Watermarking Digital Image Using Fuzzy Matrix Compositions and Rough Set

Watermarking is done in digital images for authentication and to restrict its unauthorized usages. Watermarking is sometimes invisible and can be extracted only by authenticated party. Encrypt a text or information by public – private key from two fuzzy matrix and embed it in image as watermark. In this paper we proposed two fuzzy compositions Product-Mod-Minus, and Compliment-Product-Minus. Embedded watermark using Fuzzy Rough set created from fuzzy matrix compositions. Keywords—Fuzzy Product-Mod-Minus Matrix; Fuzzy Compliment-Product-Minus Matrix; Fuzzy Rough set; Watermarking; Encrypting


INTRODUCTION
Watermarking in digital image is for authentication and restricting it for unauthorized usages.Videos and other digital contents are used often by unauthorized users.Watermarking and fingerprinting are used to find the point of leakage or user who allowed unauthorized use of the digital content.The watermarking can be visible or invisible.Visible watermarking is used for authentication whereas invisible watermarking is used for restricting unauthorized usages.The Robustness of watermark depends upon its tolerance towards its tamper or delete.It should be identified and extracted to receive information by authorized party.The amount of embedding of information in a digital content without getting identified is its Capacity.

II. ENCRYPTION METHODS
Cryptography is an art of converting a message into cipher text and send to the destination.The authorized person can decipher the text and retrieve the original message.This technology is used from very beginning of the civilization.As the days go by we have new and recent technology coming up so the cryptographic methods are also changing.Prior we had texts which are converted into cipher text by using some notion that A should read as B and B should be read as C and so on.Doing this we get a cipher text which is not easily readable unless one knows the conversion method.Public Key and Private Key method is used for cryptography.There are many methods like RSA, DES, Diffie-Hellmann and etc. for cryptography.In 1976, Martin Hellman, a professor at Stanford University, and Whitfield Diffie, a graduate student, introduced the concept of asymmetric or public key cryptography.We will use here public key and private key as two fuzzy matrices.One matrix is given by user is public key and second fuzzy matrix is randomly chosen from the database to give resultant private key fuzzy matrix.

III. FUZZY MATRIX COMPOSITIONS
The Fuzzy rules are consisting of rules defined on fuzzy set.Fuzzy set are acquired from Crisp Set using membership function.This process is known as fuzzification.Converting fuzzy set to Crisp set is called defuzzification.Fuzzy set has members which can take values 0 to 1. Thus, Fuzzy set A values like A= {0.1/x1, 0.3/x2, 0.4/x3}.This means 0.1 is membership value of x1 in set A, 0.3 membership value for x2 and 0.4 membership value for x3 in set A. Here, we will be using fuzzy matrix for encryption of the text/file which is to be used for watermarking.The encrypted file is then embedded into digital image using Fuzzy Rough sets.Fuzzy Rough set is P={ inf(A),Sup(A)} where A is the Fuzzy set.Inf(A) is 0.1 i.e. lower bound of set Fuzzy A and Sup(A) is 0.4 i.e. upper bound of set Fuzzy A. The encrypted watermark can be extracted from the digital image in unified format.The unified format is then decrypted using algorithm.The paper proposes two fuzzy matrix composition Fuzzy Product-Mod-Minus composition and Fuzzy Compliment-Product-Minus composition.Embedding will be done creating Fuzzy Rough set from these two new compositions, published Fuzzy Max-Mod-Minus composition and Fuzzy Compliment-Sum-Minus composition [29] and the Fuzzy Max-Min composition.

A. Fuzzy Product-Mod-Minus Composition
The Fuzzy Product-Mod-Minus composition is proposed rule consisting of following method.Let A, B and C are fuzzy set with A(x1, x2), B(y1,y2) and C(z1,z2).Let us say,

B. Fuzzy Compliment-Product-Minus Composition
The Fuzzy Compliment-Product-Minus composition is proposed rule consisting of following method.Let A, B and C are fuzzy set with A(x1, x2), B(y1,y2) and C(z1,z2).Let us say,

C. Fuzzy Max-Mod-Minus Composition
The Fuzzy Max-Mod-Minus composition rule [29] is consisting of following method.Let A, B and C are fuzzy set with A(x1, x2), B(y1,y2) and C(z1,z2).Let us say,

D. Fuzzy Compliment-Sum-Minus Composition
The Fuzzy Compliment-Sum-Minus composition [29] is consisting of following method.

E. Fuzzy Max-Min Composition
The Fuzzy Max-Min composition is consisting of following method.Let A, B and C are fuzzy set with A(x1, x2), B(y1,y2) and C(z1,z2).

IV. PROPOSED PUBLIC KEY-PRIVATE KEY ENCRYPTION ALGORITHM
The encryption is done using fuzzy set values.The fuzzy rules are then used to decrypt the context.The public key is given by user is a fuzzy matrix.There is also a database consisting of fuzzy matrices.The public key given by user and fuzzy matrix from database is chosen randomly and Fuzzy Compliment-Product-Minus composition is used get resultant fuzzy matrix.The encryption is done using this fuzzy matrix.

The encryption algorithm has following steps
Step 1: User chooses one Fuzzy matrix appropriate for encryption.It is public key.
Step2: Select one fuzzy matrix from database.
Step3: Generate random number using Fuzzy Compliment-Product-Minus matrix.
There are various ways of encryption.Here, 2X2 fuzzy matrices are used to obtain Compliment-Product-Minus of the fuzzy matrix.The text/files is encrypted by Compliment-Product-Minus of the fuzzy matrix.

V. PROPOSED DECRYPTION ALGORITHM
Decryption algorithm is used decrypt the cipher text file.The following algorithm is used-Step1: Collect the encrypted four parts from four different embedded region of image and combine to for one file.
Step2: Use private key fuzzy matrix key for decryption.
Step3: Retrieve the original file.

VI. EMBEDDING THE WATERMARK
The watermark embedding process is done by inserting the encrypted watermark at appropriate place.Encrypted file is divided into one, two or more parts say b1, b2 and b3.
The three encrypted files are embedded in digital image as watermark using appropriate fuzzy rule.There are many fuzzy matrix compositions like Max-Min, Max-Max, Min-Max, Max-Product, Min-Product composition and etc.The two fuzzy matrices obtained are first used for encrypting watermark.Now for embedding the various compositions of fuzzy matrices are obtained.Using five fuzzy matrix compositions of Section II we will create Fuzzy Rough set P11, P12, P21 and P22.The encrypted three parts of file are inserted at three places of digital image using the most suitable Fuzzy Rough set say P21 and P22.Fig. 4, 5 and 6 shows watermarked images using the fuzzy matrix composition and Fuzzy Rough set.