Security and Communication Networks

Volume 2017 (2017), Article ID 6729896, 12 pages

https://doi.org/10.1155/2017/6729896

## A Hybrid Chaotic and Number Theoretic Approach for Securing DICOM Images

^{1}Department of Information Technology, Thiagarajar College of Engineering, Madurai, India^{2}Department of Electronics and Communication Engineering, Thiagarajar College of Engineering, Madurai, India

Correspondence should be addressed to Jeyamala Chandrasekaran

Received 31 July 2016; Accepted 24 November 2016; Published 12 January 2017

Academic Editor: Ángel Martín Del Rey

Copyright © 2017 Jeyamala Chandrasekaran and S. J. Thiruvengadam. 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

The advancements in telecommunication and networking technologies have led to the increased popularity and widespread usage of telemedicine. Telemedicine involves storage and exchange of large volume of medical records for remote diagnosis and improved health care services. Images in medical records are characterized by huge volume, high redundancy, and strong correlation among adjacent pixels. This research work proposes a novel idea of integrating number theoretic approach with Henon map for secure and efficient encryption. Modular exponentiation of the primitive roots of the chosen prime in the range of its residual set is employed in the generation of two-dimensional array of keys. The key matrix is permuted and chaotically controlled by Henon map to decide the encryption keys for every pixel of DICOM image. The proposed system is highly secure because of the randomness introduced due to the application of modular exponentiation key generation and application of Henon maps for permutation of keys. Experiments have been conducted to analyze key space, key sensitivity, avalanche effect, correlation distribution, entropy, and histograms. The corresponding results confirm the strength of the proposed design towards statistical and differential crypt analysis. The computational requirements for encryption/decryption have been reduced significantly owing to the reduced number of computations in the process of encryption/decryption.

#### 1. Introduction

Telemedicine is essentially the remote diagnosis and treatment of patients by means of telecommunications technology. The increasing adoption and usage of internet, smart phones, mobile health care devices, and wearable health technology have significantly impacted the growth of telemedicine over the years. Telemedicine involves large volume of storage and exchange of electronic health records among physicians, patients and health care professionals for better health services. Health records involve extensive usage of multimedia especially images, which are generated from various imaging technologies like conventional X-rays, ultrasound imaging, digital mammography, Computed Axial Tomography (CT), Positron Emission Tomography (PET), and Magnetic Resonance Imaging (MRI). These medical images are highly sensitive and are to be operated in a resource constrained environment characterized by lower band width, limited processing power, and limited memory. The strong privacy requirements of medical images with the operating constraints demand secure encryption algorithms with optimal processing requirements.

Text based algorithms like Advanced Encryption Standard, Elliptic Curve Cryptography, and RC4 are not preferred for encryption of medical images because of the complicated internal structure, memory requirements, and time delay incurred in the process of key generation and encryption/decryption. Few researches in the literature focus on the customization of the text based algorithms for encrypting medical images [1–4]. However reduction in computational time is not up to the desired level.

Substantial amount of research work in the literature have focused on application of chaotic maps for cryptography due to the high sensitivity to initial conditions, nonlinearity, and random and ergodic nature of chaotic maps. Fridrich [5] proposed the initial framework for chaos based image encryption, which consists of two stages, namely, permutation/confusion and substitution/diffusion. A 2D chaotic map was used to control the parameters in both the stages. Following Fridrich, many researchers [6–11] have contributed significant enhancements for improving security in the field of chaos based image encryption.

Owing to the wide spread usage of telemedicine, many research works have been attempted to test the feasibility of chaotic maps for medical image encryption. Hu and Han [12] presented a pixel-based scrambling scheme based on simple XOR operation. The scrambling key is a true random number (TRN) sequence derived from the multiscroll chaotic attractors. Fu et al. [13] attempted to improve the efficiency of chaos based image cipher by introducing substitution in the permutation process itself. Arnold cat map and logistic map are chosen for permutation and substitution, respectively. Since pixel value mixing is contributed by both permutation and substitution stage, desired level of security could be achieved in fewer rounds. Liu et al. [14] used hash value of the pathological image and random number to generate the initial conditions of chaotic maps. Chebyshev maps are used to confuse the pixels. Since each block of the image is encrypted using the hash value of the previous block and random number, different cipher images are generated for different recipients. Praveenkumar et al. [15] proposed a trilayer cryptic system, which is a blend of Latin square image cipher, discrete Gould transform, and Rubik’s encryption for encryption of DICOM images. Confusion, diffusion, tamper proofing, permutation, randomness, and ergodicity have been fused together to enhance security. Ravichandran et al. [16] employed bioinspired crossover and mutation unit to provide confusion and diffusion of pixels in a DICOM image. Logistic, tent, and sine maps are used for the second stage of encryption. Combined logistic tent maps and combined logistic sine maps are used to generate enlarged and elevated chaotic sequences, which are further enhanced by crossover and mutation strategies. To diffuse the effect of slight change in single pixel intensity of plain image over many pixels in cipher image, logical operations such as XOR and circular rotation are proposed by Yavuz et al. [17].

Though chaotic cryptosystems provide the required confusion and diffusion properties, they do have the limitations of low cycle lengths and are computationally less efficient because of floating point arithmetic. DICOM images are highly sensitive because even a small amount of visual degradation can lead to false diagnosis. Lima et al. [18] proposed a model for medical image encryption using Cosine Number Transform (CNT) to avoid round off errors. Their model provides zero tolerance effect since it involves only integer arithmetic. Dzwonkowski et al. [19] adopted Feistel network for encryption of DICOM images, wherein special properties of quaternions are used to perform rotations in 3D space for each of the cipher rounds.

The proposed work is an attempt to improve the computational efficiency of chaotic crypto systems by generating keys based on number theory. As the system design involves modular exponentiation for key generation, it completely eliminates round off errors in decryption and provides zero tolerance effect. To further enhance the level of security, a permutation stage is included, wherein the keys generated are chaotically permuted and controlled by Henon map. Permutation of key arrays based on Henon maps generates independent key arrays for encryption of every subimage which enhances the randomness of the keys. The security of the proposed algorithm is confirmed through statistical and differential crypt analysis. While the proposed algorithm ensures a high degree of security on par with the existing algorithms reported in the literature, the computational time and resources have been reduced significantly, because of simple structure in encryption/decryption. This makes the proposed algorithm suitable for real time applications.

#### 2. Background

##### 2.1. Discrete Logarithms

###### 2.1.1. Primitive Roots

If “” is a prime and “” is any element in the residual set of and if are distinct and consist of integers within the range , then “” is called as the primitive root or generator of “” [22, 23].

###### 2.1.2. Discrete Logarithm

If “” is an arbitrary integer in and is a primitive root of “” then there exists exactly one number such that

The number “” is then called the discrete logarithm [24] of “” with respect to the base “ modulo ” and is denoted as Discrete logarithm principle has been extensively used in the design of asymmetric algorithms like ElGamal Crypto systems and Diffie Hellman Key exchange. Elliptic curve version of discrete logarithm problem forms the foundation of Elliptic Curve Cryptography. Few researchers have extended discrete logarithms for symmetric text and image encryption [25, 26]. The strength of discrete logarithm is because of the fact that the forward process of calculation of exponentials modulo prime is easier even for very larger primes using fast modular exponentiation. However, the reverse process of calculation of discrete logarithms is considered infeasible for larger primes [23].

##### 2.2. Henon Maps

The Henon map is a two-dimensional discrete dynamical system introduced by Michael Henon [27]. Henon map takes a point () in the plane and maps it to a new point () as defined by the equationsThe desired statistical properties can be obtained from the generated values if the input values are as follows: “” can be in the range of 1.16 to 1.41. “” can be in the range of 0.2 to 0.3. “” can be in the range of −1 to 1. “” can be in the range of −0.35 to 0.35. Skip value can be in the range of 80 to 1000.

Henon map is found to exhibit good chaotic behavior for values and . A minute variation in the initial parameters even in ten-millionth place value of the Henon maps could yield widely divergent results. This extreme sensitiveness to initial conditions of the Henon map is exploited in various image encryption algorithms. Henon map has been tried for various applications in cryptography such as pseudo number generation [28], encryption of satellite imagery [29], and design of substitution boxes [30].

#### 3. Proposed Methodology

The proposed methodology consists of three stages, namely,(A)generation of two-dimensional key array based on modular exponentiation,(B)permutation of key array based on Henon maps,(C)encryption/decryption of DICOM images based on bitwise XOR of subimages with permuted key arrays.

*(A) Generation of Two-Dimensional Key Array Based on Modular Exponentiation*(A1)Select a prime number “” and generate all possible primitive roots of the prime. A primitive root of a prime “” is an integer “” such that “” has multiplicative order “”.(A2)If are the primitive roots of the prime “” then a two-dimensional array “” of order “” is generated as follows: where “” refers to the number of rows = number of primitive roots for the prime “”. “” refers to the number of columns = .

*(B) Permutation of Key Matrix Based on Henon Maps*(B1)Exchange the parameters , , , , and using the public keys of the receiver for generation of permutation keys for rearranging the elements in the key pool.(i): the parameters of the Henon map , and seed values , .(ii): the number for decimal places of the mantissa that are to be supported by the calculating machines.(iii): the number of iterations after which the first value is to be picked for generating keys.(iv): the skip value to be maintained for picking successive values thereafter.(v): increment value of seed for generating different permutation keys.(B2)Iterate the two-dimensional equations of the Henon map with initial parameter settings as defined in for a predefined number of times. The number of iterations () is given by , where .(B3)Process the result in two one-dimensional arrays “” and “” of length “”.(B4)Encode the values generated by the Henon map to integer representation for locating row and column indices. where and correspond to the encoded version of and . , , , and are the minimum and maximum values of the two-dimensional Henon map equations generated in various iterations. , , , and are the minimum and maximum values of row and column indices respectively in the key matrix.(B5)Permute the two-dimensional key array using the encoded arrays () as the permutation key.(B6)Repeat steps (B2) to (B5) with updated seed value for generating different permutation keys as required. The number of permutation keys is decided by the count of subimages.

*(C) Encryption/Decryption of DICOM Images Based on Bitwise XOR of Subimages with Permuted Key Arrays*(C1)Read the input DICOM image of size × . The number of rows () in the two-dimensional key array is equal to the number of primitive roots of the prime (). The number of columns () is given by ().(C2)If is greater than and is greater than , divide the image into nonoverlapping subimages as follows.

*Case 1. *If and are exact multiples of and (i.e., and where and are integers), the image is split into subimages of order . The number of subimages () is given byThe process is illustrated by a simple example as depicted in Figure 1. Supposing the size of the two-dimensional key array is 2 × 2 and the size of the image is , the image is split into four subimages of size 2 × 2.