Discrete Logarithm Research Articles

A Matrix Multiplication Approach to Quantum-Safe Cryptographic Systems

This paper introduces a novel approach based on matrix multiplication in Fpn×n, which enables methods for public key exchange, user authentication, digital signatures, blockchain integration, and homomorphic encryption. Unlike traditional algorithms that rely on integer factorization or discrete logarithms, our approach utilizes matrix factorization, rendering it resistant to current quantum cryptanalysis techniques. This method enhances confidentiality by ensuring secure communication and facilitating user authentication through public key validation. We have incorporated a method that allows a Certification Authority to certify the public keys. Furthermore, the incorporation of digital signatures ensures nonrepudiation, while the system functions as a blockchain technology to enhance transaction security. A key innovation of this approach is its capability to perform homomorphic encryption. Our approach has practical applications in artificial intelligence, robotics, and image processing.

Collaboration failure analysis in cyber-physical system-of-systems using context fuzzy clustering

A cyber-physical system-of-systems (CPSoS) facilitates the achievement of high-level goals, such as efficient traffic management on roads, by designing and developing the collaboration of constituent CPSs. A platooning that groups autonomous vehicles in proximity is an example of collaboration. The intricate collaboration innately causes serious collaboration failures such as collisions. However, limited knowledge and complex dynamics of CPSoS cause several challenges in effectively analyzing the collaboration failures. Existing studies have applied pattern mining techniques to investigate various failures but have limitations when applied to collaboration failures: (1) absence of data model for continuous and discrete logs in CPSoS; (2) information loss problem by not considering the integrated relationship of the data; (3) dependence only on failed logs; (4) limited capability of fixed-size time windows. We propose a fuzzy clustering-based pattern mining approach that consists of a novel data model for CPSoS logs and comprehensive metrics for classifying and mining optimal collaboration failure patterns. In experiments on vehicle platooning, our approach exhibited the highest accuracy on pattern mining and clustering results. Further, we identified five collaboration failure scenarios in the empirical analysis of drone swarming results. The findings of this study can facilitate the effective analysis of CPSoS collaboration failures.

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The development of quantum computation enables exponential time complexity problems on classical computers to be solved in polynomial time on quantum computers. However, it also poses a threat to the security of classical cryptographic schemes based on integer factorization and discrete logarithms. In response to this challenge, quantum cryptographic schemes based on quantum computation and quantum communication environments have become a focal point of research. The quantum public-key cryptosystem based on the QSCDff problem stands as one of the influential schemes in the realm of quantum public-key cryptography, yet its feasibility remains unexplored in current literature. Our specific focus lies in the quantum circuit implementations and fault-tolerant construction, which serve as essential prerequisites for the physical feasibility of quantum cryptographic schemes. We provide quantum circuit implementations along with rigorous theoretical proofs for the computation of the permutation product operation and the permutation sign operation in quantum public-key cryptographic schemes. Based on the fault-tolerant quantum computation process of the aforementioned quantum circuit implementations, we propose two error-correction strategies and provide a theoretical feasibility analysis within a specified range in the ion-trap quantum computation environment, adhering to the theoretical limits of quantum computation. Rigorous proofs are presented to demonstrate the correctness and reliability of the proposed methods. Our contribution provides a theoretical foundation for the physical feasibility analysis of quantum cryptographic algorithms, offering insights into the challenges and prospects of implementing these algorithms in quantum computation environments.

A NEW DLP-BASED AUTHENTICATION ALGORITHM FOR PUBLIC KEY CRYPTOSYSTEMS

Authentication is essential to secure communication because it guarantees that communications come from authentic sources and are not altered while being transmitted. An efficient authentication algorithm based on the DLP (Discrete Logarithm Problem) is presented in this study. By ensuring the integrity and authenticity of messages, this algorithm enhances current encryption techniques to offer a better security.

Certificateless Public Integrity Checking of Group Shared Data in Cloud Storage

Cloud storage service supplies people with an efficient method to share data within a group. The cloud server is not trustworthy, so lots of remote data possession checking (RDPC) protocols are proposed and thought to be an effective way to ensure the data integrity. However, most of RDPC protocols are based on the mechanism of traditional public key infrastructure (PKI), which has obvious security flaw and bears big burden of certificate management. To avoid this shortcoming, identity-based cryptography (IBC) is often chosen to be the basis of RDPC. Unfortunately, IBC has an inherent drawback of key escrow. To solve these problems, we utilize the technique of certificateless signature to present a new RDPC protocol for checking the integrity of data shared among a group. In our scheme, user's private key includes two parts: a partial key generated by the group manager and a secret value chosen by herself/himself. To ensure the right public keys are chosen during the data integrity checking, the public key of each user is associated with her unique identity, for example the name or telephone number. Thus, the certificate is not needed and the problem of key escrow is eliminated too. Meanwhile, the data integrity can still be audited by public verifier without downloading the whole data. In addition, our scheme also supports efficient user revocation from the group. The security of our scheme is reduced to the assumptions of computational Diffie- Hellman (CDH) and discrete logarithm (DL). Experiment results exhibit that the new protocol is very efficient and feasible

Efficient quantum algorithms to break group ring cryptosystems

The security of widely-used public-key cryptographic protocols like RSA, Diffie–Hellman key exchange and the Digital Signature Algorithm (DSA) is under threat due to the emergence of quantum computers. Shor’s groundbreaking quantum algorithm poses a significant risk by efficiently factoring large integers into their prime factors, compromising RSA security. Additionally, it solves the Discrete Logarithm Problem, impacting certain Diffie–Hellman-based cryptosystems and digital signatures. Given this, it is imperative to enhance our current cryptographic tools for the post-quantum era, aiming to make it impractical, even with quantum algorithms, to breach the security of new cryptosystems. Prominent alternatives include elliptic curve and lattice-based cryptography, with exploration into other algebraic systems featuring difficult problems to ensure security. This paper establishes that systems based on the difficulty of inverting group ring elements are not quantum-resistant.

Advanced Techniques in Post-Quantum Cryptography for Ensuring Data Security in the Quantum Era

The arrival of quantum computers is a major threat to current security methods, which depend on problems like discrete logarithms and integer factorization being hard. Post-quantum cryptography (PQC) is getting to be an critical field for making secure strategies that can't be broken by quantum assaults. This exposition talks almost progressed PQC strategies, centered on the most up to date thoughts and what they cruel for securing information within the quantum age. To begin with, we see at lattice-based cryptography, which appears like a great choice since it has solid security roots and can be utilized in numerous circumstances. This incorporates strategies such as Learning With Mistakes (LWE) and Ring-LWE, which are exceptionally great at ensuring against quantum dangers and are simple to utilize. Another critical zone is code-based cryptography, which employments the trouble of breaking irregular straight codes as an case. The McEliece cryptosystem is one such framework. It is checked to see how valuable and successful these strategies are in real-life circumstances. We see into multivariate polynomial encryption, which builds security on the truth that it's difficult to illuminate sets of multivariate quadratic equations. People are particularly curious about this strategy since it may be utilized to form proficient marking plans. Another imperative strategy is hash-based cryptography, which employments hash capacities to create secure advanced marks. The consider moreover talks around blended cryptography frameworks that utilize both conventional and post-quantum strategies. These frameworks make beyond any doubt that the switch to quantum computing goes easily and give way better security. We see at the issues that come up with implementation, like additional work that has to be done on the computer and joining it with current frameworks, and come up with ways to urge around these issues. At long last, moving to post-quantum security is necessary, but it comes with a part of problems that ought to be illuminated in numerous ways.

New Proxy Signature Scheme over Elliptic Curves using Chaotic Maps Applicable during Pandemic COVID-19

Abstract: Digitalization has attracted the world to collect increasing data. Background: Proxy signature is a digital alternative for signing documents in the absence of the original signer. Methods: In this paper, we have used the mathematical methods and concepts of Chaotic maps (CMs) and elliptic curve cryptography. Results: We have proposed a new proxy signature scheme (PSS). Security of our PSS relies on "elliptic curve discrete logarithm (ECDL) and integer factorization (FAC) problems". It requires only low-complexity computation, which increases efficiency. Conclusion: It is the first PSS in such a security setting and can also be assumed to be secure in the post-quantum cryptographic world. It can be highly used digitally during thePandemic conditions like COVID-19.

Elliptic curve cryptography: Theory, security, and applications in modern network security

Abstract. Due to the swift advancement of Internet and computer technology in the 21st century, the demand for network security is increasing. Classic cryptographic algorithms like Rivest-Shamir-Adleman (RSA) and Digital Signature Algorithm (DSA) are insufficient in the face of modern network environments, while elliptic curve cryptography (ECC) has become a research hotspot due to its high security and high efficiency. The purpose of this paper is to discuss the theoretical basis, security analysis, and practical application cases of elliptic curve cryptography, to provide readers with a comprehensive understanding and trigger further research and thinking. This paper analyzes the theoretical basis of ECC, including group theory, domain theory, and the definition and properties of elliptic curves, and analyzes the application of ECC in combination with practical application cases, such as the SM2 algorithm. The article first introduces the concept of groups, the definition of domains, and the basic properties of elliptic curves. Then, the security of ECC is analyzed, especially the complexity of the Elliptic Curve Discrete Logarithm Problem (ECDLP) and the selection of key length. In addition, the security of ECC in practical applications is discussed, including digital signatures, key exchange protocols, and applications in blockchain technology. The results show that ECC provides comparable security to traditional public key algorithms at a short key length, and shows strong security and efficiency in practical applications. As technology progresses and new threats arise, research in ECC will also evolve.

Solving Elliptic Curve Discrete Logarithm Problem on Twisted Edwards Curves Using Quantum Annealing and Index Calculus Method

Abstract This paper presents an approach to solving the elliptic curve discrete logarithm problem on alternative curve models over prime fields using a quantum annealing and index calculus method. Part of the algorithm, relation searching, is transformed into the Quadratic Unconstrained Boolean Optimization (QUBO) problem and then is efficiently solved using the D-Wave computer by quantum annealing. As Faugère et al. showed, twisted Edwards curves, because of their symmetric shape, allow us to obtain solutions of relations searching step using Groebner basis faster than in the case of Weierstrass curves. Because of symmetries, a system of equations of relations searching step for twisted Edwards curves has many symmetric solutions. Using the Groebner basis and having many system solutions makes it easier to find any of them. The same is true using quantum annealing - it is easier to find any solution to the QUBO problem if many are correct. In this paper, we used this observation to find out that a properly constructed QUBO problem for the relations searching step for twisted Edwards curves allows us to find a solution faster for the same size of the base field than in the case of Weierstrass curves. Using the presented approach, we solved the discrete logarithm problem using quantum annealing and index calculus method for elliptic curve discrete logarithm problem defined on twisted Edwards curve over a field 𝔽1021 with order equal to 4 · 241. It is now the biggest field and size of the group, where the elliptic curve discrete logarithm problem was solved using quantum methods.

A Comprehensive Review of MI-HFE and IPHFE Cryptosystems: Advances in Internal Perturbations for Post-Quantum Security

The RSA cryptosystem has been a cornerstone of modern public key infrastructure; however, recent advancements in quantum computing and theoretical mathematics pose significant risks to its security. The advent of fully operational quantum computers could enable the execution of Shor’s algorithm, which efficiently factors large integers and undermines the security of RSA and other cryptographic systems reliant on discrete logarithms. While Grover’s algorithm presents a comparatively lesser threat to symmetric encryption, it still accelerates key search processes, creating potential vulnerabilities. In light of these challenges, there has been an intensified focus on developing quantum-resistant cryptography. Current research is exploring cryptographic techniques based on error-correcting codes, lattice structures, and multivariate public key systems, all of which leverage the complexity of NP-hard problems, such as solving multivariate quadratic equations, to ensure security in a post-quantum landscape. This paper reviews the latest advancements in quantum-resistant encryption methods, with particular attention to the development of robust trapdoor functions. It also provides a detailed analysis of prominent multivariate cryptosystems, including the Matsumoto–Imai, Oil and Vinegar, and Polly Cracker schemes, alongside recent progress in lattice-based systems such as Kyber and Crystals-DILITHIUM, which are currently under evaluation by NIST for potential standardization. As the capabilities of quantum computing continue to expand, the need for innovative cryptographic solutions to secure digital communications becomes increasingly critical.

TrustCNAV: Certificateless aggregate authentication of civil navigation messages in GNSS

The Global Navigation Satellite System (GNSS) is capable of accurate positioning because it can provide high-precision data. These data are transmitted to the receiver in the form of navigation messages, called civil navigation messages (CNAV). As it is transmitted in an open, transparent environment without data integrity protection mechanisms and secure data transmission measures, the CNAV is suspected to spoofing attacks. In 2023, the OPSGROUP has received approximately 50 reports of GPS spoofing activity. A spoofed plane's navigation system will show it as being in a different place - a security risk if a jet is guided to fly into a hostile country's airspace. To prevent the forging of GNSS positioning data by spoofing attacks targeting CNAV, we propose a certificateless aggregation authentication for CNAV by using the elliptic curve discrete logarithm problem and the combination of the GNAV structural characteristics, called TrustCNAV. Security proof and performance analysis indicate that this authentication scheme can resist spoofing attacks and ensure data security of CNAV, also it avoids pairing operations with high computational complexity, thus meeting security requirements without causing too much time and communication consumption.

Tighter bound for generalized multiple discrete logarithm problem via MDS matrix method

Discrete logarithm problem (DLP) is one of the fundamental hard problems used in cryptography. For 1≤k≤n, solving the k-out-of-n DLP instances is an important problem emerging in certain scenarios in public-key cryptography. Ying and Kunihiro (ACNS 2017) pioneered in studying k-out-of-n instance solutions of DLP, which is a generalized version of multiple DLP. By reducing the multiple DLP to the generalized version, they established lower bounds on the computational complexity of k-out-of-n DLP for different parameter values of k.In this paper, we further reduce the reduction complexity presented in Ying and Kunihiro's work and increase the range of k and n for the tight lower bound of k-out-of-n DLP in the generic group model, which has applications in related cryptographic schemes. To achieve the goal, the key technique is to utilize a variant of fast multipoint evaluation. We divide the discussion into two cases. In the special case when n divides p−1, by leveraging Number Theory Transform (NTT) technique, we expand k and n to a larger range. In the general case, by using a variant of fast multipoint evaluation, we increase k and n to a moderately larger range.

Discrete Logarithm Factory

The Number Field Sieve and its variants are the best algorithms to solve the discrete logarithm problem in finite fields (except for the weak small characteristic case). The Factory variant accelerates the computation when several prime fields are targeted. This article adapts the Factory variant to non-prime finite fields of medium and large characteristic. A precomputation, solely dependent on an approximate finite field size and an extension degree, allows to efficiently compute discrete logarithms in a constant proportion of the finite fields of the given approximate size and extension degree. We combine this idea with two other variants of NFS, namely the tower and special variant. This combination improves the asymptotic complexity. We also notice that combining our approach with the MNFS variant would be an unnecessary complication as all the potential gain of MNFS is subsumed by our Factory variant anyway. Furthermore, we demonstrate how Chebotarev's density theorem allows to compute the density of finite fields that can be solved with a given precomputation. Finally, we provide experimental data in order to assess the practical reach of our approach.

Mathematical Approaches to Cryptographic System Design

In the digital age, cryptographic systems are the most important part of safe communication. To protect data security, confidentiality, and validity, they need strong design frameworks. The math methods used in this paper are very important for designing and analyzing secure systems. As basic ideas, it looks at number theory, math, and complexity theory, with an emphasis on both old and new methods. Some important topics are the creation of prime numbers, modular arithmetic, elliptic curves, and finite fields, which are the basis for many encryption methods. The paper also talks about how complexity theory can be used to measure the strength of cryptography. It specifically talks about issues with discrete logarithms and integer factorization, which are at the heart of popular protocols like RSA and ECC. It also looks into lattice-based cryptography, which is seen as a strong option to quantum threats, and shows how hard it is to solve lattice issues. The study also looks at the design principles of symmetric cryptography, mainly block ciphers and stream ciphers, and how they use permutation groups and linear algebra to make sure that key plans and spread methods are safe. The paper also looks at secure hash functions, focusing on collision resistance, pre-image resistance, and how they are made using mathematics concepts such as Merkle-Damgård and sponge functions. Advanced topics like homomorphic encryption and zero-knowledge proofs show how mathematics and cryptography are increasingly coming together. They show how they can be used to make operations safe on protected data and privacy-preserving protocols. This paper gives a full picture of how mathematical theories and methods are used to build strong cryptographic systems by combining strict mathematical models with real-world cryptographic needs. The discussion stresses that the field is always changing because of new threats and improvements in computers. It also calls for constant scientific progress to make cryptography stronger against future problems.

Unforgeability of Blind Schnorr in the Limited Concurrency Setting

Blind signature schemes enable a user to obtain a digital signature on a message from a signer without revealing the message itself. Among the most fundamental examples of such a scheme is blind Schnorr, but recent results show that it does not satisfy the standard notion of security against malicious users, One-More Unforgeability (OMUF), as it is vulnerable to the ROS attack. However, blind Schnorr does satisfy the weaker notion of sequential OMUF, in which only one signing session is open at a time, in the Algebraic Group Model (AGM) + Random Oracle Model (ROM), assuming the hardness of the Discrete Logarithm (DL) problem. This paper serves as a first step towards characterizing the security of blind Schnorr in the limited concurrency setting. Specifically, we show that blind Schnorr satisfies OMUF when at most two signing sessions can be concurrently open (in the AGM+ROM, assuming DL). Our argument suggests that it is plausible that blind Schnorr satisfies OMUF for up to polylogarithmically many concurrent signing sessions. Our security proof involves interesting techniques from linear algebra and combinatorics.

A heterogeneous ring signcryption scheme with privacy protection and conditional tracing for smart grid

Smart grid develops rapidly, but there are still security risks such as user privacy leakage, power data tampering and audit data inconsistency. The existing schemes to ensure data security mainly use traceable ring signcryption, which is applied in distributed application scenarios such as smart grid. Traceable ring signcryption can ensure the anonymity, integrity, unforgeability and confidentiality of data, and can trace the real identity of anonymous users. However, the traceability of these schemes is arbitrary, any actor can trace the identity of anonymous users, and they do not resolve disputes caused by tampered or inconsistent data. To remedy these deficiencies, we combine ring signcryption with consortium blockchain technology for the first time to achieve privacy protection and conditional tracing, which can effectively avoid anonymous user identity being revealed at will. Consortium blockchain is a semi-distributed P2P network that can solve data disputes and is suitable for organizations that require certain access control mechanisms such as smart grid. In this paper, we propose a heterogeneous ring signcryption scheme with privacy protection and conditional tracing (CTHRSC) which between certificateless cryptographic system (CLC) and public key infrastructure (PKI). Besides, we prove that our scheme is secure under the discrete logarithm problem (DLP) and decisional Diffie–Hellman problem (DDHP) in random oracle model (ROM). Compared with other signature or signcryption schemes, our advantages are satisfying conditional tracing and known temporary session key security (KTSKS), requiring less computation cost and communication overhead.

A Problem as Much of Economic Theory as of Statistical Technique? The Walsh-Edgeworth Debate on Index Numbers, 1921–25

Abstract Correa Moylan Walsh (1862–1936) was a monetary economist of the early twentieth century, mostly remembered for his contributions to the theory of index numbers. Francis Ysidro Edgeworth (1845–1926) was a renowned mathematical economist, remembered as one of the great masters of the discipline. Walsh was an adamant defender of a deterministic, test approach to the theory of index numbers, whereas Edgeworth favored probabilistic approaches and thought index-number theorists should not close themselves to alternative methods. Holding contrasting views about the making of index numbers, they engaged in a fierce debate in the first half of the 1920s—a period marked by several quarrels on the subject. The main points in dispute between Walsh and Edgeworth were the existence of an ideal formula for index numbers and the use of probability in the measurement of price variations. The goal of this article is to explore this debate and its underpinnings. It is argued that the main points of contention between them relate to their conflicting perspectives on the nature of index numbers.

Blockchain-enhanced efficient and anonymous certificateless signature scheme and its application

Although the Internet of Things (IoT) brings efficiency and convenience to various aspects of people’s lives, security and privacy concerns persist as significant challenges. Certificateless Signatures eliminate digital certificate management and key escrow issues and can be well embedded in resource-constrained IoT devices for secure access control. Recently, Ma et al. designed an efficient and pair-free certificateless signature (CLS) scheme for IoT deployment. Unfortunately, We demonstrate that the scheme proposed by Ma et al. is susceptible to signature forgery attacks by Type-II adversaries. That is, a malicious-and-passive key generation center (KGC) can forge a legitimate signature for any message by modifying the system parameters without the user’s secret value. Therefore, their identity authentication scheme designed based on vehicular ad-hoc networks also cannot guarantee the claimed security. To address the security vulnerabilities, we designed a blockchain-enhanced and anonymous CLS scheme and proved its security under the Elliptic curve discrete logarithm (ECDL) hardness assumption. Compared to similar schemes, our enhanced scheme offers notable advantages in computational efficiency and communication overhead, as well as stronger security. In addition, a mutual authentication scheme that satisfies the cross-domain scenario is proposed to facilitate efficient mutual authentication and negotiated session key generation between smart devices and edge servers in different edge networks. Performance evaluation shows that our protocol achieves an effective trade-off between security and compute performance, with better applicability in IoT scenarios.

Efficient provable dual receiver hybrid and light weight public key schemes based on the discrete logarithm problem without pairings

AbstractDual‐receiver proxy re‐encryption is a cryptographic technique that enables secure data sharing among multiple authorized users or entities. It has gained significant attention for its ability to manage access permissions, data confidentiality, and streamline communication channels. These schemes have been widely used in various applications, including healthcare systems, cloud computing, Internet of Things (IoT) systems, collaborative environments, and secure communication channels. This paper aims to propose two proxy re‐encryption schemes for dual receivers without pairings. The first is dual receiver lightweight proxy re‐encryption without pairings (DR‐LWPRE‐WP), which uses a public key scheme to reduce computational complexity. The second is dual receiver hybrid proxy re‐encryption without pairings (DR‐HPRE‐WP), which incorporates public key and symmetric key schemes. Both schemes offer protection against selected plaintext attacks based on the decisional Diffie‐Hellman principle. The DR‐LWPRE‐WP scheme reduces computation by approximately 53% compared to pairing‐based schemes, making it suitable for lightweight applications like the Internet of Things. The computational efficacy of these schemes offers significant benefits for resource‐constrained environments and practical implementations.