Accelerate Performance for Elliptic Curve Scalar Multiplication based on NAF by Parallel Computing (original) (raw)

High Performance Methods of Elliptic Curve Scalar Multiplication

International Journal of Computer Applications, 2014

Elliptic curve scalar multiplication is the operation of successively adding a point along an elliptic curve to itself k times. It is used in elliptic curve cryptography (ECC) as a means of producing a trapdoor function. In this paper, algorithms to compute the elliptic curve scalar multiplication using a special form for integers will introduce, and then two types of signed digit representation will use. The signed digit form of the scalar is calculated by many types of algorithms such as binary , non adjacent form and direct recoding. The results indicate that the proposed methods perform better to compute the scalar multiplication on elliptic curves and it is more efficient than the existing methods.

Efficient algorithms for speeding up the computations of elliptic curve cryptosystems

Applied Mathematics and Computation, 2005

As we know, the performance of the elliptic curve cryptosystem (ECC) deeply depends on the computation of scalar multiplication. Thus, how to speed up the computation of the elliptic curve scalar multiplication is a significant issue. In 1994, Lim and Lee proposed a more flexible precomputation method used in wireless networks environments for speeding up the computation of exponentiation. This method can be also used for speeding up the scalar multiplication of elliptic curves. We call it LLECC method. However, the less storage is equipped with the computing devices, the less efficient it is. For this reason, we propose a more efficient algorithm than LLECCÕs in this paper. First, we modify LLECC method to reduce the storage of precomputed values, and then propose an efficient algorithm based on the nonadjacent form (NAF) representation and multidoubling method. Furthermore, the proposed algorithm can be also used for speeding up the multi-point multiplication of elliptic curves. According to the simulation results, the proposed algorithm can reduce 11% and 21% in the aspect of the computational complexity and storage cost, respectively, in an elliptic curve of size 160-bit over finite fields with characteristic greater than 3, as compared with

Classification and Comparison of Scalar Multiplication Algorithms in Elliptic Curve Cryptosystems

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.

Improvement of scalar multiplication time for elliptic curve cryptosystems

2013 11th International Symposium on Programming and Systems (ISPS), 2013

ABSTRACT Sensor nodes have limited computing power and memory sizes. Sometimes, they are used in applications that require sending rapidly secure data to a remote control center. Therefore, they require lightweight techniques to accomplish this task. In this paper, we used Elliptical Curve Cryptography (ECC) for data encryption because ECC could create smaller and more efficient cryptographic keys compared to other cryptographic techniques such as RSA. We used specific algorithms to improve scalar multiplication time in spite of energy consumption. Moreover, we proposed a distributed scheme to enhance more the data delivery time from a source node to the base station by involving neighbors in the calculation. The results of experiments on TelosB motes showed considerable improvement of data delivery time.

Parallel algorithm for multiplication on elliptic curves

Proceedings of the ENC

Abstract. 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 ...

Speeding Up the Computation of Elliptic Curve Scalar Multiplication based on CRT and DRM

Proceedings of the 6th International Conference on Information Systems Security and Privacy, 2020

In this paper, we study the parallel implementations of elliptic curve scalar multiplication over prime fields using signed binary representations. Our implementation speeds up the calculation of scalar multiplication in comparison with the standard case. We introduce parallel algorithms for computing elliptic curve scalar multiplication based on representing the scalar by the Complementary Recoding Technique (CRT) and the Direct Recording Method (DRM). Both implementations of the proposed algorithms show speed-ups reaching up to 60% in comparison with execution time for sequential cases of the algorithms. We find that ECC-DRM is faster than ECC-CRT in both parallel and sequential counterparts.

Scalar Multiplication Algorithms of Elliptic Curve Cryptography over GF (2m )

International Journal of Advanced Computer Science, 2014

Since the inception of elliptic curve cryptography by Koblitz [1] and Miller [2] for implementing public-key protocols as the Diffie-Hellman key agreement, elliptic curve cryptography has become one of the most researched area for providing one stop reliable and secure solution in the field of cryptography. The ECC covers all relevant asymmetric cryptographic primitives like digital signature (ECDSA), key exchange and agreement protocols. Point multiplication serves as the basic building block in all ECC primitives and is the computationally most expensive operation and our analysis revolves around this concept. This paper gives an introduction to Elliptic Curve Cryptography and deals with evaluation of fast scalar multiplication with parallelization of field operation and point addition/multiplication. Elliptic curve cryptography offers best optimized solution with minimum resources like Low memory, High Throughput, low power consumption and minimum key length for the same level of security as compared to its counterpart like RSA, DSA etc. in public key cryptography domain. The work is based on the extensive research work done by Julio Lopez, Ricardo Dahab, Montgomery and other pioneer scientists and academicians in the field of elliptic curve cryptography. Given the importance of Scalar multiplication , we focused ourselves on the Fast Multiplication on Elliptic Curves over finite Binary field GF(2 m) without Pre-computation whose background is set by Julio Lopez et al. in [1], because the finite field operations can be implemented very efficiently in hardware and software.

Hardware Implementation of Elliptic Curve Cryptosystem Using Optimized Scalar Multiplication

2019

This paper presents a hardware implementation of Elliptic Curve Cryptography (ECC) with optimized scalar multiplication. In elliptic curve cryptography, scalar multiplication is an important and most time-consuming operation that dominates the ECC performance. In this paper, the scalar multiplication is carried out using the Vedic multiplier for finite field multiplication operation to improvise the performance. The proposed architecture is implemented and evaluated for the performance evaluation parameters—area, delay, and power consumption. To evaluate the efficiency of the proposed design, the results are compared with Karatsuba based ECC design. The comparative results show that ECC using Vedic multiplier outperform then Karatsuba based ECC for the area, delay and power consumption. The elliptic curve cryptosystem is implemented over GF(2m) binary field for B-233 field size, which is more secured according to NIST Digital Signature Standards. The cryptosystem is designed in Veri...

A Comparison among Fast Point Multiplication Algorithms in Elliptic Curve Cryptosystem

2022

In the Elliptic Curve Cryptosystem, the multiplication of points is essential in the successful computation of any operation. Reduction of time complexity for the mathematical operations in Elliptic Curve Cryptosystems with minimum hardware resources, the methods: Addition and Subtraction, Mutual Opposite Form, and Complementary Recoding techniques proposed as fast scalar multiplication schemes. The fast-point multiplication method is always necessary for any mathematical operation on an Elliptic Curve Cryptosystem with a restricted system. This study compares the performance of fast-point multiplication algorithms in terms of computational and execution time to determine the quickest one. For any elliptic curve-related arithmetic operations, point multiplication has a vital role in reducing the idle time of hardware utilization. Rapid point multiplication is essential to minimize enhanced time complexity to determine the most suitable algorithm for mobile devices.

Elliptic Curve Scalar Multiplication Operation: a Survey Study

Journal of Kufa for Mathematics and Computer

Scalar multiplication is the fundamental operation in the elliptic curve cryptosystem. It involves calculating the integer multiple of a specific elliptic curve point. It involves three levels: field, point, and scalar arithmetic. Scalar multiplication will be significantly more efficient overall if the final level is improved. By reducing the hamming weight or the number of operations in the scalar representation, one can raise the level of scalar arithmetic. This paper reviews some of the algorithms and techniques that improve the elliptic curve scalar multiplication in terms of the third level.