The Scientific World Journal

Volume 2018, Article ID 6718653, 9 pages

https://doi.org/10.1155/2018/6718653

## On Max-Plus Algebra and Its Application on Image Steganography

^{1}Department of Mathematics, Faculty of Mathematics and Natural Science, Jember University, Kampus Bumi Tegal Boto, Jl. Kalimantan 37, Jember 68121, Indonesia^{2}Department of Mathematics, Faculty of Science and Technology, Universitas Airlangga, Kampus C, Jl. Mulyorejo, Surabaya 60115, Indonesia

Correspondence should be addressed to Fatmawati

Received 21 December 2017; Revised 22 March 2018; Accepted 28 March 2018; Published 15 May 2018

Academic Editor: Chi-Wai Chow

Copyright © 2018 Kiswara Agung Santoso 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

We propose a new steganography method to hide an image into another image using matrix multiplication operations on max-plus algebra. This is especially interesting because the matrix used in encoding or information disguises generally has an inverse, whereas matrix multiplication operations in max-plus algebra do not have an inverse. The advantages of this method are the size of the image that can be hidden into the cover image, larger than the previous method. The proposed method has been tested on many secret images, and the results are satisfactory which have a high level of strength and a high level of security and can be used in various operating systems.

#### 1. Introduction

Recently information systems are developing very quickly, especially information systems through the Internet. It happens because the Internet can be accessed by anyone, anytime, and anywhere. Access to information through the Internet does not always bring benefits but also risks to the accuracy of information. This risk is vulnerable when information is accessible by hackers.

Many efforts have been made to protect data transferred over the Internet, including encryption (protecting data before being transferred over the Internet) and authentication (verifying whether the received data is the same as the sent data). There is knowledge or art of data protecting transferred over the Internet, that is, cryptography (data encoding) and steganography (data disguise). The data to be discussed in this paper is image data.

Many steganography methods in protecting information into an image have been published. The data or information that is hidden into an image can be text data or image data. Generally, to hide text data or image data into another image, the original text and original image should be converted into binary digits (bits). Then each digit of the original image or original text is substituted into the last bit of the cover image pixel. By using this method, information could be hidden into the cover image with little difference between the image stego image and the cover image. In this algorithm, every character (for text) or pixel (for image) is hidden into three pixels of cover image. The consequence of this method is, for the text data, the maximum number of characters that can be hidden into the cover image is one-third of the total pixels of the cover image. For image data, the image size that can be hidden into the cover image is one-third the size of the cover image (either long or wide).

In modern world, all information communication is done online. It causes the security system when data transfer becomes very important. Steganography has its own mechanism in data protecting [1, 2]. In steganography, the information to be sent is hidden into other media, so that no one knows where the information is hidden. Watermarks and fingerprints are two technologies related to steganography, where steganography tends to hide data in other media [3].

Currently research on image encoding generally focuses on the following aspects: image encoding with spatial domains, image coding with domain transformation, image coding based on neural network, chaotic image coding, image coding based on cellular automata, and quantum technology [4]. In cryptography, encoding is the process of transforming information using certain algorithms that make it unreadable by anyone except the one who knows the special information, commonly called a key. The result of this process is called encrypted information [5]. Bouquard et al. have introduced the image encoding algorithm using affine transformation [6]. In this algorithm, the encryption and decryption process pass through two stages; that is, the first stage encodes the image using XOR operations with four key bits and the second stage encodes the encoded image using affine transformation. The conclusion of the study states that the correlation of pixel values between the original image and the encrypted image decreased after transforming the affine transform.

Tom has implemented data disguise using stenographic techniques. To make the technique safer they added a level of security by applying cryptography to confidential data before using steganography [7]. For cryptography, they use the Caesar algorithm while for steganography they use the adjacent pixel differences algorithm. Kulkarni and Jatgap substituted secret messages using a 14-square substitution algorithm [8]. Once the text was substituted, then this message was encoded with the RSA algorithm. The next step, this encoded message was hidden into an image by LSB (Least Significant Bit) method. This image works as a carrier file, which will be sent to the recipient. The receiver decrypts to get the original message by performing the same method but in reverse order. Here, it appears that they do two coding techniques, so the system becomes more powerful and secure in the face of hacker attacks. This technique makes it difficult for the troublemakers to manipulate the image and takes a long time to encrypt the message, so it is safe from various attacks through the Internet network.

In measuring quality of an image objectively, some data are statistically calculated to determine quality of the reconstruction image. Image quality could be seen from how close the relationship of image forming pixels or by looking at how much the difference in pixel values are statistically distributed. In general, to compare two images, one could use mean square error (MSE) and Peak to Signal Noise Ratio (PSNR) [9, 10]. Choudhary applied the optimization process to a stego image by using the LSB method, so that quality of the stego can get better with lower computational complexity [11]. MSE between stego image and cover image can be derived. Experimental results show that visually the stego image cannot be distinguished from the cover image. The results also showed improvement compared to the previous one.

In this paper, we propose a new steganography method to hide an image into another image using matrix multiplication operations on max-plus algebra. This is especially interesting because the matrix used in encoding or information disguises generally has an inverse, whereas matrix multiplication operations in max-plus algebra do not have an inverse. Another advantage of this method is the size of the image that can be hidden into the cover image which is greater than using the previous method.

#### 2. Max-Plus Algebra

Max-plus algebra can be used to model disk events related to synchronization and time delays. The application of this theory has a very strong association with production problems [12, 13].

The max-plus algebra [7] is a sequential pair , where is the set of all real numbers, whereas and are binary operations on defined as for every . Operations and are extensions of matrices and vectors in the same way as conventional linear algebra.

In the max-plus algebra [6], the matrix multiplier is defined as follows: for any matrix , , we can obtain matrix by the formulafor , . For a square matrix with degree , matrix denoted by and was defined by recursive operation on : The set of with operations and is called max-plus algebra and denoted by . As conventional algebra, operations ⊗ have a higher priority than ⊕. For example, operation has an understanding like .

Note that , where .

In addition, there is −∞ such that and . For any , there is a small number such thatLet and . In general, the system of linear equations in max-plus algebra will have no solution, if is square matrix or if the number of columns in is more than the number of rows in . Therefore, subsolutions concepts are introduced [7].

Operator ⊗ is a commutative operator. Except 0, every element has an inverse. The inverse of is denoted by or . More precisely, we denote or . multiplication could be denoted by . The operator allows it to be expanded to a matrix on [14].

Let and be two matrices of , operator ⊕, and we defineIt is not difficult to prove that the matrix exists in . Based on the triangular matrix of size , where for , it is indicated that the set of triangular matrices exists in , but the operator ⊗ is not commutative. Furthermore, not all elements in max-plus algebra have inverse [6].

#### 3. Literature Review

The image data character is very different from the text data because an image contains very large data, and all data has a very strong relationship and contains very high data loops [15].

Conceptually, the difference between text data and image data can be seen in Table 1.