Elliptic Curve Protocols

The use of public-key cryptosystems (PKC) received considerable attention. They are beneficial in encryption as well as signing which play an essential role in electronic banking and financial transactions. Elliptic Curve Cryptography (ECC) is one of best public key techniques because of its small key size and high security. Secure applications in smart cards present implementation challenges particular to the platform's memory, bandwidth, and computation constraints. ECC's unique properties make it especially well suited to smart card applications. ECC systems provide the highest strength per bit of any cryptosystem known today. The paper presents a new method for smart card implementation of elliptic curves, explaining how ECC can not only significantly reduce the cost, but also accelerate the deployment of smart cards in new applications. ECC permits reductions in key and certificate size that translate to smaller memory requirements (especially for EEPROM), which represent significant cost savings. This added functionality can play an effective role in electronic payment and online banking technologies.

In the past few years elliptic curve cryptography has moved from a fringe activity to a major challenger to the dominant RSA/DSA systems. Elliptic curves offer major advances on older systems such as increased speed, less memory and smaller key sizes. As digital signatures become more and more important in the commercial world the use of elliptic curve-based signatures will become all pervasive. This book summarizes knowledge built up within Hewlett-Packard over a number of years, and explains the mathematics behind practical implementations of elliptic curve systems. Due to the advanced nature of the mathematics there is a high barrier to entry for individuals and companies to this technology. Hence this book will be invaluable not only to mathematicians wanting to see how pure mathematics can be applied but also to engineers and computer scientists wishing (or needing) to actually implement such systems.

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

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

The data everywhere was being transferred for each second using domain of Internet of Things (IoT). To secure data using the Internet of Things (IoT) is a tedious task. When the data is being transferred using Internet of Things (IoT), more security can be provided using elliptic curve cryptography. Asymmetric cryptography is used by most of the applications for providing secure communication between two parties (Weber in Comput Law Secur Rev 26:23–30, 2010). The purpose of this type of cryptography is the requirement of huge amount of computation and storage. This is where the use of elliptic curve cryptography comes into picture, as it needs less storage and can be used in small computational devices. ECC needs smaller key sizes and provides stronger encryption compared to various asymmetric cryptographic algorithms like RSA. The usage of power required is low but the performance of the devices using ECC is high for different types of devices like IoT, sensors, etc. This paper shows how ECC must be implemented strongly for providing communication securely to encode the data on an elliptic curve, in IoT devices. The encryption of the data must be done securely while mapping the data on to the elliptic curve. This paper shows how the data is encrypted and mapped on to the curve securely. The work shows how the data can be encrypted with ECC and how it can be visible to only authorized users. In the field of cryptography, ECC is a method for asymmetric cryptography which is dependent on the algebraic structure of an elliptic curve on the finite field.

Elliptic curve cryptography (ECC) can ensure an equivalent security with much smaller key sizes. Elliptic curve scalar multiplication (ECSM) is a fundamental computation used in ECC. This paper focuses on ECSM resisting simple power attack and safe error attack of side-channel attack specifically. Elliptic curve complete addition (CA) formulae can achieve secure ECSM algorithms but are inefficient from memory and computational cost perspectives. Another secure ECSM, which uses (extended) affine, is more efficient for both memory and computational costs. However, it scans input scalars from right to left. In this paper, our developed scalar multiplication algorithms also use their extended affine, but scan from left to right (LR). We also prove the security of our LR ECSM algorithms and analyze them both theoretically and experimentally. Our new LR ECSM algorithms can reduce the amount of memory by \(37.5\%\) and reduce the computational time by more than \(40\%\) compared to Joye’s regular 2-ary LR algorithm with CA formulae.

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.

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.

Elliptic curves are some specific type of curves known as hyper elliptic curves.Compared to the integer factorization problem(IFP) based systems, using elliptic curve based cryptography will significantly decrease key size of the encryption.Therefore, application of this type of cryptography in systems that need high security and smaller key size has found great attention.Hyperelliptic curves help to make key length shorter.Many investigations are done with regard to improving computations, hardware and software implementation of these curves, their security and resistance against attacks.This paper studies and analyzes researches done about security and efficiency of hyperelliptic curves.

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

Elliptic Curve Cryptography (ECC) is a promising public key cryptography, probably takes the place of RSA. Not only ECC uses less memory, key pair generation and signing are considerably faster, but also ECC's key size is less than that of RSA while it achieves the same level of security. However, the magic behind RSA and its friends can be easily explained, is also widely understood, the foundations of ECC are still a mystery to most of us. This paper's aims are to provide detailed mathematical foundations of ECC, especially, the subgroup and its generator (also called base point) formed by one elliptic curve are researched as highlights, because they are very important for practical ECC implementation. The related algorithms and their implementation details are demonstrated, which is useful for the computing devices with restricted resource, such as embedded systems, mobile devices and IoT devices.

Elliptic curves are beautiful mathematical objects that again and again appear in the most surprising places. Their history certainly originates at least in ancient Greece, whereas the study of arithmetic properties of elliptic curves as objects in algebra, geometry, and number theory traces back to the nineteenth century. Curiously, the earliest use of the term “elliptic curve” in the literature seems to have been by James Thomson in 1727 in “A Poem sacred to the Memory of Sir Isaac Newton”: “He, first of Men, with awful Wing pursu’d the Comet tro’ the long Elliptic Curve.” In 1985, Koblitz and Miller independently proposed to use elliptic curves in cryptography which can only be described as a magnificent and practical application of elliptic curves. This paper intends to mostly present a low-brow introduction of elliptic curves and their use in real-world applications of public-key cryptography.

Cloud computing represents a significant challenge for securing data due to its inherently distributed model. Elliptic Curve Cryptography (ECC) will provide substantial security with smaller key sizes, creating a strong candidate for cloud-based environments. This review critically examines ECC based security architectures in cloud computing with the aim of reducing the cipher space for efficiency and scalability. A comprehensive examination of recent ECC implementations guided this review, including identity-based and lightweight ECC Architectures. Specific consideration was given to cipher space reduction techniques incorporating point compression and optimized scalar multiplication. The review provided evidence of a gap in the literature regarding the ability to counterbalance strength of security with the compactness of the cipher across a range of cloud scenarios and use cases. Most models considered provide little flexibility for real-time or post-quantum alternatives. None of the literature considered cipher space optimization for ECC to date. The ongoing optimization of cipher space is essential to progressing security based on ECC implementations for cloud scenarios. Future research efforts must centre on adaptive, efficient, quantum resilient ECC schemes, in ways that yield high security with minimum impacts on the operational scale, and the associated computational loads.

The current era has an explosive growth in communications. Many applications like internet banking, personal digital assistants, mobile communication, smart carts need for security in resource-constrained environments. Elliptic curve cryptography (ECC) used as an excellent tool for cryptographic, because of the security and smaller key sizes when compared with other public key algorithms. The efficiency is largely affected by the underlying arithmetic primitives. This paper explores FPGA designs for two of the most important field primitives namely multiplication and inverse. The smallest programmable entity in an FPGA is the lookup table. A novel finite field multiplier based on the recursive Karatsuba algorithm is proposed. This proposed multiplier combines two variants of Karatsuba. The general Karatsuba multiplier has a large gate count but for small sized multiplications is compact because it utilizes LUT resources efficiently. For largely sized multiplications, the simple Karatsuba is efficient as it requires lesser gates. This proposed hybrid multiplier uses a simple algorithm for initial recursion and small-sized final multiplication has been performed using the general algorithm. While comparing with reported literature this multiplier obtains the best area time product.

Privacy Preservation in Data Mining (PPDM) including for Privacy Preserving Association Rule Mining (PPARM) has attracted lots of attention in recent research and practice. However, the current method or approach still have drawbacks in the sense that there are trade-offs between efficiency and privacy preservation. This paper describes our work towards providing a new efficient PPARM protocol. We reviewed current literature on PPARM and mapped the methods or approaches involved. As previous research showed that Elliptic Curve Cryptography (ECC) perform better than the other Public Key systems such as RSA and Diffie-Hellman, we will utilize ECC for reducing the computational cost of the new PPARM protocol. In choosing good elliptic curves for ECC, we measured the running time of the key generation for various group of recommended elliptic curves i.e. Brainpool curves (by Brainpool), Prime, C2pnb, C2tnb curves (by ANSI X9.62), Secp curves (by SECG), and PrimeCurve curves (by CDC Group). As the result, Secp curves outperformed all of the other curves on overall average ratio of running time and key size of key generation by 4.4% up to 357.6%.