Discrete Logarithm Research Articles

A Like ELGAMAL Cryptosystem But Resistant To Post-Quantum Attacks

The Modulo 1 Factoring Problem (M1FP) is an elegant mathematical problem which could be exploited to design safe cryptographic protocols and encryption schemes that resist to post quantum attacks. The ELGAMAL encryption scheme is a well-known and efficient public key algorithm designed by Taher ELGAMAL from discrete logarithm problem. It is always highly used in Internet security and many other applications after a large number of years. However, the imminent arrival of quantum computing threatens the security of ELGAMAL cryptosystem and impose to cryptologists to prepare a resilient algorithm to quantum computer-based attacks. In this paper we will present a like-ELGAMAL cryptosystem based on the M1FP NP-hard problem. This encryption scheme is very simple but efficient and supposed to be resistant to post quantum attacks.

Secure and privacy-preserving D2D communication in fog computing services

Device-to-device (D2D)-aided fog network has proven to be a viable means to improve the performance of fog computing technology. To exploit the full potentials of the D2D-supported fog computing architecture, it is imperative to protect the information transmitted among collaborating fog nodes against unauthorized access as failure to do so may have catastrophic consequences. However, this critical issue has not been investigated in most of the existing works. Therefore, this paper proposes a secure and privacy-aware trust-based communication strategy for fog-to-fog collaborative services. An elliptic curve cryptography (ECC)-based two-party authentication and key agreement protocol is designed to allow two collaborating fog nodes generate a unique shared secret session key. A multi-colony head selection strategy is proposed using a multi-criteria decision-making method (MCDM), where multiple fog nodes are selected as colony head. A new efficient group key distribution and updating mechanism is designed for ensuring secured group communication. Consequently, ECC-based message dissemination approaches are designed for intra- and inter-colony trustworthy communications, where the generated secret session and group keys are used for data encryption. Moreover, we demonstrate that the proposed scheme guarantees data confidentiality and unforgeability against both Type I and Type II adversaries in the random oracle model (ROM) based on the intractability of the Computational Diffie–Hellman Problem (CDHP) and Discrete Logarithm Problem (DLP), respectively. An extensive performance analysis shows that the proposed scheme is very efficient and suitable for practical deployment. Comparative analysis reveals that the proposed scheme reports an improvement of about 33.15%–93.88%, 18.87%–67.91% , and 38.5%–93.8% in computation cost, communication overhead, and energy consumption, respectively, over the state-of-the-art related schemes.

An initial step toward a quantum annealing approach to the discrete logarithm problem

The discrete logarithm problem over a multiplicative group of integer modulo n is the key ingredient in the ElGamal encryption system. This problem has been proven to be tractable in polynomial time on a quantum computer by Shor's algorithm. This algorithm makes use of basic quantum gates, circuits, measurements and it has been experimented in the circuit-gate framework of quantum computing. On the other hand, the adiabatic framework of quantum computing with the quantum annealers such as the D-Wave machine that simulates the adiabatic process has become very popular, showing its potential for making a practical pathway to achieve quantum speedup. Furthermore, much progress has been reported recently on the tractability of the integer factoring problem, which is the other widely-used encryption method, on quantum annealers. In this context, it is important to explore the tractability of the discrete logarithm problem too using quantum annealers. In this work we present a first step made in this direction, by converting the discrete logarithm problem over a multiplicative group into the problem of minimizing a binary quadratic form, a standard problem format acceptable to a quantum annealer. Our formulation was tested for small-scale problem instances using the quantum-classical hybrid platform PyQUBO and we discuss the computational challenges and propose several potential improvements to the formulation.

Two-Round n-out-of-n and Multi-Signatures and Trapdoor Commitment from Lattices

Although they have been studied for a long time, distributed signature protocols have garnered renewed interest in recent years in view of novel applications to topics like blockchains. Most recent works have focused on distributed versions of ECDSA or variants of Schnorr signatures; however, and in particular, little attention has been given to constructions based on post-quantum secure assumptions like the hardness of lattice problems. A few lattice-based threshold signature and multi-signature schemes have been proposed in the literature, but they either rely on hash-and-sign lattice signatures (which tend to be comparatively inefficient), use expensive generic transformations, or only come with incomplete security proofs. In this paper, we construct several lattice-based distributed signing protocols with low round complexity following the Fiat–Shamir with Aborts (FSwA) paradigm of Lyubashevsky (Asiacrypt 2009). Our protocols can be seen as distributed variants of the fast Dilithium-G signature scheme and the full security proof can be made assuming the hardness of module SIS and LWE problems. A key step to achieving security (unexplained in some earlier papers) is to prevent the leakage that can occur when parties abort after their first message—which can inevitably happen in the Fiat–Shamir with Aborts setting. We manage to do so using homomorphic commitments. Exploiting the similarities between FSwA and Schnorr-style signatures, our approach makes the most of observations from recent advancements in the discrete log setting, such as Drijvers et al.’s seminal work on two-round multi-signatures (S&P 2019). In particular, we observe that the use of commitment not only resolves the subtle issue with aborts, but also makes it possible to realize secure two-round n-out-of-n distributed signing and multi-signature in the plain public key model, by equipping the commitment with a trapdoor feature. The construction of suitable trapdoor commitment from lattices is a side contribution of this paper.

Constructing New Collective Signature Schemes Base on Two Hard Problems Factoring and Discrete Logarithm

In network security, digital signatures are considered a basic component to developing digital authentication systems. These systems secure Internet transactions such as e-commerce, e-government, ebanking, and so on. Many digital signature schemes have been researched and published for this purpose. In this paper, we propose two new types of collective signature schemes, namely i) the collective signature for several signing groups and ii) the collective signature for several individual signings and several signing groups. And then we used two difficult problems factoring and discrete logarithm to construct these schemes. To create a combination of these two difficult problems we use the prime module p with a special structure: p = 2n +1. Schnorr's digital signature scheme is used to construct related basic schemes such as the single signature scheme, the collective signature scheme, and the group signature scheme. The proposed collective signature schemes are built from these basic schemes. The proposed signature scheme is easy to deploy on existing PKI systems. It can support PKIs in generating and providing a unique public key, a unique digital signature, and a unique digital certificate for a collective of many members. This is essential for many collective transactions on today's Internet.

Text Cryptography via Special Polynomial Technique

Discrete cryptographic such as RSA, knapsack, and discrete logarithms are the oldest and best cryptographic techniques. They are worked during finite field. In this paper, an attempt to another branch of cryptography was introduced. In this proposed method, the interpolation branch of mathematics is used for text cryptography. Proposed method was introduced to cipher the plaintext message as word by word. In encryption part, three steps are implemented for product cipher word. First, each letter of the word transformed to decimal number, then to binary. Second, the binary codeword transformed to decimal. Third, the finally cipher word is represented as triple. So it transformed to three numbers by descending the original number. A quadratic polynomial is constructed where the three numbers are represented in the coefficients of the polynomial. By choosing of three temp small independent values, three dependent values are calculated as the code word. For decryption part, the special polynomial technique is used for recover the quadratic polynomial. The rest steps are deduced conversely with the encryption part procedure

A Parallel Strategy for Solving Sparse Linear Systems Over Finite Fields

In this paper we describe a number of parallel techniques that were applied to the problem of finding the null-spaces of thousands of large sparse matrices. This collection of matrices were derived from the discrete logarithm problem attack over the finite field $\F_{3^{6\cdot 509}}$ recently carried out by Adj et al. in~\cite{Adj2017}. Our software library was mainly executed in the supercomputer ABACUS~\cite{abacus}, where in total $21,870$ large sparse linear algebra systems were processed. Solving those linear algebra problems involved a computational effort of over $138$ core-years, requiring a memory space of over 645 gigabytes to store the corresponding vector solutions.

Type-II polyadic constacyclic codes over finite fields

Polyadic constacyclic codes over finite fields have been of interest due to their nice algebraic structures, good parameters, and wide applications. Recently, the study of Type-I m-adic constacyclic codes over finite fields has been established. In this paper, a family of Type-II m-adic constacyclic codes is investigated. The existence of such codes is determined using the length of orbits in a suitable group action. A necessary condition and a sufficient condition for a positive integer s to be a multiplier of a Type-II m-adic constacyclic code are determined. Subsequently, for a given positive integer m, a necessary condition and a sufficient condition for the existence of Type-II m-adic constacyclic codes are derived. In many cases, these conditions become both necessary and sufficient. For the other cases, determining necessary and sufficient conditions is equivalent to the discrete logarithm problem which is considered to be computationally intractable. Some special cases are investigated together with examples of Type-II polyadic constacyclic codes with good parameters.

A New Key Exchange Protocol Based on Infinite Non-Abelian Groups

In order to resist quantum attacks, a key exchange protocol based on infinite non-abelian groups is proposed in this paper. For the purpose, by the composition of twice the operation of a semidirect product, we construct a shared secret key which contains two hard problems of equivalent decomposition problem (EDP) and discrete logarithm problem (DLP). Then, two methods，algebra attack and brute force attack, were employed to verify the antiattack for the proposed protocol. By a sound mathematical inference, it demonstrates that the proposed protocol possesses security positively. Finally, we analyzed the computational complexity and bit complexity when the protocol being implemented on braid groups, and furthermore, the complexity data confirm the feasibility of establishing the key exchange protocol there. Thus, in any case, security or complexity, the actual use of the proposed protocol means achievable in practice.

Multi-Recipient encryption with keyword search without pairing for cloud storage

With the rapid development of cloud computing technology and communication technology, cloud storage has become a tool used by people in daily life. Cloud storage service enables users to outsource data to cloud servers and retrieve desired document efficiently. Individual privacy in outsource data are very sensitive and should be prevented from any leakage. Public-key encryption with keyword search (PEKS) scheme resolves this tension, while public-key authentication encryption with keyword search (PAEKS) scheme improve its keyword guessing attacks problem potentially. Whereas, the loss of keyword privacy, the limitation of single user interaction and low efficiency make PEKS/PAEKS schemes far from enough in practical applications.In this paper, we develop a multi-recipient public key encryption scheme with keyword search without pairing (MREKS) for cloud storage under public key infrastructure. The proposed scheme has the merits of supporting multi-recipient keyword search way as well as requiring no expensively bilinear pairing operations under standard model. We present a concrete and efficient construction of MREKS, and prove its security based on discrete logarithm assumptions. Furthermore, we embed the algorithm of data plaintext encryption and decryption into the scheme, which makes the scheme more practical. We show that our scheme enjoys much more efficiency than previous PEKS/PAEKS scheme in the simulation experiment, especially the keyword encryption is optimized by 79.5%.

A Note on the Insecurity of Cryptosystems Based on Chebyshev Polynomials

The security of cryptosystems based on Chebyshev recursive relation, [Formula: see text], relies on the difficulty to find the large degree [Formula: see text] of Chebyshev polynomial for given parameters. But this relation cannot be used to evaluate [Formula: see text], if [Formula: see text] is very large. We investigate three other methods: matrix-multiplication-based evaluation, halve-and-square evaluation, and root-extraction-based evaluation. We find they have the same complexity [Formula: see text] over finite field [Formula: see text]. The result shows that the hardness of some cryptosystems based on modular Chebyshev polynomials is almost equivalent to the general discrete logarithm. This partially answers the question why such chaos-based cryptosystems have rarely been put into practice.

ECLAS: An Efficient Pairing-Free Certificateless Aggregate Signature for Secure VANET Communication

Vehicular ad hoc networks (VANETs) have become an important part of the intelligent transportation system that aims to promote communication among vehicles to ensure vehicular safety and improve the driving experience. In VANETs, signed messages need to be authenticated by roadside units in a very short time. Certificateless aggregate signature (CLAS) scheme has been considered as one of the promising solutions to solve the problem of limited network bandwidth and computing power in VANETs environment. In this article, we propose an efficient pairing-free CLAS, named eCLAS, which is suitable for communication between vehicle to infrastructure. The aggregated signature allows individual signature on different messages from different vehicles to be aggregated into one short signature. In the random oracle model, the security of the scheme is proved by the adaptive selection message attack and the difficulty of computation of discrete logarithm problem on the elliptic curve. Additionally, the security analysis shows that the proposed scheme can meet security requirements in VANETs. Compared with existing schemes, the result demonstrates that our scheme has obvious advantages in signature verification, thereby granting it better applicability.

The State of the Art in Integer Factoring and Breaking Public-Key Cryptography

In this column, we will review the current state of the art of cryptanalysis for three number-theoretic problems using classical (nonquantum) computers, including, in particular, our most recent computational records for integer factoring and prime-field discrete logarithms.

Number-Theoretical Methods of Factoring Composite Numbers and Calculating the Discrete Logarithm

Number-Theoretical Methods of Factoring Composite Numbers and Calculating the Discrete Logarithm

Implementation Of Hybrid Cryptography Algorithm RSA-CRT And Play Fair Cipher In Image File Security

The development of technology currently allows everyone to exchange information without any limitations of time and distance. The possibility that will happen is the existence of data leakage during the process of information exchange conducted. Therefore in the data transmission, especially image, security aspects, confidentiality and efficiency of data storage are necessary. If the important information falls into the wrong hands, it will cause undesirable things, for example image manipulation with a negative shape and can harm the image owner. One of the methods used to maintain the security and confidentiality of the data one of the techniques used is RSA-CRT (Rivest Shamir Adleman-Chinese Remainder Theorem). The strength of this algorithm lies in the difficulty of calculating discrete logarithms in prime integer groups in which multiplication operations are performed. In this study, the authors combined between Playfair Cipher to encrypt image encryption and RSA-CRT Algorithm to encrypt the keys of Playfair Cipher. The image was first encrypted using the Playfair Cipher Algorithm, then the Playfair key was encrypted using the RSA- CRT Algorithm. Through testing, it can be concluded that the result of image encryption using Playfair Cipher algorithm is random and can secure image file then the Playfair key was encrypted using the RSA- CRT Algorithm. Through testing, it can be concluded that the result of image encryption using Playfair Cipher algorithm is random and can secure image file then the Playfair key was encrypted using the RSA- CRT Algorithm. Through testing, it can be concluded that the result of image encryption using Playfair Cipher algorithm is random and can secure image file

The discrete logarithmic Minkowski problem for q-capacity

The logarithmic Minkowski problem for q-capacity asks for necessary and sufficient conditions for a finite Borel measure on the unit sphere so that it is the logarithmic q-capacitary measure of a convex body. This paper solves the case of discrete measures whose supports are in general position.

Two-Party Signing For ISO/IEC Digital Signature Standards

Abstract Two-party signing can be used to provide a high level of key protection especially in the blockchain systems where the safety of money relies on the safety of the signing key. With a two-party signing protocol, the signing key is distributed among two devices, thus the funds are safe as long as one device remains uncorrupted. In this paper, we study the two-party signing protocols for all ISO/IEC signature standards. The mechanisms based on elliptic curve discrete logarithm in ISO/IEC can be divided into three types: Schnorr-type, Elliptic Curve Digital Signature Algorithm (ECDSA)-type and SM2-type. There have already been efficient two-party protocols based on Schnorr signature scheme which can be easily extended into all Schnorr-type standards. However, it is particularly hard to construct efficient distributed SM2-type and ECDSA-type protocols due to their nonlinear signing equations. In this paper, we present the first secure and efficient two-party protocol over SM2-type signature standard. We prove its security in the generic group model. We then construct a more efficient two-party ECDSA protocol that is secure in the generic group model and outperforms all previous works.

How did women\u2019s education and differential family planning policies shape transition to second and third births in historical China? New evidence from micro data

The influence of China’s family planning policies on fertility transition is widely acknowledged in research studies. However, little is known about how improvements in women’s education have shaped reproductive decisions of Chinese women across different family planning regimes, particularly at micro level. This study uses retrospective pooled birth history data from five consecutive population and family planning surveys collected over the period 1982–2006 to systematically examine the interrelationship between family planning policies and women’s education, and their interactive effect on the second and third birth transitions. We hypothesize that family planning policies had a differential influence on educational groups in reducing the transition to second and third births. The results from discrete time complementary log–log survival models provide strong evidence of differential reproductive behavior of education groups across time in China, and the simultaneous influence of women’s education and family planning policies in lowering risks to higher parities. The rates of progression to second and third births tend to be lower after the introduction of rigid family planning policies, and more importantly, the policy impact persisted even after adjusting for demographic and socioeconomic factors. The increase in women’s education overall had a depressing effect on transition to higher parities, and family planning policies implemented overtime have had differential effects on women from different educational groups. The findings show that both family planning policies and women’s education have been instrumental in shaping fertility behavior in China.

A constant-size lattice-based partially-dynamic group signature scheme in quantum random oracle model

A group signature allows a group member who signs the message on behalf of the other group members. Any recipient also verifies the signature without knowing the signer. However, it is possible to reveal the signer’s identity since the group signatures have the tracing mechanism. In general, the group signature size increases the number of group members. However, the schemes are also proposed with the constant signature size. On the other hand, the security of some schemes are based on the discrete logarithm and factorization problems while some schemes which are secure against quantum attacks are proposed. To the best of our knowledge, the signature scheme in the group concept has not been proposed on lattices in quantum oracles. In this paper, we propose a new constant-size and partially-dynamic group signature scheme on lattice assumptions. We combine the Dilithium signature scheme, a Stern-like zero-knowledge scheme and Regev’s encryption scheme. We give a zero-knowledge argument system for our proposed scheme and security analysis in the quantum random oracle model. According to the comparison results, the proposed scheme has better size complexity in asymptotic concept.

A Hybrid Design of Linkable Ring Signature Scheme with Stealth Addresses

Blockchain is a transformational technology which affects finance, Internet, and politics. However, many privacy protection problems for blockchain are waiting to be solved. In this study, we propose a novel linkable ring signature scheme with stealth addresses, which enables the payer and payee of the transaction to be anonymous and unlinkable in the cryptocurrency. The scheme is combined with an elliptic curve discrete logarithm (ECD logarithm)-based key encapsulation mechanism (KEM) stage and a lattice-based signature stage. The master public key and master secret key are much smaller compared with the previous scheme. Complete secure proof of the scheme is also presented in this study.