Elliptic curve cryptosystems (ECCs) are utilised as an alternative to traditional public-key cryptosystems, and are more suitable for resource-limited environments because of smaller parameter size. In this study, the authors carry out a thorough investigation of side-channel attack aware ECC implementations over finite fields of prime characteristic including the recently introduced Edwards fo...

For public key cryptosystems multiplication on elliptic curves can be used instead of exponentiation in finite fields. One attack to such a system is: embedding the elliptic curve group into the multiplicative group of a finite field via weilpairing; calculating the discrete logarithm on the curve by solving the discrete logarithm in the finite field. This attack can be avoided by constructing ...

Three decades ago, Montgomery introduced a new elliptic curve model for use in Lenstra’s ECM factorization algorithm. Since then, his curves and the algorithms associated with them have become foundational in the implementation of elliptic curve cryptosystems. This article surveys the theory and cryptographic applications of Montgomery curves over non-binary finite fields, including Montgomery’...

Elliptic Curve Cryptosystems (ECC) have recently received significant attention by researchers due to their high performance such as low computational cost and small key size. In this paper a novel untraceable blind signature scheme is presented. Since the security of proposed method is based on difficulty of solving discrete logarithm over an elliptic curve, performance of the proposed scheme ...

1 I n t r o d u c t i o n Elliptic curves can be applied to public-key cryptosystems, and as such several schemes have been proposed [3, 4, 5, 6, 9, 11]. There are two typical elliptic curve cryptosystems: E1Gamal-type scheme [4, 11] and RSA-type schemes [3, 5, 6]. The security of the EIGamal-type elliptic curve cryptosystem is based on the difficulty of solving a discrete logarithm over ellipt...

Constructing non-supersingular elliptic curves for pairing-based cryptosystems have attracted much attention in recent years. The best previous technique builds curves with ρ = lg(q)/lg(r) ≈ 1 (k = 12) and ρ = lg(q)/lg(r) ≈ 1.25 (k = 24). When k > 12, most of the previous works address the question by representing r(x) as a cyclotomic polynomial. In this paper, we propose a new method to find m...

We apply power analysis on known elliptic curve cryptosystems, and consider an exact implementation of scalar multiplication on elliptic curves for resisting against power attacks. Our proposed algorithm does not decrease the computational performance compared to the conventional scalar multiplication algorithm, whereas previous methods did cost the performance or fail to protect against power ...

We prove lower bounds on the degree of polynomials interpolating the Diffie–Hellman mapping for elliptic curves over finite fields and some related mappings including the discrete logarithm. Our results support the assumption that the elliptic curve Diffie–Hellman key exchange and related cryptosystems are secure.

The use of elliptic-curve groups in cryptography, suggested by Miller [1] and Koblitz [2] three decades ago,provides the same level of security for the Discrete Logarithm Problem as multiplicative groups, with much smallerkey sizes and parameters. The idea was refined two years later by Koblitz, who worked with the group formed bythe points of the Jacobian of hyperelliptic curve...

Elliptic curves over number elds with CM can be used to design non-isogenous elliptic cryptosystems over nite elds e ciently. The existing algorithm to build such CM curves, so-called the CM eld algorithm, is based on analytic expansion of modular functions, costing computations of O(2 5h=2 h 21=4 ) where h is the class number of the endomorphism ring of the CM curve. Thus it is e ective only i...