Use Of Elliptic Curves Research Articles

FOR PSEUDORANDOM SEQUENCES BASED ON ELLIPTIC CURVE ISOGENIES GENERATING METHOD

The pseudorandom sequences generation is a cryptographic systems fundamental aspect that affects cryptographic strength. One of these sequences advanced generating methods involves the use of elliptic curves (ECs), in particular by exploiting the isogeny properties of ECs. This approach not only improves the security features of cryptographic algorithms, but also ensures efficiency and reliability in the generation process. The use of isogenic transformations - morphisms between elliptic curves that preserve their group structure - further enriches the technique by introducing complex algebraic operations that are difficult to solve. Recent research has detailed the effectiveness of pseudorandom sequence generators based on elliptic curves. Methods have been developed that exploit the properties of elliptic curves over finite fields to generate sequences with low correlation and high linear complexity. There is also another approach that uses linear shift feedback registers (LFSRs) in combination with elliptic curve points to generate pseudorandom sequences. The new obtained method makes it possible to increase the number of internal states of the Dual_EC_DRBG generator by √n times, where n is the number of cyclic subgroups of simple order of the initial curve. This increases the complexity of disclosing the law of formation of the DRBG by an attacker. The application of the developed method also allows to avoid the existing disadvantages of Dual_EC_DRBG The article investigates the use of EC isogenies in the generation of pseudorandom sequences, considering their potential for improving cryptographic strength. By means of a detailed analysis of the algebraic structure and properties of these transformations, a method for PSPs generating is developed that can potentially provide advantages over existing methods in terms of security and efficiency in the transition period to post-quantum cryptography.

Стандартизація еліптичних кривих: аналіз та впровадження в криптографічні протоколи

The purpose of the article is to consider the current state of elliptic cryptography, the prerequisites for its use, as well as the requirements of modern standards related to the use of elliptic cryptography The use of elliptic curves in cryptography is considered one of the most promising areas of development of modern security algorithms. This mathematical approach is based on the complexity of solving the discrete logarithm problem in a group of points of an elliptic curve over a finite field. The use of cryptography on elliptic curves allows you to ensure the security of data exchange using effective encryption algorithms and the creation of digital signatures (DI). This study examines elliptic curves for cryptographic purposes, and provides basic operations on the point group of elliptic curves. Special attention is paid to Elliptic curve Diffie-Hellman (ECDH) and Elliptic Curve Digital Signature Algorithm (ECDSA) key exchange algorithms. The standards regulating the use of elliptic curves in cryptographic systems are also analyzed, and the advantages of this cryptographic paradigm compared to the main asymmetric algorithms are considered. Potential threats and vulnerabilities of cryptographic algorithms based on elliptic curves are investigated. Examples of popular standardized curves recommended by relevant organizations, such as NIST, used in real-world cryptographic applications are also provided. Elliptic curve cryptography (ECC) is currently one of the foundations for the development of modern public-key cryptographic algorithms. ECC has gained recognition in cryptography for providing a high level of security with shorter key lengths (compared to other cryptographic approaches), high speed, resource savings, and versatility, giving it an advantage over other methods such as RSA and others. It provides a secure network connection, generates secret keys for TLS servers and their clients, and is also used to create digital signatures that guarantee the authenticity of transactions in cryptocurrency systems.

ELLIPTIC CURVE CRYPTOGRAPHY AND ITS PRACTICAL APPLICATION

Elliptic curves are one of the most promising tools for constructing modern cryptographic algorithms. The security of elliptic curve cryptography is based on the complexity of solving the discrete logarithm problem in the group of points of the elliptic curve over a finite field. Elliptic curve cryptography enables two parties communicating over public channel using elliptic curve encryption and signing algorithms. Elliptic curves allow to achieve the same level of security with small key sizes than other asymmetric cryptographic algorithms. The article describes the mathematical apparatus of elliptic curves used for cryptographic purposes, the basic operations in the group of points of elliptic curves, such as addition of points, doubling of a point, and scalar multiplication of a point by a number are given. The steps and principles of the Diffie-Hellman key exchange algorithm (ECDH) and the digital signature scheme (ECDSA) on elliptic curves are considered. An overview of standards establishing recommendations and requirements for the use of elliptic curves in cryptographic systems is provided. The advantages of elliptic curve cryptography compared to traditional asymmetric algorithms, such as smaller key sizes, computational speed, and efficient use of resources, are analyzed. Potential threats and vulnerabilities of cryptographic algorithms based on elliptic curves are discussed. The main practical application areas of cryptographic algorithms on elliptic curves, including network security, cryptocurrency operations, message exchange, the Internet of Things, and government institutions are investigated. Examples of popular standardized curves (Curve25519, Curve448, secp256k1) that have been tested and recommended by specialized organizations such as NIST are given.

Building cryptographic schemes based on elliptic curves over rational numbers

The possibility of using elliptic curves over the rational field of non-zero ranks in cryptographic schemes is studied. For the first time, the construction of cryptosystems is proposed the security of which is based on the complexity of solving the knapsack problem on elliptic curves over rational numbers of non-zero ranks. A new approach to the use of elliptic curves for cryptographic schemes is proposed. A few experiments have been carried out to estimate the heights characteristic of points on elliptic curves of infinite order. A model of a cryptosystem resistant to computations on a quantum computer and based on rational points of an infinite order curve is proposed. A study of the security and effectiveness of the proposed scheme has been carried out. An attack on the secret search in such a cryptosystem is implemented and it is shown that the complexity of the attack is exponential. The proposed solution can be applied in the construction of real cryptographic schemes as well as cryptographic protocols.

A New Fail-Stop Group Signature over Elliptic Curves Secure against Computationally Unbounded Adversary

If an adversary has unlimited computational power, then signer needs security against forgery. Fail Stop signature solves it. If the motive of the signature is to hide the identity of the signer who makes signature on behalf of the whole group then solution is Group signature. We combine these two features and propose “A new Fail Stop Group Signature scheme (FSGSS) over elliptic curves”. Security of our proposed FSGSS is based on “Elliptic curve discrete logarithm problem” (ECDLP). Use of elliptic curve makes our proposed FSGSS feasible to less bandwidth environment, Block chains etc. Due to security settings over elliptic curves, efficiency of proposed scheme increases in terms of computational complexity.

Design of Nonlinear Components Over a Mordell Elliptic Curve on Galois Fields

Elliptic curve cryptography ensures more safety and reliability than other public key cryptosystems of the same key size. In recent years, the use of elliptic curves in public-key cryptography has increased due to their complexity and reliability. Different kinds of substitution boxes are proposed to address the substitution process in the cryptosystems, including dynamical, static, and elliptic curve-based methods. Conventionally, elliptic curve-based S-boxes are based on prime field but in this manuscript; we propose a new technique of generating S-boxes based on mordell elliptic curves over the Galois field . This technique affords a higher number of possibilities to generate S-boxes, which helps to increase the security of the cryptosystem. The robustness of the proposed S-boxes against the well-known algebraic and statistical attacks is analyzed to classify its potential to generate confusion and achieve up to the mark results compared to the various schemes. The majority logic criterion results determine that the proposed S-boxes have up to the mark cryptographic strength.

Number of rational points of elliptic curves

In the modern theory of elliptic curves, one of the important problems is the determination of the number of rational points on an elliptic curve. The Mordel–Weil theorem [T. Shioda, On the Mordell–Weil lattices, Comment. Math. University St. Paul. 39(2) (1990) 211–240] points out that the elliptic curve defined above the rational points is generated by a finite group. Despite the knowledge that an elliptic curve has a final number of rational points, it is still difficult to determine their number and the way how to determine them. The greatest progress was achieved by Birch and Swinnerton–Dyer conjecture, which was included in the Millennium Prize Problems [A. Wiles, The Birch and Swinnerton–Dyer conjecture, The Millennium Prize Problems (American Mathematical Society, 2006), pp. 31–44]. This conjecture uses methods of the analytical theory of numbers, while the current knowledge corresponds to the assumptions of the conjecture but has not been proven to date. In this paper, we focus on using a tangent line and the osculating circle for characterizing the rational points of the elliptical curve, which is the greatest benefit of the contribution. We use a different view of elliptic curves by using Minkowki’s theory of number geometry [H. F. Blichfeldt, A new principle in the geometry of numbers, with some applications, Trans. Amer. Math. Soc. 15(3) (1914) 227–235; V. S. Miller, Use of elliptic curves in cryptography, in Proc. Advances in Cryptology — CRYPTO ’85, Lecture Notes in Computer Science, Vol. 218 (Springer, Berlin, Heidelberg, 1985), pp. 417–426; E. Bombieri and W. Gubler, Heights in Diophantine Geometry, Vol. 670, 1st edn. (Cambridge University Press, 2007)].

Design optimization of hydrodynamic coupling applying response surface method combined with CFD technique

Hydrodynamic coupling is a type of fluid machinery that uses fluid kinetic energy to transmit power. In this paper, meridional profile optimization of fully filled fluid coupling was performed to improve transmitted torque. The response surface method, a global optimization method, combined with CFD was introduced to find optimum meridional profile of blades. The torque coefficient was applied as an objective function. The response surface method was adopted with three design variables: hydraulic diameter ratio of blades, central core position, and meridional profile of the hub. Results showed that the best diameter ratio occurred at d/D =0.67. To specify an optimal position for the central core, a dimensionless parameter (b/d) was introduced. With regression analysis, it was found that the optimal point occurs at about b/d =0.37. It was confirmed with the results of contour and vector plots. Three types of curves (circular, elliptical, and cubic Bezier curve) were employed to optimize the meridional shape of blades. The circular curve was applied in the design of the hub in the reference geometry. Relative to the circular curve hub, the use of elliptic curve had no significant effect on the objective function. Two dimensionless parameters, l1/d and l2/d, were defined to investigate the effect of Bezier curves on the performance of hydrodynamic coupling. The optimum values for design variables of l1/d and l2/d were obtained 0.80 and 0.88, respectively.

Implementation of Double Fold Text Encryption Based on Elliptic Curve Cryptography (ECC) with Digital Signature

As the internet provides access to millions of communications in every second around the world, security implications are tremendously increasing. Transfer of important files like banking transactions, tenders, and e commerce require special security and authenticated mechanism in its journey from the sender to the receiver. Recent attention of cryptography is mainly focused on use of elliptic curves in public key cryptosystems. The present paper explains an innovative public key cryptographic scheme for protecting sensitive of critical information using elliptic curve over finite field. This mechanism besides providing the robustness of the cipher contributes the authentication of the message with digital signature.

НОВІ АЛГОРИТМИ ЗНАХОДЖЕННЯ БАЗОВОЇ ТОЧКИ НА ЕЛІПТИЧНИХ КРИВИХ У ФОРМІ ЕДВАРДСА

Transformations on  elliptic  curves which are  used in the national digital signature standard DSTU 4145 2002, satisfy  modern requirements. However, the fast development of computer technologies and a significant interest in cryptology worldwide have led to an increase in research, the constant emergence of new powerful cryptanalysis methods and, as a consequence, to the possible shortening of the lifetime of existing and new algorithms. This article addresses the current scientific and practical problem of investigating the properties of elliptic curves in the Edwards form over a finite field , p ≠ 2, suitable for use in asymmetric cryptosystem, in particular, digital signature algorithms. Based on the research completed, new ways of determination of a base point on Edwards curves were outlined and described.  Three new algorithms were proposed  for determination of the base point for constructing a cryptosystem on the full and twisted Edwards  curves. In this work the comparative analysis of the performance of the developed algorithms of the Edwards curves base point determination and the performance of crypto-algorithms on the non-perpendicular elliptic curves in the Weierstrass form over the fields of characteristic 2 was carried out. The analysis shows that proposed algorithms are faster than the standard Weirstrass digital signature curve algorithm – respectively, the first algorithm – 180 times, the second–times, and the third algorithm– (times. It is proved that the use of elliptic curves in the form of Edwards over finite  field , instead of Weierstrass curves, can increase the speed of operations of adding  points in asymmetric cryptosystems. The results of the work can be applied to the analysis of existing problems and creation of new algorithms and standards of asymmetric cryptography.

Elliptic-Curves Cryptography on High-Dimensional Surfaces

We discuss the use of elliptic curves in cryptography on high-dimensional surfaces. In particular, instead of a key exchange protocol written in the form of a bi-dimensional row, where the elements are made up with 256 bits, we propose a Diffie-Hellman key exchange protocol given in a matrix form, with four independent entries each of them constructed with 64 bits. Apart from the great advantage of significantly reducing the number of used bits, this methodology appears to be immune to attacks of the style of Western, Miller, and Adleman, and at the same time it is also able to reach the same level of security as the cryptographic system presently obtained by the Microsoft Digital Rights Management. A nonlinear differential equation (NDE) admitting the elliptic curves as a special case is also proposed. The study of the class of solutions of this NDE is in progress.

Secure and Efficient Secret Sharing Scheme with General Access Structures Based on Elliptic Curve and Pairing

Secret sharing techniques allow a secret to be shared among n participants in such a way that a specified subset of participant can reconstruct the secret by combining their shares. Secret sharing techniques are now the building blocks of several security protocols. Shamir's (t, n) threshold secret sharing scheme is one in which t or more participant can join together to retrieve the secret from the set of n participants. Traditional single secret sharing schemes are modified and generalized to share multiple secrets. Use of Elliptic Curve and Pairing in secret sharing is gaining more importance. The use of Elliptic Curve helps to improve the security and also the computational complexity is reduced. In this paper we propose a secret sharing scheme with a monotone generalized access structure. The scheme makes use of Shamir's scheme and Elliptic Curve pairing. The shares are chosen by the participant itself. So the consistency of the shares are ensured in this scheme. The participant shares remain secret during the reconstruction phase and this provides multi use facility where the same share can be used for the reconstruction of multiple secret. The shared secret, access structure or the participant set can be modified without updating the secret shadow of the participant. This provides dynamism and adds more flexibility to the scheme. The combiner can also verify the shares submitted by the participants during the reconstruction phase in order to identify the cheaters. The cheating detection and cheater identification is done by using bilinear pairing. This proposed scheme is simple and easy to implement compared with other generalized multi secret sharing scheme with extended capabilities using pairing.

A Literature Review on Scalar Recoding Algorithms in Elliptic Curve Cryptography

Elliptic Curve Cryptosystem (ECC) is a type of public key cryptography (PKC) based on the algebraic structure of elliptic curve over finite fields. In mid 80s Neal Koblitz and Victor Miller independently proposed the use of elliptic curves in cryptography. For a smaller key size, ECC is able to provide same level of security with RSA. This feature made ECC one of the most popular PKC algorithms today. Scalar multiplication is known as the fundamental operation in ECC algorithm and protocols. The efficiency of ECC is critically depends on efficiency of scalar multiplication operation. Scalar multiplication involves with three levels of computations: scalar arithmetic, point arithmetic and field arithmetic. Improving the first two levels will lead to significant increment in efficiency of scalar multiplication. Scalar arithmetic level can improve by employing an enhanced scalar recoding algorithm that can reduce the Hamming weight or de-crease the number of operations in the scalar representation process. This paper reviews some of the recoding algorithms and techniques.

A Pairing-free ID-based Key Agreement Protocol with Different PKGs

This paper proposes an identity based key agreement protocol based on elliptic curve cryptography (ECC) between users of different networks with independent private key generations (PKGs). Instead of bilinear pairings which commonly used for contracting identity based schemes, the proposed protocol makes use of elliptic curves to obtain more computational efficiency. The proposed protocol develops Cao et al’s protocol for situations that two users of independent organizations or networks with separate servers (that in this article, are named PKGs, based on their main duty, generating private keys for the users) want to share a secret key via an insecure link. The main novelty of this paper is security proof of the proposed protocol in the random oracle model. The security proof argues the security attributes of the proposed protocol.

Assessing the effectiveness of artificial neural networks on problems related to elliptic curve cryptography

Cryptographic systems based on elliptic curves have been introduced as an alternative to conventional public key cryptosystems. The security of both kinds of cryptosystems relies on the hypothesis that the underlying mathematical problems are computationally intractable, in the sense that they cannot be solved in polynomial time. In this paper, we study the performance of artificial neural networks on the computation of a Boolean function derived from the use of elliptic curves in cryptographic applications.

Factoring Integers with Elliptic Curves

This paper is devoted to the description and analysis of a new algorithm to factor positive integers. It depends on the use of elliptic curves. The new method is obtained from Pollard's (p - 1)-method (Proc. Cambridge Philos. Soc. 76 (1974), 521-528) by replacing the multiplicative group by the group of points on a random elliptic curve. It is conjectured that the algorithm determines a non-trivial divisor of a composite number n in expected time at most K( p)(log n)2, where p is the least prime dividing n and K is a function for which log K(x) = /(2 + o(1))log x log log x for x -x o. In the worst case, when n is the product of two primes of the same order of magnitude, this is