Improvement of Menezes-Vanstone Elliptic Curve Cryptosystem Based on Quadratic Bézier Curve Technique

Abstract

Cryptography is one of the most important applications and widely used in our life especially in the information security that needed by many government institutions, banks, communications and others to keep data over internet and other transportations that it is ensure safety of transfers between the sender and the recipient. The most important system in cryptography is public key cryptography and the mostly used is the elliptic curves cryptosystem, because of it is very efficient and secure and difficult to solve the discrete logarithm problem and find the secret key. In this study a new method is introduced using the Menezes-Vanstone Elliptic Curve Cryptosystem (MVECC) for data based on quadratic Bézier curve techniques. The purpose of this proposal is to increase the security of this cryptosystem. We will apply this proposed method to all measurements of National Institute of Standards and Technology (NIST) tests and running time and compared it with the original method.

Bézier Coefficients Matrix for ElGamal Elliptic Curve Cryptosystem

“…This simulation uses Wolfram Mathematica (Version 12.3.1 for Mac OS X x86/ 64-bit). The results of ElGamal ECC [1] and Ghadi and Al-Rammahi's MVECC (GAMVECC) in [6] will be compared.…”

“…As mentioned in [1], cryptosystem offers different implementation times depending on the input size, length of algorithms or code, as well as the type of language and processors used throughout the encryption and decryption. 2 illustrates the implementation time on GAMVECC in [6] using different platforms. Based on Tables 1 and 2, the time taken for encryption and decryption on Wolfram Mathematica (Version 12.3.1 for Mac OS X x86/ 64-bit) is much faster than MATLAB (version 7.10.0.499/ 32bits) by Al-Saffar and MATLAB R2018b (version 9.5.0.944444/ 64-bits) in [6].…”

“…Many studies and methods had been developed by researchers to increase the level of security in cryptography such as the application of Bézier curve in cryptosystem by Ghadi and Al-Rammahi (2020) [6], Srividya and Akhila (2014) [16], and Abdul Wahab and Satter Jaber (2016) [19]. However, these studies lack important information on how to embed the Bézier curve coefficient into the cryptosystem to improve its security level.…”

“…In 2016, Abdul Wahab and Satter Jaber applied Bézier curve with a chaotic system technique to generate an encryption key for a secure communication [19]. Meanwhile, Ghadi and Al-Rammahi manipulated the formula of Menezes-Vanstone Elliptic Curve Cryptosystem (MVECC) with the mathematical notation of quadratic Bézier curve as their proposed method in [6].…”

“…This study compares the implementation time and mathematical operations of the suggested method with the original ElGamal ECC. The proposed method also compares the method introduced in [6] since both methods used Bézier curve for encryption and decryption process. This paper is structured as follows.…”