Efficient Algorithm in Projective Coordinates for EEC Over (original) (raw)
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2010
As one of the most secure and well-known public key cryptography schemes; Elliptic Curve Cryptography (ECC) which heavily based in its computations on modular inversion arithmetic which is known to be most costly operation in the modular arithmetic. Many solutions tried to decrease the cost of the inversion operations for ECC Cryptosystem over Galois Field GF (p) problem. In this paper, we propose to use a new projective coordinates system, which is (X/Z2, Y/Z2), instead of the usual systems (X/Z, Y/Z), (X/Z, Y/Z2), and (X/Z2, Y/Z3). In addition, we will focus on the design and implementation of a new hardware algorithm and Architecture for ECC Coprocessor in GF (p) based on efficient projective coordinates systems. Many projective coordinates were proposed to compute inversion operations for ECC. Our Proposed work uses new projective coordinates systems for two different elliptic curves, which are Standard Elliptic Curves and Edwards Elliptic Curves over GF (p). We found that applying the new projective coordinates system will enhance the use of Edwards’s elliptic curves by 20% in terms of area and speed and it gives comparable results regarding the standard elliptic curves.
2012
Elliptic Curve Cryptography (ECC) is a public key cryptosystem that is considered among the most important schemes in information security. ECC computations points on the curve and suffers from modular inversion operation, which is well known to be very expensive operation. The use of projective coordinates to represents the point on the Elliptic Curves instead of affine coordinates replaced the inversion operations by several multiplication operations. Many types of projective coordinates have been proposed for the elliptic curve E: y2=x3 + ax + b which is defined over prime finite fields: GF (p). In this paper, we studied new projective coordinates systems to achieve higher performance. The selected coordinates were tested by using parallel multipliers to obtain maximum gain. The experiment showed competitive results when using Tripling Oriented and Montgomery Curves. Montgomery curves gave the best results regarding area and time when applied for AT measure. These findings makes these curves a good choice for efficient EC Cryptoprocessor design. The results for the FPGA implementation for EC design using these curves is also proposed in this paper.
2015
Elliptic curve cryptosystem (ECC) is being used nowadays more than ever to fulfill the need for public key cryptosystem because of its ability to use shorter keys lengths and computationally more efficient algorithms than anther public key cryptosystems such as Rivest-Shamir- Adleman (RSA), Digital Signature Algorithm (DSA) and ElGamal. The most time consuming operation in ECC is elliptic curve scalar multiplication (ECSM). Many research have been carried out to accelerate this operation. The structure of the ECSM involves three mathematical levels: finite field arithmetic, point arithmetic and scalar arithmetic. The purpose of this work is to study different issues that arise in the efficient implementation of ECSM over prime field, specifically targeting the point and scalar arithmetic levels over elliptic curve. At the point arithmetic level, we introduce the 4- dimensional Jacobian coordinates system (4 - DJC), where a point (X; Y;Z; T) with Z 6= 0 and T = Z2, corresponds to aff...
A Survey on Single Scalar Point Multiplication Algorithms for Elliptic Curves over Prime Fields
2016
Elliptic Curve Cryptography (ECC) is an attractive field of research since it requires a shorter key length compared to other public-key cryptosystems such as RSA. A shorter key reduces the required computations, power consumption, and storage. The major time-consuming operation in ECC is the point multiplication, kP . Therefore, a lot of research has been carried out to improve the efficiency of ECC implementations. Composite Elliptic Curve (EC) operations and recoding methods are two factors that affect the efficiency of EC scalar multiplication. Deciding which composite EC operation to be used in an ECC system helps to improve the computational efficiency. In addition, finding a method that accelerates the EC computations, which depends on a new recoding method and employing the most efficient composite operations, is considered a pressing need. In this a research, a survey of EC single scalar multiplication methods is introduced. Therefore, a comprehensive information related to...