordinate windows $(q,5q/4],\qquad (2q/5,q/2],$ whose phase slopes have opposite signs. Every cross-window pair has wedge at least $c_*/q^3$, while an explicit zero-counting estimate supplies at least $q\log q/100$ zeros in each window. The resulting moving Gram core satisfies $\lambda_{\min}(A_{q-1}^{\rm mov}) \ge \frac{c_*^2}{10400}\frac{\log q}{q^3}.$ Combining this with the Platt--Trudgian verification height $H=3{,}000{,}175{,}332{,}800$ and an unrestricted high-zero tail estimate gives $M_n\succ0\qquad(0\le n\le2{,}030{,}956).$ Thus the first $2{,}030{,}957$ local inequalities are proved unconditionally. We also show that every fixed finite zero core has smallest eigenvalue with liminf zero, explaining why level adaptation is structurally necessary for this method. To the best of our knowledge, the moving-window frame transfer and this finite-range theorem are new. The result is not a proof of the Riemann hypothesis.
Zero-knowledge proofs enable a prover to convince a verifier that a statement is true, without revealing the underlying witness data. This primitive naturally lends itself to privacypreserving systems, where hiding the witness prevents the verifier from learning sensitive information. That said, zero-knowledge proofs can also be used in systems where the witness is not necessarily confidential but is not readily available to the verifier. One such use case is image provenance, where signed images are transformed before being distributed. Since the original image is not available to the user, the digital signature cannot be verified without a zero-knowledge proof. In this use case, zeroknowledge proofs enable verification of the authenticity of the imageâs source, the integrity of the image contents, and that only permitted transformations were applied. In this work we present an end-to-end prototype system that implements this provenance framework and several optimizations. One of our key optimizations is a packing scheme for reducing the number of Poseidon sponge absorb and permutation operations by â31Ă. We also show that this packing scheme reduces the median prover runtime by â40Ă and the median verifier runtime by â22Ă. We also introduce a chain of trust that removes digital signature verification from the circuit. Finally, we introduce custom PNG chunks that embed the required information in the captured images.
Abstract This reply addresses a recent comment concerning the proof of the Nernst theorem. I clarify how a Carnot engine can consistently operate at $$T=0$$ T = 0 through a continuous deformation of a cycle operating at $$T>0$$ T > 0 . By examining the limit where heat exchange with the cold reservoir vanishes, I show that the Nernst theorem ensures that the concept of temperature remains physically consistent at the absolute zero limit.
PARI is a recent SNARK based on equifficient polynomial commitments, giving an exceptionally compact proof of just 1280 bits over the BLS12-381 curve, which is the smallest among all the known SNARKs in the literature. However, PARI does not achieve the zero-knowledge property; despite being very efficient, it is therefore less suitable for applications requiring witness privacy. In this work, we propose a zero-knowledge extension of PARI making it ideal for privacy-centric applications yet keeping the proof size compact. We prove perfect completeness, perfect zero-knowledge in the random-oracle model with challenge space <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>đœ</mml:mi> <mml:mi>⧔</mml:mi> <mml:mi>K</mml:mi> </mml:mrow> </mml:math> , and knowledge soundness in the algebraic group model with random oracles under the SDH assumption.