Homomorphic-encryption blockchain frameworks for IoT sensor aggregation generally rely on classical cryptographic hardness assumptions and seldom account for network topology in liveness and performance analysis. This work introduces Phi-PHE-BC, a topology-aware homomorphic blockchain architecture for secure and privacy-preserving IoT sensor data aggregation. The framework combines threshold Paillier decryption with graph-parameterized security and performance analysis, linking protocol behavior to the validator graph. On-chain Paillier ciphertexts support homomorphic aggregation while providing IND-CPA confidentiality under the Decisional Composite Residuosity assumption, and authentication signatures provide EUF-CMA transaction integrity. Threshold partial-decryption shares are protected by a noise-flooding wrapper that provides information-theoretic privacy under the configured statistical-hiding condition. Under partial synchrony and Byzantine fault-tolerance assumptions, liveness requires validator connectivity kappa(Gv) >= f+1. We derive topology-dependent throughput bounds for tree, star, mesh, and scale-free networks, together with a per-block communication-cost model. A game-theoretic analysis shows that honest validator participation is a dominant strategy under the stated utility model, yielding an all-honest Nash equilibrium. Experiments on Hyperledger Fabric 2.5 show lower end-to-end latency than the selected traditional PHE-blockchain baseline while maintaining controllable threshold-decryption overhead. Results across topology scaling, validator sensitivity, threshold decryption, and Byzantine-load experiments indicate that Phi-PHE-BC is a practical architecture for secure, privacy-preserving, and topology-aware IoT sensor aggregation.
The Moroccan Ministry of Justice aims to enhance the process of the judicial system. Through digitalization, given the sensitive information and the complexity of managing this volume of data, along with the multiple electronic materials exchanged, several challenges regarding the security, integrity, and confidentiality of personal data are presented that indicate difficulties in confirming authenticity. Using blockchain technology, the Ministry of Justice can exchange data and knowledge in a secure and transparent way. The goal of the proposed method is to automate the procedure for generating birth certificates to strengthen trust, security, and operational efficiency within the Moroccan judicial system.
This research presents a novel decentralized identity management system leveraging Ethereum smart contracts and cryptographic protocols to enable self-sovereign digital identities. Traditional centralized identity solutions pose risks related to data privacy, security, and user autonomy. To address these limitations, we propose a blockchain-based architecture integrating smart contracts for identity governance, IPFS for decentralized attribute storage, and threshold cryptography for private data recovery. Our dual-contract model (Identity Contract and Recovery Contract) facilitates secure identity creation, verifiable attribute attestation, and robust key recovery through social consensus. We introduce a privacy-preserving mechanism using encryption-key splitting among trusted peers to enable recovery of encrypted off-chain data. Implementation uses Web3.js, Solidity, and QR-code-based communication, abstracting cryptographic complexities from end-users. Security analysis addresses replay, man-in-the-middle, Sybil, and multi-user compromise attacks with mitigations including challenge-response authentication and time-delayed contract execution. This work contributes a cryptographically secure, user-centric identity framework ensuring data sovereignty, recoverability, and interoperability within the Ethereum ecosystem.
PQ-Sortition is a post-quantum cryptographic sortition protocol constructed from the NTRU lattice hardness assumption and instantiated using Falcon-512 (FN-DSA). The construction uses deterministic Falcon signing to obtain a reproducible, publicly verifiable proof and combines it with a consensus-layer commit-then-reveal mechanism to address the lack of unconditional uniqueness inherent in GPV-style lattice signatures. The work introduces NTRU-Sortition, a many-time lattice-based verifiable random function construction, and provides formal analyses of third-party uniqueness, pseudorandomness under the NTRU-SIS assumption in the Random Oracle Model, and provability. The paper further defines PQ-Sortition as a post-quantum proof-of-stake leader-election protocol using a historical randomness beacon, stake-weighted sortition, adaptive difficulty, equivocation slashing, and grinding resistance. The Falcon-512 instantiation provides a 32-byte output and proofs of up to 666 bytes. The paper also presents concrete performance measurements, security parameters, consensus integration details, comparisons with prior post-quantum VRF constructions, and open research problems.
Secure financial transactions require more than just an immutable record — they also demand privacy-preserving identity assurance (which enables secure, trusted and transparent communication), adaptive fraud intelligence (to detect fraudulent transactions), policy-aware execution (so organizations can set their own rules for data use), resilient consensus (enables multiple parties to agree on data use), and auditable records within a single low-latency pipeline. Current permissioned-blockchain solutions often have independent optimizations for authentication, access control, fraud detection, consensus and auditing; as such, these separate areas lead to fragmented security decision making, unnecessary disclosure, static endorsement policies and throughput–latency tradeoffs. The research presented here describes FinTrust-X, a cross-layer risk-adaptive permissioned blockchain architecture where the security state created by each layer is used to create the next. A Zero-Knowledge Context Adaptive Role and Trust Authentication System (ZK-CARTA) provides zero knowledge context adaptive role and trust authentication to enable verifiable credentials to be selectively disclosed based on user device/session context and dynamically authorize users to minimize identity exposure and privilege abuse. Users are provided authenticated evidence to feed a Temporal Graph Transformer (TRiG-FraudFormer) that models joint transactional, account, device, merchant, beneficiary and trust relationships to produce a calibrated fraud-risk assessment along with counter-factual explanations. Risk is converted into adaptive smart contract paths, confidence levels and endorsement requirements to minimize unnecessary verification overheads. Safety constrained reinforcement learning is applied in RA-BFTune to adaptively optimize batching, ordering and Byzantine fault tolerant consensus based on transaction risk and network-states. Continuous cryptographic audit evidence is produced in PQ-AuditTwin utilizing immutable provenance, Merkle verification and ML-DSA-based post-quantum signature generations. Feedback regarding changes/drift in previous layer inputs is returned to those layers. Targeted validation results show ROC-AUC values of .96-.98 and F1 values of .92-.95 were achieved in addition to achieving authentication times less than 30ms., 1500-2000 TPS, P95 response time < 700ms, and greater than a 90% reduction in unnecessary disclosure of sensitive data from users indicating significant improvements in confidentiality, fraud-resilience, authorization-efficiency, scalability and auditability when compared against multi-organization Fabric workloads that included injected fraud and Byzantine faults.
Blockchain data is immutable and publicly visible a sensitive signature is placed on-chain, anyone can attempt to verify it. This openness may lead to unintended information exposure. The standard ring signature cannot fully address all the privacy, selective-verification and time -controlled disclosure requirements that arise in modern secure systems. The proposed algorithm presents a Blockchain-based Ring Signature with Multi-Designated Verifier and Zero-Knowledge Proof (BRSMDV-ZKP) enables a signer anonymously authenticate a transaction with a group of public keys while ensuring that only designated verifiers can verify the signature, The scheme incorporates a challenge–response mechanism, randomized commitments, and encrypted verifiers specific data to ensure signer anonymity, trace resistance, and verifier exclusivity. A Zero-Knowledge Proof (ZKP) is employed to prove correct decryption of the signature without revealing the verifier′s private key. The time-lock puzzle enforces a predefined delay, preventing early verification and enabling reward–penalty mechanisms for verifier compliance. This approach reduces the risk of key leakage, preserves privacy in decentralized systems and multi-party environments, and supports secure applications such as confidential e-voting, sealed-bid auctions, and legal document verification on blockchain platforms.
With the rapid expansion of digital communication and data storage, cybersecurity has become a critical concern for organizations and individuals. Cryptographic methods play a vital role in ensuring data confidentiality, integrity, and authentication. This study explores the mathematical foundations of encryption, blockchain security, and post-quantum cryptography. Traditional encryption methods such as symmetric and asymmetric encryption rely on number theory and complex mathematical problems like integer factorization and discrete logarithms. Blockchain security is reinforced by cryptographic hashing and digital signatures, ensuring tamper-proof transactions. However, the advent of quantum computing poses a significant threat to existing cryptographic protocols, necessitating the development of post-quantum cryptographic methods. This research provides an in-depth analysis of current cryptographic techniques, evaluates their effectiveness, and discusses future advancements in quantum-resistant cryptography.
C. O. Enuma, Matthias D., V.I.E. Anireh, Bennett E.O.
Abstract Cloud computing has become the preferred platform for deploying blockchain-enabled smart contracts because of its scalability and flexibility. However, existing access control mechanisms such as Role-Based Access Control (RBAC), Attribute-Based Access Control (ABAC), and conventional blockchain authentication expose sensitive user information during authentication, rely on static authorization policies, and lack intelligent mechanisms for detecting evolving cyber threats. This study proposes an Intelligent Privacy-Preserving Access Control Framework for Cloud-Based Smart Contracts that integrates Modified Groth16 Zero-Knowledge Proofs (ZKP), Secure Multi-Party Computation (SMPC), Trusted Execution Environments (TEE), Federated Learning, Differential Privacy, GraphSAGE Graph Neural Networks, Autoencoder-based anomaly detection, Proximal Policy Optimization (PPO), and Blockchain Smart Contracts. The framework enables credential-free authentication, confidential collaborative computation, adaptive authorization, intelligent threat detection, and immutable blockchain auditing without compromising user privacy. The proposed framework was implemented and evaluated using the CICIDS2017 cybersecurity dataset. Experimental results achieved 96.4% privacy preservation, 94.1% security strength, 99.0% execution integrity, 98.7% auditability, 90.3% scalability, 88.6% computational performance, 86.9% cost efficiency, 98.91% validation accuracy, 99.62% ROC-AUC, 94.90% Macro F1-Score, and an overall system fitness of 94.23%. Comparative evaluation against Hawk, Zether, Ekiden, and a Federated Learning-only IDS demonstrated superior performance across all evaluation metrics. The proposed framework therefore provides an intelligent, scalable, and privacy-preserving access control solution suitable for next-generation cloud-based smart contract systems. Keywords: Privacy-Preserving Access Control; Smart Contracts; Cloud Computing; Zero-Knowledge Proof; Secure Multi-Party Computation; Trusted Execution Environment; Federated Learning; Blockchain.
The rapid growth of digital education and online recruitment has significantly increased the demand for reliable academic credential verification. Conventional certificate verification methods are often centralized, time-consuming, and susceptible to document forgery, unauthorized modification, and administrative delays. To address these challenges, this paper presents a Blockchain-Enabled Decentralized Framework for Secure Academic Certificate Issuance and Real-Time Verification. The proposed framework utilizes Ethereum blockchain technology through Solidity smart contracts to establish an immutable and transparent repository of certificate records, ensuring that issued credentials cannot be altered without detection. A SHA-256 cryptographic hashing mechanism is employed to generate unique digital fingerprints for each certificate, while Firebase Authentication and Cloud Firestore provide secure identity management and efficient off-chain metadata storage. The user interface is developed using React.js, enabling educational institutions to issue certificates and allowing employers, universities, and other stakeholders to verify credentials instantly through a simple web-based platform. During verification, the system recomputes the certificate hash and compares it with the blockchain record to detect tampering and validate authenticity in real time. Experimental evaluation on a local Ethereum network demonstrates reliable certificate issuance, rapid verification with sub-second response times, secure transaction handling, and effective resistance against certificate forgery. The proposed framework enhances transparency, trust, and operational efficiency while minimizing manual verification efforts. Furthermore, its modular architecture facilitates future migration to public blockchain networks and decentralized storage platforms, making it suitable for scalable deployment across educational institutions and digital credential ecosystems.
MRS‑AUTH is a novel authentication framework that achieves deniability even against an active verifier who may adaptively query candidate credentials both before and after receiving a challenge. Unlike ring signatures or zero‑knowledge proofs – where the prover holds a single secret witness that can be extracted under coercion – MRS‑AUTH exploits the multiplicative structure of linear Diophantine equations. Through recursive decomposition, it generates a Diophantine forest of exponentially many syntactically valid credential chains. The authentic chain is sampled uniformly from this forest and committed together with k‑1 indistinguishable aliases using a fixed‑shape Merkle tree with dummy leaves, eliminating structure‑ and length‑based side‑channel leakage. The Forest Symmetry Theorem proves that all chains are structurally information‑theoretically indistinguishable. However, the full index‑anonymity against an active verifier is computational and bounded in Theorem 6.6 by k · ε_SHA3 + ε_coll + negl(λ). For cryptographic scales N ∼ 10⁴², the Ehrhart‑based continuous‑volume approximation yields an effective entropy exceeding 371 bits, with a statistical distance to the perfect uniform distribution of Δ ≤ 2⁻¹³⁵ – well below the 128‑bit security threshold. Empirical validation via exact enumeration and a chi‑squared test (χ²/dof ≈ 0.985) confirms the uniformity. A constant‑time Rust implementation, leveraging the subtle and zeroize crates, exhibits an execution time of approximately 0.12 ms across four orders of magnitude of N, demonstrating practical deployability. The work also formalises the Active Verifier Game model, a new adversarial definition that quantitatively captures coercion resistance in a post‑quantum setting.
Open access
2 source records
Cryptography and Data Security
Physical Unclonable Functions (PUFs) and Hardware Security
Stateful cryptographic schemes—exemplified by the hash-based signatures XMSS (RFC 8391) and LMS (RFC 8554)—require the signer to advance a local state monotonically; any rollback is catastrophic, yet a verifier has no way to check it. IETF guidance on state and backup management for hash-based signatures states explicitly that the verifier must simply trust the signer not to have reused state. We define verifiable monotone chains (VMC), a primitive that makes such state discipline cryptographically verifiable: state evolves along a finite poset (S, ⪯) under inflationary monotone operators, every transition carries a zero-knowledge proof, and a public commitment to the state provides an audit trail. We formalize two security notions: monotone-unforgeability (MU), which captures that an external adversary cannot certify an illegal or rolled-back transition, and auditability (AUD), which captures that signer rollback cannot be hidden from a public root history. Both notions reduce, with explicit advantage bounds, to position binding of the underlying vector commitment and knowledge soundness of the proof system. We instantiate VMC as RSEP-XMSS, in which each XMSS signature carries a proof that the signed leaf advanced along the chain FRESH ≺ USED ≺ SPENT in a Poseidon-based state Merkle tree, and we give a complete algorithmic specification with a concrete circuit design (~6041 R1CS constraints estimated, Groth16 proving time estimated at 5–15 ms, signature overhead of about 1–3 KB). RSEP-XMSS is one-way compatible with standard XMSS: legacy verifiers verify the core signature, while enhanced verifiers reject unprotected signatures, preventing downgrade attacks.
The disclosure duty of arbitrators does not carry a big stick: the consequence attached to its breach is near to none. The duty is carried out through a black-box judgment call; no reasoning for how the arbitrators weed out what not to disclose is made available to the parties. There may be circumstances falling under a grey area but have ended up undisclosed due to, for instance, confidentiality obligations. Exhaustive disclosure and minimum revelation of sensitive information would serve the greatest benefit of all stakeholders. Zero-knowledge proofs (ZKP) – a class of cryptographic protocols – may make this possible.
The subject matter of the article is the cryptographic integrity of digital authentication systems facing quantum computing threats, specifically focusing on post-quantum alternatives and efficient authenticated data structures. The goal is to design and formally analyze VERKLE-FRI—a hybrid architecture synthesizing Verkle tree proof-size reduction with FRI-based quantum-resistant commitments, establishing a scalable, stateless, and quantum-secure framework. The tasks are: analyze limitations of hash-based signatures and Merkle trees; evaluate polynomial commitment schemes (KZG, Bulletproofs, FRI, lattice-based); propose a hybrid Verkle-FRI design; develop a formal security proof against classical and quantum adversaries; execute complexity analysis with concrete implementation parameters. The methods used are: theoretical cryptographic analysis, formal security modeling via reductionist proofs, algebraic methods over finite fields, polynomial interpolation, random oracle model, FRI protocol with DEEP-FRI optimization, Merkle trees, vector commitments, and asymptotic complexity analysis. The following results were achieved: a novel architecture where Verkle node vectors are polynomial-encoded, committed via Merkle trees over FRI codewords, and verified through FRI with out-of-domain sampling. A formal proof establishes λ-bit quantum security using 2λ-bit hash functions. Complexity yields proof size O(λ log² N), prover time O(λ N log N), and verifier time O(λ log N). Concrete 128-bit quantum parameters include SHA3-512, field size ≈2²⁵⁵, branching factor 256, and 128 FRI rounds, achieving soundness error ≤2⁻¹²⁷. For a concrete benchmark authenticating 2²⁶ elements, a traditional Merkle proof requires ≈0.8 KB, whereas our VERKLE-FRI proof requires ≈180 KB. While larger, this provides quantum resistance and eliminates the trusted setup, a critical trade-off for long-term security. Conclusions. Scientific novelty consists in: 1) the first hybrid Verkle-FRI architecture replacing pairing-based assumptions with hash-based proximity testing; 2) a formal security proof reducing security to hash collision resistance and FRI soundness; 3) quantified efficiency-security trade-offs; 4) a viable pathway for quantum-resistant infrastructure in blockchains, software distribution, and government communications.
Open access
Cryptographic Implementations and Security
Cryptography and Data Security
Physical Unclonable Functions (PUFs) and Hardware Security
Secure distributed systems offer reliability and privacy guarantees that are crucial across applications ranging from blockchains and cloud computing to fault-tolerant distributed Cyber-Physical Systems (CPS). These protocols enable groups of mutually distrusting parties to collaborate and execute tasks at scale while maintaining robust security guarantees against faulty and adversarial behavior. Blockchains demonstrate that the reliability half of this promise is achievable in practice, with deployments spanning hundreds of parties over geo-distributed testbeds. The privacy half has {\it not} kept pace: despite rapidly growing demand from applications such as anonymous networks and privacy-preserving AI, systems at blockchain scale have been unable to offer privacy guarantees. At the other end of the spectrum, the reliability techniques that succeeded in the blockchain setting are far too expensive for emerging distributed CPS applications, where hardware and network conditions are substantially weaker. In both settings, existing solutions are too slow and resource-intensive to be deployed in practice. This thesis asks whether both guarantees can be delivered at the scale their applications demand, on the hardware those applications actually run on.The first half of this thesis builds Multi-Party Computation (MPC) protocols for systems with a hundred or more parties over real-world geo-distributed networks, motivated by modern blockchains. MPC enables $n$ mutually distrusting parties to jointly compute any function over their private inputs. We identify computationally expensive heavyweight cryptography based on number-theoretic hardness assumptions as the central scalability bottleneck and address it by designing protocols entirely using \emph{lightweight} cryptography such as symmetric-key encryption and cryptographic Hash functions. These tools are two orders of magnitude cheaper than heavyweight operations and additionally offer post-quantum security. We present three works in this line: HashRand, a random beacon protocol, Velox, an MPC protocol achieving fairness, and Aeternum, a framework for guaranteed output delivery in asynchronous MPC and dynamic proactive secret sharing. We implement and evaluate all three, showing that they outperform prior work by two orders of magnitude and scale to $100$ or more parties on geo-distributed testbeds with practical latency and communication costs.The second half turns to Asynchronous Approximate Agreement (AAA) for distributed CPS with a hundred or more parties, motivated by robot and drone swarms. Unlike randomized Byzantine Agreement (BA) protocols, which depend on expensive heavyweight cryptography to produce common coins, AAA protocols are deterministic and avoid these tools. These protocols still have a high cubic communication cost, which is unaffordable in the low-bandwidth CPS setting. We introduce \emph{Relaxed Validity}, an approximate validity property that allows nodes to trade the accuracy of the protocol's output for sub-cubic communication.Leveraging this property, we design SensorBFT and Delphi, both AAA protocols with sub-cubic communication overhead. We apply both to agreement problems in the CPS domain and experimentally demonstrate their scalability relative to prior works. Both consume an order of magnitude less energy than prior protocols based on randomized BA, a decisive metric on resource- and power-constrained sensor devices.
Murugeshwari B., Kayalvizhi S, Daya Florance D, Saranya S
To deploy deep learning-based diagnostic systems in a clinical setting, they need to have not only a high degree of predictive accuracy, but also an unbroken cryptographic chain to prove that the model parameters have not been changed from one inference to the next. This incompatibility arises because softmax, GELU activation, and layer normalization involve transcendental operations to produce the final output. Transcendental operations cannot be represented over the prime finite fields that are necessary when utilizing Rank-1 Constraint Systems (R1CS). The work here provides a mathematically sound approach to resolving the incompatibility by providing three formally defined polynomial approximations: a degree-3 Taylor series softmax approximation (ε ≤ 2.47E-03 per weight); a degree-5 composite polynomial GELU (ε ≤ 1.80E-03); and a squared witness hint reformulation of layer normalization that results in exact constraint satisfaction. The accumulating polynomial approximation errors across 12 transformer encoder blocks have a total approximation error defined as (ε total ≤ 0.0512) and results in a loss of less than one-fifth of a percent in accuracy (94.1% vs. 94.3%) on the RSNA Pneumonia Detection benchmark. The entire ViT-B/16 inference model was compiled into an R1CS form of approximately 2.3×10⁸ R1CS constraints and instantiated as a Groth 16 zk-SNARK. The integrity of the model is confirmed by hashing the parameters using SHA-256 onto an immutable Polygon zkEVM smart contract, allowing for on-chain verification of inference without revealing any of the proprietary model weights. Additionally, this system produces an area under the receiver operating characteristic curve (AUC-ROC) of 0.961, a mean latency for proof generation of 2.84 seconds, an end-to-end verification time of 5.07 seconds, and an average cost for on-chain verification of 0.012ETH, all within the operational constraints of typical radiology workflows.
TT-G41: A Hybrid Post-Quantum Cryptosystem with Ly-Algebraic Quasi-Equivalence Index and Symmetry-Modulated Padding Chloe J. Tully Independent Researcher https://doi.org/10.5281/zenodo.21860133 Orcid: https://orcid.org/0009-0007-5661-7332 Version: 1.1 August 2026 ======================================== Abstract TT-G41 is a hybrid post-quantum cryptosystem that unifies a five-dimensional Ly-Algebraic Quasi-Equivalence Index (QEI) with an NTRU-style lattice layer. A novel symmetry-modulated padding mechanism injects structured noise scaled by s = exp(-alpha × QEI), establishing a direct causal link between the geometric coherence of the input and the entropy of the ciphertext. Empirical evaluation over 4000 trials yields a logistic security bound P(fail) = (1 + exp[15.57(QEI - 0.209)])^(-1) with R-squared = 0.980. A deterministic hard gate at QEI = 0.12 converts geometric incoherence into an immediate, deterministic decryption rejection, providing an active anti-tamper primitive resilient to partial-message side channels. The construction demonstrates that Ly-Algebraic geometric coherence can serve as a measurable quantum-resilient agent for cryptographic failure probability, establishing a new class of symmetry-gated post-quantum protocols. Keywords: post-quantum cryptography, Lie algebra, Quasi-Equivalence Index, NTRU, symmetry-modulated padding, geometric security bound, anti-tamper encryption ======================================== 1. Introduction Most post-quantum constructions treat geometric or algebraic structures solely as a source of hardness assumptions. TT-G41 inverts this relationship: it elevates a continuous geometric measure, the Quasi-Equivalence Index (QEI) derived from a graded Lie algebra, into an active security control surface. The system combines three elements: 1. A five-dimensional graded algebra with golden-ratio expansion (the Ly-Algebra core). 2. An NTRU-style lattice public-key layer with trusted circulant-matrix inversion. 3. A symmetry-modulated padding that scales ciphertext noise according to the QEI of the supplied input vector. The result is a hybrid scheme in which low geometric coherence effectively raises the noise floor until decryption fails, and a deterministic hard gate rejects decryption entirely once QEI falls below a calibrated threshold. This yields both a probabilistic security bound and a deterministic anti-tamper mechanism. ======================================== 2. Preliminaries 2.1 Ly-Algebra and Quasi-Equivalence Index The Ly-Algebra is a five-dimensional graded construction whose product is defined by a mapping from integer matrices L_i over F_11 (or R) weighted by golden-ratio coefficients. Given an input vector v in R^5, the Quasi-Equivalence Index is computed as: QEI(v) = max(0, 1 - sigma_distortion / sigma_identity) where sigma_distortion is the weighted Euclidean norm of the graded square Lv. High QEI indicates that v lies close to the preferred symmetry locus of the algebra; low QEI indicates structural distortion. 2.2 NTRU-Style Lattice Layer The lattice component follows the classical NTRUEncrypt paradigm: - Private key: ternary polynomial f with controlled weight parameter d_f. - Public key: h = f^(-1) × g (mod q), where inversion is performed via the circulant matrix of f over Z/qZ. - Encryption: e = r × h + m (mod q). - Decryption: recover a = f × e (mod q), then multiply by the inverse of f modulo p and center to obtain m. The parameter set used in this work is n = 17, q = 2048, p = 3, d_f = 3 (a convenience configuration) with compressed configurations exploring the boundary of reliable recovery. ======================================== 3. TT-G41 Construction 3.1 Hybrid Architecture TT-G41 operates in two modes: - Pure Ly-Algebra mode: computes QEI and reports the result only. - NTRU-enhanced mode: performs full key generation, encryption, and decryption, optionally modulated by the supplied input vector. 3.2 Symmetry-Modulated Padding (Coupling Mechanism 3) When an input vector v is supplied at encryption, the system computes: s = exp(-alpha × QEI(v)) and adds deterministic noise of amplitude proportional to s to the message polynomial. The same vector (hence the same QEI) must be supplied at decryption to subtract the matching noise pattern. A mismatch leaves residual noise that destroys the plaintext. Two operating regimes are defined: - Hard mode (amplitude s × 1.8): produces active anti-tamper behavior. - Soft mode (amplitude s × 0.55): scientific characterization of the failure curve. 3.3 Hard Gate In production (hard mode), the decryption program first evaluates QEI. If QEI < 0.12, decryption is rejected with the exception: ValueError: structurally incoherent (QEI = ... < 0.12). Decryption rejected by hard gate. No partial plaintext is ever returned. This eliminates the common side-channel leak associated with error-correcting or soft-decision decoders. ======================================== 4. Empirical Security Bound A soft-diagnostic campaign of 4000 encrypt/decrypt trials was performed across a radial drift of the input vector that systematically lowers QEI. Failure probability was recorded at each point. Three models were fitted: Simple exponential: P(fail) = exp(-alpha × QEI), alpha = 3.612, R-squared = 0.945 Shifted exponential: P(fail) = exp(-alpha × max(QEI - q0, 0)), alpha = 23.55, q0 = 0.168, R-squared = 0.976 Logistic (best fit): P(fail) = (1 + exp[beta × (QEI - Q_mid)])^(-1), beta = 15.57, Q_mid = 0.209, R-squared = 0.980 The logistic model provides the highest fidelity. At the operational threshold QEI = 0.12, the mean observed failure rate is 0.963; above the threshold it falls to 0.323. The hard gate therefore sits safely on the high-failure shoulder of the empirically determined curve. ======================================== 5. Discussion The central claim of TT-G41 is that a continuous geometric invariant of a graded algebra can be turned into a practical cryptographic control surface. The symmetry-modulated padding realises a causal chain: geometric distortion -> elevated noise -> decryption failure while the hard gate converts the continuous measure into a binary, side-channel-resistant decision. Because the QEI is computed from a public or shared input vector, the anti-tamper property can be applied to any data source whose structural integrity is expected to remain high (sensor streams, physical-system state vectors, authenticated configuration parameters, etc.). A shift in that source immediately invalidates the cryptographic layer. Limitations of the present study include the modest lattice dimension (n = 17) used for the statistical campaign and the still-sharp transition of the underlying QEI landscape. Both are engineering parameters that can be refined without altering the architectural principle. ======================================== 6. Future Work and Research Directions Building upon the foundations established in this work, several promising extensions are identified for subsequent investigation: 6.1 Scaling Lie Algebra Dimensions The current construction relies on a five-dimensional Lie algebra. Exploring higher-dimensional Lie algebras, such as higher-rank semisimple algebras or structures analogous to E8, could provide a broader entropy space and create more complex geometric invariants for the Quasi-Equivalence Index. This would enhance the system's robustness against adversarial vector manipulation attacks. 6.2 Adapting the NTRU Layer to NIST Post-Quantum Standards The lattice dimension n = 17 was employed in the initial statistical campaign to explore operational boundaries. It is of significant interest to test how the logistic security bound behaves when scaling the NTRU layer to align with standard NIST dimensions, such as n = 503, 701, or 821, and to study whether the symmetry-modulated padding maintains computational efficiency at these substantially larger dimensions. 6.3 Adaptive Hard-Gate Thresholding Rather than relying on a fixed failure threshold at QEI = 0.12, an adaptive algorithm could be designed to dynamically adjust this threshold based on the statistical variance of the input vector stream. This extension would render the system suitable for Internet of Things applications or industrial control systems where natural structural noise levels vary over time. 6.4 Integration with Zero-Knowledge Proofs The geometric coherence represented by the Quasi-Equivalence Index could serve as the foundation for a novel zero-knowledge proof protocol. A prover could demonstrate possession of a structurally coherent vector without revealing the actual data, leveraging the continuous property of the geometric invariant as a geometric hash function. 6.5 Hardware Implementation and Side-Channel Analysis Implementing the hard-gate logic and symmetry-modulated padding mechanisms on FPGA platforms would enable evaluation of actual resistance to side-channel attacks, such as power consumption and electromagnetic emissions. The deterministic rejection of decryption may exhibit a unique power signature worthy of study to ensure no information leakage occurs via a side channel when the hard mode is activated. 6.6 Integration with Quantum Entropy Incorporating Quantum Random Number Generators into the symmetry-modulated padding mechanism would inject true quantum entropy into the noise vector, adding an additional layer of protection that directly bridges lattice-based cryptography and quantum mechanics. ======================================== 7. Conclusion TT-G41 demonstrates that Ly-Algebraic geometric coherence can be elevated from a passive diagnostic into an active post-quantum security primitive. The combination of
IoT sensor deployments increasingly export measurement streams to edge gateways and content-addressed storage such as IPFS, but access control decisions must be enforced without disclosing sensor owner policies, requester attributes, or stale data versions. Existing blockchain, CP-ABE, and zero-knowledge approaches reduce parts of this leakage, yet they can still expose public policy structure, accept stale Merkle proofs after sensor stream updates, overload provers when policies grow, or leave IPFS gateways vulnerable to bandwidth abuse. This paper proposes zk-Guard-R, a policy-hidden and replay-safe zk-SNARK access control framework for privacy-preserving IoT sensor data sharing. zk-Guard-R replaces public sparse policy matrices with MiMC-Merkle policy commitments verified inside the proof, separates long-lived logical sensor policy roots from frequently updated physical IPFS data roots, binds every proof to an on-chain nonce, and decouples attribute possession from policy interpretation through a bounded stack-based policy interpreter. Numeric sensor-access predicates are represented through committed values and range check gadgets, while an off-chain verification gateway couples accepted proofs with payment channel vouchers before releasing encrypted IPFS chunks. The design contribution is separated from the measured prototype: the full protocol specifies a bounded policy interpreter, whereas the present gnark prototype evaluates the core committed policy, committed attribute, range check, data root, nonce, Solidity verifier, and gateway-metering mechanisms. We implement a gnark BN254/Groth16 research prototype and benchmark it against a matrix-public zk-Guard prototype, a blockchain ABAC baseline, an IoT token/HMAC baseline, and a CP-ABE-style cryptographic-work proxy. For 128 attributes, the zk-Guard-R prototype with MiMC-Merkle commitments uses 425,574 R1CS constraints, generates proofs in 3.12 s, verifies in 0.73 ms, and uses 641 MB peak Go heap allocation. A three-run repeat of the 128-attribute configuration gives a proof-generation mean of 2.80 s with a 0.54 s standard deviation on the same local host, illustrating the runtime variability of prover measurements. We also deploy the generated Solidity verifier on a local Anvil EVM and measure 241,942 gas for a successful verification transaction, and we evaluate a local Kubo/IPFS gateway under valid, replayed, and voucher-limited flood requests. The results show that zk-Guard-R shifts substantial but measurable work to the prover while improving policy confidentiality, freshness, and gateway metering for IPFS-backed IoT sensor data sharing.
Online voting platforms that rely on classical cryptography and centralized trust anchors face escalating challenges as the demand for secure and transparent digital elections grows. Such systems remain exposed to quantum-era threats, insider manipulation, and delayed audit mechanisms, which together can undermine public confidence and electoral legitimacy. To counter these risks, a quantum-resistant, multi-layer blockchain architecture has been developed to enable remote voting with continuous verifiability and resilience. This architecture resolves key weaknesses through five integrated layers. Quantum-Resistant Distributed Ledger Initialization (QR-DLI) embeds lattice-based cryptography, specifically Kyber and Dilithium variants, directly within the genesis block, ensuring the ledger is tamper-proof from inception and immune to quantum brute-force attacks. The Self-Adaptive Smart Contract Governance Engine (SASCG) introduces dynamic, participation-aware rule adjustments, allowing principled governance without manual overrides and ensuring that voting periods and eligibility rules adapt securely in real time. Homomorphic Vote Encryption with Multi-Authority Shard Key Distribution (HVE-MASKD) guarantees ballot confidentiality and authenticity by combining fully homomorphic encryption with distributed key shares, eliminating single points of trust. The Zero-Knowledge Proof–Based Real-Time Audit Layer (ZKP-RTAL) continuously validates ballot integrity while concealing vote content, creating a public and immutable audit trail. Finally, the Federated Performance & Threat Intelligence Optimizer (FPTIO) aggregates live telemetry and historical attack data to proactively tune consensus parameters and predict potential intrusions without interrupting the election process. Collectively, these layers achieve sub-second cryptographic operations, transaction throughput exceeding 1,500 TPS, over 99 % fraud detection accuracy, and strong scalability. The model provides a future-ready, auditable replacement for current e Voting infrastructures, strengthening digital democracy through post-quantum security, adaptive governance, and intelligent, continuous optimizations.
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 kB—over an order of magnitude smaller than previous constructions—while 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 CØCØ, 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, Scott–Guillevic, Gasnier–Guillevic, 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 Diffie–Hellman-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 matrix–vector 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-Sasson–Chiesa–Spooner (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 4.6–7.2 ms and verifies them in 0.46–0.52 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 20.9–30.4 ms with Metal acceleration and 59.2–229.2 ms on CPU, while verification takes 5.09–5.79 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.
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 image’s 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 ≈31×. We also show that this packing scheme reduces the median prover runtime by ≈40× and the median verifier runtime by ≈22×. 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.