区块链或者中国剩余定理或者零知识证明方向

Federated Learning (FL) enables distributed model training without centralizing data, but it remains vulnerable to corrupted or adversarial clients that can degrade the global model through malicious weight update. This paper proposes a modular aggregation mechanism using the Chinese Remainder Theorem (CRT) to secure model aggregation. The CRT encodes model weights under multiple modular bases, allowing consistent and tamper-resilient reconstruction of global weights. Experimental analysis demonstrates that the modular arithmetic based secure aggregation significantly mitigates the effect of corrupted clients, improving accuracy and stability compared to standard Federated Averaging (FedAvg). The modular CRTFed proposed achieves a 53.76 % reduction in the maximum validation loss. This is a great quantitative measure of improved robustness, indicating the method effectively prevents the model from diverging severely due to malicious or noisy data updates.

Blinding has been one of the most effective approaches to resist power analysis attacks on asymmetric cryptosystems like RSA. Blinding is similar to masking in symmetric cryptosystems, but masking can be implemented in various ways like Boolean, affine, polynomial masking, etc. However, for asymmetric cryptosystems with modular exponentiation as a fundamental operation, arithmetic masking or simply blinding has been extremely popular. In this paper, we have presented a secured approach for modular exponentiation in RSA and CRT-RSA cryptosystems with dual blinding. Through dual blinding, we have masked both secret exponent and message twice before executing the fundamental operations. We have also injected two ineffectual instructions between the fundamental operations and blinded the intermediate results to felicitate hiding and resist simple power analysis. The implementation results shows that with a nominal penalty, RSA and CRT-RSA with dual blinding can effectively resist some popular simple power analysis and differential power analysis attacks to a significant extent.

In order to get an efficient comprehensive analysis on Doppler estimation in RADAR; need an enhanced arithmetic formulation procedure for density, power and latency optimisations. Modular adders and multipliers are very crucial components in the performance of residue number system-based applications.  The Residue Number System (RNS) is a non-positional number system that allows parallel computations without transfers between digits. However, some operations in RNS require knowledge of the positional characteristic of a number. Among these operations is the conversion from RNS to the positional number system. The methods of reverse conversion for general form moduli based on the Chinese remainder theorem and the mixed-radix conversion are considered, as well as the optimized methods for special form moduli. A modified New CRT-I & New CRT-II with conjugate moduli set is considered to implement adder, multipliers and subtractions with optimised algorithms. This paper mainly deals with the conversion of numbers from binary to RNS as well RNS to binary with the specific modulo {2^n±k} which proves this new method. Modified Radix16 booth encoding algorithm and square carry bypass adder are used in implementation of RNS system to reduce parameter constraints.

Montgomery modular multiplication (MMM) in residue number systems (RNS) uses a base extension (BE) technique. This is to avoid division, which is hard, slow and costly in RNS. It is somewhat less costly and faster than the reverse conversion, via Chinese remainder theorem (CRT) and reduction factor method. However, it is used one after the other, for each of the equally large bases. In this work, we modify the conventional RNS-MMM algorithm via replacing the two unparalleled BE undertakings with three parallel CRT-like operations with the same complexity, as BE. As for the reduction factors, we use a special case of the Kawamura’s algorithm that leads to definitive result. The proposed RNS-MMM method allows for squaring the working dynamic range, or halving the bit-width of the balanced residue channels. Moreover, the common practice of dynamically changing the working moduli set in security and crypto applications is less critical due to doubled size of the pool of available moduli. The proposed circuits are simulated, tested and synthesized via Synopsys Design Compiler on the TSMC 65-nm technology, to show 69% less delay and 28% less area-time-product at the cost of 14% more energy consumption, with respect to the most relevant reference work.

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In this paper, we consider one of the key problems in modular arithmetic. It is known that scaling in the residue number system (RNS) is a rather complicated non-modular procedure, which requires expensive and complex operations at each iteration. Hence, it is time consuming and needs too much hardware for implementation. We propose a novel approach to power-of-two scaling based on the Chinese Remainder Theorem (CRT) and rank form of the number representation in RNS. By using minimal redundancy of residue code, we optimize and speed up the rank calculation and parity determination of divisible integers in each iteration. The proposed enhancements make the power-of-two scaling simpler and faster than the currently known methods. After calculating the rank of the initial number, each iteration of modular scaling by two is performed in one modular clock cycle. The computational complexity of the proposed method of scaling by a constant Sl=2l associated with both required modular addition operations and lookup tables is estimeted as k and 2k+1, respectively, where k equals the number of primary non-redundant RNS moduli. The time complexity is log2k+l modular clock cycles.

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The homomorphic encryption over the torus (TFHE) is a promising fully homomorphic encryption (FHE) scheme that allows arbitrary homomorphic computations with the programmable bootstrapping (PBS) algorithm. However, PBS suffers from prohibitive computation complexity and latency, which hinders the practical applications of TFHE. To address these challenges, we propose ALT, a field-programmable gate array (FPGA) accelerator for PBS that exhibits high area efficiency and low latency. Our approach involves modifying the parameters of the PBS algorithm to strike a balance between the computation complexity and the decryption failure rate (DFR). In addition, we leverage the Chinese residue theorem (CRT) to exploit the inherent parallelism and construct the primes to eliminate the need of CRT process and facilitate fast modular arithmetic. The ALT design comprises several carefully designed computation units, including inverse CRT (ICRT), divide-and-round (DR) operation, and monomial number theoretic transform (MNTT). We employ algorithmic and architectural co-optimization techniques to optimize these units. Notably, ALT features a low-complexity MNTT module, enabling the utilization of the bootstrapping key unrolling (BKU) technique with reduced latency and minimal hardware resources. Furthermore, all submodules of ALT are parameterized and scalable, allowing the entire design to be configurable according to varying requirements across different application scenarios. Experimental results on FPGA demonstrate that ALT significantly outperforms a similar configurable work in terms of latency, throughput, and efficiency. In comparison with the fastest FPGA implementation, ALT can realize lower latency while reducing digital signal processor (DSP) reduction by over 50%, leading to enhanced area efficiency and energy efficiency.

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In digital systems, the Residue Number System (RNS) represents an interesting alternative to the traditional two’s complement representation. Its performance and low-power properties have attracted significant research interest over the years. In this paper, RNS is used to estimate the angular position of a multi-trace electrical encoder (EE), an electro-mechanical device to measure angles at high precision widely used, for example, in antennas on-board satellites. The model of this system presents cyclic characteristics and, consequently, allows efficient use of modular arithmetic for its description. The RNS is applied to EEs equipped with more than two plates, and the absolute angle reconstruction is performed by using the Chinese Remainder Theorem (CRT). Furthermore, the use of RNS allows detection and mitigation solutions for errors due to encoders’ non-idealities and electrical noise. In this noisy context, we provide a detailed analysis of the performance of the system and propose a more robust, flexible, and easy-to-implement solution compared with the traditional methods. The results show that the RNS-based system can attenuate the noise, measure accurately the angles, and improve the overall performance.

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With the shift in storage paradigm, there is an increasing need for privacy of dataset and also for an encryption scheme that permits computation on encrypted data. Paillier cryptosystem is a good example of such a homomorphic encryption scheme. To improve the efficiency of the Paillier homomorphic encryption scheme in terms of its decryption speed and overall computational cost, we propose an improved decryption process. Specifically, the inclusion of a variable $k$ to reduce the modular multiplicative arithmetic. The variable $k$ is combined with the L function and CRT recombination method, to arrive at a fast and improved decryption process, showing the mathematical correctness of the decryption algorithm. Experimental results validate that our scheme is significantly efficient in its decryption speed.

This paper proposes a novel residue-to-binary converter for residue number system (RNS) based on the moduli set $\{2^{n+\mathrm{k}},\ 2^{2_{n}+1}-1,2^{n}+1,2^{n}-1\}$. By adopting the new Chinese Remainder Theorem II (CRT II) and the properties of modulo 2k-1 arithmetic, the research herein presented shows the possibility of designing RNS residue-to-binary converters free of modular adders. By using exclusively regular binary adders, one can leverage the design of efficient residue-to-binary converters based on the large range of optimizations proposed to those regular arithmetic units. Experimental results show over 37% energy-improvement in comparison to conventional designs, which make use of modulo adders.

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Secret image sharing (SIS) belongs to but differs from secret sharing. In general, conventional (k,n) threshold SIS has the shortcoming of "all-or-nothing". In this article, first we introduce ramp SIS definition. Then we propose a (k1,k2,n) ramp SIS based on the Chinese remainder theorem (CRT). In the proposed scheme, on the one hand, when we collect any k1 or more and less than k2 shadows, the secret image will be disclosed in a progressive way. On the other hand, when we collect any k2 or more shadows, the secret image will be disclosed losslessly. Furthermore, the disclosing method is only modular arithmetic, which can be used in some real-time applications. We give theoretical analyses and experiments to show the effectiveness of the proposed scheme.

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Integer and modular arithmetic is a fundamental area of mathematics, with extensive applications in computer science, and is essential for cryptographic protocols, error correction, and algorithm efficiency. However, students often struggle to understand its abstract nature, especially when transitioning from theoretical knowledge to practical applications. To address these issues, interactive educational software can play a crucial role in supporting the learning process. In this study, we present ArtEM (Integer and Modular Arithmetic), a Java‐based tool that was developed to assist both students and instructors in exploring topics related to integer and modular arithmetic. ArtEM offers an intuitive approach, allowing users to experiment and to gain a deeper understanding of the subject. A survey conducted among students at the University of Alicante (Spain) assessed their satisfaction with ArtEM, which enabled a detailed analysis of key aspects, including its usability, content quality, and didactic effectiveness. These dimensions were crucial in evaluating how well ArtEM supports learning and enhances the overall educational experience. The results indicated that students found ArtEM user‐friendly, providing a seamless learning experience. Feedback on content quality emphasised its relevance and clarity, while participants noted that the tool effectively supported their learning objectives. Data analysis also demonstrated that ArtEM improved students' understanding of complex concepts, encouraged independent learning, and contributed to improved academic performance. A remarkable improvement in students' ability to apply theoretical knowledge to practical scenarios was observed, underscoring the effectiveness of ArtEM in fostering deeper learning and better educational outcomes.

This paper introduces a new method for designing modular arithmetic units, for reduction (<inline-formula> <tex-math notation="LaTeX">$X (mod\ P)$ </tex-math></inline-formula>), multiplication (<inline-formula> <tex-math notation="LaTeX">$(A\cdot B) (mod\ P)$ </tex-math></inline-formula>), and multiplication by a constant (<inline-formula> <tex-math notation="LaTeX">$(constant\cdot A)(mod \ P)$ </tex-math></inline-formula>). The proposed method employs a divide and conquer strategy, splitting input vectors in the first stage and merging the results in a second stage. A framework has been developed to automate the process. Experimental results obtained from implementing the proposed modular arithmetic designs on reconfigurable devices (FPGAs) demonstrate that our approach outperforms standard FPGA design tools, achieving up to <inline-formula> <tex-math notation="LaTeX">$30\times $ </tex-math></inline-formula> better area efficiency (in LUT usage) and up to <inline-formula> <tex-math notation="LaTeX">$2.9\times $ </tex-math></inline-formula> higher speed for inputs up to 500 bits.

The authors study the features and effectiveness of modular arithmetic within a quadratic range, including the implementation of basic arithmetic operations on commodity computers. The research aim is to analyze the structure and features of performing various arithmetic operations in the modular arithmetic within quadratic range and compare their time complexity with similar operations in traditional positional systems. To meet this objective, the authors investigated the structure and characteristics of modular arithmetic within a quadratic range. They implemented arithmetic operations in Python, performed experiments, and assessed the time complexity of operations. The research methods include a theoretical study of the basis of the modular arithmetic within quadratic range, the creation of algorithms for performing operations in Python, experimental testing and analysis of the results. The research result is the creation of an algorithm for performing arithmetic operations in the modular arithmetic within quadratic range, revealing a significant performance gain compared to positional number systems, confirmed experimentally. Therefore, this study proves that modular arithmetic can improve task productivity that requires speed and resources.

Measurement-based uncomputation (MBU) is a technique used to perform probabilistic uncomputation of quantum circuits. We formalize this technique for the case of single-qubit registers, and we show applications to modular arithmetic. Using MBU, we reduce Toffoli count and depth by 10% to 15% for modular adders based on the architecture of [1], and by almost 25% for modular adders based on the architecture of [2]. Our results have the potential to improve other circuits for modular arithmetic, such as modular multiplication and modular exponentiation, and can find applications in quantum cryptanalysis.

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Public key cryptography is an asymmetric cryptographic scheme that consists of two key pairs: a public key, which is publicly known, and a private key, which is kept secret. Public keys are typically constructed based on the properties of number theory, and one example is polynomial congruence. Karyadi’s public key algorithm is an example of a public key algorithm based on polynomial congruence. Cryptanalysis is the method of analyzing a cryptographic system by attempting to attack it, including public key cryptosystems. Cryptanalysis of public key systems can focus on obtaining the encrypted message or the secret private key through various attack methods. In this paper, cryptanalysis of Karyadi’s public key algorithm and Rabin Cryptosystem are conducted with the aim of obtaining the original message from the encrypted message without knowing the trapdoor information. The cryptanalysis results demonstrate that it is possible to easily recover the original plaintext of the encrypted message in Karyadi’s public key algorithm, Rabin Cryptosystem, and other polynomial congruence-like cryptosystems under specific conditions without requiring the trapdoor information. Furthermore, a general method is presented for cryptanalysis to obtain the plaintext without knowing the trapdoor information in public key systems based on polynomial congruence using the Chinese Remainder Theorem. Additionally, measures to avoid the impact of this cryptanalysis method on polynomial congruence-like cryptosystems are provided.

RSA exponent reduction and AES S-box inversion share a hidden commonality: both are governed by the same impartial combinatorial principle, which we call a Product-Congruence Game (PCG). A Product-Congruence Game tracks play via the modular or finite-field product of heap values, providing a single invariant that unifies the algebraic cores of these two ubiquitous symmetric and asymmetric cryptosystems. We instantiate this framework with two companion games. First, $\phi$-MuM, in which a left-associated"multi-secret"RSA exponent chain compresses into the game of Multiplicative Modular Nim, PCG($k,\{1\}$), where $k = ord_N(g)$. The losing predicate then factorizes via the Chinese remainder theorem, mirroring RSA's structure. Second, poly-MuM, our model for finite-field inversion such as the AES S-box. For poly-MuM we prove the single-hole property inside its threshold region, implying that the Sprague-Grundy values are multiplicative under disjunctive sums in that region. Beyond these instances, we establish four structural theorems for a general Product-Congruence Game PCG($m,R$): (i) single-heap repair above the modulus, (ii) ultimate period $m$ per coordinate, (iii) exact and asymptotic losing densities, and (iv) confinement of optimal play to a finite indeterminacy region. An operation-alignment collapse principle explains why some variants degenerate to a single aggregate while MuM, $\phi$-MuM and poly-MuM retain rich local structure. All ingredients (multiplicative orders, the Chinese remainder theorem, finite fields) are classical; the contribution is the unified aggregation-compression viewpoint that embeds both RSA and AES inside one impartial-game framework, together with the structural and collapse theorems.

The Chinese Remainder Theorem for the integers says that every system of congruence equations is solvable as long as the system satisfies an obvious necessary condition. This statement can be generalized in a natural way to arbitrary algebraic structures using the language of Universal Algebra. In this context, an algebra is a structure of a first-order language with no relation symbols, and a congruence on an algebra is an equivalence relation on its base set compatible with its fundamental operations. A tuple of congruences of an algebra is called a Chinese Remainder tuple if every system involving them is solvable. In this article we study the complexity of deciding whether a tuple of congruences of a finite algebra is a Chinese Remainder tuple. This problem, which we denote CRT, is easily seen to lie in coNP. We prove that it is actually coNP-complete and also show that it is tractable when restricted to several well-known classes of algebras, such as vector spaces and distributive lattices. The polynomial algorithms we exhibit are made possible by purely algebraic characterizations of Chinese Remainder tuples for algebras in these classes, which constitute interesting results in their own right. Among these, an elegant characterization of Chinese Remainder tuples of finite distributive lattices stands out. Finally, we address the restriction of CRT to an arbitrary equational class $\mathcal{V}$ generated by a two-element algebra. Here we establish an (almost) dichotomy by showing that, unless $\mathcal{V}$ is the class of semilattices, the problem is either coNP-complete or tractable.

Certain famous combinatorial sequences, such as the Catalan numbers and the Motzkin numbers, when taken modulo a prime power, can be computed by finite automata. Many theorems about such sequences can therefore be proved using Walnut, which is an implementation of a decision procedure for proving various properties of automatic sequences. In this paper we explore some results (old and new) that can be proved using this method.

This paper presents a comprehensive study on modular congruences and the Chinese Remainder Theorem (CRT), considering its historical importance and practical applications in solving mathematical and computational problems. The paper aims to investigate and demonstrate how modular congruence, based on Number Theory, can serve as an effective tool to support problem solving, presenting its computational implementation and applications in everyday situations. To this end, a pedagogical intervention on "Modular Congruence in High School" is carried out in a state school in Fortaleza-CE, using qualitative methodology and action research, in which the exposition has pedagogical and legal support. Thus, the analysis is carefully based on the National Common Curricular Base (BNCC, 2017), in addition to the development of computational implementations of the CRT in C language. Thus, it is observed that the theorem demonstrated versatility in several applications, from RSA encryption to barcode verification systems, with efficient computational implementation of complexity O(n log m). The research also revealed significant pedagogical benefits in contextualizing mathematics through practical examples. This allows us to conclude that TCR and its computational implementation constitute valuable tools in both theoretical and practical terms, promoting the development of logical reasoning and offering efficient solutions to contemporary problems in areas such as digital security and data validation.

Raccoon is an additional digital signature scheme currently under evaluation in the NIST Post-Quantum Cryptography (PQC) standardization process. Its security relies on standard lattice assumptions and is specifically designed for application scenarios resistant to side-channel attacks. In Raccoon, the modulus q of the Number Theory Transform (NTT) is a 49-bit non-prime number, rendering existing hardware designs relying on prime modulus reduction algorithms ineffective. This paper presents the first hardware design of a non-prime modular multiplication leveraging the Chinese Remainder Theorem (CRT), decomposes the non-prime modulus q=q1×q2. It includes three sub-modules: a parallel modular multiplication module to maximize the parallelism of DSPs, a prime modular reduction module that integrates the K-RED algorithm and congruence relation properties for the prime moduli q1 and q2, and a restoration module that utilizes the improved Barrett algorithm to obtain the correct result for the non-prime modulus q. Moreover, the hardware implementation of the pipelined NTT concluding the proposed modular multiplication is presented to verify functional correctness and enable benchmarking against state-of-the-art solutions. Evaluated on the Xilinx Artix-7 platform, the proposed NTT architecture utilizes 35678 LUTs, 15946 FFs, 48 DSPs, and 1.5 BRAMs, achieving an operating frequency of 98 MHz with a total latency of 6.59 μs. This represents a 4× speedup over the official software implementation.

Aiming at the problems of privacy leakage risks and third-party dependence existing in crowdsourced localization, this paper proposes a privacy-preserving algorithm named CRT-ADL that integrates the chinese remainder theorem (CRT) and zero-sum noise. This algorithm is developed based on the accurate and distributed localization (ADL) framework and enhances security through a three-stage refinement process. Firstly, a multi-party secret sharing scheme based on CRT is designed. The localization information of the anchor node is encoded as a system of congruence equations under relatively prime moduli. Combined with the dynamic perturbation mechanism of zero-sum noise, it realizes data privatization while eliminating the cumulative effect of noise. Secondly, the encryption process is completed through the distributed interaction among anchor nodes, avoiding the need for the intervention of a third-party server. Finally, a multidimensional theoretical verification framework covering the correctness, privacy, and efficiency of the algorithm is constructed. Simulation experiments show that in a typical indoor scenario, the CRT-ADL algorithm can not only completely maintain the localization accuracy of the original ADL algorithm, but also significantly reduce the communication overheads compared with mainstream privacy-preserving schemes. This study provides a new paradigm of high-precision and low-loss privacy protection for decentralized crowdsourced localization scenarios.

The t,n secret sharing scheme is used to protect the privacy of information by distribution. More specifically, a dealer splits a secret into n shares and distributes them privately to n participants, in such a way that any t or more participants can reconstruct the secret, but no group of fewer than t participants who cooperate can determine it. Many schemes in literature are based on the polynomial interpolation or the Chinese remainder theorem. In this paper, we propose a new solution to the system of congruences different from Chinese remainder theorem and propose a new scheme for t,n secret sharing; its secret reconstruction is based upon Euler’s theorem. Furthermore, our generalized conclusion allows the dealer to refresh the shared secret without changing the original share of the participants.

: We propose an algorithm to solve general linear Diophantine equations and an algorithm to solve linear congruence problems efficiently using LU decomposition, which means unsafety of cryptography systems based on linear congruence equations. Thus, we focus on the generalization of the argument for a specific reduction of the Learning with Error (LWE) problem established in a previous work ([BLP13]) so that LWE can accommodate for more general choices of matrices. More specifically, we relaxed [BLP13]'s constraint on the choice of the identity matrix to general diagonal matrices. Two examples are presented here to show the validity of our results further.

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The rapid growth of instant messaging platforms has exposed critical security vulnerabilities, including data interception, unauthorized access, and privacy breaches. This research develops a secure instant messaging framework using a hybrid cryptographic model that integrates the Affine Cipher for message encryption with RSA enhanced by the Chinese Remainder Theorem (RSA CRT) for secure key exchange. The Affine Cipher operates over ASCII characters 32-126 with a modulus of 95. Experimental validation demonstrates that RSA CRT encrypts Affine keys in an average of 0.003076 seconds and decrypts them in 0.0044 seconds. Message encryption time using Affine Cipher increases linearly with message length, while maintaining efficient performance for typical instant messages. The RSA CRT implementation significantly reduces computational overhead compared to standard RSA through the use of smaller exponents and modular operations. Testing showed minimal variation in execution times, with encryption ranging from 0.00187 to 0.0043 seconds and decryption from 0.003 to 0.007 seconds. Based on the running time analysis and the functionality evaluation, the integrated approach successfully demonstrates that the hybrid cryptographic model achieves optimal balance between robust security measures and practical operational efficiency, making it highly suitable for real-time messaging applications where both security and performance are critical requirements.

RSA algorithm is one type of algorithms in public-key cryptographic systems where the private keys are kept secret and the public keys can be disseminated. Beaufort Cipher algorithm is a type of classical symmetric algorithm. Meanwhile, the RSA-CRT is one variant of RSA cryptosystem. The RSA-CRT is not efficient for encrypting large files such as image files, since it uses modular exponentiation, which is slow, and the resulting ciphertext would be very much larger than the message. To avoid this problem, a hybrid cryptosystem scheme using RSA-CRT and Beaufort algorithm is designed to encrypt images. Beaufort Cipher algorithm is used for encryption and decryption of digital images, while RSA-CRT algorithm for decryption and decryption of beaufort cipher keys. The results showed that the size of the image (*.BMP) files and the encrypted image files are the same.

RSA cryptosystem is a widely-used public-key cryptographic algorithm in TLS/SSL and IPSec protocols. Fault-injection attack has a powerful threat on the CRT-based implementation of RSA cryptosystem. In 2016, Y. Choi et al. proposed a new right-to-left square-always exponentiation algorithm and a test-based CRT-RSA exponentiation algorithm to defeat the fault-injection attack. In this paper, we propose a fault-injection attack on Y. Choi et al.'s test-based CRT-RSA exponentiation algorithm. By inducing a permanent fault in the computation process of CRT-RSA cryptosystem, the attacker can obtain a faulty RSA signature and then recover the RSA private key. Furthermore, we give an improved CRT-RSA exponentiation algorithm to fix the security flaw. The security analysis shows that the improved algorithm can resist the fault-injection attack.

Data security is a critical aspect in the digital era, especially in securing text-based messages. This research compares the performance of three asymmetric cryptography algorithms-RSA, RSA-CRT, and Rabin-in terms of the speed and efficiency of the encryption and decryption process. The RSA (Rivest-Shamir Adleman) algorithm, one of the most popular cryptographic methods, uses two different keys (public and private) to encrypt and decrypt data, but has a weakness in decryption time efficiency on large data. RSA-CRT, as a modification of RSA with the Chinese Remainder Theorem approach, speeds up the decryption process by dividing large operations into smaller ones. Rabin's algorithm, which is based on prime factorization, shows high time efficiency for both encryption and decryption processes, although it produces four possible plaintexts that require additional selection. All three algorithms were tested using text data with varying character sizes, ranging from 100 to 10,000 characters. The test results show that RSA has the fastest encryption time, RSA-CRT excels in decryption, and Rabin offers the best overall efficiency for securing text-based data. This research provides algorithm recommendations according to system needs, taking into account the balance between efficiency and security.

This work presents a novel, black-box software-based countermeasure against physical attacks including power side-channel and fault-injection attacks. The approach uses the concept of random self-reducibility and self-correctness to add randomness and redundancy in the execution for protection. Our approach is at the operation level, is not algorithm-specific, and thus, can be applied for protecting a wide range of algorithms. The countermeasure is empirically evaluated against attacks over operations like modular exponentiation, modular multiplication, polynomial multiplication, and number theoretic transforms. An end-to-end implementation of this countermeasure is demonstrated for RSA-CRT signature algorithm and Kyber Key Generation public key cryptosystems. The countermeasure reduced the power side-channel leakage by two orders of magnitude, to an acceptably secure level in TVLA analysis. For fault injection, the countermeasure reduces the number of faults to 95.4 % in average.

: Over the years, the Base64 cryptographic algorithm has been a crucial component of information security employed in various security protocols and applications including digital signature schemes, random number generation and Message Authentication Codes, to guarantee data integrity and authenticate the origin of data. However, research has identified security vulnerabilities in the algorithm due to its non-availability of key. The research aimed to develop a novel cryptosystem that uses Residue Number System (RNS) to enhance the Base64 (B64) algorithm's performance alongside with the efficiency of the algorithm. The developed cryptosystem employs the approach of a modern encryption algorithm with the adoption of length three moduli set to design an efficient forward conversion for the encryption algorithm and reverse conversion using Chinese Reminder Theorem (CRT) for decryption algorithm. The algorithmic process design was implemented using dart, flutter technology and android studio. The research examines various cryptographic algorithms while considering several evaluation metrics such as encryption time, decryption time, storage overhead and algorithm type. A secured cryptosystem called Residue Number System Base64 Algorithm (RNS-B64) was developed. In terms of encryption and decryption time performance, the result shows that the RNS-B64 cryptosystem has 0.0005 and 0.0002 respectively while the existing cryptosystem has 0.0037 and 0.0029 respectively on sixteen bytes textual data. The findings indicate that this research outperformed the previous work by enhancing the security level and reducing the encryption and decryption time thus increasing computational efficiency of the developed cryptosystem.

The Internet of Things (IoT) introduces new security considerations because connected devices typically have limited computational resources, making traditional cryptographic algorithms inefficient. RSA, a widely used public-key cryptosystem, is particularly resource-intensive due to its large key sizes and heavy arithmetic operations. This study investigates the inefficiency of standard RSA in resource-constrained IoT environments and evaluates optimization strategies aimed at improving performance without compromising security. Enhancements include key size reduction, the use of the Chinese Remainder Theorem (CRT) for decryption, and precomputation techniques. Performance was assessed in a simulated IoT environment, measuring execution time, memory consumption, and energy efficiency. Security considerations, including potential vulnerabilities of CRT-based RSA such as differential fault attacks, were addressed through redundancy and verification mechanisms. Comparative insights with Elliptic Curve Cryptography (ECC) and post-quantum lightweight schemes (e.g., Ascon, Kyber-SLH-DSA) were provided to contextualize the results. The findings show that optimized RSA achieves significant reductions in computational overhead and energy consumption while maintaining correctness, demonstrating its feasibility for low-power IoT devices. Hybrid cryptography approaches, combining RSA key exchange with symmetric AES payload encryption, are recommended for future implementations. These results reinforce that efficient and practical public-key encryption is achievable in constrained IoT systems while preserving strong security.

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Nowadays, the transmission of private statistics through the computer network needs protection. In a broad range of services, protection is equally troubling. In providing data protection against mischievous attacks, cryptographic algorithms play an important role. RS A has a set of rules widely used in renowned Public Key infrastructure implementations. An efficient RSA method with many public keys was implemented in this article. Two exceptional keys for cryptography, also referred to as Public Key cryptography, are used for asymmetrical key cryptography, also referred to as Public Key cryptography. One key is used to encrypt and another one is used to decrypt, where only the alternative equivalent key can be used. No other key, not even the initial key used for encryption, can decrypt the message now. Under this enhanced RSA Encryption Algorithm, several prosecutions have been carried out to make the use of four keys quicker and more efficient than the original encryption and decryption process. This article gives the implementation of the operation of continuous subtraction operation in the place of division operation. By applying this technique, a high computational speed can be achieved and decreased the complexities of mathematical steps.

Abstract The approximate greatest common divisor problem (ACD) and its variants have been used to construct many cryptographic primitives. In particular, the variants of the ACD problem based on Chinese remainder theorem (CRT) are being used in the constructions of a batch fully homomorphic encryption to encrypt multiple messages in one ciphertext. Despite the utility of the CRT-variant scheme, the algorithms that secures its security foundation have not been probed well enough. In this paper, we propose two algorithms and the results of experiments in which the proposed algorithms were used to solve the variant problem. Both algorithms take the same time complexity 2O~(γ(η−ρ)2) $\begin{array}{} \displaystyle 2^{\tilde{O}(\frac{\gamma}{(\eta-\rho)^2})} \end{array}$ up to a polynomial factor to solve the variant problem for the bit size of samples γ, secret primes η, and error bound ρ. Our algorithm gives the first parameter condition related to η and γ size. From the results of the experiments, it has been proved that the proposed algorithms work well both in theoretical and experimental terms.

This research proposes a watermarking method that combines Integer Haar Wavelet Transform (IHWT) and Chinese Remainder Theorem (CRT) algorithms on RGB images that are converted to YCbCr color space. The goal is to get a watermarking method that is robust, imperceptible and secure. Robust because with IHWT done on component Y as a place to insert a watermark done on the frequency domain that is resistant to attack. Whereas CRT is an insertion method in a spatial domain that is relatively more imperceptible to human vision, CRT also more secure because CRT is a cryptographic algorithm. Based on the results of the test and bookkeeping of the imperceptibility quality, the average PSNR value is above 48dB and the average SSIM is above 0.99. Whereas the robustness test proves that by using the YCbCr color space the increase in resistance occurs in Gaussian noise, JPEG compression and JPEG 2000 compression with an average NC value above 0.8.

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One of the most crucial requirements in this digital age is data security. The number of data usage increased drastically now a days, but how far the data is secured is the very big problem, though we have enough cryptographic algorithms for securing real time applications, but the level of the security against modern attacks is not determined. Elliptic Curve based Cryptography (ECC) is the most important cryptographic algorithm for confidentiality and authentication, providing high security level with small length keys when compared to other asymmetric algorithms like RSA, Diffie-Hellman, etc. The real time system usage of ECC is very less due to computational complexity. So, to increase the real time system usage we propose the novel method of combining the ECC with the Chinese Remainder Theorem (CRT), to reduce the larger values to the smaller one, so that the complexity of constructing ECC points can be reduced nearly 40% when compared to the existing ECC based algorithms. Also, its proved that the level of security getting increased and can be used as the fundamental component in real time communication system.

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Cloud has become one of the most demanding services for data storage. On another hand, the security of data is one of the challenging tasks for Cloud Service Provider (CSP). Cryptography is one of the ways for securing the storage data. Cryptography is not a new approach instead of the efficient utilization of cryptographical algorithms is greatly needed. In this work, we proposed a Secure Hidden Layer (SHL) and Application Programming Interface (API) for data encryption. The SHL is consisting of two major modules (i) Key Management Server (KMS) and (ii) Share Holder Server (SHS) which is used for storing and sharing of cryptographic key. For this purpose, we proposed a server-side encryption algorithm, which is based on the asymmetric algorithm (RSA and CRT) for providing end-to-end security of multimedia data. The experimental results of text and video are evidence that the size of file is not much affected after the encryption and effectively stored at Cloud Storage Server (CSS). The parameters like ciphertext size, encryption time and throughput are considered for performance evaluation of the proposed encryption technique.

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Asymmetric cryptographic algorithm (e.g., RSA and Elliptic Curve Cryptography) implementations on Graphics Processing Units (GPUs) have been researched for over a decade. The basic idea of most previous contributions is exploiting the highly parallel GPU architecture and porting the integer-based algorithms from general-purpose CPUs to GPUs, to offer high performance. However, the great potential cryptographic computing power of GPUs, especially by the more powerful floating-point instructions, has not been comprehensively investigated in fact. In this paper, we fully exploit the floating-point computing power of GPUs, by various designs, including the floating-point-based Montgomery multiplication/exponentiation algorithm and Chinese Remainder Theorem (CRT) implementation in GPU. And for practical usage of the proposed algorithm, a new method is performed to convert the input/output between octet strings and floating-point numbers, fully utilizing GPUs and further promoting the overall performance by about 5%. The performance of RSA-2048/3072/4096 decryption on NVIDIA GeForce GTX TITAN reaches 42,211/12,151/5,790 operations per second, respectively, which achieves 13 times the performance of the previous fastest floating-point-based implementation (published in Eurocrypt 2009). The RSA-4096 decryption precedes the existing fastest integer-based result by 23%.

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Message security is an important thing to prevent the interference from third parties. RSA cryptographic algorithm is believed to be a powerful algorithm in securing message. However, RSA computing process takes a long time so it takes several variants of RSA, namely R prime RSA and Multi-factor RSA which can reduce time and computing cost on the encryption and decryption side. In this paper, the author use three prime numbers to improve better security than only use two prime numbers and minimize the value of private key “d” is by using Chinese Reminder Theorm (CRT).

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This research was designed to provide an idea for choosing the best two equations that can be used to finish the RSA decryption process. In general, the four strategies suggested to accelerate this procedure are competitors. Chinese Remainder Theorem (CRT) is among four rivals. The remains are improved algorithms that have been adjusted from CRT. In truth, the primary building block of these algorithms is CRT, but the sub exponent of CRT is substituted with the new value. Assuming the modulus is obtained by multiplying two prime numbers, two modular exponentiations must be performed prior to combining the results. Three factors are chosen to determine the optimal equation: modular multiplications, modular squares, and modular inverses. In general, the proposed method is always the winner since the optimal equation is selected from among four methods. The testing findings show that the proposed technique is consistently 10-30% faster than CRT.

Message security and authenticity is a very important issue which cannot neglect in wireless network. This paper explains how the RSA algorithm can be used to achieve both, over a wireless network. The RSA-CRT technique was premeditated for data decryption and operative illustration of cryptograpghy using the Chinese Remainder Theorem (CRT) for message security which is nearly four times faster.

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A reversible image secret sharing algorithm is proposed based on quantum logistic mapping and Chinese remainder theorem. In this algorithm, the hash value of the original image is generated using the SHA-3 algorithm, and then encrypted using the RSA algorithm to obtain the encrypted hash value and the public key, which are the initial values in the quantum logistic mapping equations. Next, the quantum logistic mapping algorithm will be utilized to generate the chaotic sequence. After scrambling the original image matrix with the chaotic sequence, the pre-encryption part will be completed. Finally, the cover images are binarized and then the pre-encrypted secret image is embedded into them, adding the Chinese remainder theorem. According to the simulation results, the present algorithm improves the key sensitivity of Chinese remainder theorem-based image secret sharing and has high robustness to salt and pepper noise and cut-off attack.

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In this paper, we present the design and implementation of a systolic RSA cryptosystem based on a modified Montgomery's algorithm and the Chinese Remainder Theorem (CRT) technique. The CRT technique improves the throughput rate up to 4 times in the best case. The processing unit of the systolic array has 100% utilization because of the proposed block interleaving technique for multiplication and square operations in the modular exponentiation algorithm. For 512-bit inputs, the number of clock cycles needed for a modular exponentiation is about 0.13M to 0.24M. The critical path delay is 6.13 ns using a 0.6 /spl mu/m CMOS technology. With a 150 MHz clock, we can achieve an encryption/decryption rate of about 328 to 578 Kb/s.

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A recent work of Harn and Fuyou presents the first multilevel (disjunctive) threshold secret sharing scheme based on the Chinese Remainder Theorem. In this work, we first show that the proposed method is not secure and also fails to work with a certain natural setting of the threshold values on compartments. We then propose a secure scheme that works for all threshold settings. In this scheme, we employ a refined version of Asmuth-Bloom secret sharing with a special and generic Asmuth-Bloom sequence called the anchor sequence. Based on this idea, we also propose the first multilevel conjunctive threshold secret sharing scheme based on the Chinese Remainder Theorem. Lastly, we discuss how the proposed schemes can be used for multilevel threshold function sharing by employing it in a threshold RSA cryptosystem as an example.

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We introduce DM-RSA (Dual Modulus RSA), a variant of the RSA cryptosystem that employs two distinct moduli symmetrically to enhance security. By leveraging the Chinese Remainder Theorem (CRT) for decryption, DM-RSA provides increased robustness against side-channel attacks while preserving the efficiency of classical RSA. This approach improves resistance to partial compromise of a modulus and integrates easily into existing infrastructures.

Abstract This research introduces an algorithm for signing and verifying digital signatures on electronic official documents to establish the authenticity of a person’s identification. The proposed method focuses on minimizing processing. The process begins by converting the electronic document into a digital image. The image is converted into a black-and-white format to reduce the quantity of detail. Then, only specific regions containing handwritten data are extracted for processing. In fact, the proposed method effectively chooses and handles only the black pixels in the extracted image parts. Additionally, the verification process utilizes the best equations, including Chinese remainder theorem, to further optimize processing time. The experimental results are separated into two sections. The first phase involves determining the average duration of the entire system using the proposed method, which falls within the range of 33–35 seconds. The second part is the comparison between the proposed method (utilizing data extraction from selected black pixels) and a method processing all pixel data for verification. The result shows that the proposed method demonstrates a substantial reduction in processing time. Furthermore, the proposed method significantly provides reliable identity verification even though only a small portion of the pixel data is extracted.

Cipher security is becoming an important step when transmitting important information through networks. The algorithms of cryptography play major roles in providing security and avoiding hacker attacks. In this work two hybrid cryptosystems have been proposed, that combine a modification of the symmetric cryptosystem Playfair cipher called the modified Playfair cipher and two modifications of the asymmetric cryptosystem RSA called the square of RSA technique and the square RSA with Chinese remainder theorem technique. The proposed hybrid cryptosystems have two layers of encryption and decryption. In the first layer the plaintext is encrypted using modified Playfair to get the cipher text, this cipher text will be encrypted using squared RSA to get the final cipher text. This algorithm achieved higher security to data but suffers from a long computational time. So Chinese remainder theorem has been used in the second hybrid cryptosystem to obtain less encryption and decryption time. The simulation results indicated that using the modified Playfair with the proposed square RSA has improved security. Moreover, using the Chinese remainder theorem achieved less encryption and decryption time in comparison to our first proposed and the standard algorithms.

We present a practical realization of Rivest-Shamir-Adleman (RSA) with a 2048-bit key on MSP430, a widely used microcontroller in wireless sensor network and Internet of things applications, and show that 2048-bit RSA is feasible on a constrained microcontroller. We exploit several methods for acceleration, e.g. Montgomery modular multiplication, subtractive Karatsuba-Ofman and Chinese remainder theorem (CRT) based modular exponentiation, and achieve RSA encryption and decryption with a 2048-bit key on MSP430 in just 0.14 s and 7.56 s, respectively. Our implementation on the low-end MSP430 microcontroller achieves 2048-bit RSA significantly faster (<inline-formula> <tex-math notation="LaTeX">$\times 2.9$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\times 2.4$ </tex-math></inline-formula> for encryption and decryption) with respect to the existing implementation in the literature on the comparable ATmega128 microcontroller. While our implementation is secure against the brute force attack due to its 2048-bit key, and thus 112-bit security level, it also includes the necessary side-channel countermeasures, e. g. message and key blinding, to help mitigate implementation attacks such as simple power analysis and differential power analysis.

: Encryption involves every aspect of working with and learning about codes. Over the last 40 years, it has grown in prominence to become a prominent scholarly discipline. Because most interactions now take place online, people require secure means of transmitting sensitive information. Several modern cryptosystems rely on public keys as a crucial component of their architecture. The major purpose of this research is to improve the speed and security of the RSA algorithm. By employing Linear Congruential Generator (LCG) random standards for randomly generating a list of large primes; and by employing other selected algorithms, such as the Chinese Remainder Theorem (CRT) in decryption, exponent selection conditions, the Fast Exponentiation Algorithm for encryption, and finally, a comparison of the enhanced RSA versus the normal RSA algorithm that shows an improvement will be provided.

This research introduces a novel RSA encryption methodology tailored to accelerate encryption and decryption processes in peer-to-peer networks. Leveraging the RSA algorithm, the study addresses the critical need for robust data security in dynamic P2P environments. By employing both public and asymmetric key cryptographic techniques, the system ensures secure data communication. The public key, disseminated openly, enables data encryption, while the private key remains concealed for decryption. This approach safeguards data integrity across diverse digital communication channels. Employing advancements in the RSA Encryption Algorithm, the study achieves a notable enhancement in speed and efficiency. Empirical evidence demonstrates encryption and decryption within 42 milliseconds, fortifying data security through the incorporation of the Chinese Remainder Theorem. The integration of four public key pairs within the RSA approach significantly bolsters data security while streamlining encryption and decryption operations. This research makes a substantial contribution to secure data transmission in peer-to-peer networks and broader digital communication contexts.

Industrial Internet of Things (IIoT) has strict requirements on the performance and security of devices. Public-key cryptography, as a kind of computing resource-consuming algorithm, is widely used in the digital signature, key exchange, and so on. The embedded graphics processing units (GPUs) are now rapidly achieving extraordinary computing power, such as NVIDIA Tegra K1/X1/X2/Xavier, which are also treated as edge computing devices. They are widely used in IIoT environments, such as intelligent manufacturing, smart cities, and vehicle-mounted systems. The performance advantages endow embedded GPUs with the possibility of accelerating cryptography that also requires high-density computing. This article implements an efficient Tegra-based embedded GPU RSA acceleration server-oriented IIoT, named TEGRAS. Various optimization methods are employed to promote efficiency, including multithreaded Montgomery multiplication and Chinese Remainder Theorem implementation on the resource-constricted embedded GPUs. With about 40–50 W of power consumption, TEGRAS can deliver 34 kops/s of RSA2048 signature generation and 1007 kops/s of RSA signature verification, which outperforms implementations in the desktop GPUs and embedded CPUs in the perspective of performance-to-power ratio. To evaluate TEGRAS in real-world scenarios, we additionally build a network stack to deliver digital signature services, which can provide more than 34 and 978 kops of signature generation and signature verification, respectively. In a word, based on the embedded GPU, we provide a high-throughput, low-latency, and ready-to-use RSA accelerator-oriented IIoT.

With the rise in Internet and Network applications in the last decade, there is a need for making communication over the internet to be more secure and strong in order to avoid cyber-attacks. The use of digital signatures has been widely used and there is a need to strengthen the existing public key algorithms. RSA is among the most common public key algorithms but is prone to security risks and attacks due to the advancing computing technology. This paper suggests a new algorithm based on RSA which increases the randomness and diffusion of the RSA algorithm making it more secure from cyber-attacks than RSA. The suggested approach employs four prime integers to generate two sets of public and private keys and involves double encryption and decryption. This modified form of the RSA method adds to the complexity of the encryption process while simultaneously decreasing the decryption time by leveraging the Chinese Remainder theorem.

Rivest–Shamir–Adleman (RSA) is one of the widely deployed public-key algorithms. Yet, its decryption facet is very time consuming for resource-constrained Internet-of-Thing (IoT) devices, as it is based on the modular exponentiation of a large number. Although several variants of RSA have been designed to accelerate decryption, the outcomes have been far from satisfactory. Therefore, it is of imminent importance to investigate how to securely outsource RSA decryption to computational powerful parties as an alternative solution. In this article, we introduce the first efficient and secure outsourcing scheme for RSA decryption in IoT. Though RSA decryption is achieved via modular exponentiation, existing secure outsourcing schemes for modular exponentiation either assume the modulus to be prime and are not applicable to RSA or incur massive computation costs and are heavy laden in practice. To address these issues, we have designed our scheme based on the Chinese remainder theorem (CRT). In our scheme, the private keys (including the exponent and the modulus) and the plaintext are concealed concurrently, and the proposed scheme is highly efficient for both client and cloud. In addition, our scheme enables the client to detect any misbehavior of the cloud server with a probability of 99.17%. To validate the effectiveness of our proposed scheme, we provide rigorous proofs of security and verifiability, as well as efficiency analysis. The effectiveness and efficiency of our scheme are further confirmed based on experimental results.

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Internet-of-Things (IoT) devices have grown in popularity over the past few years. The RSA public-key cryptographic primitive is time consuming for resource-constrained IoT. Recently, Zhang et al. proposed a two-party outsourcing protocol between a client and a server for RSA decryption in IoT. It relies on the Chinese remainder theorem as proposed by Quisquater and Couvreur in 1982 and is very efficient. We show that their protocol does not achieve the claimed security guarantees: 1) the (secret) decryption exponent, the plaintext, and the factorization of the RSA modulus are revealed to a passive adversary and 2) a malicious server can make the client accept an (invalid) value of its choice as the result of the delegated computation.

As the most widely applied public-key cryptographic algorithm, RSA is now integrated into many low-cost devices such as IoT devices. Due to the limited resource, most low-cost devices only ship a 2048-bit multiplier, making the longest supported private key length as 2048 bits. Unfortunately, 2048-bit RSA keys are gradually considered insecure. Utilizing the existing 2048-bit multiplier is challenging because a 4096-bit message cannot be stored in the multiplier. In this paper, we perform a thorough study of RSA and propose a new method that achieves the 4096-bit RSA cryptography with the existing hardware. We use the Montgomery modular multiplication and the Chinese Remainder Theorem to reduce the computational cost and construct the necessary components to compute the RSA private key operation. To further validate the correctness of the method and evaluate its performance, we implement this method on a micro-controller and build a testbed named CanoKey with three commonly used cryptography protocols. The result shows that our method is over 200x faster than the naive method, a.k.a., software-based big number multiplications.

In this paper, we propose two new attack algorithms on RSA implementations with CRT (Chinese remainder theorem). To improve the attack efficiency considerably, a clustering collision power attack on RSA with CRT is introduced via chosen-message pairs. This attack method is that the key parameters dp and dq are segmented by byte, and the modular multiplication collisions are identified by k-means clustering. The exponents dp and dq were recovered by 12 power traces of six groups of the specific message pairs, and the exponent d was obtained. We also propose a second order clustering collision power analysis attack against RSA implementation with CRT, which applies double blinding exponentiation. To reduce noise and artificial participation, we analyze the power points of interest by preprocessing and k-means clustering with horizontal correlation collisions. Thus, we recovered approximately 91% of the secret exponents manipulated with a single power curve on RSA-CRT with countermeasures of double blinding methods.

Chinese remainder theorem (CRT) is widely applied in cryptography, coding theory, and signal processing. It has been extended to the multidimensional CRT (MD-CRT), which reconstructs an integer vector from its vector remainders modulo multiple integer matrices. This paper investigates a generalized MD-CRT for multiple integer vectors, where the goal is to determine multiple integer vectors from multiple vector residue sets modulo multiple integer matrices. Comparing to the existing generalized CRT for multiple scalar integers, the challenge is that the moduli in MD-CRT are matrices that do not commute and the corresponding uniquely determinable range is multidimensional and the inclusion relationship is much more complicated. In this paper, we address two fundamental questions regarding the generalized MD-CRT. The first question concerns the uniquely determinable range of multiple integer vectors when no prior information about them is available. The second question is about the conditions under which the maximal possible dynamic range can be achieved. To answer these two questions, we first derive a uniquely determinable range without prior information and accordingly propose an algorithm to achieve it. A special case involving only two integer vectors is investigated for the second question, leading to a new condition for achieving the maximal possible dynamic range. Interestingly, this newly obtained condition, when the dimension is reduced to 1, is even better than the existing ones for the conventional generalized CRT for scalar integers. These results may have applications for frequency detection in multidimensional signal processing.

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In edge-fog computing networks, it is very crucial to ensure that group keys are secure and reliable. In recent years, there has been much interest in group key updating schemes based on Chinese remainder theorem (CRT). It is mostly necessary to utilize the extended Euclidean algorithm to compute the inverse element in conventional schemes. But it does not always exist an inverse element. It leads to the computation being complicated and infeasible. Thus, we propose a scheme for updating group keys and batch verification based on Euler function and Chinese remainder theorem (EF-CRT). Group keys can be constructed directly using Euler function without requiring inverse elements. In addition, we design a batch verification scheme using the updated group key. The proposed EF-CRT scheme accomplishes dynamic group key updates, message integrity authentication, forward and backward security, and conditional privacy protection. Simultaneously, we prove that the security of EF-CRT is existentially unforgeable under the chosen message attack. In experiments with the java pairing-based cryptography library, the proposed scheme shows efficiency both in computational and communication overhead.

— A new method of lossless Secure Data Aggregation for Wireless Sensor Network is presented. Secure Data Aggregation is achieved using the popular Chinese Remainder theorem. Here, an ‘Augmented Chinese Remainder System’ is introduced that incorporates additional features to enforce a higher level of security to the aggregated data. The scheme provides inbuilt signature verification and eliminates the need for separate data validation algorithms. The method achieves data integrity and authentication simultaneously in addition to lossless data aggregation for the data forwarded from the Cluster Head to the Base Station. The aggregate contains the entire individual data from sensors in the encrypted form and the receiver de-aggregates it to get the original data in full without any loss. The Augmented Chinese Remainder System can be extended to secure Multi-level Data Aggregation for WSN.

Extended secret image sharing (SIS) has meaningful shadow, the meaningful shadow will decrease encryption suspiciousness when we transfer secret image via public channels. Chinese Remainder Theorem (CRT) based SIS (CRTSIS) has the advantages of lossless recovery and low recovery computation complexity. Quick Response Code (QR) has become more and more common in our daily life. In this paper, based on CRT and QR, we propose a SIS algorithm for (k, n) threshold with meaningful share and lossless recovery. We choose QR code as the carrier to output valid QR code, which can be scanned to decode. When we collect any k or more shares out of total n shares, the secret image will be recovered losslessly by CRT. We carry out three experiments to illustrate the feasibility and effectiveness of our algorithm.

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This paper proposes a number of theorems and algorithms for the Chinese Remainder Theorem, which is used to solve a system of linear congruences, and the extended Rabin cryptosystem, which accepts a key composed of an arbitrary finite number of distinct primes. This paper further proposes methods to relax the condition on the primes with trade-offs in the time complexity. The proposed algorithms can be used to provide ciphertext indistinguishability. Finally, this paper conducts extensive experimental analysis on six large data sets. The experimental results show that the proposed algorithms are asymptotically tight to the existing decryption algorithm in the Rabin cryptosystem with the key composed of two distinct primes while maintaining increased generality.

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A polynomial remainder code, derived from the Chinese remainder theorem, is a class of linear block codes, where the Reed–Solomon (RS) code is a special case. In this letter, an extended version of polynomial remainder codes is introduced, where the class of doubly extended RS codes is a special case. Furthermore, the extended version of Chinese remainder codes is also presented. The erasure decoding methods for both the codes are proposed. Finally, an application of the extended polynomial remainder codes is discussed.

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In this work we extend the Construction $\pi_{A}$ lattices proposed in [1], to Hurwitz quaternion integers. This construction is provided by using an isomorphism from a version of the Chinese remainder theorem applied to maximal orders in contrast to natural orders in prior works. Exploiting this map, we analyze the performance of the resulting multilevel lattice codes and show via computer simulations their notably reduced computational complexity provided by the multistage decoding.

Multi-key full homomorphic encryption (MKFHE) can perform arbitrary operations on encrypted data under different public keys (users), and the final ciphertext can be jointly decrypted by all involved users. Therefore, MKFHE has natural advantages and application value in security multi-party computation (MPC). The MKFHE scheme based on Brakerski-Gentry-Vaikuntanathan (BGV) inherits the advantages of BGV FHE scheme in aspects of encrypting a ring element, the ciphertext/plaintext ratio, and supporting the Chinese remainder theorem (CRT)-based ciphertexts packing technique. However some weaknesses also exist such as large ciphertexts and keys, and complicated process of generating evaluation keys. In this paper, we present an efficient BGV-type MKFHE scheme. First, we construct a nested ciphertext extension for BGV and separable ciphertext extension for Gentry-Sahai-Waters (GSW), which can reduce the size of the extended ciphertexts about a half. Second, we apply the hybrid homomorphic multiplication between RBGV ciphertext and RGSW ciphertext to the generation process of evaluation keys, which can significantly reduce the amount of input/output ciphertexts and improve the efficiency. Finally, we construct a directed decryption protocol which allows the evaluated ciphertext to be decrypted by any target user, thereby enhancing the ability of data owner to control their own plaintext, and abolish the limitation in current MKFHE schemes that the evaluated ciphertext can only be decrypted by users involved in homomorphic evaluation.

Multi-baseline (MB) phase unwrapping (PU) is a key step of MB synthetic aperture radar (SAR) interferometry (InSAR). Compared with the traditional single-baseline (SB) PU, MB PU is applicable to the area where topography varies violently without obeying the phase continuity assumption. A two-stage programming MB PU approach (TSPA) proposed by H. Yu. builds the link between SB and MB PUs, so many existing classical SB PU methods can be transplanted into the MB domain. In this paper, an extended PU max-flow/min-cut (PUMA) algorithm for MB InSAR using the TSPA, referred to as TSPA-PUMA, is proposed, consisting of a two-stage programming procedure. In stage 1, phase gradients are estimated based on Chinese remainder theorem (CRT). In stage 2, a Markov random field (MRF) model of PUMA is designed for modeling local contextual dependence based on the phase gradients obtained by stage 1. Subsequently, the energy of the MRF model is minimized by graph cuts techniques. The experiment results illustrate that the TSPA-PUMA method can drastically enhance the accuracy of the original PUMA method in the rugged area, and is more efficient than the original TSPA method. In addition, the noise robustness of TSPA-PUMA can be improved through adding more interferograms with different baseline lengths.

We revisit the finite Abelian hidden subgroup problem (AHSP) from a mathematical perspective and make the following contributions. First, by employing amplitude amplification, we present an exact quantum algorithm for the finite AHSP, our algorithm is more concise than the previous exact algorithm and applies to any finite Abelian group. Second, utilizing the Chinese Remainder Theorem, we propose a distributed exact quantum algorithm for finite AHSP, which requires fewer qudits, lower quantum query complexity, and no quantum communication. We further show that our distributed approach can be extended to certain classes of non-Abelian groups. Finally, we develop a parallel exact classical algorithm for finite AHSP with reduced query complexity; even without parallel execution, the total number of queries across all nodes does not exceed that of the original centralized algorithm under mild conditions.

Carrier phase-based techniques offer wavelength-level precision in range estimation, making them highly attractive for Integrated Sensing and Communication (ISAC). However, their effectiveness is constrained by the system’s maximum unambiguous velocity, beyond which integer phase ambiguities cause significant range errors. To overcome this challenge, we propose a high-precision range estimation method. First, we mathematically model the velocity ambiguity problem, analyze the solution structure based on the Extended Chinese Remainder Theorem (ECRT), and design an ambiguity resolution algorithm that leverages multiple subcarrier phase measurements to estimate the integer number of phase cycles, thereby extending the unambiguous velocity range. Second, a robust path extraction strategy is introduced by combining the Channel Impulse Response (CIR) and the Extended Cancellation Algorithm (ECA), enabling accurate isolation of the target’s reflection path under multipath conditions. Simulation results demonstrate that the proposed method maintains millimeter-level single-step error and decimeter-level cumulative error even when the target velocity exceeds the maximum unambiguous velocity. Experimental validation on an indoor mmWave ISAC platform shows that the single-step estimation error remains below 5 mm, and the cumulative error along a one-way trajectory stays under 0.3 m, validating the robustness and practicality.

Privacy-preserving multidimensional data aggregation (MDA) aggregates the data of all different users into a single value, preventing the leakage of personal data while ensuring its availability. However, most current MDA schemes only consider sum operations, i.e., the message of each dimension in the aggregation result is the sum of the corresponding dimensional messages of all individual message vectors. We propose an MDA scheme with fine-grained linear homomorphism, called MDA-FLH. First, we construct a fine-grained linear homomorphic encryption scheme which can assign different weights to each dimension of user’s data and maintain the linear homomorphic property in each dimension. We combine the Chinese remainder theorem (CRT) and Paillier encryption to encode the multidimensional data with CRT and assign different weights to each dimension of user’s data in the Paillier ciphertext. Second, our scheme has the property of fault tolerance. In conjunction with the extended Shamir’s threshold secret-sharing scheme, a security-enhanced and fault-tolerant data aggregation method has been designed so that it is resistant to internal attacks such as the control center (CC) access to individual private data if given the corresponding ciphertext. Finally, two practical schemes are designed based on MDA-FLH: 1) fine-grained electricity price statistics scheme and 2) multistep electricity price statistics scheme. Security analysis shows that our scheme can achieve privacy, confidentiality, integrity, and source authentication. Performance analysis shows that our scheme is efficient, especially that smart meters (SMs) are computationally economic, which makes our scheme more suitable for resource-constrained SM. Also, our scheme can provide linear homomorphism operations on each dimension, which further expands its applications.

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We present a 3D array construction with application to video watermarking. This new construction uses two main ingredients: an extended rational cycle (ERC) as a shift sequence and a Legendre array as a base. This produces a family of 3D arrays with good auto and cross-correlation. We calculate exactly the values of the auto correlation and the cross-correlation function and their frequency. We present a unified method of obtaining multivariate recursion polynomials and their footprints for all finite multidimensional arrays. Also, we describe new results for arbitrary arrays and enunciate a result for arrays constructed using the method of composition. We also show that the size of the footprint is invariant under dimensional transformations based on the Chinese Remainder Theorem.

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Smart grid is a combination of traditional power system engineering and information and communication technology. Smart grid provides users with convenient services through real-time data updates. Multi-dimensional data aggregation can be more flexible for statistical analysis of electricity information. However, most of the existing multi-dimensional data aggregation schemes require the participation of a trusted third party and do not support fault tolerance. In this paper, we propose a fault-tolerant data aggregation scheme supporting fine-grained linear operations in smart grid. Firstly, we used the Chinese remainder theorem to encode the user’s multi-dimensional data and the corresponding weights. Secondly, we construct a privacy-preserving data aggregation scheme without a trusted third party, by combining paillier homomorphic encryption scheme and a secure key agreement protocol. Finally, we use the extended Shamir secret sharing scheme to construct a fault-tolerant data aggregation scheme that supports the reuse of shared key shares. Security analysis results show that our scheme satisfies semantic security and user data privacy protection. Experimental results show that compared with the existing multidimensional data aggregation schemes that require a trusted third party, our scheme does not increase additional computation and communication overhead.

Secure online consultations can provide convenient medical services to patients who require experts from different regions. Moreover, this process can save time, which is critical in emergency cases, and cut medical costs. However, medical services need a high level of privacy protection that advances the difficulty of a construction method. It is a good idea to construct a virtual private chain through public networks by means of cryptology and identity verification. For this purpose, novel protocols are proposed to finish the package layout, secure transmission, and authorization. By mining the special characteristics of this application, two different kinds of encryption channels were designed to support the proposed protocol to ensure the secure transmission of data. And Hash values and multiple checking were employed in the transmission package to find the incompleteness of data related to network errors or attacks. Besides the secure communication of medical information, the Extended Chinese Remainder Theorem was utilized to finish the approval during a change in committee in emergency situations. Finally, example case was used to verify the effectiveness of the total methods.

Ranked keyword search has gained Ranked keyword search has gained traction due to its attractive properties such as flexibility and accessibility. However, most existing ranked keyword search schemes ignore the semantic associations between the documents and queries. To solve this challenging issue in cloud-assisted edge computing, we first design the Semantic-aware Ranked Multi-keyword Search (SRMS) scheme by adopting the Latent Dirichlet Allocation (LDA) topic model and the Chinese Remainder Theorem (CRT)-based secret sharing mechanism. Considering that the cloud server may be malicious, we implement a basic verification mechanism in SRMS to verify the correctness and completeness of search results and extend this verification mechanism in cloud-assisted edge computing scenarios. Formal security analysis proves that SRMS and extended result verification mechanisms are secure in both the known ciphertext model and the known background model. Extensive experiments using the real-world dataset demonstrate that SRMS is efficient and practical.

Summary In this article we formalize some number theoretical algorithms, Euclidean Algorithm and Extended Euclidean Algorithm [9]. Besides the a gcd b, Extended Euclidean Algorithm can calculate a pair of two integers (x, y) that holds ax + by = a gcd b. In addition, we formalize an algorithm that can compute a solution of the Chinese remainder theorem by using Extended Euclidean Algorithm. Our aim is to support the implementation of number theoretic tools. Our formalization of those algorithms is based on the source code of the NZMATH, a number theory oriented calculation system developed by Tokyo Metropolitan University [8].

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Confidential data security is associated with the cryptographic primitives, asymmetric encryption, elliptic curve cryptography, homomorphic encryption, cryptographic pseudorandom sequence generators based on an elliptic curve, etc. For their efficient implementation is often used Residue Number System that allows executing additions and multiplications on parallel computing channels without bit carrying between channels. A critical operation in Residue Number System implementations of asymmetric cryptosystems is base extension. It refers to the computing a residue in the extended moduli without the application of the traditional Chinese Remainder Theorem algorithm. In this work, we propose a new way to perform base extensions using a Neural Network of a final ring. We show that it reduces 11.7% of the computational cost, compared with state-of-the-art approaches.

Modulo inverse is an important arithmetic operation. Many famous algorithms in public key cryptography require to compute modulo inverse. It is argued that the method of DaYan deriving one of Jiushao Qin provides the most concise and transparent way of computing modulo inverse. Based on the rule of taking the least positive remainder in division, this paper presents a more precise algorithmic description of the method of DaYan deriving one to reflect Qin's original idea. Our form of the algorithm is straightforward and different from the ones in the literature. Some additional information can be revealed easily from the process of DaYan deriving one, e.g., the invariance property of the permanent of the state, natural connection to continued fractions. Comparison of Qin'a algorithm and the modern form of the Extended Euclidean algorithm is also given. Since DaYan deriving one is the key technical ingredient of Jiushao Qin's DaYan aggregation method (aka the Chinese Remainder Theorem), we include some explanation to the latter as well.

The Residue Number System is a widely used non-positional number system. Residue Number System can be effectively used in applications and systems with a predominant proportion of addition, subtraction and multiplication operations, due to the parallel execution of operations and the absence of inter-bit carries. The reverse conversion of a number from Residue Number System to positional notation requires the use of special algorithms. The main focus of this article lies in introducing the new conversion method, which incorporates Chinese Remainder Theorem, Akushsky Core Function and rank of number. The step-by-step procedure of the conversion process is detailed, accompanied by numerical examples. The proof of the relationship between the ranks of positional characteristics using the Chinese Remainder Theorem is presented. Through careful analysis and comparison with existing transformation methods, it is concluded that the presented approach takes on average 8 % less time than the Approximate Method.

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The new focus of cryptographic research is on encryption schemes that can withstand cyber-attacks, with the arrival of cloud computing. The widely used public key encryption system designed by Taher El Gamal based on the discrete logarithm problem has been used in many sectors such as internet security, E-voting systems, and other applications for a long time. However, considering the potential data security threats in cloud computing, cryptologists are developing new and more robust cryptographic algorithms. To this end, a new robust homomorphic encryption scheme based on Paillier, Residue Number system (RNS), and El Gamal (PRE), is proposed in this paper., which is expected to be highly effective and resistant to cyber-attacks. The proposed scheme is composed a three-layer encryption and a three-layer decryption processes thereby, making it robust. It employs an existing RNS moduli set {2n + 1, 2n, 2n − 1, 2n−1} − 1}, having passed it through the Paillier encryption process for forward conversion and then the El Gamal cryptosystem to encrpyt any data. The decryption process is a reversal of these processes starting from the El Gamal through a reverse conversion with the same moduli set using the Chinese Remainder Theorem (CRT). The simulation results shows that the proposed scheme outperforms similar existing schemes in terms of robustness and therefore, making it more secured which however, trades off with the time of execution in similar comparison.

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The Residue Number System (RNS) offers significant advantages in parallel, carry-free arithmetic for highperformance computing but remains critically vulnerable to errors during transmission and computation. Traditional error detection approaches rely on post-processing reverse conversion, which introduces substantial latency and undermines the inherent speed of RNS, making them unsuitable for real-time, reliability-critical applications. This paper proposes a novel architecture for in-situ error detection and correction that operates directly within the residue domain, eliminating the need for costly full reverse conversion. Using an optimized moduli set {2n+1 − 1, 2n + 1, 2n, 2n − 1, 2n−1 − 1}, we develop a hybrid algorithm that combines the Modulus Computation Method (MCM) for rapid reverse estimation with Hamming distance-based majority voting for robust syndrome analysis. The method guarantees single-residue error detection and correction by systematically evaluating residue triples to identify consistent values. Experimental simulations demonstrate a 99.9% correction success rate for realistic fault probabilities (p ≤ 10−4) while maintaining a low-latency, hardwareefficient pipeline. Comparative analysis against state-of-the-art techniques confirms superior performance in area utilization (complexity 5n + 1), detection latency (3 cycles), and flexibility across generalized moduli sets. These results advance the design of fault-tolerant RNS architectures for applications such as cryptography and digital signal processing.

In the work methods for building systems of distributed data storage based on the system of residual classes are considered. The use of direct conversion of data from the positional system of calculation to the residue number system will have a large computational complexity, the use of modules of a particular type allows you to solve this problem. The operation of scaling and expansion of the base system, which is necessary to restore the number of stored parts in case of failure of one or more cloud servers, is considered.

The paper presents algorithms for the generation of Residue Number System (RNS) triples with <inline-formula> <tex-math notation="LaTeX">$SQ=2^{k}-1$ </tex-math></inline-formula> and quadruples with <inline-formula> <tex-math notation="LaTeX">$SQ=2^{k}$ </tex-math></inline-formula> for some k. Triples and quadruples allow us to design efficient hardware implementations of non-modular operations in RNS such as division, sign detection, comparison of numbers, reverse conversion with using of a diagonal function from requiring division with the remainder by the diagonal module SQ. Division with a remainder in the general case is the most complex arithmetic operation in computer technology. However, the consideration of special cases can significantly simplify this operation and increase the efficiency of hardware implementation. We show that there are 5573 good RNS triples (2301 even and 2372 odd) with elements less than 10 000, as the values of SQ vary from <inline-formula> <tex-math notation="LaTeX">$2^{5}-1$ </tex-math></inline-formula> to <inline-formula> <tex-math notation="LaTeX">$2^{27}-1$ </tex-math></inline-formula>. In contrast, RNS quadruples with <inline-formula> <tex-math notation="LaTeX">$SQ=2^{k}$ </tex-math></inline-formula> seem to be quite rare. Restricting our search to sums of the elements in a quadruple less than 4000 we find that exactly 31 such quadruples exist. Their values of SQ vary between 2²⁰ and 2³⁰ with always even exponent. We suggest the measure of RNS balance and find perfectly balanced RNS among triples according to this measure. We demonstrate the advantages of more balanced quadruples by means of hardware implementation.

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We live in a world where technological advances are continually creating more data than what we can deal with. Machine learning algorithms, in particular Deep Neural Networks (DNNs), are essential to process such large data. Computation of DNNs requires loading the trained network on the processing element and storing the result in memory. Therefore, running these applications need a high memory bandwidth. Traditional cores are memory limited in terms of the memory bandwidth. Hence, running DNNs on traditional cores results in high energy consumption and slows down processing speed due to a large amount of data movement between memory and processing units. Several prior works tried to address data movement issue by enabling Processing In-Memory (PIM)using crossbar analog multiplication. However, these designs suffer from the large overhead of data conversion between analog and digital domains. In this work, we propose RNSnet, which uses Residue Number System (RNS)to execute neural network completely in the digital domain in memory. RNSnet simplifies the fundamental neural network operations and maps them to in-memory addition and data access. We test the efficiency of the proposed design on several popular neural network applications. Our experimental result shows that RNSnet consumes 145.5x less energy and obtains 35.4x speedup as compared to NVIDIA GPU GTX 1080. In addition, our results show that RNSnet can achieve 8.5 x higher energy-delay product as compared to the state-of-the-art neural network accelerators.

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In this paper, we deal with the critical problems in residue arithmetic. The reverse conversion from a Residue Number System (RNS) to positional notation is a main non-modular operation, and it constitutes a basis of other non-modular procedures used to implement various computational algorithms. We present a novel approach to the parallel reverse conversion from the residue code into a weighted number representation in the Mixed-Radix System (MRS). In our proposed method, the calculation of mixed-radix digits reduces to a parallel summation of the small word-length residues in the independent modular channels corresponding to the primary RNS moduli. The computational complexity of the developed method concerning both required modular addition operations and one-input lookup tables is estimated as Ok2/2, where k equals the number of used moduli. The time complexity is Olog2k modular clock cycles. In pipeline mode, the throughput rate of the proposed algorithm is one reverse conversion in one modular clock cycle.