Discrete Dynamics in Nature and Society

Volume 2018 (2018), Article ID 4585083, 24 pages

https://doi.org/10.1155/2018/4585083

## Image Encryption Technology Based on Fractional Two-Dimensional Triangle Function Combination Discrete Chaotic Map Coupled with Menezes-Vanstone Elliptic Curve Cryptosystem

^{1}Department of Mathematics, Shanghai University, Shanghai 200444, China^{2}Science and Technology on Communication Security Laboratory, Chengdu, Sichuan 610041, China

Correspondence should be addressed to Tiecheng Xia; nc.ude.uhs@ctaix

Received 7 December 2017; Revised 10 February 2018; Accepted 22 February 2018; Published 23 April 2018

Academic Editor: Youssef N. Raffoul

Copyright © 2018 Zeyu Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A new fractional two-dimensional triangle function combination discrete chaotic map (2D-TFCDM) with the discrete fractional difference is proposed. We observe the bifurcation behaviors and draw the bifurcation diagrams, the largest Lyapunov exponent plot, and the phase portraits of the proposed map, respectively. On the application side, we apply the proposed discrete fractional map into image encryption with the secret keys ciphered by Menezes-Vanstone Elliptic Curve Cryptosystem (MVECC). Finally, the image encryption algorithm is analysed in four main aspects that indicate the proposed algorithm is better than others.

#### 1. Introduction

Nowadays, image encryption plays a significant role with the development of security technology in the areas of network, communication, and cloud service. Multifarious chaos-based image encryption algorithms have been developed up to now, such as in [1–6]; however a few of them have referred to the image encryption algorithm based on fractional discrete chaotic map accompanied with Elliptic Curve Cryptography (ECC).

The theory of the fractional difference has been developed for decades [7–13]. Recently, Wu et al. [14–16] made a contribution to the application of the discrete fractional calculus (DFC) on an arbitrary time scale, and the theories of delta difference equations were utilized to reveal the discrete chaos behavior.

ECC is a widely used technology in data security and communication security; it can achieve the same level of security with smaller key sizes and higher computational efficiency [17]. Many famous public-key algorithms, such as Diffie-Hellman, EIGamal, and Schnorr, can be implemented by means of elliptic curves over finite fields. MVECC is one of the popular elliptic curve public-key cryptosystems [18] and we adopt it in our cryptosystem.

Many encryption methods based on fractional derivatives have been proposed in recent time, like fractional logistic maps [19], fractional-order chaos systems [20], and fractional form of hyperchaotic system [21].

In [22], a new image encryption algorithm based on one-dimensional fractional chaotic time series within fractional-order difference has been proposed; however, the two-dimensional discrete chaotic map has seldom been used in image encryption except [23, 24].

Our main purpose is to introduce a new two-dimensional discrete chaotic map based on fractional-order difference and apply it in image encryption. The rest of this paper is organized as follows. In Section 2, the definitions and the properties of the DFC are introduced. After that, the definitions and operation of ECC are given. Then, the working principle of MVECC is described in the next section. In Section 5, we give the fractional 2D-TFCDM on time scales from the discrete integral expression. The bifurcation diagrams and the phase portraits of the map are presented while the difference orders and the coefficients are changing; the largest Lyapunov exponent plots are also displayed. Afterwards, we apply the proposed map into image encryption and show several examples. In Section 7, the performance of the proposed image encryption method is analysed systematically. Finally, we have come to some conclusions.

#### 2. Preliminaries

The definitions of the fractional sum and difference are given as follows. Let denote the isolated time scale and ( fixed). Within the DFC, the function is changed as a sequence . The difference operator is defined as .

*Definition 1 (see [25]). *Let : and be given. The th fractional sum is defined byNote that is the starting point; is the falling function defined as

*Definition 2 (see [26]). *For , , and defined on , the -order Caputo fractional difference is defined by

Theorem 3 (see [27]). *For the delta fractional difference equationthe equivalent discrete integral equation iswhereThe complex difference equation with long-term memory is obtained here. It can reduce to the integer order one with the difference order but the integer one does not hold the discrete memory. From (3) to (5), the domain is shifted from to and the function is preserved to be defined on the isolated time scale in the fractional sums.*

#### 3. Introduction to Elliptic Curve

*Definition 4. *An elliptic curve (EC) over a prime field denoted by refers to the set of all points () that satisfy the equationtogether with a special point at infinity, where , and [28, 29].

##### 3.1. Elliptic Curve Operations

If , ; then if but , ; that is, [29]. where

The scalar multiplication over is defined by where is an integer.

*Definition 5. *The order of an EC is defined by the number of points that lie on the EC denoted by [29].

*Definition 6. *Set , and then is called a generator point if ( is the smallest positive integer that makes ) [29].

#### 4. Menezes-Vanstone Elliptic Curve Cryptosystem (MVECC)

MVECC is one of most significant extensions of ECC; the working principle of MVECC is as follows.

If Andy wants to encrypt and send the message to Bob, they should do the step as mentioned hereunder:

Andy and Bob make an agreement on an elliptic curve and the base point .

Bob firstly selects a private key to compute the public key ().

If Andy wants to send a message to Bob, he firstly chooses a random private key () and then computes his public key . On the other hand, Andy calculates the secret key byHe should compute the ciphered message afterwards by

Then the ciphertext is sent to Bob. When Bob wants to get the plaintext , firstly, he computes the secret key , and then he computes by to get the plaintext [18].

Any adversary that only has and without the private keys and very difficultly breaks the MVECC to get the plaintext . What is more, if have only one big prime divisor, the EC is called a safe EC [29]; then, the MVECC can become an more efficient and secure cryptosystem.

#### 5. Fractional 2D-TFCDM

From [14–16], we notice the application of the DFC in fractional generalizations of the discrete chaotic maps. Recently [30], the following 2D-TFCDM was proposed: Now, consider the fractional generalization of ; the map was also studied in [31]:From Theorem 3, we have the following equivalent discrete numerical formula for the variable : with :

Let = 1, , , , and be fixed. In what follows, Figure 1 is the bifurcation diagram where the step size of is 0.002. Figure 2 is the same bifurcation diagram except for = 0.8.