Elliptic curves cryptosystems for ecommerce applications

Over the past few years, the percentage of customers using electronic commerce (Ecommerce) is increasing rapidly. E-commerce transaction security is a major concern for E-commerce websites along with its customers. The basic requirements for any Ecommerce transaction are privacy, authentication, integrity and non-repudiation. To fulfil the E-commerce security requirement mentioned above, Rivest, Shamir, and Adelman (RSA) cryptography algorithm is widely used. In case of RSA with security requirement, key size is increasing proportionally. Large key size in RSA makes it inappropriate in environments where processing speed, storage, bandwidth, or power consumption is a major constraint. Due to these limitations of RSA algorithm, a new public key cryptography scheme known as the Elliptic Curve Cryptography (ECC) is emerging as a good option for RSA. In this paper ECC performance is analyzed in terms of computation time taken by elliptic curve when used for the ECC application. For the analysis of ECC different key sizes are considered. This paper explains by comparison how ECC is better than the traditional RSA. To improve the security in E-commerce business general security model is proposed by considering the ECC. The proposed E-commerce system is suitable in the constrained applications which have computation speed, bandwidth and storage as its major constraint. In proposed security model along with the payment information, customers purchase order information is also secured.

International Journal of Innovative Technology and Exploring Engineering, 2019

Cryptography is a mathematical science which permits only the authorized users to access the data. Today, it is necessary for our routine life to safeguard online data like credit card numbers, bank transactions, etc. Many cryptographic algorithm-based data security mechanisms have been introduced by various researchers for protecting the online / M-Commerce data. Even though, no data security algorithms achieved the required security level with less time. For overcoming these issues, we propose a new data security algorithm called Elliptic-Curve Cryptography and Diffie-Hellman based data security algorithm for securing the M-Commerce data. Here, the key size of the Elliptic Curve Cryptography is less than RSA that will be helpful to improve the efficiency and reduce the storage space. This algorithm is able to establish a secure session key through a server over an insecure channel and also handle various illegitimate users. In addition, this new algorithm is more secure, efficient...

Electronic Commerce Research, 2010

The use of e-commerce has been associated with a lot of skepticism and apprehension due to some crimes associated with e-commerce and specifically to payment systems. The secure socket layer (SSL) protocol is trusted in this regard to secure transactions for sensitive applications like e-commerce. Unfortunately, the use of SSL protocol causes slow response time on the server which is a major cause of frustration for on-line shoppers. In this paper, we propose a secured credit-debit card payment systems based on Elliptic Curve Cryptosystem (ECC). We first examined ECC algorithm over prime fields GF(p), implement our proposed method using a typical transaction involving credit/debit card numbers and compared the performance with RSA cryptosystem. Our result shows that ECC is faster in terms of response to transaction request and occupies less memory space than equivalent RSA system. Thus, these makes it more suitable public Key cryptography scheme for application in a constraint open environment like payment system where fast operations are needed.

International Journal of Computer Applications, 2018

The purpose of this research is to enhance the cryptographic system called the Elliptic Curve. Elliptic Curve cryptosystem (ECC) is a technique of public-key encryption, which is rooted on the arithmetical construction of elliptic curves over finite fields. Elliptic Curve Cryptographic System necessitates smaller keys compared to non-ECC cryptography to offer equal security. The security of RSA is based on the computational task of considering extensive numbers leading to an increase in encryption computation time, slower connection of the SSL handshake and increase in CPU usage during handshakes. Therefore, there should be a new way of solving this problem, which is ECC encryption. Elliptic curves are effective for digital signatures, key agreement, generators, pseudo-random and other related tasks. The first phase of the project involves understanding the key exchange of Diffie-Hellman and applying the properties of the Elliptic Curves. It is terminated with key facts that the Elliptic Curve Cryptography has a shorter key length, saves bandwidth, which facilitates key generation during the encryption/decryption of data, also the assurance of faster encryption and decryption, and notwithstanding its efficiency and efficacy in small devices.

The most popular public-key cryptography systems nowadays are RSA and Elliptic Curve Cryptography (ECC). ECC is a type of public-key cryptosystem which uses the additive group of points on a nonsingular elliptic curve as a cryptographic medium. The basic operation in most elliptic curve cryptosystems is a scalar multiplication. Scalar Multiplication is the costliest operation among all in ECC which takes 80% of key calculation time on Elliptic curve calculation. Hence Scalar multiplication is the most time-consuming operation in ECC protocols. Scalar multiplication (or point multiplication) is the operation of calculating an integer multiple of an element in additive group of elliptic curve. in this paper, we classify and compare proposed scalar multiplication algorithms and compute their executing time.

Undergraduate Texts in Mathematics, 2014

Elliptic curves have been intensively studied in number theory and algebraic geometry for over 100 years and there is an enormous amount of literature on the subject. To quote the mathematician Serge Lang: It is possible to write endlessly on elliptic curves. (This is not a threat.) Elliptic curves also figured prominently in the recent proof of Fermat's Last Theorem by Andrew Wiles. Originally pursued for purely aesthetic reasons, elliptic curves have recently been utilized in devising algorithms for factoring integers, primality proving, and in public-key cryptography. In this article, we aim to give the reader an introduction to elliptic curve cryptosystems, and to demonstrate why these systems provide relatively small block sizes, high-speed software and hardware implementations, and offer the highest strength-per-key-bit of any known public-key scheme.

Journal of Internet Technology and Secured Transaction, 2012

We present in this paper an important area of information security emerged in the last decades, namely Elliptic Curves Cryptosystems (ECC). Compared to traditional public-key cryptosystems like RSA or Diffie-Hellman, ECC offers equivalent security with smaller key sizes; these result in faster computations, lower power consumption, as well as memory and bandwidth savings. ECC are more and more considered as an attractive public-key cryptosystem for mobile/wireless environments. ECC are especially useful for mobile devices, which are typically limited in terms of their CPU, power and network connectivity. ECC are the next frontier in the use of security mechanisms by providing good security margins with lower computational cost. ECC's domain is an important field emerged in information security. The elliptic curves (EC) are used for conceiving efficient factorization algorithms and for proving the primality. They are used in public key cryptosystems and in pseudorandom bit generators, too. The elliptic curves were also applied in Codes Theory, where they were used to create very good error protected codes. In this paper, our aim is to examine the security, implementation and performance of ECC applications on various mobile devices. Also, our goal is to compare ECC and conventional PKC performances. Doing these, we want to prove that ECC could become the next-generation of PKC.

International Journal of Electrical and Computer Engineering (IJECE), 2011

The paper presents an extensive and careful study of elliptic curve cryptography (ECC) and its applications. This paper also discuss the arithmetic involved in elliptic curve and how these curve operations is crucial in determining the performance of cryptographic systems. It also presents different forms of elliptic curve in various coordinate system , specifying which is most widely used and why. It also explains how isogenenies between elliptic curve provides the secure ECC. Exentended form of elliptic curve i.e hyperelliptic curve has been presented here with its pros and cons. Performance of ECC and HEC is also discussed based on scalar multiplication and DLP.

Proceedings of the …, 2006

In recent years, elliptic curve cryptography (ECC) has gained widespread exposure and acceptance, and has already been included in many security standards. Engineering of ECC is a complex, interdisciplinary research field encompassing such fields as mathematics, computer science, and electrical engineering. In this paper, we survey ECC implementation issues as a prominent case study for the relatively new discipline of cryptographic engineering. In particular, we show that the requirements of efficiency and security considered at the implementation stage affect not only mere low-level, technological aspects but also, significantly, higher level choices, ranging from finite field arithmetic up to curve mathematics and protocols.

Cryptography is the technique of transforming an intelligible message into unintelligible format so that the message can't be read or understood by an unauthorized person during its transmission over the public networks. A number of cryptographic techniques have been developed over the centuries. With technological advancement, new techniques have been evolved significantly. Public key cryptography offers a great security for transmitting data over the public networks such as Internet. The popular public key cryptosystems like RSA and Diffie-Hellman are becoming slowly disappearing because of requirement of large number of bits in the encryption and decryption keys. Elliptic Curve Cryptograph (ECC) is emerging as an alternative to the existing public key cryptosystems. This paper describes the idea of Elliptic Curve Cryptography (ECC) and its implementation through two dimensional (2D) geometry for data encryption and decryption. This paper discusses the implementation of ECC over prime field. Much attention has been given on the mathematics of elliptic curves starting from their derivations.