Mohammad Reehan Nawaz, Mohammad Afaque, Anzer Hussain, Dr. Anand Prakash
The emergence of quantum computing poses a significant threat to classical cryptographic mechanisms such as RSA and Elliptic Curve Cryptography that are widely used to secure email communication. Traditional secure email systems rely on classical public-key infrastructure and therefore lack resilience against quantum attacks. This paper presents QuMail, a quantum-secure email client that integrates BB84-based Quantum Key Distribution (QKD) simulation, CRYSTALS-Kyber post-quantum cryptography (PQC), and blockchain-based audit logging within a unified architecture. The proposed system operates entirely at the application layer and remains compatible with existing email infrastructures using standard SMTP and IMAP protocols without requiring any server-side modification. A modular prototype was implemented using IBM Qiskit for quantum key generation and hybrid cryptographic techniques for secure message transmission. Experimental evaluation demonstrates an average latency of 120–180 ms for QKD key generation and 20–30 ms for Kyber-based encryption while maintaining minimal overhead for email transmission. The results demonstrate the feasibility of integrating quantum-resilient security mechanisms into existing email systems and highlight the potential of hybrid QKD–PQC architectures for next-generation secure communication platforms.
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
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.
Rohan Goyal, Venkatesan Guruswami, Yihang Sun, Mary Wootters
Proximity gaps are a property of error correcting codes that arise in the study of Interactive Oracle Proofs (IOPs) and Succinct Non-interactive Arguments of Zero Knowledge (SNARKs). Recent work of Goyal and Guruswami has established near-optimal proximity gaps for many families of codes, including subspace design codes, as well as random ensembles like random linear codes, Reed-Solomon codes with random evaluation points, and Gallager's ensemble of LDPC codes (Goyal & Guruswami, 2025). However, the parameters for these latter randomized ensembles are worse than the parameters for subspace design codes, and degrade as the degree ell increases. In this work, we obtain improved proximity gaps for random ensembles of codes, including random linear codes, Reed-Solomon codes with random evaluation points, and Gallager's ensemble. Quantitatively, our results for these random ensembles match the results that Goyal and Guruswami attained for subspace design codes. In fact, our techniques are a black-box transference from subspace design codes: any progress on subspace design codes will automatically lead to analogous progress for these random ensembles. To obtain our results, we extend the Local Coordinate-wise Linear (LCL) property framework developed by Levi, Mosheiff, and Shagrithaya and by Brakensiek, Chen, Dhar, and Zhang to a \textit{row-span constrained} version (Levi, Mosheiff & Shagrithaya, 2025; Brakensiek, Chen, Dhar & Zhang, 2025). This allows us to cast \textit{curve-decodability} -- a property that implies proximity gaps -- directly as a row-span constrained LCL property, and make use of that machinery. In contrast, because curve-decodability is not obviously a vanilla LCL property, prior work had worked with a proxy property instead, leading to the aforementioned parameter losses.
Modular exponentiation is among the most demanding computational operations in cryptographic systems. Effective computation of modular exponentiation is most beneficial for public-key cryptography. The computational complexity and the growing number of bits of the key size, as required by increasingly stringent security demands in the RSA, the Diffie–Hellman key exchange and the Zero-Knowledge Proof (ZKP) protocols, have become a top research priority in terms of algorithmic efficiency. This study proposes a novel triple modular exponentiation algorithm based on the Improved Common-Multiplicand-Multiplication (ICMM) framework. The exact complexity formula was obtained through systematic probabilistic analysis of eight mutually exclusive bit-level states. The efficiency of modular exponentiation is primarily determined by the number of modular multiplications and exponentiation squares required. It is observed that improved common-multiplicand multiplication efficiently minimizes the computational complexity of the triple modular exponentiation by reducing the number of modular multiplications. The overall computational complexity of triple modular exponentiation is 1.875j, where j is the bit length of the exponent. This represents a reduction of approximately 16.7% in total multiplication count relative to double modular exponentiation, corresponding to a 44.4% reduction on a per-exponent basis, and a reduction of 58.3% relative to three independent binary exponentiations. This study concludes that the proposed decomposition reduces the average-case computational complexity of triple modular exponentiation to 1.875j modular multiplications for a j-bit exponent. The proposed triple modular exponentiation algorithm is shown to have lower number of multiplications per bit length of exponent as compared to double modular exponentiation. This result demonstrates the potential of proposed algorithm to reduce the computational cost of triple modular exponentiation in cryptographic protocols where it is a recurring operation, such as interactive ZKP identification schemes.
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.
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.
Modern cryptographic primitives have evolved from supporting basic to more advanced functionalities, and such schemes are now getting more practical. In this thesis, we identify and rectify some limitations of such cryptographic constructions and their proofs of security. Specifically, we work with functional encryption, secure aggregation, and threshold signature schemes, and observe key functional or security limitations in prior work. Our first focus is functional encryption (FE), which enables function evaluation on encrypted messages using a functional secret key. A different primitive named function-revealing encryption (FRE) allows one to compute a fixed function of the underlying messages using their ciphertexts only. We give formal definitions and construct an inner-product FRE scheme. We also analyze the relationship between FE and FRE. Our second contribution considers secure aggregation, a classic problem that has numerous applications in privacy preserving machine learning. Secure aggregation lets many clients contribute data for aggregation without revealing their individual data. Existing practical protocols either have multiple rounds of interaction between clients and the server or rely on heavyweight cryptographic primitives. We build a non-interactive secure aggregation protocol using a novel combination of inner-product FE and a fully-linear probabilistically checkable proof (FLPCP) system. For this protocol, we use an existing FLPCP system [BBCGI’19] that we prove satisfies soundness and zero-knowledge properties even when reused for multiple proof instances. Finally, we address a pressing open question: achieving fully adaptive security for the Sparkle+ [CKM’23] threshold signature scheme. Threshold schemes require t signers to provide partial signatures to form a valid one. Fully adaptive security prevents adversaries from forging signatures even when corrupting up to t-1 signers. While Sparkle+ is secure against static corruption and a limited number of adaptive corruptions, a previous proof of fully adaptive security was shown to be incorrect. We propose a novel hardness assumption under which Sparkle+ satisfies this notion with a tight reduction. We establish hardness of this assumption in the elliptic-curve generic-group model. Our contributions close important gaps in prior work and push advanced cryptographic primitives closer to practice.
Abstract In recent years, progress in practical applications of multi-party computation (MPC), fully homomorphic encryption (FHE), and zero-knowledge proofs (ZKP) motivates people to explore symmetric-key cryptographic algorithms, as well as corresponding cryptanalysis techniques (such as differential cryptanalysis, linear cryptanalysis), over finite Abelian groups or prime fields $${\mathbb {F}}_p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:math> for large p . In this paper, we establish the links between linear cryptanalysis and differential cryptanalysis over general finite Abelian groups. As the first application, we revisit linear cryptanalysis and give general results of linear approximations over arbitrary finite Abelian groups. More precisely, we consider the linearity , which is the maximal non-trivial linear approximation, to characterize the resistance of a function against linear cryptanalysis. This thereby generalizes the work of Pott in 2004 and completes the generalization of Sidelnikov–Chabaud–Vaudenay’s bound from $${\mathbb {F}}_2^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>F</mml:mi> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:msubsup> </mml:math> to finite Abelian groups. As the second application, we give an exact expression for the correlation of differential-linear approximations over arbitrary finite Abelian groups ( $${\mathbb {F}}_p^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>F</mml:mi> <mml:mi>p</mml:mi> <mml:mi>n</mml:mi> </mml:msubsup> </mml:math> ) under the sole assumption that the two parts of the cipher are independent of each other. In particular, we completely generalize the differential-linear cryptanalysis from $${\mathbb {F}}_2^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>F</mml:mi> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:msubsup> </mml:math> to arbitrary finite Abelian groups ( $${\mathbb {F}}_p^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>F</mml:mi> <mml:mi>p</mml:mi> <mml:mi>n</mml:mi> </mml:msubsup> </mml:math> ).
We explore the design and analysis of post-quantum cryptographic primitives with an emphasis on two angles. First, diversity of assumptions, by building and analysing cryptography that does not rely on structured lattices. Second, diversity of functionalities, focusing on various primitives that extend beyond standard encryption. Motivated by the rapid development of quantum computing and the consequent threat to classical public-key cryptography, we therefore consider two families of post-quantum assumptions: isogeny-based and code-equivalence problems from which we build and analyse various primitives. In the first part, we focus on isogeny-based cryptography. We start our study with digital signatures derived from SIDH squares and investigate their security by analysing a range of attack vectors, including higher-dimensional attack strategies (known as SIDH attacks), and consequently propose suitable parameters. Building on the powerful constructive tools that the SIDH attacks became, we design an updatable public-key encryption scheme from a variant of FESTA, a public-key encryption scheme by Basso, Maino and Pope, enabling an efficient instantiation of this primitive that allows an unbounded number of updates. Finally, we investigate isogeny-based group actions and propose a framework for general-purpose zero-knowledge proofs. The second part focuses on code-based cryptography, particularly tensor group actions and code-equivalence problems. We first analyze and break a commitment scheme based on a structured tensor isomorphism problem, and we propose a secure fix. Then, we study variants of code-equivalence problems that underlie the security of two blind signature schemes. For one variant, we provide a reduction to a standard problem, whilst for another one we show that its security has been largely over-estimated. Overall, this thesis contributes to the development of a diverse suite of post-quantum primitives by providing new constructions, security analyses, and insights into the use of alternative assumptions beyond lattice-based systems.
List decoding of codes can be seen as the generalization of unique decoding of codes while list decoding over finite fields has been extensively studied, extending these results to more general algebraic structures such as Galois rings remains an important challenge. Due to recent progress in zero knowledge systems, there is a growing demand to investigate the proximity gap of codes over Galois rings in Yizhou Yao(2025). The proximity gap is closely related to the decoding capability of codes. It was shown in Eli Ben-Sasson(2020) that the proximity gap for RS codes over finite field can be improved to $1-\sqrt{r}$ if one consider list decoding instead of unique decoding. However, we know very little about RS codes over Galois ring which might hinder the development of zero knowledge proof system for ring-based arithmetic circuit. In this work, we first extend the list decoding procedure of Guruswami and Sudan to Reed-Solomon codes over Galois rings, which shows that RS codes with rate $r$ can be list decoded up to radius $1-\sqrt{r}$. Then, we investigate the list decoding of folded Reed-Solomon codes over Galois rings. We show that the list decoding radius of folded Reed-Solomon codes can reach the Singlton bound as its counterpart over finite field. We also extend the deterministic pruning method of Vikrant Ashvinkumar(2026) to Galois rings, showing how to prune the affine free module obtained from the linear-algebraic decoder and recover the candidate codewords. Finally, we improve the list size of our folded Reed-Solomon code to $O(1/\varepsilon^2)$ by extending recent work in Shashank Srivastava(2025) to Galois Rings. By developing the recent work of Yeyuan Chen(2025), we show that folded Reed-Solomon codes over Galois rings satisfy the relaxed generalized Singleton bound in the average-radius sense with optimal list size $O(1/\varepsilon)$.
The dawn of the disruptive quantum computing scenario marks a serious threat to the existence of traditional cryptosystems. With laws such as Shor’s, capable of factoring large integers in polynomial time, and Grover’s, able to speed up brute-force key searches, these attacks make conventional public-key infrastructures increasingly vulnerable, whereas even symmetric ciphers lose good measure of their strength. In this article, we focus on an elaborative description of a patented method for quantum-secure key generation, wherein Qabbalah (QBLH) complexity is utilized in the geometric-symbolic realm, in conjunction with magic number squares, phi/pi coordinate weighting, and tetrahedral trinary state encoding. The proposed system of TriGate QBLH Quantum-Safe Encryption converts seed inputs to multidimensional keys that resist linear algebraic attacks owing to non-linear permutations, irrational constant weighting, and topological complexity. Normally, pseudo-random number generators spatialize entropy in Euclidean geometry, as opposed to the present technique that places entropy in a completely non-Euclidean domain, where classical as well as quantum adversaries find it hard to traverse. We describe the method in detail, present its benefits over lattice- and hash-based post-quantum schemes, and walk through an example of its implementation. Consideration is also given to its potential integration with PQC standards, blockchain authentication, and decentralized finance applications. The system fuses symbolic mathematics, such as the 231 Gates of QBLH, with trinary logic mapped onto tetrahedral states to not only create encryption keys but also verifiable geometric signatures. This represents a paradigm shift toward geometric cryptography, which may be a viable method to realize scalable and trustworthy digital infrastructure in a quantum-threatened environment.
Evgeniya Ishchukova, Sergei Petrenko, A. Petrenko, Konstantin Gnidko · 5 authors
Today, blockchain technologies are a separate, rapidly developing area. With rapid development, they open up a number of scientific problems. One of these problems is the problem of reliability, which is primarily associated with the use of cryptographic primitives. The threat of the emergence of quantum computers is now widely discussed, in connection with which the direction of post-quantum cryptography is actively developing. Nevertheless, the most popular blockchain platforms (such as Bitcoin and Ethereum) use asymmetric cryptography based on elliptic curves. Here, cryptographic primitives for blockchain systems are divided into four groups according to their functionality: keyless, single-key, dual-key, and hybrid. The main attention in the work is paid to the most significant cryptographic primitives for blockchain systems: keyless and single-key. This manuscript discusses possible scenarios in which, during practical implementation, the mathematical foundations embedded in the algorithms for generating a digital signature and encrypting data using algorithms based on elliptic curves are violated. In this case, vulnerabilities arise that can lead to the compromise of a private key or a substitution of a digital signature. We consider cases of vulnerabilities in a blockchain system due to incorrect use of a cryptographic primitive, describe the problem, formulate the problem statement, and assess its complexity for each case. For each case, strict calculations of the maximum computational costs are given when the conditions of the case under consideration are met. Among other things, we present a new version of the encryption algorithm for data stored in blockchain systems or transmitted between blockchain systems using elliptic curves. This algorithm is not the main blockchain algorithm and is not included in the core of modern blockchain systems. This algorithm allows the use of the same keys that system users have in order to store sensitive user data in an open blockchain database in encrypted form. At the same time, possible vulnerabilities that may arise from incorrect implementation of this algorithm are considered. The scenarios formulated in the article can be used to test the reliability of both newly created blockchain platforms and to study long-existing ones.
Abstract: Algebraic geometry offers a powerful and elegant mathematical framework for the design and analysis of modern cryptographic protocols. This research paper investigates the application of algebraic geometry methods—such as elliptic curves, abelian varieties, and projective algebraic structures—in enhancing the security, efficiency, and scalability of cryptographic systems. By bridging advanced algebraic structures with cryptographic primitives, the study demonstrates how algebraic geometry enables the construction of secure public key protocols, zero-knowledge proofs, and post-quantum resilient schemes. Through theoretical modeling, performance benchmarking, and comparative analysis with classical cryptographic approaches, the paper illustrates the advantages of algebraic geometry in terms of computational hardness assumptions, structural integrity, and potential for innovation in secure communications. The findings contribute to the evolving landscape of cryptography by positioning algebraic geometry as a foundational tool in next-generation cryptographic protocol design. Keywords: algebraic geometry, cryptographic protocols, elliptic curves, public key cryptography, post-quantum cryptography, projective varieties, zero-knowledge proofs, secure communication, mathematical cryptography, abelian varieties
This article presents a comprehensive framework for implementing privacy-preserving credit card reward systems using Secure Multi-Party Computation (SMPC) technologies. Traditional reward architectures require extensive sharing of sensitive transaction data across multiple entities, creating significant privacy risks, security vulnerabilities, and regulatory compliance challenges. It leverages cryptographic advances to enable card issuers, payment networks, and merchant partners to collaborate on reward calculations,fraud detection, and personalized offers without revealing sensitive transaction details to one another. The article explores the evolution of privacy-preserving technologies in financial systems, comparing Fully Homomorphic Encryption, Zero-Knowledge Proofs, and SMPC approaches. A detailed case study of a travel rewards program implementation demonstrates how this framework ensures data remains protected throughout the entire process while maintaining the performance characteristics necessary for production deployment. The system provides comprehensive privacy protection, enhances fraud detection capabilities through secure collaboration, and facilitates compliance with evolving privacy regulations.Performance evaluations confirm the practical viability of the article, with minimal latency impact, strong scalability characteristics, and robust security guarantees. It contributes to the growing field of privacy-enhancing technologies for financial services and offers a viable solution to balance analytical utility with privacy protection in consumer-facing applications.
Elvira Albert, Samir Genaim, Daniel Kirchner, Enrique Martin-Martin
Program optimization is a key factor for green software. In the context of the Ethereum blockchain, optimization is particularly relevant because there is a fee to pay for each EVM (Ethereum Virtual Machine) instruction executed and also there exist bytecode-size limitations for deploying the software on the blockchain. Still, optimization of EVM code is not as widely spread as one could imagine. This is at least partly due to the lack of trust in the correctness of the tools, as security is even more relevant than efficiency in the blockchain context in which bugs may cause huge economical losses. This article develops a formal verification framework using Coq to ensure the security of EVM optimizations performed on jump-free sequences of EVM bytecode. By means of Coq’s theorem proving capabilities, we are able to automatically verify/certify that an optimized jump-free sequence of EVM opcodes is semantically equivalent to a given original one. We also present an extension to our framework that can handle inter-block optimizations that propagate global information across blocks. We have applied our tool to successfully prove the security of peephole optimizations performed by the standard Solidity compiler, and also to existing EVM superoptimization tools (namely GASOL and Superstack) in which we have found bugs that have been reported and fixed.
Recently, the practical applications of advanced cryptographic protocols, such as Multi-Party Computation (MPC), Fully Homomorphic Encryption (FHE), and Zero Knowledge Proofs (ZKP), have spurred the development of a series of new symmetric encryption primitives. These novel symmetric encryption primitives, referred to as Arithmetization-Oriented (AO) ciphers, aim to minimize the number of field multiplications in large finite fields, including 𝔽2n or 𝔽p. In order to evaluate the algebraic degrees of AO ciphers over 𝔽2n, the general monomial prediction technique was proposed by Cui et al. at ASIACRYPT 2022. However, when using their searching tool to evaluate the algebraic degrees of AO ciphers with complex affine layers, the efficiency is low, preventing solutions within a predetermined timeframe. In this study, we extend the propagation rules of monomials for field-based operations and present an automatic searching tool based on Mixed Integer Linear Programming (MILP) and Boolean Satisfiability (SAT) Problem for evaluating the upper bound of the algebraic degrees. Moreover, to accurately calculate the algebraic degrees of monomials in the SAT model, we improve the sequence encoding method, enabling it to accurately determine whether the monomials of degree d exist in the output. We apply our new searching tool to various AO ciphers, including Chaghri, MiMC, and Ciminion. For Chaghri, we compare our results with the Coefficient Grouping technique proposed by Liu et al. at EUROCRYPT 2023, and our results yield tighter upper bounds compared to Liu et al.’s findings. Additionally, we evaluate the algebraic degrees of Chaghri and MiMC with arbitrary complex affine layers and obtain tighter bounds compared to the results from Liu et al. at CRYPTO 2023. Regarding Ciminion, we have observed that starting from the 4-th round, the upper bound on the algebraic degrees for each round actually 1 degree lower than the previous bound. Our searching tool enables a more precise evaluation of the algebraic degrees of AO ciphers, contributing to a deeper understanding of the design and analysis of such primitives.
At a time when post-quantum cryptography is more and more present in the cryptographic landscape, it is of great interest to find new hard problems on which we can rely. Here, we present a new problem, the vector space factorization problem, and use it to build a signature scheme. The idea of factorizing subspaces of a finite field is used in rank metric codes, most notably in the decoding of LRPCs. In this context, one of the subspaces is known to factorize. Factorizing without the knowledge of both subspaces appears in the signature scheme Murave, in which the rank support basis decomposition problem is introduced from a coding theory in rank metric point of view. In Bro's thesis, the SquareSpace problem is introduced, where one wants to find the 'square root' of a subspace. We generalize here this problem into the vector space factorization problem, which is the same as the rank support basis decomposition problem introduced in Murave, the difference being we do not look at it from a coding theory point of view, but really from a vector subspace one. We use it here to build a zero-knowledge proof of knowledge. The scheme uses the MPCitH paradigm, and especially the TCitH framework, which is an efficient way to build ZK proofs. We study the difficulty of solving the vector space factorization problem by detailing the combinatorial attacks on the problem, analyzing their complexity, and describing an algebraic model to solve the problem. We then explain the MPC protocol used to build the signature scheme. Finally, this construction allows us to obtain sizes of signature of 8.9 to 10.9 kB for the first security level defined by NIST, which is reasonable as MPC-in-the-Head signatures typically range from 2.5 kB for an MQ instance to 14 kB for lattice-based instances.
Zero-knowledge proof (ZKP) is an attractive cryptographic paradigm that allows a party to prove the correctness of a given statement without revealing any additional information. It offers both computation integrity and privacy, witnessing many celebrated deployments, such as computation outsourcing and cryptocurrencies. Recent general-purpose ZKP schemes, e.g., zero-knowledge succinct non-interactive argument of knowledge (zk-SNARK), suffer from time-consuming proof generation, which is mainly bottlenecked by the large-scale number theoretic transformation (NTT) and multi-scalar point multiplication (MSM). To boost its wide application, great interest has been shown in expediting the proof generation on various platforms like GPU, FPGA and ASIC.So far as we know, current works on the hardware designs for ZKP employ two separated data-paths for NTT and MSM, overlooking the potential of resource reusage. In this work, we particularly explore the feasibility and profit of implementing both NTT and MSM with a unified and high-performance hardware architecture. For the crucial operator design, we propose a dual-precision, load-balanced and fully-pipelined Montgomery multiplier (LBFP MM) by introducing the new mixed-radix technique and improving the prior quotient-decoupled strategy. Collectively, we also integrate orthogonal ideas to further enhance the performance of LBFP MM, including the customized constant multiplication, truncated LSB/MSB multiplication/addition and Karatsuba technique. On top of that, we present the unified, scalable and highperformance hardware architecture that conducts both NTT and MSM in a versatile pipelined execution mechanism, intensively sharing the common computation and memory resource. The proposed accelerator manages to overlap the on-chip memory computation with off-chip memory access, considerably reducing the overall cycle counts for NTT and MSM.We showcase the implementation of modular multiplier and overall architecture on the BLS12-381 elliptic curve for zk-SNARK. Extensive experiments are carried out under TSMC 28nm synthesis and similar simulation set, which demonstrate impressive improvements: (1) the proposed LBFP MM obtains 1.8x speed-up and 1.3x less area cost versus the state-of-the-art design; (2) the unified accelerator achieves 12.1x and 5.8x acceleration for NTT and MSM while also consumes 4.3x lower overall on-chip area overhead, when compared to the most related and advanced work PipeZK.
Abstract We propose a generic compiler that can convert any zero-knowledge (ZK) proof for SIMD circuits to general circuits efficiently, and an extension that can preserve the space complexity of the proof systems. Our compiler can immediately produce new results improving upon state of the art. By plugging in our compiler to Antman, an interactive sublinear-communication protocol, we improve the overall communication complexity for general circuits from $$\mathcal {O}(C^{3/4})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo>/</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> to $$\mathcal {O}(C^{1/2})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . Our implementation shows that for a circuit of size $$2^{27}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mn>2</mml:mn> <mml:mn>27</mml:mn> </mml:msup> </mml:math> , it achieves up to $$83.6\times $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>83.6</mml:mn> <mml:mo>×</mml:mo> </mml:mrow> </mml:math> improvement on communication compared to the state-of-the-art implementation. Its end-to-end running time is at least $$70\%$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>70</mml:mn> <mml:mo>%</mml:mo> </mml:mrow> </mml:math> faster in a 10Mbps network. Using the recent results on compressed $$\varSigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Σ</mml:mi> </mml:math> -protocol theory, we obtain a discrete-log-based constant-round zero-knowledge argument with $$\mathcal {O}(C^{1/2})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> communication and common random string length, improving over the state of the art that has linear-size common random string and requires heavier computation. We improve the communication of a designated n -verifier zero-knowledge proof from $$\mathcal {O}(nC/B+n^2B^2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mi>C</mml:mi> <mml:mo>/</mml:mo> <mml:mi>B</mml:mi> <mml:mo>+</mml:mo> <mml:msup> <mml:mi>n</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:msup> <mml:mi>B</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> to $$\mathcal {O}(nC/B+n^2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mi>C</mml:mi> <mml:mo>/</mml:mo> <mml:mi>B</mml:mi> <mml:mo>+</mml:mo> <mml:msup> <mml:mi>n</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . To demonstrate the scalability of our compilers, we were able to extract a commit-and-prove SIMD ZK from Ligero and cast it in our framework. We also give one instantiation derived from LegoSNARK, demonstrating that the idea of CP-SNARK also fits in our methodology.
This paper investigates the use of threshold signature schemes in isogeny-based cryptosystems. By leveraging the distinct features of isogeny graphs, we propose a protocol that offers high security and practical efficiency, making it ideal for distributed ledger technologies and secure multi-party computations. Our scheme is resistant to quantum attacks and maintains minimal computational and communication overheads. We present an extensive analysis of the security and performance of our protocol, emphasizing its relevance to real-world cryptographic systems. MSC 2020: 94A60, 11G07, 68M07.