Discrete Logarithm Research Articles

An Efficient Privacy-Enhancing Cross-Silo Federated Learning and Applications for False Data Injection Attack Detection in Smart Grids

Federated Learning is a prominent machine learning paradigm which helps tackle data privacy issues by allowing clients to store their raw data locally and transfer only their local model parameters to an aggregator server to collaboratively train a shared global model. However, federated learning is vulnerable to inference attacks from dishonest aggregators who can infer information about clients’ training data from their model parameters. To deal with this issue, most of the proposed schemes in literature either require a non-colluded server setting, a trusted third-party to compute master secret keys or a secure multiparty computation protocol which is still inefficient over multiple iterations of computing an aggregation model. In this work, we propose an efficient cross-silo federated learning scheme with strong privacy preservation. By designing a double-layer encryption scheme which has no requirement to compute discrete logarithm, utilizing secret sharing only at the establishment phase and in the iterations when parties rejoin, and accelerating the computation performance via parallel computing, we achieve an efficient privacy-preserving federated learning protocol, which also allows clients to dropout and rejoin during the training process. The proposed scheme is demonstrated theoretically and empirically to provide provable privacy against an honest-but-curious aggregator server and simultaneously achieve desirable model utilities. The scheme is applied to false data injection attack detection (FDIA) in smart grids. This is a more secure cross-silo FDIA federated learning resilient to the local private data inference attacks than the existing works.

Identity-Based Edge Computing Anonymous Authentication Protocol

With the development of sensor technology and wireless communication technology, edge computing has a wider range of applications. The privacy protection of edge computing is of great significance. In the edge computing system, in order to ensure the credibility of the source of terminal data, mobile edge computing (MEC) needs to verify the signature of the terminal node on the data. During the signature process, the computing power of edge devices such as wireless terminals can easily become the bottleneck of system performance. Therefore, it is very necessary to improve efficiency through computational offloading. Therefore, this paper proposes an identity-based edge computing anonymous authentication protocol. The protocol realizes mutual authentication and obtains a shared key by encrypting the mutual information. The encryption algorithm is implemented through a thresholded identity-based proxy ring signature. When a large number of terminals offload computing, MEC can set the priority of offloading tasks according to the user’s identity and permissions, thereby improving offloading efficiency. Security analysis shows that the scheme can guarantee the anonymity and unforgeability of signatures. The probability of a malicious node forging a signature is equivalent to cracking the discrete logarithm puzzle. According to the efficiency analysis, in the case of MEC offloading, the computational complexity is significantly reduced, the computing power of edge devices is liberated, and the signature efficiency is improved.

A method for constructing public-key block cipher schemes based on elliptic curves

The article proposes a method for constructing public-key block cipher schemes based on the difficulty of the discrete logarithm problem on elliptic curves. The schemas are construc according to the proposed method and can simultaneously perform security functions and authenticate the origin and integrity of the encrypted message. In addition, a shared secret key is established between the sender/encryptor and the receiver/decryptor for each encrypted message based on public key cryptography which also improves the security of these cipher schemes.

A NEW METHOD FOR CONSTRUCTING DIGITAL SIGNATURE SCHEME BASED ON NEW HARD PROBLEM

Research in the field of public key cryptography in general and digital signatures in particular is often considered and evaluated at two levels: The first level is considering the mathematical basis for constructing cryptographic and digital signature algorithms, especially these difficult problems: factorizing a large integer into prime factors, finding root problem, discrete logarithm problem, discrete logarithm problem on elliptic curve... The second level is constructing cryptographic and digital signatures algorithms on these hard problems including RSA, DSA, Schnorr, GOST R34.10-94. At the first level, research focuses mainly on improving algorithms for finding large primes, structures for strong primes and algorithms to attack these problems efficiently. At the second level, the research concentrates on improving the existing algorithms to enhance the safety or effective performance of the algorithm. In this article, the authors propose a solution to improve the security of digital signature schemes, implemented in both levels of digital signatures. At the first level, the authors propose a new hard problem - different from the hard problems used before and hasn’t been solved by anyone so far (except by "brute force attack" method). At the second level, the authors propose a method to construct new digital signature algorithms that can help create not only one but also a family of new digital signature algorithms with high security level for practical applications.

SOME VARIANTS OF THE SCHNORR SIGNATURE SCHEMA ON THE FINITE FIELD AND THE ELLIPTIC CURVE

Schnorr signature schemas and the version on its elliptic curve (EC-Schnorr) are signature schemas that are not only highly secure but are also rated as having the highest performance among signature schemas based on discrete logarithm problems (on the Fp finite field and on the elliptic curve). The Schnorr and EC-Schnorr schemas have been incorporated into ISO/IEC 14888-3 and the BSI standard of the Federal Republic of Germany. Currently, the development of signature schemas based on Schnorr and EC-Schnorr schemas for BlockChain applications and digital currencies is of great interest. In this article the authors suggest several variations of the Schnorr schema on the finite field and the elliptic curve. The analyses in the article shows that the new schemas proposed here have the same security and performance as the Schnorr/EC-Schnorr signature schema in practical applications.

A new construction method of digital signature scheme based on the discrete logarithm combining find root problem on the finite field F_p

The article proposes a method to build a signature scheme based on a new hard problem, called the logarithm problem with roots on the finite field . Now, this is a hard problem belonging to the class of unsolvable problems, except for the “brute force” method. Therefore, building a digital signature scheme based on the difficulty of this problem will most likely allow improving the security of the digital signature algorithm according to the proposed new method. In addition, the method of building signature schema here can be applied to develop a class of signature algorithms suitable for applications with high requirements for security in practice applications.

Improved and Provably Secure ECC-Based Two-Factor Remote Authentication Scheme with Session Key Agreement

The remote authentication scheme is a cryptographic protocol incorporated by user–server applications to prevent unauthorized access and security attacks. Recently, a two-factor authentication scheme using hard problems in elliptic curve cryptography (ECC)—the elliptic curve discrete logarithm problem (ECDLP), elliptic curve computational Diffie–Hellman problem (ECCDHP), and elliptic curve factorization problem (ECFP)—was developed, but was unable to address several infeasibility issues while incurring high communication costs. Moreover, previous schemes were shown to be vulnerable to privileged insider attacks. Therefore, this research proposes an improved ECC-based authentication scheme with a session key agreement to rectify the infeasible computations and provide a mechanism for the password change/update phase. The formal security analysis proves that the scheme is provably secure under the random oracle model (ROM) and achieves mutual authentication using BAN logic. Based on the performance analysis, the proposed scheme resists the privileged insider attack and attains all of the security goals while keeping the computational costs lower than other schemes based on the three hard problems. Therefore, the findings suggest the potential applicability of the three hard problems in designing identification and authentication schemes in distributed computer networks.

MODIFICATION OF POLLARD RHO ALGORITHM USING NEGATION MAPPING

El Gamal encryption was introduced in 1985 and is still commonly used today. Its hardness is based on a discrete logarithm problem defined over the finite abelian cyclic group group chosen in the original paper was but later it was proven that using the group of Elliptic Curve points could significantly reduce the key size required. The modified El Gamal encryption is dubbed its analog version. This analog encryption bases its hardness on Elliptic Curve Discrete Logarithm Problem (ECDLP). One of the fastest attacks in cracking ECDLP is the Pollard Rho algorithm, with the expected number of iterations where is the number of points in the curve. This paper proposes a modification of the Pollard Rho algorithm using a negation map. The experiment was done in El Gamal analog encryption of elliptic curve defined over the field with different values of small digit . The modification was expected to speed up the algorithm by times. The average of speed up in the experiment was 1.9 times.

Privacy-Preserved Image Protection Supporting Different Access Rights

The boom in cloud computing and social networking has led to a large number of online users in the networks. It is necessary to use appropriate privacy protection mechanisms to prevent personal privacy leakage. In general, image privacy protection techniques proceed with the whole image. However, the image may contain multiple users’ information, and different people’s data may require different privacy protection. In addition, cloud servers are semi-trusted instead of trusted service providers due to unknown security vulnerabilities and the possibility of grabbing sensitive data from the cloud server. In order to provide image protection with different privacy-preserved access rights and to prevent the cloud server from retrieving sensitive information from the image, we propose a privacy-preserved scheme that supports different privacy-preserved access rights in a single image based on the difficulty of solving the factorization problem and the discrete logarithm problem. The cloud server performed the image management method without knowing the privacy information contained in the protected images. By working out the proposed scheme in detail, we experimentally validate the scheme and discuss various options of access rights in practice.

Shaping Postquantum Cryptography: The Hidden Subgroup and Shift Problems

The security of popular public key-cryptographic protocols, such as RSA, Diffie-Hellman key exchange and the Digital Signature Algorithm (DSA), is endangered by the advent of quantum computers. Shor brought a big breakthrough with his quantum algorithm that can be used to factor an arbitrarily large integer into the product of its prime factors, hence jeopardizing the security of RSA, and that at the same time also solves the Discrete Logarithm Problem, which raises issues for certain Diffie-Hellman based cryptosystems and digital signatures. It is hence crucial to upgrade our current tools for post-quantum cryptography: it should be infeasible, even using quantum algorithms, to break the new cryptosystems. Popular candidates include for example elliptic curve or lattice-based cryptography, but they share something in common: they are specific cases of the more general Hidden Subgroup and connected Hidden Shift Problem.

RC2PAS: Revocable Certificateless Conditional Privacy-Preserving Authentication Scheme in WBANs

Wireless body area networks (WBANs) are an essential part of health-care system. It collects the patient’s biomedical data through sensors and sends the data to the remote doctor. Based on these data, remote doctors can provide the patient with a detailed diagnosis and suitable treatment. However, in wireless communication environment, WBANs may be affected by various attacks, which leads to privacy leakage. Before WBANs come into application, data confidentiality and patient privacy are the main challenges for us. Therefore, we proposed a revocable certificateless authentication scheme that effectively overcomes these problems. In this scheme, to realize large-scale applications, we divided WBANs into several independent areas, and each area is responsible for the patients in its own area. Once a malicious user appears, revocation center can change the key of its own area to achieve efficient revocation. This approach disperses the management burden of the network manager and realizes conditional privacy-preserving. Besides, bilinear pairing is not used in this scheme, which greatly reduces the computational cost. It is especially important for clients with limited capabilities in WBANs. It is proved that the security of this scheme is equivalent to the assumption of elliptic curve discrete logarithm in a random oracle.

Вдосконалення методів генерації ключів шифрування за допомогою дивних атракторів

The urgency of the work is determined by the need to transfer confidential information through open communication channels. Such information can be of two types: symmetric encryption keys and directly informational messages that are encrypted with encryption keys. The article deals with the problem of improving the transmission of closed information over open channels using the Diffie- Hellman algorithm. The improvement is due to the introduction of a new type of one-sided function based on the numerical solution of the system of ordinary differential equations describing the dynamics of the phase coordinate movement of the strange attractor. For this purpose, the classic Diffie-Hellman algorithm based on the one-sided function of the discrete logarithm was considered. The required properties of one-sided functions in the general case were considered. Next, the peculiarities of algorithm modification in the case of transition to a one-sided function based on the use of a strange attractor were considered. It is assumed that at the beginning of the operation of the modified algorithm, through a secret channel, the exchange parties (agents) exchange information regarding the properties of the strange attractor to be used, namely, the definition of the differential equations describing the dynamics of a strange attractor, the values of the parameters of the equations, the initial integration conditions and the integration step (for methods with a constant step of integration). After that, all exchanges are conducted exclusively through open channels. The paper also considers the case of information exchange between more than two agents, in particular, the approach of hiding the number of agents participating in the exchange. Approbation of the method is carried out and intermediate and final results of the one-sided function based on strange attractors are given. Possibilities regarding partial disclosure by agents of certain parameters of the use of one-way functions are discussed. But at the same time, the safety of revealing such information is justified in the general case (both in the classical and in the modified Diffie-Hellman method). It was determined that depending on the needs of users, the complexity of the encryption keys can be increased by changing the initial parameters of the attractor, which will also allow controlling speed of key generation and encryption in general. The software that implements the proposed modified algorithm is implemented in three programming languages, C#, Python, and MatLab. This made it possible to perform a comparative analysis of the results and consciously choose the programming language of individual parts of the software to optimize the encryption key generation process for specific conditions.

IPChain: Blockchain-Based Security Protocol for IoT Address Management Servers in Smart Homes

The dynamic host configuration protocol (DHCP) servers are forms of an Internet of Things (IoT) address management server (IoTAMS) that gives network configuration settings to newly connected hosts. Administrators of a network may save time by setting DHCP servers instead of every network node. However, the absence of a more robust authentication method for DHCP servers makes hosts susceptible to attacks since neither the server nor the users are able to check the other’s authenticity during DHCP connections. These concerns result in both internal and external threats to the system that have the potential to impair network services. Among these threats are malicious DHCP servers and DHCP starvation. This paper aims to provide a novel approach for tackling these issues and protect the DHCP protocol. The proposed model uses the Diffie–Hellman key exchange mechanism, the elliptic curve discrete logarithm problem (ECDLP), a one-way hash function, blockchain technology, and a smart contract. In addition, registration and validation processes provide support for the proposed model in combating DHCP risks for both internal and external system threats. Results from this study show that the proposed model has an average of 21.1% more resistance to a growing number of adversaries than the benchmark models, thus revealing that the model is better suited for the security of IoT address management servers in smart homes, thereby enhancing resilience against related threats and the success of IP address management.

A Modified Wei-Hua-He Digital Signature Scheme Based on Factoring and Discrete Logarithm

A symmetric cipher such as AES in cryptography is much faster than an asymmetric cipher but digital signatures often use asymmetric key ciphers because they provides the sender’s identity and data integrity. In this paper, a modified-He digital signature scheme is proposed using a one-way hash function. The proposed scheme, unlike the He signature technique, employs Euclid’s Division Lemma with large prime moduli p. Its security is built on large integer factoring, discrete logarithms and expanded root problems. The time complexity of the proposed scheme is O(log3p). The proposed modified-He scheme is efficient, as evidenced by the analytical results with key lengths greater than 512 bits.

Implementing cryptographic-based DH approach for enterprise network

The network is emerging as it involves multiple heterogeneous devices that are interconnected and able to communicate through certain information. Here security plays an important role in overcoming certain problems like loss of information, data traffic and protecting data activity. To the above security issues, the cryptography concept will enhance the security and also be able to protect the network resource that is transferred. To ensure data protection and privacy, here we recommend proposing a cryptographic-based DH approach to ensure the security enhancement consumes a lot of resources, power, memory, and CPU utilization time; here go for the modified key exchange and the encryption process. Initially, the modified Encryption process, where encryption is applied between the client and server and the encryption is processed at the client side to enhance the data privacy and integrity. Then, user authentication performs Modified Diffie Hellman to enhance the authentication process while sharing the data and avoid unauthorized users; finally, by overcoming certain attacks like Man in Middle and Discrete Logarithm attacks with the key exchange process, Modified Zero Knowledge Proof (ZKP) algorithm is applied. The analysis will be performed based on key aspects of size, key generation, encryption time, decryption time, and execution time.

New Constructions of Existential Unforgeable Aggregate Signature Scheme from CSP

In future, hundreds of years of mathematical problems that the security of public key cryptography algorithms rely on may be defeated by quantum algorithms. How can a digital signature scheme gracefully balance security and efficiency? This study uses the conjugate search problem and the left self-distributive system to combine and uses the RSA-like algorithm as the underlying structure to propose a new aggregated signature scheme. We, through the EUF game, under the random metaphor model, prove that the security of the scheme satisfies the adaptation unforgeability under selective message attack, the scheme can be finally reduced to the discrete logarithm problem or large prime number decomposition problem. In addition, we can achieve antiquantum attack and exhaustive attack by performing matrix calculations on the message, defining and changing the structure of the matrix by encoding, and setting thresholds for the matrix dimension and the length of the private key. In terms of efficiency, the message signature implementation is linear compared with the expansion rate in terms of storage and computing overhead, and the generation and verification of the final signature pair have nothing to do with the number of users. In addition, the length of the signature is fixed and the size is only the length of a single group, which effectively reduces the generation of public and private key pairs and saves a lot of storage space. The storage space and computational complexity are also effectively improved compared with other solutions.

DIVRS: Data integrity verification based on ring signature in cloud storage

Nowadays, cloud service is becoming an indispensable part of people’s lives and a growing number of users store data on servers. Although this way can reduce storage and maintenance costs, data owners lose the control and possession of the data, which will result in some additional security issues, such as data corruption. It means that cloud server providers may delete or tamper with the infrequently used data to economize on storage space. Therefore, it is essential to provide an audit service to verify integrity of data stored on the cloud. Most of the existing audit schemes are based on the difficult problems of large integer decomposition and discrete logarithms. With the development of quantum computers, these schemes will face the security threat, because quantum computers can easily solve these difficult problems. In short, it is essential to design a data audit scheme that is resistant to quantum attacks. Therefore, we design a novel data audit scheme to achieve post-quantum security, in which the ring signature based on the problem of learning with errors is employed. In addition, two data possession games are designed to prove the security of the DIVRS. Experiment results show that the communication and computing costs are acceptable, and the time spent in each phase of the DIVRS is shorter than that of the comparison schemes.

LAMAS: Lightweight anonymous mutual authentication scheme for securing fog computing environments

Fog computing uses near-user edge devices that are usually smart devices with very limited storage and computation resources; hence, the authentication process should be lightweight to be suitable for fog computing environment. In this paper, a Lightweight Anonymous Mutual Authentication Scheme (LAMAS) for securing fog computing is proposed based on elliptic curve cryptography. The proposed scheme succeeds to achieve mutual authentication between fog users and the fog servers even if the fog server is added after user registration, without need to re-register or do any more computation. Additionally, the fog user's identity is totally hidden and cannot be predicted due to the Elliptic Curve Discrete Logarithm Problem (ECDLP). Moreover, formal and informal security analysis is conducted to prove the security of the proposed scheme. Furthermore, comparisons with related schemes are carried out to demonstrate the superiority and efficiency of the proposed scheme in terms of computation cost and storage requirements.

Development of a hardware cryptosystem based on a random number generator with two types of entropy sources

In modern software, crypto-algorithms are widely used for both data encryption tasks, and authentication and integrity checks. There are well-known and proven crypto-algorithms. Their cryptoresistance is either mathematically proven or based on the need to solve a mathematically complex problem (factorization, discrete logarithm, etc.). On the other hand, in the computer world, information constantly appears about errors or «holes» in a particular program (including one that uses crypto-algorithms) or that it was broken (cracked). This creates distrust both in specific programs and in the possibility to protect something in general by cryptographic methods not only from special services, but also from ordinary hackers. A promising direction of research in this field is the implementation of a hybrid random number generator with two types of entropy sources in cryptosystems. The method and means of implementing a hybrid random number generator with two types of entropy sources: external – based on Zener diode noise and internal – based on the uncertainty state of the transistor-transistor logic structure are presented. One of the options for the practical implementation of a random number generator is presented, where two sources are used as a source of entropy: an external source – Zener diode noise and an internal source – the undefined state of the transistor-transistor logic structure. The functional diagram of the proposed random number generator with two types of entropy sources is given. The MATLAB/Simulink model of the proposed random number generator is built, the results of the statistical analysis of the generated random sequences by the NIST SP 800-22 test package are given.

Lightweight and Secure IoT-Based Payment Protocols from an Identity-Based Signature Scheme

After the great success of mobile wallets, the Internet of Things (IoT) leaves the door wide open for consumers to use their connected devices to access their bank accounts and perform routine banking activities from anywhere, anytime, and with any device. However, consumers need to feel safe when interacting with IoT-based payment systems, and their personal information should be protected as much as possible. Unlike what is usually found in the literature, in this paper, we introduce two lightweight and secure IoT-based payment protocols based on an identity-based signature scheme. We adopt a server-aided verification technique to construct the first scheme. This technique allows to outsource the heavy computation overhead on the sensor node to a cloud server while maintaining the user’s privacy. The second scheme is built upon a pairing-free ECC-based security protocol to avoid the heavy computational complexity of bilinear pairing operations. The security reduction results of both schemes are held in the Random Oracle Model (ROM) under the discrete logarithm and computational Diffie–Hellman assumptions. Finally, we experimentally compare the proposed schemes against each other and against the original scheme on the most commonly used IoT devices: a smartphone, a smartwatch, and the embedded device Raspberry Pi. Compared with existing schemes, our proposed schemes achieve significant efficiency in terms of communication, computational and storage overheads.