As computing and communication devices are equipped with increasingly versatile wireless connection capabilities, the demand for security increases. Cryptography provides a method for securing and authenticating the transmission of information over the insecure channels. Elliptic Curve [EC] Cryptography is a public key cryptography which replaces RSA because of its increased security with lesse...

The MOV and FR algorithms, which are representative attacks on elliptic curve cryptosystems, reduce the elliptic curve discrete logarithm problem (ECDLP) to the discrete logarithm problem in a finite field. This paper studies these algorithms and introduces the following three results. First, we show an explicit condition under which the MOV algorithm can be applied to non-supersingular ellipti...

In 1990, Menezes, Okamoto and Vanstone proposed a method that reduces EDLP to DLP, which gave an impact on the security of cryptosystems based on EDLP. But this reducing is valid only when Weil pairing can be defined over the m-torsion group which includes the base point of EDLP. If an elliptic curve is ordinary, there exists EDLP to which we cannot apply the reducing. In this paper, we investi...

Power analysis is a serious attack to implementation of elliptic curve cryptosystems (ECC) on smart cards. For ECC, many power analysis attacks and countermeasures have been proposed. In this paper, we propose a novel power analysis attack using differential power between modular multiplication and modular squaring. We show how this difference occurs in CMOS circuits by counting the expectation...

The system we propose is a mathematical problem with the necessary properties to define public key cryptosystems. It is based on the Elliptic Curve Discrete Logarithm Problem (ECDLP) and polynomial matrices. In this way, we achieve to increase the possible number of keys and, therefore, we augment the resolution complexity of the system. Also, we make a cryptanalisys of the system detecting its...

From a practical point of view, a cryptosystem should require a small key size and less running time. For this purpose, we often select its definition field in such a way that the arithmetic can be implemented fast. But it often brings attacks which depend on the definition field. In this paper, we investigate the definition field F on which elliptic curve cryptosystems can be implemented fast,...

We present new candidates for quantum-resistant public-key cryptosystems based on the conjectured difficulty of finding isogenies between supersingular elliptic curves. The main technical idea in our scheme is that we transmit the images of torsion bases under the isogeny in order to allow the two parties to arrive at a common shared key despite the noncommutativity of the endomorphism ring. Ou...

This paper shows that many of elliptic curve cryptosystems over quartic extension fields of odd characteristics are reduced to genus two hyperelliptic curve cryptosystems over quadratic extension fields. Moreover, it shows that almost all of the genus two hyperelliptic curve cryptosystems over quadratic extension fields of odd characteristics come under Weil descent attack. This means that many...

Given a positive integer n and a point P on an elliptic curve E, the computation of nP , that is, the result of adding n times the point P to itself, called the scalar multiplication, is the central operation of elliptic curve cryptosystems. We present an algorithm that, using p processors, can compute nP in time O(log n+H(n)=p+ log p), where H(n) is the Hamming weight of n. Furthermore, if thi...

A method is described to represent points on elliptic curves over F 2 n , in the context of elliptic curve cryptosystems, using n bits. The method allows for full recovery of the x and y components of the point. This improves on the naive representation using 2n bits, and on a previously known compressed representation using n + 1 bits. Since n bits are necessary to represent a point in the gen...