Multimed Tools Appl DOI 10.1007/s11042-016-3504-1 Securing color images using Two-square cipher associated with Arnold map Sachin Kumar1 · Rajendra K. Sharma1 Received: 27 May 2015 / Revised: 27 February 2016 / Accepted: 29 March 2016 © Springer Science+Business Media New York 2016 Abstract This paper presents an image encryption scheme using modified Two-square cipher associated with Arnold map. The traditional Two-square cipher is modified to make it more secure and applicable on the image data. A new block-based scheme is considered for Arnold map to handle the images of any size. The proposed scheme is structured into a substitution-permutation framework such that it has an excessively huge key space, and the correct decoding is highly sensitive to the correct keys with their correct order. The exper- imental results are given to validate the feasibility and robustness of the proposed scheme. Further, the superiority of the proposed scheme is analyzed by comparing with the related work. Keywords Image encryption · Image decryption · Arnold map · Two-square cipher 1 Introduction Secure transmission of digital images over the open networks is an important area being researched widely. The image data may be subject to various types of passive and active attacks such as interception, modification and substitution etc. The cryptographic functions applied to the images assure the authenticity and privacy of the image data from unautho- rized access and receiving by the intended user only. Since image data has some intrinsic characteristics such as bulk data, high data redundancy and strong correlation of adjacent pixels, the traditional algorithms like Data Encryption Standard (DES), Advanced Encryp- Sachin Kumar [email protected] Rajendra K. Sharma [email protected] 1 Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, 110016, India Multimed Tools Appl tion Standard (AES) and Rivest Shamir Adleman (RSA) may not be ideal candidates for real-time image encryption as demanding the high computational power [17]. Thus, study of image encryption is necessary to have the techniques satisfying the requirements and characteristics of the image data. In the literature, several methods are developed to han- dle the image data based on various design techniques and primitives in combination. The fourier, discrete cosine, wavelet, gyrator, affine, arnold transforms, and the chaotic maps etc. are some of the common primitives used to determine the image encryption methods [1, 3, 5–19, 21–26]. In [5, 12, 15, 21, 23, 24], the image encoding and decoding methods are presented based on Arnold map associated with other different primitives. Guo et al. [5] proposed an image encryption method, where a color image in Red-Green-Blue (RGB) space is first trans- formed to intensity-hue-saturation (IHS) space and then encrypted in IHS space by using Arnold map and discrete fractional random transform. Liu et al. [15] presented a technique, where the pixel values of RGB components of the original image are scrambled by Arnold Map, and then translated using a matrix defined by a random angle. Further, the discrete cosine transform is employed to the scrambled and translated image. Sui and Gao [21]pre- sented the application of Arnold map associated with chaotic maps and gyrator transform. All these mentioned schemes [5, 15, 21] bring the nice application of Arnold map in con- junction with other primitives but these are limited to handle the images of square size, and also have the key sensitivity on the correct keys not on their arrangements. Tang and Zhang [23] used Arnold map with three random strategies to define a nice scheme for the images of any size. In the first strategy, image is divided into random overlapping blocks of square size followed by generating random iterative numbers in the second strategy. In the third strategy, a random encryption order is obtained for processing the overlapping blocks by Arnold map. All these strategies are governed by separate keys. In Tang and Zhang’s scheme, the random division can result in adjacent squares having overlapping regions where some pixels in overlapping regions can be processed several times by Arnold map. Since Arnold map is periodic in nature, these overlapping regions can have reduced effect or even no effect of iteration numbers. Thus, iteration numbers requires to be chosen carefully such that periodicity is not exhaust. Recently, Tang et al. [24] have proposed an efficient image encryption scheme using block shuffling and chaotic map, where Arnold map is exploited to generate a set of secret matrices used to make final encryption. The schemes [23, 24] have abilities to handle the images of any size and also the key sensitivity on the arrangements of the keys, but lack having the robustness validation against the many desired properties such as correlation, diffusion characteristics, resistance against cryptanalytical attacks etc. There are several methods for image encryption using Arnold map with different primi- tives in combination but some of these lack in having good security against brute force attack (i.e., huge key space), and some in having key sensitivity on the arrangements of the keys. None of these has complete validation of robustness as does not provide security analysis against attacks like differential, known plaintext, chosen plaintext and chosen ciphertext. The high security of the image data in transmitting over the open network has become an important agenda of the present era. This motivates us to design an image encryption scheme having high security and complete validation of robustness against the cryptana- lytical attacks. To meet our aim, we explore the combination of widely used Arnold map with a new primitive from the books of classical cryptography, i.e., traditional Two-square cipher. The traditional Two-square cipher is modified to make it applicable on images and more secure. A new block-based scheme for Arnold map is also designed to have no size Multimed Tools Appl limitation of the input images, which is combined with modified Two-square finally to have encrypted images. Further, the experimental results are conduced to validate the strength of the scheme against statistical analysis and attacks including its time complexity, key sen- sitivity and space analysis. We are able to have an image encryption scheme which has a complete validation of robustness and an extensively huge key space with security not only in terms of the keys but also in their arrangements. The scheme has ability to resist most of the statistical and cryptanalytical attacks. The proposed scheme is also compared with the related image encryption schemes. Compared to the related schemes, the proposed scheme is highly robust and secure, and has better performance. The rest of this paper is organized as follows. In Section 2, the traditional Two-square cipher and Arnold map are discussed briefly. Section 3 describes the proposed encryption and decryption algorithm for color images. Section 4 presents the experimental results and robustness validation. In Section 5, key sensitivity and key space are analyzed. In Section 6, the proposed scheme is compared with the related work. Section 7 draws the concluding remarks. 2 Traditional Two-square cipher and Arnold map This section briefly discusses the encryption algorithms using the traditional Two-square cipher and the Arnold map. 2.1 Traditional Two-square cipher The Two-square cipher was developed to have a classical system stronger than Playfair cipher and less cumbersome than Four-square cipher. It is a digraphic system, where two letters of plaintext (in English) are replaced with two letters of ciphertext in each encipher- ment. It consists two 5 by 5 squares either next to each other (horizontal Two-square) or one top of each other (vertical Two-square). Normally, both squares contain mixed sequences preferably using two different keywords. An example of horizontal and vertical Two-square using keywords “ENCRYPT” and “DECRYPT” is given in Fig. 1. Fig. 1 An illustration: a Horizontal Two-square; b Vertical Two-square Multimed Tools Appl For encipherment or decipherment, first the text message is divided into digraphs. The following rule is used for enciphering the plaintext digraph p1p2 into the ciphertext digraph c1c2. a) Locate the first letter of the digraph within the first square (left/top) and the second letter in the second square (right/bottom) b) Form the rectangle with these two letters at from the opposite corners c) Select the ciphertext or plaintext digraph from the other two corners of the rectangle In horizontal Two-square, if the letters of a digraph to be enciphered are in same row then these are replaced with the same letters in reverse order. In vertical Two-square, whenever the letters to be enciphered are in same column, however, the letters become their own equivalents. In Two-square cipher, the decipherment is handled in exactly the similar way as encipherment. Figure 2 illustrates the encipherment using horizontal and vertical Two- square as given in Fig. 1. The digraphs in ciphertext which are same as plaintext or in reverse are called transparen- cies or reverse transparencies respectively. The traditional Two-square cipher has weakness that in long run, about 20 % of the digraphs will be transparencies. We propose some mod- ifications in traditional Two-square cipher to remove this weakness as well as to make it suitable for encrypting the images. 2.2 Arnold map It is also known as cat map face, which was proposed by V. Arnold in the research of ergodic theory [2], and is defined as x 11 x = (mod 1) (1) y 12 y where the point (x, y) in unit square is transformed to another point (x,y). In respect to the image of size N × N, the Arnold map is defined as x 11 x = (mod N) (2) y 12 y The pixel at position (x, y) in the original image is mapped to the position (x,y) after one iteration of Arnold map in (2).