Shor's algorithm represents the main threat of quantum computers to cryptography. In order to precisely understand its feasibility, many authors have worked towards reducing its costs, either at the logical level (assuming a fault-tolerant architecture), or at the physical level (taking into account the constraints of envisioned hardware). In particular, recent works by Chevignard et al. (CRYPTO 2024) and Gidney (arXiv 2025) used improved arithmetic to significantly reduce the qubit cost of factoring RSA public keys. Even more recently, Babbush et al. (arXiv 2026) improved the cost of computing elliptic curve discrete logarithms, with a reduction of a factor 2 to 3 in gate count and qubit count compared to a previous work by Litinski (arXiv 2023). Their result relies on optimized point addition circuits on elliptic curves over prime fields. However they did not reveal their logical quantum circuits, relying instead on a zero-knowledge proof. In this paper, we detail a quantum logical circuit architecture which gives similar results as Babbush et al., with a slightly higher number of qubits (around 1.5% increase) and a slightly smaller Toffoli gate count (between 6.5% and 10% reduction) for the curve secp256k1. We also give gate counts for a generic variant of the circuit, which is valid for any prime field.
Quantum computing poses significant challenges to traditional zero-knowledge proof schemes based on number-theoretic assumptions. As a result, code-based cryptography has attracted increasing attention for its resistance against quantum computing. In this paper, we study the Rank Syndrome Decoding problem (RSD) and investigate its ZK proof formulation within the MPC-in-the-Head framework. To prove the possession of a secret witness, we reformulate the secret witness as a mixed-field matrix multiplication preserving the rank constraint, and then obtain a representation that aligns naturally with the local-view paradigm of MPC-in-the-Head. Utilizing this value-to-calculation technique, we introduce the RSD relation into a ZKBoo-style (2, 3)-secret-sharing MPC-in-the-Head framework and obtain an RSD-based zero-knowledge proof scheme via mixed-field secret sharing. The resulting scheme reduces the proof size relative to generic formulations while preserving completeness, soundness, and zero-knowledge for the interactive protocol. The Fiat–Shamir non-interactive extension is analyzed only in the classical random oracle model; we do not claim QROM security for this variant.
With the widespread adoption of cryptocurrencies, the ability to conduct continuous offline payments has increasingly become a critical technological requirement. In network-constrained scenarios, current dual-offline payment technologies are useful for single transactions. However, their limitations in continuous payment scenarios have become increasingly evident, making them unable to meet real-world application needs. This has prompted the industry to demand more urgent innovations in research on continuous offline payment capabilities. To address these challenges, this paper proposes a continuous dual-offline payment system capable of supporting multiple continuous payments. The system integrates elliptic curve cryptography (ECC) and zero-knowledge proof (ZKP) technology to generate secure asset credentials, ensuring both immutability and privacy credentials throughout the offline payment lifecycle. A dynamic credential decomposition mechanism enables the splitting of input credentials into change credentials and receipt credentials, facilitating uninterrupted dual-offline payments between hardware wallets. Additionally, it incorporates a batch verification scheme based on smart contracts, utilizing zero-balance verification and chained hash tracing to ensure payment uniqueness and prevent double-spending attacks, thereby guaranteeing the verifiability and validity of payment settlements. Experimental evaluations demonstrate that the proposed system reduces gas consumption per payment and improves execution efficiency during batch processing, combining high security with strong performance. This research provides a feasible solution for the application of digital currencies in offline scenarios, carrying significant theoretical value and practical significance for driving technological innovation and application expansion in the cryptocurrency field. In addition to cryptocurrency payments, the proposed system is also applicable to IoT and sensor network environments. Many IoT devices operate in disconnected or network-limited areas and require secure micro-transactions. Our dual-offline payment mechanism supports such scenarios, as the main cryptographic operations are lightweight enough for typical IoT hardware. This further extends the practical value of our system beyond traditional cryptocurrency payments.
Polynomial multiplication serves as a fundamental computational primitive in modern cryptography–including fully homomorphic encryption and zero-knowledge proofs –as well as in digital signal processing. Its performance optimization has become increasingly critical amid the rapid development of privacy-preserving computation and blockchain technologies. To address the limitations of traditional algorithms in meeting the demands for high throughput and low latency, this study proposes a high-performance polynomial multiplication accelerator based on the collaborative optimization of GPU-NTT and the Karatsuba algorithm. The method deeply integrates the asymptotically optimal complexity of NTT with the constant-factor efficiency of Karatsuba at moderate scales, and fully exploits the parallel computing power of GPUs to construct a modular, multi-stage pipelined acceleration framework. The divide-and-conquer nature of the Karatsuba algorithm is leveraged for coarse-grained parallelism, splitting large polynomial multiplications into subproblems handled by GPU thread blocks in parallel, while each subproblem is solved with fine-grained parallelism using GPU-accelerated NTT kernels. An innovative zero-padding strategy is introduced to enhance the generality of the NTT kernels, and shared memory caching is employed to alleviate GPU memory bandwidth bottlenecks. Experimental results on the NVIDIA RTX 4060 GPU demonstrate that the proposed method achieves a stable speedup of 1.43 \(\times \) to 1.49 \(\times \) over the baseline GPU-NTT for lower-dimensional polynomials, and outperforms the KNTT algorithm by up to 2.44 \(\times \) for higher dimensions (e.g., \(\log _2 n = 14\) ), showing superior scalability and robustness. Kernel execution time analysis further confirms that the method benefits from efficient kernel fusion and balanced workload distribution, which effectively avoids pipeline stalls and ensures high-throughput execution. This research provides a significant performance optimization solution for the practical deployment of advanced cryptographic technologies such as FHE and ZKP.
Information-Theoretic Deniability and the Forest Analogy This paper establishes information-theoretic deniability through a precise structural argument: the Forest Analogy. The central result is a symmetry property showing that an attacker who possesses a valid alibi chain stands in exactly the same position as an attacker who knows only the public parameter N. The verification function cannot distinguish the real authentication chain from any alibi chain. This property holds without cryptographic assumptions — it is information-theoretic, not computational. The authentication chain is built on the linear Diophantine system N = 19A + 9B, in which every number N has approximately N/171 valid representations per layer. A three-layer Matryoshka structure produces a multiplicatively structured search space of approximately 10¹¹² operations. The key insight is that this search space is symmetric: it is equally large and equally unnavigable from outside the system as from within it. Uniform Sampling and the Generation Algorithm Uniform sampling means here that every valid chain — consisting of A, B, a1, b1, a2, b2 satisfying 19A + 9B = N, 19a1 + 9b1 = A, 19a2 + 9b2 = a1, and the digit-root conditions — has exactly equal probability of being generated as the real authentication chain. The generation algorithm achieves this through a hierarchical weighted CDF sampler (cumulative distribution function), operating as follows: 1. The distribution over all valid A-values is computed, where the weight assigned to a given A equals the exact number of valid chains on layers 1 and 2 that can be extended from that A. 2. A is sampled according to this weighted distribution. 3. Given A, the value a1 is sampled in the same manner, with weights determined by the exact number of valid (a2, b2)-pairs beneath that a1. 4. Finally, (a2, b2) is chosen uniformly at random from all solutions to 19a2 + 9b2 = a1. All weights are computed exactly. This hierarchical weighted sampling procedure therefore induces a provably uniform distribution over all complete chains, without requiring the generator to enumerate or store them explicitly. Proposition (Uniform Chain Distribution): Let N be a fixed public parameter. Let Omega(N) denote the set of all valid chains C = (A, B, a1, b1, a2, b2) satisfying: - 19A + 9B = N - 19a1 + 9b1 = A - 19a2 + 9b2 = a1 - and the digit-root conditions on each layer. Then the hierarchical weighted CDF sampler generates each chain C in Omega(N) with probability exactly 1 / |Omega(N)|. Proof. Define the following counting functions: - w2(a1) = |{(a2, b2) : 19a2 + 9b2 = a1, digit-root conditions satisfied}| - w1(A) = sum over all valid a1 of w2(a1), where the sum runs over all a1 satisfying 19a1 + 9b1 = A for some valid b1 - w0 = sum over all valid A of w1(A) = |Omega(N)| The sampler proceeds in three steps: Step 1. A is drawn with probability P(A) = w1(A) / w0. Step 2. Given A, a1 is drawn with probability P(a1 | A) = w2(a1) / w1(A). Step 3. Given a1, the pair (a2, b2) is drawn uniformly with probability P(a2, b2 | a1) = 1 / w2(a1). The joint probability of generating the complete chain C = (A, B, a1, b1, a2, b2) is: P(C) = P(A) * P(a1 | A) * P(a2, b2 | a1) = w1(A) / w0 * w2(a1) / w1(A) * 1 / w2(a1) = 1 / w0 = 1 / |Omega(N)| Since this holds for every C in Omega(N), the sampler induces the uniform distribution over Omega(N). "Remark" B and b1 are uniquely determined once A, a1, a2, and b2 are fixed, via the Diophantine equations. They do not need to be sampled separately. The uniformity therefore holds over all complete chains, including these values. Theorem (Forest Symmetry): Let N be a public parameter and let C = (A, B, a1, b1, a2, b2) be any valid verification chain. Then for every valid alibi chain C': P(C is authentic | verification succeeds, C') = P(C is authentic | verification succeeds) Proof: The verification function evaluates solely against N. By Proposition (Uniform Chain Distribution), every valid chain is generated with identical probability 1 / |Omega(N)|. The alibi C' therefore carries no information about which chain is authentic. Formally, C' and the event "C is authentic" are conditionally independent given that verification succeeds. The posterior probability is thus unaffected by knowledge of C'. The Forest Analogy: The Forest Analogy makes the symmetry precise. The public parameter N is the forest. Each representation (A, B) on layer zero is a tree. Each sub-representation on layer one is a branch. Each leaf is a layer-two representation. All leaves are identical — every chain verifies correctly against N. An attacker holding a valid alibi leaf cannot determine which leaf is the real one, because the verification function provides no distinguishing information. *This work is licensed under CC BY-SA 4.0. Commercial licensing (without ShareAlike) available on request: elissa_oui@outlook.com*
Anonymous digital credentials allow a user to prove possession of an attribute that has been asserted by an identity issuer without the user revealing any extra information about themselves. For example, a user who has received a digital passport credential can prove their “age is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>></mml:mo> <mml:mn>18</mml:mn> </mml:mrow> </mml:math> ” without revealing any other attributes such as their name or date of birth. Despite their clear application to privacy-preserving authentication, anonymous credential schemes have been difficult to deploy at scale. Part of the difficulty arises because schemes in the literature, such as BBS+, use new cryptographic primitives that require system-wide changes to existing issuer infrastructure. In addition, issuers often require digital identity credentials to be device-bound by incorporating the device’s secure element into the presentation flow. As a result, schemes like BBS+ require updates to the hardware on every user's device. We propose new ZK techniques which enable the construction of an anonymous credential scheme for the legacy Elliptic Curve Digital Signature Algorithm (ECDSA) signature scheme. By adding efficient ZK arguments for statements about SHA-256 and document parsing for ISO-standardized identity formats, we construct the first ZK proof of posession of a credential that can be deployed without changing any issuer processes, without changes to mobile devices, and without requiring non-standard cryptographic assumptions. Furthermore, our proof system itself only relies on SHA-256 as its complexity assumption. Producing ZK proofs about ECDSA signatures has been a bottleneck for other ZK proof systems because standardized curves such as P256 use finite fields which do not support efficient number theoretic transforms. We overcome this bottleneck by designing a ZK proof system around sumcheck and the Ligero argument system, by designing efficient methods for Reed-Solomon encoding over the required fields, and by designing specialized circuits for ECDSA. Our proofs for ECDSA can be generated in as little as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>≈</mml:mo> <mml:mn>20</mml:mn> </mml:mrow> </mml:math> ms. When incorporated into a fully standardized identity protocol such as the ISO MDOC standard, our system can generate a zero-knowledge proof for the MDOC presentation flow in a few hundred ms on mobile devices. These advantages make our scheme a promising candidate for privacy-preserving digital identity applications.
This is Paper 6 of a series of formally-verified analyses of masked NTT hardware for post-quantum cryptography; Paper 1 [1] established structural dependency analysis of the QANARY platform, and Paper 2 [2] quantified security margins under partial NTT masking. Boolean masking composition is well-understood through NI, SNI, and PINI. Arithmetic masking over $\mathbb{Z}_q$ for prime $q$, the foundation of NTT-based post-quantum cryptography, has lacked an analogous theory. We prove, to our knowledge, the first machine-checked composition theorems for arithmetic masking over prime fields. Our key insight is the renewal argument: when a fresh random mask is applied between two pipeline stages, the intermediate wire becomes perfectly uniform regardless of Stage 1's security parameter. For two PF-PINI gadgets with parameters $k_1$ and $k_2$, the composed two-stage pipeline with fresh masking satisfies PF-PINI($k_2$), Stage 1's multiplicity is completely erased from the composed output. Without fresh masking, intermediate wires have multiplicity up to $k_1$, creating a necessary condition for differential power analysis. We formalize both theorems in Lean 4 with 18 machine-checked proofs and zero sorry stubs. We formally bridge the algebraic and hardware-faithful arithmetic models of Barrett reduction, and instantiate the theorems to formally diagnose Microsoft's Adams Bridge PQC accelerator: its absence of fresh inter-stage masking leaves Barrett output wires non-uniform under the first-order probing model, the same architectural flaw that two independent empirical analyses [3, 4] and our own prior structural analysis [1] identified. Computational evidence further suggests the 1-Bit Barrier is universal across Barrett and Montgomery reductions.
Abstract Multi-Scalar Multiplication (MSM) is the primary computational bottleneck in zero-knowledge (ZK) proof generation for decentralized networks. This research accelerates MSM by solving the memory bandwidth constraints inherent in high-dimensional elliptic curve cryptography. We introduce Modular Hypercube Chunking, a novel microarchitectural approach that partitions high-dimensional algebraic precomputations into smaller, orthogonal blocks. Specifically, we divide a 12-dimensional workload into three separate 4D hypercubes, restricting the entire memory footprint to 31.1 KB. This geometric partitioning ensures perfect residency within the ultra-fast L1 cache of modern processors. By employing shared doubling across these blocks, the algorithm processes twelve scalars simultaneously with a single elliptic curve duplication, bypassing slow RAM access entirely. Empirical evaluations conducted on an ARM Snapdragon 8 Gen 2 mobile processor demonstrate a peak 5.37× speedup compared to optimized sequential baselines, reducing the computational cost to 18.44 microseconds per scalar. These findings prove that geometric data partitioning within strict L1 cache boundaries significantly outperforms traditional arithmetic-heavy optimizations. The implications of this work provide a highly scalable architecture capable of executing server-grade ZK-Rollup proof generation on resource-constrained edge devices, while establishing a highly efficient blueprint for future multicore hardware accelerators. Furthermore, initial stress-tests of a 12D monolithic architecture (68 MB footprint) yielded an anomalous 8.88× peak speedup. This finding reveals a novel sparse-access memory optimization path, which we introduce as an open architectural challenge.
The modular invariance and automorphism group rigidity of vertex operatoralgebras provide a profound mathematical foundation for constructing novel postquantum cryptographic systems. However, a significant theoretical and engineeringgap exists between mathematical theorems and deployable cryptosystems. Thispaper does not propose new cryptographic protocols but rather systematicallyexamines the core challenges encountered in engineering vertex operator algebracryptography: the discrete selection of parameter spaces and their quantitativerelationship with security strength, the computational resource requirements ofcandidate algebraic families (lattice vertex operator algebras, WZW models, andmoonshine vertex operator algebras), the assessment of security boundaries underquantum attack models, and the practical overhead of auxiliary mechanisms suchas zero-knowledge proofs. The objective is to provide a clear problem inventoryand a feasibility analysis framework for future research, rather than to claim anyimmediately usable security parameters. The article concludes by summarizing thecurrent technology readiness levels and identifying the key breakthroughs requiredto advance from a theoretical framework toward a practical system.
Digital signature schemes derived from non-interactive zero-knowledge (NIZK) proofs are rapidly gaining prominence within post-quantum cryptography. CROSS is a promising new code-based post-quantum digital signature scheme based on the NIZK framework. It is currently in the second round of the NIST’s additional call for standardization for post-quantum digital signatures. However, CROSS’s reference implementation has a substantially large memory footprint. This makes its deployment on resource-constrained platforms prohibitively difficult.In particular, we identified the most memory-intensive areas such as Merkle and GGM tree structures, and zero-knowledge proof commitment generation. We propose several novel algorithms and implementation strategies to reduce the memory requirement of these components. Apart from these, we also propose several memory optimization techniques, such as just-in-time hashing and execution flow analysis. As a result, our implementation reduces the memory footprint of Key Generation, Signature Generation, and Verification of the CROSS reference code by as much as 95%, 92%, and 85%, respectively. This results in a suite of implementations in which all variants are under 128kB (for all security levels of KeyGen/Sign/Verify) and six variants under 32kB. Our memory optimization techniques are not specific to CROSS, but can be applied to other NIZK-based signature schemes.Regarding efficiency, matrix multiplications are crucial to the performance of CROSS. We show how the Digital Signal Processing (DSP) instructions on ARM Cortex-M4, specifically packing and multiplying, can be utilized to efficiently implement matrix operations over finite fields. The DSP optimizations combined with the memory reductions improve the efficiency of CROSS by up to 32% and 33% in Signature Generation and Verification respectively.
Florian Krieger, Christian Dobrouschek, Florian Hirner, Sujoy Sinha Roy
We present the first high-performance SIMD software implementation of Spielman codes for their use in polynomial commitment schemes and zero-knowledge proofs. Spielman codes, as used in the Brakedown framework, are attractive alternatives to Reed-Solomon codes and benefit from linear-time complexity and field agnosticism. However, the practical deployment of Spielman codes has been hindered by a lack of research on efficient implementations. The involved costly finite-field arithmetic and random memory accesses operate on large volumes of data, typically exceeding gigabytes; these pose significant challenges for performance gains. To address these challenges, we propose several computational and memory-related optimizations that together reach an order-of-magnitude performance improvement in software. On the computation side, we propose SIMD optimizations using the AVX-512-IFMA instruction set and introduce a lazy reduction method to minimize the modular arithmetic cost. On the memory side, we implement a cache-friendly memory layout and a slicing technique, which exploit the CPU memory hierarchy. Finally, we present our multithreading approach to improve throughput without saturating memory bandwidth. Compared to prior Spielman software, our optimizations achieve speedups of up to 21.9x and 20.6x for single- and multi-threaded execution, respectively. In addition, instantiating our software with 64 threads on a high-end CPU even outperforms a recent FPGA accelerator by up to 4.3x for small and mid-sized polynomials. Our improvements make Spielman codes competitive with well-optimized Reed-Solomon codes on software platforms.
Zero-knowledge proof (ZKP) provers remain costly because multi-scalar multiplication (MSM) and number-theoretic transforms (NTTs) dominate runtime as they need significant computation. AI ASICs such as TPUs provide massive matrix throughput and SotA energy efficiency. We present MORPH, the first framework that reformulates ZKP kernels to match AI-ASIC execution. We introduce Big-T complexity, a hardware-aware complexity model that exposes heterogeneous bottlenecks and layout-transformation costs ignored by Big-O. Guided by this analysis, (1) at arithmetic level, MORPH develops an MXU-centric extended-RNS lazy reduction that converts high-precision modular arithmetic into dense low-precision GEMMs, eliminating all carry chains, and (2) at dataflow level, MORPH constructs a unified-sharding layout-stationary TPU Pippenger MSM and optimized 3/5-step NTT that avoid on-TPU shuffles to minimize costly memory reorganization. Implemented in JAX, MORPH enables TPUv6e8 to achieve up-to 10x higher throughput on NTT and comparable throughput on MSM than GZKP. Our code: https://github.com/EfficientPPML/MORPH.
This repository/dataset presents the Matrix Cryptographic Tree (MCT), a novel cryptographic primitive based on the special linear group SL(2, \mathbb{Z}). Unlike traditional hash-based Merkle Trees, MCT is fully reversible and structure-preserving, allowing for unique algebraic verification without information loss. The core of this research is the integration of Unified Zero-Knowledge Proofs (ZKP). MCT enables advanced privacy-preserving operations, such as proving data similarity (L1 norm distance), prefix/suffix matching, and existence proofs, all without revealing the underlying data.
The security of modern public-key cryptography generally relies on computational intractability assumptions, such as integer factorization and discrete logarithm problems. This paper proposes a fundamentally different foundation for security: the intrinsic mathematical properties of tensor product categories—globalentanglement, rigid decomposition, and sensitivity amplification—are directly employed as security resources of the cryptosystem. Within the modular representation category of finite simple groups over finite fields, the private key correspondsto an irreducible modular representation, while the public key was originally conceived as the character vector of a tensor product of that representation. However, this paper reveals a fatal structural vulnerability: because the character ofthe base representation is public, an adversary can fully recover the private keycharacter through trivial division, causing the original security assumption to collapse completely. To address this, the paper accomplishes a paradigm shift from“character-exposure cryptography” to “structure-commitment cryptography,” redefining the public key as a cryptographic commitment to the multiplicity vectorof the tensor product decomposition. Building upon this, the commitment-basedrepresentation recognition problem and the commitment-based tensor product decomposition problem are formalized, and their hardness is argued under both classical and quantum computational models. At the protocol level, it is pointed outthat non-interactive key exchange faces a fundamental obstacle due to the lack ofrepresentation-category homomorphic commitments; consequently, the research focus is shifted to digital signature schemes. The proposed TC-Sig scheme bridges thegap between commitment hiding and multiplicity verification using zero-knowledgeproof techniques, with security reduced to the commitment-based representationrecognition problem in the random oracle model. A feasibility assessment indicatesthat, for candidate groups such as the Mathieu group M12, key generation and commitment computation can be completed within milliseconds, while the introductionof zero-knowledge proofs increases latency to the order of seconds or minutes, making the scheme suitable for low-frequency, high-security scenarios. The security ofthis framework rests on three cornerstones: the classification rigidity of finite simple groups, the one-wayness of commitment schemes, and the non-abelian quantumcomputing barrier, thereby offering a new pathway for post-quantum cryptographyrooted in pure mathematical structure.
Recent advancements, specifically the 2026 whitepaper by Google Quantum AI, Stanford University, and the Ethereum Foundation (arXiv:2603.28846), have demonstrated the resource feasibility of breaking secp256k1 elliptic curve cryptography using fault-tolerant quantum computation (≤ 1200 logical qubits and ≤ 90 million Toffoli gates). While their work validates this capability via zero-knowledge STARK proofs without disclosing explicit circuits, we provide the continuous operator-theoretic framework that explains the exact physical collapse mechanism underlying their discrete resource results. By modeling cryptographic hardness as a stable, invariant computational manifold, we show that quantum vulnerability is a manifestation of a Birman-Schwinger instability. We prove that, within this model, the introduction of a transverse quantum operator (e.g., Shor's algorithm implemented via Quantum Phase Estimation) forces a resolvent singularity in the classical generator when the resource perturbation parameter crosses a critical threshold (μ_c). We establish a strict Hardness Phase Transition, demonstrating that cryptographic security is equivalent to the point 1 remaining outside the spectrum of the Birman-Schwinger kernel. Furthermore, we formalize zero-knowledge proofs (such as the Groth16-wrapped STARK artifacts published by Babbush et al.) as highly constrained Boolean projectors. We show that these proofs trigger an epistemic spectral collapse via Zeno stabilization, certifying the non-invertible regime without decohering the raw computational state into the public domain. The manuscript includes an exact analytic toy model demonstrating bound-state collapse into the continuum, explicitly mapping the destruction of exponential cryptographic isolation to a polynomial scattering state. This formalization transitions cryptographic failure from a domain of discrete computational estimates to a continuous framework of operator-theoretic necessity.
The Legendre Pseudorandom Function (PRF) is a highly efficient cryptographic primitive built upon the Legendre symbol, valued for its low multiplicative complexity in Multi-Party Computation (MPC) and Zero-Knowledge Proof (ZKP) protocols. While its security over prime fields $\mathbb{F}_p$ is well-documented, recent interest has shifted toward instantiations over extension fields $\mathbb{F}_{p^r}$. This paper presents the first comprehensive cryptanalysis of the single-degree Legendre PRF operating over $\mathbb{F}_{p^r}$. First, we analyze polynomial input encoding under a standard passive threat model (sequential additive counter queries). We demonstrate that while the absence of polynomial carry-overs causes an asynchronous "no-carry fracture" that neutralizes classical sliding-window collision attacks, the fracture itself is deterministically periodic. By introducing a novel "Differential Signature" bucketing technique, we prove that an adversary can systematically group fractured sequences by their structural shapes to bypass this defense, recovering the secret key in $\mathcal{O}(U \cdot p^r/M)$ operations, where $U$ is the unicity distance. Second, we evaluate the PRF under an active Chosen-Query threat model. We demonstrate that an adversary can circumvent the additive fracture by evaluating the PRF along a geometric sequence generated by a primitive polynomial. This structure invokes strict multiplicative homomorphism over $\mathbb{F}^*_{p^r}$, permitting a direct generalization of state-of-the-art table collision attacks to extract the key in $\mathcal{O}(p^r/M)$ operations. Finally, we establish the cryptographic boundaries of these attacks, formally proving the necessity of higher-degree key variants ($d \ge 2$) to achieve exponential security against structural reduction in extension fields.
Modern digital financial systems continuously face the fundamental trilemma of transaction confidentiality, computational efficiency, and cryptographic security. While existing blockchain technologies have made significant progress in security and decentralization, they are limited in ensuring confidentiality of financial information due to their public distributed ledger structure. To overcome these limitations, we present Epsilon, an innovative protocol that fundamentally addresses this complexity by directly combining Elliptic Curve-based Partially Homomorphic Encryption with parallel processing architectures. This paper introduces the Cryptora Protocol, a complete end-to-end operational framework that implements a two-tiered architecture where transaction processing is performed off-chain in a peer-to-peer manner, with results recorded on smart contract-based on-chain distributed ledgers. Our experimental results demonstrate that Epsilon achieves 500,000 TPS with end-to-end latency of 50-100ms while maintaining strong cryptographic guarantees including semantic security, unlinkability, and double-spending resistance.
Contemporary digital currency systems face fundamental challenges in achieving optimal balance between transaction privacy, computational efficiency, and cryptographic security. While zero-knowledge proof systems have dominated privacy-preserving cryptocurrency research, their practical implementations often involve prohibitive computational overhead that limits real-world deployment. This paper presents a comprehensive analysis of the Elliptic Homomorphic Token (EHT) protocol, which leverages elliptic curve-based partially homomorphic encryption combined with parallel processing architecture to enable privacy-preserving peer-to-peer transactions without the computational complexity of zero-knowledge constructions. Our theoretical analysis demonstrates strong privacy guarantees under standard cryptographic assumptions, while experimental evaluation shows that EHT achieves 500,000 transactions per second with parallel processing and 50-100ms latency. The protocol eliminates the need for complex zero-knowledge proofs by directly utilizing elliptic curve cryptographic primitives, resulting in performance improvements exceeding 1000× over existing privacy-focused systems while maintaining equivalent security properties through formally proven cryptographic guarantees.
The rapid evolution of digital currency systems has consistently faced the fundamental challenge of achieving an optimal balance between transaction privacy, computational efficiency, and cryptographic security. This comprehensive research paper introduces the Elliptic Homomorphic Token (EHT), a groundbreaking cryptographic protocol that revolutionizes privacy-preserving peer-to-peer transactions through the innovative integration of elliptic curve-based partially homomorphic encryption mechanisms and advanced digital signature schemes. Unlike conventional zero-knowledge proof systems that have dominated the privacy-focused cryptocurrency landscape, EHT takes a fundamentally different approach by directly leveraging the underlying cryptographic primitives that form the mathematical foundation of these complex systems. The protocol implements a sophisticated pre-transaction mechanism followed by distributed block recording, achieving remarkable performance metrics of 1000 transactions per second (TPS) with consistently low latency ranging from 50 to 100 milliseconds. Our comprehensive approach systematically addresses the significant computational overhead challenges that were extensively documented during Central Bank Digital Currency (CBDC) implementation projects, while simultaneously providing a robust and practical framework for privacy-preserving digital transactions that maintains the highest standards of cryptographic security. The EHT protocol represents a paradigm shift in how we conceptualize and implement privacy-preserving digital currency systems, offering a more direct, efficient, and mathematically elegant solution compared to existing approaches. Through extensive theoretical analysis, rigorous security proofs, and comprehensive performance evaluations, this paper demonstrates that EHT not only meets but exceeds the requirements for next-generation digital currency systems in terms of privacy, efficiency, scalability, and security.
This comprehensive paper presents a thorough analysis of Identity-Based Cryptography (IBC), focusing on both encryption (IBE) and signature (IBS) schemes, along with novel contributions in threshold implementations and zero-knowledge proofs. We provide detailed mathematical formulations and practical implementations of major IBE schemes including Boneh-Franklin, Boneh-Boyen, and Sakai-Kasahara, all enhanced with Fujisaki-Okamoto transformations for achieving IND-CCA2 security. For IBS systems, we examine Barreto, Cha-Cheon, Galindo-Garcia, Hess, and ShangMi schemes, demonstrating their EUF-CMA security properties. A significant contribution of this work is the development of novel non-interactive zero-knowledge proofs (ZKPs) tailored for identity-based systems. We introduce two distinct ZKP constructions: a generalized approach for IBE/IBS systems based on Schnorr-type proofs adapted for bilinear pairings, and a specialized ZKP for the ShangMi/Barreto scheme offering alternative security properties. These ZKPs enable efficient verification of private key knowledge without disclosure, leveraging the Fiat-Shamir heuristic for non-interactive operation. Furthermore, we present a practical threshold IBE/IBS implementation that distributes the Private Key Generator (PKG) functionality across multiple servers using Shamir's Secret Sharing. This enhancement mitigates single points of failure and strengthens resilience against key compromise while maintaining compatibility with existing identity-based cryptographic primitives. All schemes are implemented using the BLS12-381 elliptic curve and are accompanied by complete mathematical correctness proofs. The implementations are publicly available through Go Playground links, providing practical reference implementations for researchers and practitioners. This work bridges theoretical cryptography with practical implementation, offering valuable insights for privacy-preserving authentication systems, decentralized identity management, and secure delegation protocols in identity-based infrastructure.
We present EHT (Elliptic Homomorphic Token), a generalized cryptographic framework that bridges the gap between theoretical homomorphic encryption and practical, verifiable encrypted computation. EHT is built on an elliptic-curve–based partial homomorphic encryption scheme (EC-ElGamal) and extends it with verifiable digital signatures (EHDSA) and zero-knowledge policy proofs (zk-FIDNA), enabling both confidentiality and integrity in distributed execution environments.Unlike lattice-based fully homomorphic encryption, which suffers from high computational cost and ciphertext expansion, EHT preserves constant-size ciphertexts and achieves O(1) amortized complexity per operation, allowing real-time encrypted computation even in large-scale systems. The proposed four-layer architecture separates cryptographic primitives from domain-specific semantics, enabling seamless interoperability across heterogeneous applications such as encrypted databases, federated learning, web authentication, and blockchain transaction networks.Through its tokenized abstraction, EHT allows operations—query execution, aggregation, verification—to be performed directly on ciphertexts while maintaining verifiability through EHDSA and zk-FIDNA proofs.Experimental results demonstrate sub-millisecond elliptic-curve operations, achieving over 8,000 homomorphic additions per second on commodity hardware with less than 2% overhead relative to baseline elliptic-curve performance. EHT thus represents a cryptographically lightweight yet distributedly scalable homomorphic framework: compact enough for real-time use, verifiable enough for regulatory and enterprise environments, and extensible enough to support post-quantum and cross-domain adaptations. By unifying encryption, verification, and computation into a single token-based execution model, EHT advances the state of privacy-preserving technology toward a truly encrypted, interoperable, and verifiable computation fabric.
We introduce TorusDB, the first database engine supporting practical SQL query processing over ciphertexts using a fully homomorphic encryption scheme derived entirely from elliptic curve cryptography. Unlike prior approaches based on lattice FHE or zero-knowledge proofs, TorusDB preserves the elliptic curve group structure and extends additive homomorphism via a formal multiplicative construction and rational extension, enabling full homomorphic evaluation without decryption. We formalize the underlying EC-based FHE scheme, prove its security under standard elliptic curve assumptions (ECDLP, DDH, BDH), and present a query execution model supporting selection, projection, aggregation, and grouping. Our implementation demonstrates that encrypted query execution incurs only 77% overhead relative to plaintext execution, marking a substantial improvement over existing homomorphic database systems which typically exhibit 10-100× overhead.
This paper addresses the centralized trust problem inherent in the Elliptic Curve Homomorphic Digital Signature Algorithm (EHDSA), where the critical security parameter t is traditionally generated and held by a single trusted authority, creating a significant single point of failure and raising concerns about trust and security. To overcome this fundamental limitation, we propose MPC-EHDSA, a novel and practical protocol that leverages Multi-Party Computation (MPC) to securely distribute the generation and management of the parameter t among multiple independent participants. Our approach ensures that no individual party ever gains knowledge of the secret value of t, thereby eliminating centralized trust assumptions and significantly enhancing the overall security and robustness of the system. The protocol combines Shamir secret sharing with the well-established BGW MPC framework, augmented with homomorphic encryption techniques and zero-knowledge proofs to provide strong cryptographic guarantees and resistance against semi-honest and malicious adversaries. Through rigorous theoretical analysis and extensive performance evaluations, we demonstrate that MPC-EHDSA not only preserves the full functionality and security properties of the original EHDSA scheme but also achieves practical efficiency that enables deployment in real-world decentralized environments such as blockchain systems and distributed ledgers.
This paper introduces the Elliptic Curve Homomorphic Digital Signature Algorithm (EHDSA), a novel digital signature scheme that enhances security by leveraging homomorphic encryption. Unlike traditional ECDSA, which generates signatures using the x-coordinate of elliptic curve points, EHDSA employs a homomorphic mapping between elliptic curves and Zn. This mapping conceals the original elliptic curve point information, providing increased security. EHDSA is particularly advantageous in resource-constrained environments due to its reduced signature size, computational speed, and security compared to RSA. Additionally, this paper explores the ω protocol, which utilizes ElGamal Encryption and a Common Reference Domain Set (CRDS) to perform secure zero-knowledge proofs. The protocol’s arithmetic circuit is transformed into a Linear Form Arithmetic Program (LFAP), ensuring efficient proof creation. We also discuss the use of digital signatures for polynomial commitments, ensuring the integrity and authenticity of the commitment process. The integration of EHDSA into the ω protocol significantly enhances the overall security and efficiency of digital signatures and zero-knowledge proofs, addressing fundamental privacy vulnerabilities in traditional ECDSA while maintaining computational efficiency through J-invariant-based curve classification and signature-integrated commitment schemes.