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 $O(1/\varepsilon)$.
Abstract: Homomorphic encryption (HE) enables secure computations on encrypted data without decryption, offering a transformative solution for privacy-preserving computation. This review presents a ten-year retrospective (2014–2024) on HE’s evolution since Gentry’s 2009 fully homomorphic encryption (FHE) scheme, which introduced the concept of performing arbitrary computations on ciphertexts. Early schemes were hindered by inefficiencies like computational overhead and noise accumulation. Over the past decade, significant advancements have addressed these barriers. Schemes such as BGV, BFV, and CKKS have been developed for efficient integer and approximate real-number computations. Algorithmic innovations like optimized bootstrapping and improved noise management have reduced complexity. Hardware acceleration using GPUs and FPGAs has enhanced performance, while integration with secure multi-party computation and zero-knowledge proofs has broadened HE’s applicability. Applications now span privacy-preserving machine learning, genomic data analysis, and financial analytics. Toolkits such as SEAL, HElib, and PALISADE have improved accessibility for developers and researchers. Despite progress, challenges remain, including balancing efficiency and security, and improving usability for non-experts. The article also explores HE’s reliance on lattice-based problems like Learning With Errors (LWE) and Ring-LWE, which provide quantum resistance. As hybrid cryptographic models emerge, HE is increasingly recognized as a key component in securing sensitive data in the postquantum era. This review highlights HE’s maturation from a theoretical concept to a practical solution, demonstrating its potential as a cornerstone for secure, privacy-preserving computing across industries.
Emerging applications in cloud computing, big data, and the Internet of things have driven the advancement and implementation of security protocols, including secure multi-party computation, fully homomorphic encryption, and zero-knowledge proofs, to meet heightened security demands. Designing cryptographic permutations and block ciphers using a partial substitution-permutation network (P-SPN) approach, where the nonlinear part does not cover the entire state, has recently gained attention due to favorable implementation characteristics in various scenarios. For the word-oriented P-SPN schemes with a fixed linear layer, the choice of the maximum distance separable (MDS) matrix significantly affects the security level provided by P-SPN designs. If the MDS matrix is chosen weak, it will allow for extremely maximum invariant subspace that pass the entire rounds without activating any non-linear operation. Firstly, we investigate the properties of a special block matrix with circulant block, specifically utilized within the linear layer matrix of P-SPN structure schemes. Subsequently, our investigation extends to present the annihilating polynomial of low degree for these specific type of matrices, as well as to put forward the range of determining their minimal polynomial degree. Finally, this study articulates a lower bound estimated for the dimension of the maximum invariant subspace within the P-SPN structure schemes when integrated with the aforementioned matrix type. In scenarios where the S-box number$s$= 1 in the P-SPN structure schemes, we achieve a precise determination of the dimension of maximum invariant subspace. Conversely, for cases with$s$> 1, with some certain specific conditions, our research establishes more compact lower bound for the dimension of the maximum invariant subspace. The research results of this paper offer valuable design guidance for the development of matrices within the linear layer of P-SPN architecture schemes.
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.
This paper presents a comprehensive examination of advanced cryptographic primitives and their instantiations, focusing on signature schemes, zero-knowledge proofs, and threshold encryption schemes. The foundational properties required for a secure and complete digital signature scheme are first outlined, with an emphasis on existential unforgeability. Zero-knowledge proofs are then examined in detail, including definitions of completeness, zero-knowledge, soundness, and simulation extractability, along with a discussion of universal versus non-universal proof systems. The core contribution lies in the design and analysis of a threshold encryption scheme based on Shamir’s secret sharing and a CLT encryption framework. The construction of a (t, P)-threshold encryption scheme is defined, introducing the concept of partial decryption simulability. The framework ensures that any (t + 1)-sized subset of parties can decrypt the ciphertext while maintaining security against adversarial attempts. Additionally, the implementation of universal zero-knowledge proof systems is discussed, highlighting the trade-offs between universal and specific SRS-based proofs. The instantiation of pseudorandom functions and their weak robustness properties is also examined, ensuring secure key management and resistance to adversarial key collisions. Through detailed analysis and construction, this work provides a solid foundation for building secure cryptographic systems with efficient threshold encryption and zero-knowledge proofs, contributing to the advancement of cryptographic protocols and their applications in secure communications and data protection.