SPECIFICS OF IMPLEMENTATION OF THE ASYMMETRIC ENCRYPTION ALGORITHM ON ELLIPTIC CURVES

Elliptic curves cryptography was introduced independently by Victor Miller (Miller, 1986) and Neal Koblitz (Koblitz, 1987(Koblitz, ) in 1985. .At that time elliptic curve cryptography was not actually seen as a promising cryptographic technique.As time progress and further research and intensive development done especially on the implementation side, elliptic curve cryptography is now being implemented widely.Elliptic curves cryptography offers smaller key size, bandwidth savings and faster in implementations when compared to the RSA (Rivest-Shamir-Adleman) cryptography which based its security on the integer factorization problem.The most interesting feature of the elliptic curves is the group structure of the points generated by the curves, where points on the elliptic curves form a group.The security of elliptic curves cryptography relies on the elliptic curves discrete logarithm problem.The elliptic curve discrete logarithm problem is analogous to the ordinary algebraic discrete logarithm problem, l = g x , where given the l and g, it is infeasible to compute the x.Elliptic curve discrete logarithm problem deals with solving for n the relation P = nG.Given the point P and the point G, then it is very hard to find the integer n.To implement the discrete logarithm problem in elliptic curve cryptography, the main task is to compute the order of group of the curves or in other words the number of points on the curve.Computation to find the number of points on a curve, has given rise to several point counting algorithms.The Schoof and the SEA (Schoof-Elkies-Atkin) point counting algorithms will be part of the discussion in this chapter.This chapter is organized as follows: Section 2, gives some preliminaries on elliptic curves, and in section 3, elliptic curve discrete logarithm problem is discussed.Some relevant issues on elliptic curve cryptography is discussed in section 4, in which the Diffie-Hellman key exchange scheme, ElGamal elliptic curve cryptosystem and elliptic curve digital signature scheme are discussed here accompanied with some examples.Section 5 discussed the two point counting algorithms, Schoof algorithm and the SEA (Schoof-Elkies-Atkin) algorithm.Following the discussion in section 5, section 6 summaries some similarities and the differences between these two algorithms.Section 7 gives some brief literature on these two point counting algorithms.Finally, section 8 is the concluding remarks for this chapter. Elliptic curvesElliptic curves obtained their name from their relation to elliptic integrals that arise from the computation of the arc length of ellipses (Lawrence & Wade, 2006).Elliptic curves are www.intechopen.com Cryptography and Security in Computing1 1 2 3 4 2 2 4 Fig. 1.1. 2(1 ) (1 ) yx x x   www.intechopen.com

Elliptic curve cryptography (ECC) has shorter key length than other asymmetric cryptography algorithms such as RSA with the same security level. Existing faults in cryptographic computations can cause faulty results. If a fault occurs during encryption, false information will be sent to the destination, in which case channel error detection codes are unable to detect the fault. In this paper, we consider the error detection in elliptic curve scalar multiplication point, which is the most important operation in ECC. Our technique is based on non-linear error detection codes. We consider an algorithm for scalar multiplication point proposed by Microsoft research group. The proposed technique in our methods has less overhead for additions (36.36%) and multiplications (34.84%) in total, compared to previous works. Also, the proposed method can detect almost 100% of injected faults.

In today’s advanced network and digital age, the Internet of Things network is experiencing a significant growing trend and, due to its wide range of services and network coverage, has been able to take a special place in today’s technology era. Among the applications that can be mentioned for this network are the field of electronic health, smart residential complexes, and a wide level of connections that have connected the inner-city infrastructure in a complex way to make it smart. The notable and critical issue that exists in this network is the extent of the elements that make up the network and, due to this, the strong and massive data exchanges at the network level. With the increasing deployment of the Internet of Things, a wide range of challenges arise, especially in the discussion of establishing network security. Regarding security concerns, ensuring the confidentiality of the data being exchanged in the network, maintaining the privacy of the network nodes, protecting the identity of the network nodes, and finally implementing the security policies required to deal with a wide range of network cyber threats are of great importance. A fundamental element in the security of IoT networks is the authentication process, wherein nodes are required to validate each other’s identities to ensure the establishment of secure communication channels. Through the enforcement of security prerequisites, in this study, we suggested a security protocol focused on reinforcing security characteristics and safeguarding IoT nodes. By utilizing the security features provided by Elliptic Curve Cryptography (ECC) and employing the Elliptic Curve Diffie–Hellman (ECDH) key-exchange mechanism, we designed a protocol for authenticating nodes and establishing encryption keys for every communication session within the Internet of Things. To substantiate the effectiveness and resilience of our proposed protocol in withstanding attacks and network vulnerabilities, we conducted evaluations utilizing both formal and informal means. Furthermore, our results demonstrate that the protocol is characterized by low computational and communication demands, which makes it especially well-suited for IoT nodes operating under resource constraints.

A promising direction for constructing cryptographically stable pseudo-random sequence generators is an application of transformations in a group of points of elliptic and hypereliptic curves. This will allow building evidence-stable crypto algorithms, the problem of finding the private key in which is associated with solving a theoretically complex elliptic curve discrete logarithm problem. This paper proposes a method for generating pseudo-random sequences of the maximal period using transformations on elliptic curves. This method consists in the application of recurrent transformations with sequential formation of elements of points group of elliptic curves. This allows providing the maximum period of pseudo-random sequences with the reduction of the problem of finding the private key to the solution of the theoretically complex elliptic curve discrete logarithm problem. The block diagram of the device for generating pseudo-random sequences and the scheme for generating the internal states of the generator are given. We also present the results of statistical testing of some generators, which show that the generated sequences are indistinguishable in a statistical sense from truly random ones.KeywordsPseudo-random sequencesElliptic curvesMaximum period

Wireless sensor networks (WSNs) are largely deployed in different sectors and applications, and Elliptic Curve Cryptography (ECC) is proven to be the most feasible PKC for WSN security. ECC is believed to provide same level of security such as RSA with a much shorter key length, and thus they seem to be ideal for applications with small resources such a sensor network, smartcard, RFID, etc. However, like any other cryptographic primitive, ECC implementations are vulnerable to Power Analysis Attacks (PAAs) that may reveal the secret keys by exploiting leaked power consumption from running cryptographic devices (e.g. smart cards, mobile phones etc.). In this paper, we present a comprehensive study of major PAAs and its countermeasures on ECC cryptosystems. In addition, this paper describes critical concerns to be considered in designing PAAs on ECC particular for WSNs, and illustrates the need to conduct, in the near future, intensive researches for the development of these specific PAAs.

This paper examines the cryptographic security of fixed versus random elliptic curves over the field GF(p). Its basic assumption is that a large precomputation to aid in breaking the elliptic curve discrete logarithm problem (ECDLP) can be made for a fixed curve. We take this into account when examining curve security as well as considering a variation of Pollard's rho method where computations from solutions of previous ECDLPs can be used to solve subsequent ECDLPs on the same curve. We present a lower bound on the expected time to solve such ECDLPs using this method, as well as an approximation of the expected time remaining to solve an ECDLP when a given size of precomputation is available. We conclude that adding 5 bits to the size of a fixed curve to avoid general software attacks and an extra 6 bits to avoid attacks on special moduli and a parameters provides an equivalent level of security.

There are many threats from virus, hacker, electronic eavesdropper, and electronic deceiver in the network. The growth of the computer system and network makes the organizations and personal users to depend on the information that circulate in these systems more and more. This makes us be aware of that protecting the data and resources from leak, insuring the reliability of data and information, and protecting the system from suffering the hit from network are very important.Submit a cryptography that is affirmative by people to reach the demand on the safety of SET system. The algorithm submitted is public, and everyone can use and to verify its safety, using Elliptic Curve Cryptography to execute the key exchange and encryption decryption, and making sure its performance of SET is better while the safety of SET is same as before, and the overload of compute and transmission is yet lower.

The elliptic curve cryptography is almost used in the literature to develop the public key cryptosystems. A new general-purpose symmetric key cryptosystem using elliptic prime curves is proposed in this paper. For a given prime number P, there are many elliptic prime curves on which each point in the quadrant (0,0) to (P-1,P-1) lays. The set of minimum number of eligible elliptic prime curves for a given prime number P constitutes a lookup table that is the secret key of the proposed cryptosystem. The sender can pick up one eligible elliptic curve for each 2n-bit plaintext block. The sender sends two integers cipher for each 2n-bit plaintext block with encryption rate = 0.8. Moreover, a small prime number can be used efficiently with high security and good robustness against brute force attack.

This article mainly discusses the history of progress regarding the legislation and updates through scrutiny, its current architecture, and its standards applied around the world, such as eIDAS and eIDAS 2.0, which are used not only as technical but also as purely political standards. The penetration of the digital economy has largely promoted the evolution of cyber platforms for public and private use, so that e-signatures have become one of the pillars in the national IT architecture. 1) Legislation The e-signature operations in Türkiye are regulated by Electronic Signature Law No. 5070 and its secondary regulations. However, today, the infrastructure groundwork relies heavily on RSA-based cryptography. RSA is attracting greater and more debate these days, with concerns about whether it will continue to be useful given the increasing importance of security requirements for mobile devices, performance expectations, and the threat posed by quantum computing. This article also examines the extent to which Turkish law is compatible with the EU regulation of compliance. In this paper, we study whether the Turkish legislation complies with the European legislation. This covers security, performance, key size, and support for RSA and Elliptic Curve Cryptography (ECC) algorithms. This paper discusses technical, institutional, and legal requirements for the transition of the national e-signature infrastructure to ECC. Furthermore, ECC’s contribution is evaluated in terms of Post-Quantum Cryptography (PQC), and a roadmap is suggested for transitioning to hybrid (ECC + PQC) architectures. Finally, policy and technical suggestions are proposed to facilitate international electronic and mobile signature interoperability under eIDAS 2.0.

The need to establish safer communication channels in a world where technological development is progressing in leaps and bounds is indispensable. Thus, implementing cryptographic algorithms, which are more complex to compromise, improves the possibilities of securing our sensitive data. In this paper, the authors analyze the algorithmic foundations and perform a comparative analysis of the traditional public-key cryptographic algorithms (e.g., RSA, ElGamal, Schnorr, DSA) and elliptic curve cryptography with NIST recommended curves. In the study, they focus on six different security strengths: 80-, 96-, 112-, 128-, 192-, and 256-bit key sizes. Moreover, this study provides a benchmark among different curves (NIST, SEC2, and IEFT Brainpool) that can be used with various security levels. The authors study and compare the characteristics and performance of the traditional asymmetric algorithms and the elliptic curve algorithms for information security. The results obtained in this study will be graphically visualized through statistical graphs and tables with quantification response times.

The reach of algebraic curves in cryptography goes far beyond elliptic curve or public key cryptography yet these other application areas have not been systematically covered in the literature. Addressing this gap, Algebraic Curves in Cryptography explores the rich uses of algebraic curves in a range of cryptographic applications, such as secret sh

Elliptic curve cryptography was invented by Neal Koblitz and Victor Miller in 1985. However, for many years its security was viewed with suspicion by cryptographers. These fears were not entirely meritless, especially in light of the subsequent (and ongoing) development of innovative algorithms for solving special instances of the elliptic curve discrete logarithm problem (ECDLP) and the discrete logarithm problem in the divisor class groups of other families of algebraic curves. Surprisingly, cryptographers seem to have cast aside their fears after the invention of pairing-based cryptography in 2000. I will overview progress made on algorithms for solving the ECDLP and related problems, and will examine the long and curious struggle for ECC acceptance.

Secure audio transmission is crucial in military, telemedicine, and IoT multimedia, where confidentiality and integrity are vital. Traditional methods, such as RSA, are computationally intensive, while DNA cryptography is efficient but less secure. This paper proposes an audio encryption and decryption scheme using Elliptic Curve Cryptography (ECC). In the proposed method, a pair of audio signals is taken as plaintext, and corresponding sample points are combined into ordered pairs. Some points fall outside the elliptic curve and cannot be directly decrypted; these are preserved in a sparse vector. The encrypted vector and sparse vector are transmitted through a noiseless channel, and at the receiving end, they are combined to ensure complete recovery of the original signals. Appropriate ECC parameters are selected to minimize the number of ambiguous points, allowing for 100% accurate recovery. The performance of ECC is compared with RSA and DNA cryptography in terms of processing time and security level. Additionally, encryption rigidity is evaluated using cross-correlation, discrete wavelet transform (DWT) spectral analysis, and fuzzy entropy. Results demonstrate that ECC achieves the strongest security among the three approaches, albeit with higher processing complexity, whereas DNA is most suitable for real-time applications due to its efficiency.

The Residue Number System (RNS) is a numeral representation enabling for more efficient addition and multiplication implementations. However, due its non-positional nature, modular reductions, required for example by Elliptic Curve (EC) Cryptography (ECC), become costlier. Traditional approaches to RNS modular reduction resort to the Montgomery algorithm, underpinned by large basis extensions. Recently, Hybrid-Positional Residue Number Systems (HPRs) have been proposed, providing a trade-off between the efficiency of RNS and the flexibility of positional number representations. Numbers are represented in a positional representation with the coefficients represented in RNS. By crafting primes of a special form, the complexity of reductions modulo those primes is mitigated, relying on extensions of smaller bases. Due to the need of crafting special primes, this approach is not directly extensible to group operations over currently standardised elliptic curves. In this paper, the Hybrid-Polynomial Residue Number System (HyPoRes) is proposed, enabling for improved modular reductions for any prime. Experimental results show that the modular reduction of HyPoRes, although at most 1.4 times slower than HPR for HPR-crafted primes, is up to 1.4 times faster than a generic RNS approach for primes of ECC standards.

Due to lack of effective trust management mechanism, heaps of deceptions exist in peer-to-peer (P2P) e-commerce (EC) environments, which seriously damage authenticity and availability of the systems. There are various challenges that are faced in the open autonomous environments. These existing challenges are mostly owing to the scarcity of correct authentication in P2P networks. Generally speaking, structured P2P networks are assumed to be accessible by allowing heterogeneous nodes or clients to interact and share one another. The identity (ID) authentication problem for this type of network has now become important. The paper presents a specific authentication key exchange scheme for P2P networks. Theoretic analysis shows that, a robust evaluation scheme based on elliptic curve cryptography (ECC) is proposed for P2P e-commerce networks with better secure identifying-and-authenticating features.