Public Key Cryptography Research Articles

An Improved Public Key Cryptographic Algorithm Based on Chebyshev Polynomials and RSA

Due to its very desirable properties, Chebyshev polynomials are often used in the design of public key cryptographic systems. This paper discretizes the Chebyshev mapping, generalizes the properties of Chebyshev polynomials, and proposes an improved public key encryption algorithm based on Chebyshev chaotic mapping and RSA, i.e., CRPKC−Ki. This algorithm introduces alternative multiplication coefficients Ki, the selection of which is determined by the size of TrTdxmod N=TdTrxmod N, and the specific value selection rules are shared secrets among participants, overcoming the shortcomings of previous schemes. In the key generation and encryption/decryption stages, more complex intermediate processes are used to achieve higher algorithm complexity, making the algorithm more robust against ordinary attacks. The algorithm is also compared with other RSA-based algorithms to demonstrate its effectiveness in terms of performance and security.

Algorithms of the Möbius function by random forests and neural networks

The Möbius function μ(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu (n)$$\\end{document} is known for containing limited information on the prime factorization of n. Its known algorithms, however, are all based on factorization and hence are exponentially slow on logn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\log n$$\\end{document}. Consequently, a faster algorithm of μ(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu (n)$$\\end{document} could potentially lead to a fast algorithm of prime factorization which in turn would throw doubt upon the security of most public-key cryptosystems. This research introduces novel approaches to compute μ(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu (n)$$\\end{document} using random forests and neural networks, harnessing the additive properties of μ(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu (n)$$\\end{document}. The machine learning models are trained on a substantial dataset with 317,284 observations (80%), comprising five feature variables, including values of n within the range of 4×109\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$4\ imes 10^9$$\\end{document}. We implement the Random Forest with Random Inputs (RFRI) and Feedforward Neural Network (FNN) architectures. The RFRI model achieves a predictive accuracy of 0.9493, a recall of 0.5865, and a precision of 0.6626. On the other hand, the FNN model attains a predictive accuracy of 0.7871, a recall of 0.9477, and a precision of 0.2784. These results strongly support the effectiveness and validity of the proposed algorithms.

Entanglement-based quantum digital signatures over a deployed campus network.

The quantum digital signature protocol offers a replacement for most aspects of public-key digital signatures ubiquitous in today's digital world. A major advantage of a quantum-digital-signatures protocol is that it can have information-theoretic security, whereas public-key cryptography cannot. Here we demonstrate and characterize hardware to implement entanglement-based quantum digital signatures over our campus network. Over 25 hours, we collect measurements on our campus network, where we measure sufficiently low quantum bit error rates (<5% in most cases) which in principle enable quantum digital signatures at over 50 km as shown through rigorous simulation accompanied by a noise model developed specifically for our implementation. These results show quantum digital signatures can be successfully employed over deployed fiber. Moreover, our reported method provides great flexibility in the number of users, but with reduced entanglement rate per user. Finally, while the current implementation of our entanglement-based approach has a low signature rate, feasible upgrades would significantly increase the signature rate.

Public-Key Cryptosystems and Signature Schemes from $$p$$-Adic Lattices

Public-Key Cryptosystems and Signature Schemes from $$p$$-Adic Lattices

Deep-Learning-based Cryptanalysis through Topic Modeling

Neural cryptography is a technique that uses neural networks for secure data encryption. Cryptoanalysis, on the other hand, deals with analyzing and decrypting ciphers, codes, and encrypted text without using a real key. Chosen-plaintext cryptanalysis is a subfield of cryptanalysis where both plain text and ciphertext are available and the goal is either to find the encryption technique, the encryption key, or both. This study addresses chosen plaintext cryptanalysis within public key cryptography, to categorize topics of encrypted text. Using a fixed encryption technique and key, the focus was placed on creating a framework that identifies the topic associated with ciphertext, using diverse plaintexts and their corresponding cipher texts. To our knowledge, this is the first time that chosen-plaintext cryptanalysis has been discussed in the context of topic modeling. The paper used deep learning techniques such as CNNs, GRUs, and LSTMs to process sequential data. The proposed framework achieved up to 67% precision, 99% recall, 80% F1-score, and 71% AUPR on a dataset, showcasing promising results and opening avenues for further research in this cryptanalysis subarea.

Enhanced Security in Public Key Cryptography: A Novel Approach Combining Gaussian Graceful Labeling and NTRU Public Key Cryptosystem

This research explores an encryption system that combines the Nth-degree Truncated Polynomial Ring Unit (NTRU) public key cryptosystem with Gaussian Graceful Labeling. This process assigns distinct labels to a graph's vertices, resulting in successive Gaussian integers. The NTRU method offers enhanced security and efficient key exchange. The communication encryption process depends on integers P, a, and b, with P being the largest prime number in the vertex labeling. The original receivers are the vertex labeling with the largest prime number coefficient, while all other receivers receive messages from the sender. A polynomial algebraic mixing system and a clustering principle based on the abecedarian probability proposition are used in NTRU encryption and decryption. The choice of relatively prime integers p and q in NTRU plays a role in the construction of polynomial rings used for encryption and decryption, with specific choices and properties designed to ensure scheme security.

Single-Trace Side-Channel Attacks on NTRU Implementation

Most of the currently used cryptosystems are not secure in the presence of cryptographically relevant quantum computers. As the research in quantum technologies proceeds, a need for quantum-safe cryptography is imminent. NTRU is a post-quantum public-key cryptosystem based on lattices and was a finalist in the 3rd round of the post-quantum standardization process organized by the National Institute of Standards and Technology (NIST). This paper aims to study the implementation security of the cryptosystem with respect to an attacker with access to power leakage. Such a threat model is relevant especially, but not only, for embedded devices. We studied a countermeasure implementation of the NTRU decryption algorithm from An et al. (Appl Sci https://doi.org/10.3390/app8112014, 2018) that claimed its security against power attacks. This paper revisits an attack presented in as reported by Rabas (In: Proceedings of the9th International Conference on Information Systems Security and Privacy,ICISSP 2023, Lisbon, 2023) that shows it is in fact vulnerable even in the case of just a single trace available to the enemy for extracting the key. We then describe a new profiling template attack on the implementation and show experimental results of the attack using the same datasets, resulting in a comparison of these two methods and further confirmation of the vulnerability of the algorithm even to generic profiling attacks. Several possible types of countermeasures are discussed.

A new application to cryptology using generalized Jacobsthal matrices, inner product and self-adjoint operator

In this paper, we first examine the properties of the matrices of the generalized [Formula: see text]-step Jacobsthal sequence, and we develop a public key cryptosystem using key exchange based on the relationship between inner product and orthogonality. Then we create encryption and decryption algorithms using this key exchange, self-adjoint operator, and generalized [Formula: see text]-step Jacobsthal sequence, and we provide a detailed example of how encryption and decryption algorithms work. Finally, a security strength and complexity analysis of these algorithms will be made.

A Noval and Efficient Three-Party Identity Authentication and Key Negotiation Protocol Based on Elliptic Curve Cryptography in VANETs

In the process of vehicles transitioning from conventional means of transportation to mobile computing platforms, ensuring secure communication and data exchange is of paramount importance. Consequently, identity authentication has emerged as a crucial security measure. Specifically, effective authentication is required prior to the communication between the On-Board Unit (OBU) and Roadside Unit (RSU). To address vehicle identity authentication challenges in the Internet of Vehicles (VANETs), this paper proposes a three-party identity authentication and key agreement protocol based on elliptic curve public key cryptography. Considering issues such as vehicle impersonation attacks, RSU impersonation attacks, and vehicle privacy breaches in existing schemes within wireless mobile environments, this protocol introduces a trusted registry center that successfully enables mutual authentication between OBU and RSU. The proposed protocol not only enhances the VANETs system’s ability to withstand security threats but also improves the credibility and efficiency of the authentication process.

Average case error estimates of the strong Lucas test

AbstractReliable probabilistic primality tests are fundamental in public-key cryptography. In adversarial scenarios, a composite with a high probability of passing a specific primality test could be chosen. In such cases, we need worst-case error estimates of the test. However, in many scenarios, the numbers are randomly chosen and thus have a significantly smaller error probability. We are hence interested in average-case error estimates. In this paper we establish such bounds for the strong Lucas primality test, as there exist only worst-case, but no average-case error bounds. This allows us to use this test with more confidence. Let us examine an algorithm that draws odd k-bit integers uniformly and independently, runs t independent iterations of the strong Lucas test with randomly chosen parameters, and outputs the first number that passes all t consecutive rounds. We attain numerical upper bounds on the probability that a composite is returned. Moreover, we examine a slight modification of this algorithm that only considers integers that are not divisible by small primes, yielding improved bounds. In addition, we classify the numbers that contribute most to our estimate.

Resilient Intermediary-Based Key Exchange Protocol for IoT

Due to the limited resources of Internet of Things (IoT) devices, Symmetric Key Cryptography (SKC) is typically favored over resource-intensive Public Key Cryptography (PKC) to secure communication between IoT devices. To utilize SKC, devices need to execute a key exchange protocol to establish a session key before initiating communication. However, existing SKC-based key exchange protocols assume that communication devices have a pre-shared secret or there are trusted intermediaries between them; neither is always realistic in IoT. We introduce a new SKC-based key exchange protocol for IoT devices. While also intermediary based, our protocol fundamentally departs from existing intermediary-based solutions in that intermediaries between two key exchange devices may be malicious, and moreover, our protocol can detect cheating behaviors and identify malicious intermediaries. We prove our protocol is secure under the universally composable model, and show that it can detect malicious intermediaries with probability 1.0. We implemented and evaluated our protocol on different IoT devices. We show that our protocol has significant improvements in computation time and energy cost. Compared to the PKC-based protocols ECDH, DH, and RSA, our protocol is 2.3 to 1,591 times faster on one of the two key exchange devices and 0.7 to 4.67 times faster on the other.

A blind signature scheme for IoV based on 2D-SCML image encryption and lattice cipher

Today’s Internet of Vehicles (IoV) faces many security risks in the data transmission process, and image data is more vulnerable to security threats in the transmission process due to its special characteristics such as large amounts of information and high visibility. Therefore, to guarantee the dependability of data transmission in the IoV environment, this paper designs a blind signature scheme for IoV based on two-dimensional sine cosine cross-chaotic mapping (2D-SCML) image encryption and lattice cipher (BSS-IoV). The innovation of this scheme is that it aims at blind signature of image information, blinds it before sending the information, and combines the lattice public key encryption algorithm to better ensure the safe and reliable transmission of information and reduce the risk of information disclosure. To further ensure the security of the scheme, an image encryption algorithm based on 2D-SCML and pixel splitting (2PS-IEA) is proposed, which is used to blind the information and thus reduce the risk of information leakage on the one hand, on the other hand, it is used in the signature process to ensure the security of the signed information. The 2D-SCML is derived from the cross-model structure proposed in this paper. Through simulation results and experimental analysis, the values of NPCR and UACI, respectively, 99.6094% and 33.4635%, are close to ideal values. And 50% of the cut image can also recover the rough information, which indicates that the signature scheme has the security against differential attacks, cut attacks and noise attacks. Moreover, the security analysis shows that the scheme has the anti-tamper, anti-repudiation and traceability.

Continued Fractions Applied to the One Line Factoring Algorithm for Breaking RSA

The RSA (Rivest–Shamir–Adleman) cryptosystem is an asymmetric public key cryptosystem popular for its use in encryptions and digital signatures. However, the Wiener’s attack on the RSA cryptosystem utilizes continued fractions, which has generated much interest in developing competitive factoring algorithms. A general-purpose integer factorization method first proposed by Lehmer and Powers formed the basis of the well-known Continued Fraction Factorization (CFRAC) method. Recent work on the one line factoring algorithm by Hart and its connection with Lehman’s factoring method have motivated this paper. The emphasis of this paper is to explore the representations of PQ as continued fractions and the suitability of lower ordered convergences as representations of ab. These simpler convergences are then prescribed to Hart’s one line factoring algorithm. As an illustration, we demonstrate the working of our approach with two numbers: one smaller number and another larger number occupying 95 bits. Using our method, the fourth convergence finds the factors as the solution for the smaller number, while the eleventh convergence finds the factors for the larger number. The security of the RSA public key cryptosystem relies on the computational difficulty of factoring large integers. Among the challenges in breaking RSA semi-primes, RSA250, which is an 829-bit semi-prime, continues to hold a research record. In this paper, we apply our method to factorize RSA250 and present the practical implementation of our algorithm. Our approach’s theoretical and experimental findings demonstrate the reduction of the search space and a faster solution to the semi-prime factorization problem, resulting in key contributions and practical implications. We identify further research to extend our approach by exploring limitations and additional considerations such as the difference of squares method, paving the way for further research in this direction.

The Application of Fermat’s Little Theorem in Cryptography

This article aims to demonstrate the application of Fermat’s theorem in cryptography. With the development of technology, we have access to more and more information, and the necessity of encryption has also increased. As a mainstream public key cryptography algorithm, RSA encryption will demonstrate the application of Fermat’s theorem in RSA encryption in this article. To achieve this, I will introduce cryptography and Fermat’s theorem separately. Then, I will explain how the RSA encryption algorithm and Fermat’s theorem are applied to the RSA encryption algorithm. Finally, I will demonstrate it using Python. The theoretical proof and encryption process were elaborated on through empirical Python programming examples. Then, I will discuss whether RSA encryption is safe. Finally, I will give my conclusion.

New public-key cryptosystem algorithm based on function composition of matrix

Experts argue that public key cryptography systems based on commutative groups (abelian groups) are quite vulnerable to quantum algorithms attacks. For this reason, experts have developed a non-commutative group-based public-key cryptosystem. In this study, we propose a public-key cryptosystem using the non-commutative property of the function composition of the matrix. The results of this study indicate that the proposed public-key cryptosystem can work well. Furthermore, the proposed cryptosystem is also safe from various attempted mathematical attacks on the key exchange process at the sender and receiver of the message.

Non-Cascaded and Crosstalk-Free Multi-Image Encryption Based on Optical Scanning Holography Using 2D Orthogonal Compressive Sensing.

We propose a non-cascaded and crosstalk-free multi-image encryption method based on optical scanning holography and 2D orthogonal compressive sensing. This approach enables the simultaneous recording and encryption of multiple plaintext images without mechanical scanning, while allows for independent retrieval of each image with exceptional quality and no crosstalk. Two features would bring about more substantial security and privacy. The one is that, by employing a sequence of pre-designed structural patterns as encryption keys at the pupil, multiple samplings can be achieved and ultimately the holographic cyphertext can be obtained. These patterns are generated using a measurement matrix processed with the generalized orthogonal one. As a result, one can accomplish the differentiation of images prior to the recording and thus neither need to pretreat the pending images nor to suppress the out-of-focus noise in the decrypted image. The other one is that, the non-cascaded architecture ensures that different plaintexts do not share sub-keys. Meanwhile, compared to 1D orthogonal compressive sensing, the 2D counterpart makes the proposed method to synchronously deal with multiple images of more complexity, while acquire significantly high-quality decrypted images and far greater encryption capacity. Further, the regularities of conversion between 1D and 2D orthogonal compressive sensing are identified, which may be instructive when to manufacture a practical multi-image cryptosystem or a single-pixel imaging equipment. A more general method or concept named synthesis pupil encoding is advanced. It may provide an effective way to combine multiple encryption methods together into a non-cascaded one. Our method possesses nonlinearity and it is also promising in multi-image asymmetric or public key cryptosystem as well as multi-user multiplexing.

О генерической сложности проблемы вычисления функции Эйлера

We study the generic complexity of the problem of the Euler function computation. This problem has important applications in modern cryptography. For example, the cryptographic strength of the famous public key encryption system RSA is based on the assumption of its hardness. We prove that under the condition of worst-case hardness and P = BPP there is no polynomial strongly generic algorithm for this problem. For a strongly generic polynomial algorithm, there is no efficient method for random generation of inputs on which the algorithm cannot solve the problem. Thus, this result justifies the application of the problem of computing the Euler function in public key cryptography. To prove this theorem, we use the method of generic amplification, which allows us to construct generically hard problems from the problems that are hard in the classical sense. The main feature of this method is the cloning technique, which combines the input data of a problem into sufficiently large sets of equivalent input data. Equivalence is understood in the sense that the problem is solved in a similar way for them.

E-Governance Security using Public Key Cryptography With special focus on ECC

E-Governance Security using Public Key Cryptography With special focus on ECC

Inverse matrices with applications in public-key cryptography

The applications of non-square binary matrices span many domains including mathematics, error-correction coding, machine learning, data storage, navigation signals, and cryptography. In particular, they are employed in the McEliece and Niederreiter public-key cryptosystems. For the parity check matrix of these cryptosystems, a systematic non-square binary matrix [Formula: see text] with dimensions [Formula: see text], [Formula: see text], [Formula: see text], there exist [Formula: see text] distinct inverse matrices. This article presents an algorithm to generate these matrices as well as a method to construct a random inverse matrix. Then it is extended to non-square matrices in arbitrary fields. This overcomes the limitations of the Moore-Penrose and Gauss-Jordan methods. The application to public-key cryptography is also discussed.

Subsets of groups in public-key cryptography

Subsets of groups in public-key cryptography