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Procedia Computer Science 2 (2010) 242-247 ICEBT 2010

Procedia Computer Science

www.elsevier.com/locate/procedia

Privacy Protection of Biometric Traits using Modified Hill Cipher with Involutory Key and Robust Cryptosystem

Bibhudendra Acharya*, Mukul Dhar Sharma, Sourabh Tiwari, Vinay Kumar Minz

Department of E & TC, NIT Raipur, Chhattisgarh-492010, India

Abstract

In this paper we have proposed a technique for securing biometric traits using the modified Hill Cipher with an involutory key and a robust cryptosystem. The Modified Hill Cipher solves the drawbacks of conventional Hill ciphers by using iterations and interlacing. Interlacing of the binary bits of the plaintext image creates confusion while iterations create complexity to the cipher image. The Modified Hill cipher is secure to known plaintext. Binary conversion is only possible for integer elements so an integer involutory key matrix is used. As the key matrix is involutory, which eliminates necessity of matrix inverse while decryption. We have used a robust cryptosystem algorithm to send the cipher matrix safely. The cryptanalysis and histogram graphs show the variation of image property before and after encryption.

© 2010 Published by Elsevier Ltd

Keywords-Encryption; Decryption; Involutory matrix; Hill Cipher; Modified Hill Cipher; Robust cryptosystem

1. Introduction

Biometric-based authentication applications include workstation, network, and domain access, single sign-on, application logon, data protection, remote access to resources, transaction security and web security. Trust in these electronic transactions is essential to the healthy growth of the global economy. Hence the security of the biometric templates is of utmost importance. Biometric templates stored in the system database are the heart of the identification process which can be tempered with using hacking or unauthorized breach to the database. The following risks concerning security and privacy are to be prevented:

1. Identity theft

2. Cross matching attack

3. Disclosure of sensitive information

To prevent such adversary and ensure a healthy recognition system, the biometric templates are encrypted before storing in database. Different cryptographical techniques are used for encrypting and decrypting the templates depending upon the level of optimization between security and speed of response of the biometric system. Modified Hill Cipher with interlacing and iteration is one of strongest cryptographic algorithm to secure image templates [1, 3, 4, 12].

This paper presents a method which combines the Modified Hill cipher using involutory key matrix and the robust cryptosystem algorithm for encryption and decryption of biometric template efficiently.

Following the introduction, the basic concept of Hill Cipher is outlined in section 2. Section 3 presents the modified Hill cipher with iterations and interlacing. Section 4 describes the method to generate involutory key matrix. Robust cryptosystem algorithm is mentioned in section 5. Proposed technique is presented in section 6. Results are presented in section 7. Finally conclusion is discussed in section 8.

2. Hill Cipher

The Hill cipher is a famous symmetric cryptosystem from the early days, which was invented by Lester S. Hill [1, 2, 5]. Hill

* Corresponding author. Tel.: +91-9907445868; fax: +91-771-2254600. E-mail address: bacharya.etc@nitrr.ac.in.

1877-0509 © 2010 Published by Elsevier Ltd doi:10.1016/j.procs.2010.11.031

cipher requires inverse of the key matrix while decryption. In fact that not all the matrices have an inverse and therefore they will not be eligible as key matrices in the Hill cipher scheme. Furthermore, due to its linear nature, the basic Hill cipher succumbs to known-plaintext attacks. In Hill Cipher algorithm to encrypt plaintext block of size n, we need key matrix (Kn x n) with entries are between (0, p -1) included, but the determinant must be relatively prime to p, each entry in the plaintext block is between (0, p-1) included each block of plaintext is then an n -dimensional vector X. We encrypt vector X. simply to produce the cipher text vector C using the following linear algebra equation

C = K x X mod( p) (1)

To decrypt cipher text vector C, we need first to find the inverse matrix K-1 to K, where that matrix must be invertible then can calculate X from the mathematical model

X = Kx C mod( p) (2)

This Hill cipher technique is not secure by known plaintext attack. By knowledge of P and C key can be generated. Other problem is the Key must have to be invertible. The inverse of K is required for decryption. Modification in Hill cipher technique can reduce the chances of known plaintext attack. Use of involutory key will diminish the invertible key problem. Same key can be used in both encryption and decryption.

3. Modified Hill Cipher

V.U.K. Sastry and N. Ravi Shankar, in their research paper "Modified Hill Cipher for a Large Block of Plaintext with Interlacing and Iteration" [6] have proposed interlacing in the process of encryption and decomposition in the process of decryption. Use of iterations in both encryption and decryption makes the process complex and secure. Interlacing means the interchange of binary bits of message matrix element. Iteration means to execute this interlacing process in definite loop. Here, it is to be noted that decomposition is a reverse process to that of interlacing.

3.1. Algorithm for Interlacing and Decomposition:

(1) Convert all elements of matrix into 8 bit binary form.

(2) New dimension is n x 8n . Divide this new matrix into two matrixes with dimension n x 4n .

(3) Interchange (2m) th column of first matrix with (2m -1) th column of second matrix with m varies from 1 to 2n.

(4) Combine two matrixes to create new n x 8n matrix.

(5) Convert 8 bit elements into decimal numbers and to get nx n matrix.

(6) This matrix is interlaced and different from the original matrix.

Advantage of modified Hill cipher technique is that it is secure from known plaintext attack. By the knowledge of plaintext, key cannot be determined as encryption takes place many n number of times in the iteration loop. Cryptanalysis and Avalanche effect have proved the above advantage.

V.U.K. Sastry and N. Ravi Shankar have proposed modified Hill cipher technique for a data block. While using this we get following problems:

(1) If the plaintext matrix, key matrix or inverse of key matrix contains elements with decimal values then the elements cannot be changed into binary form and process will not be correct.

(2) The key matrix used must have to be invertible, so that decryption can take place. We have proposed solution to above problems in this paper of image encryption:

(1) All the elements should be pure integer so that they can be changed into binary form.

(2) An involutory matrix with integer elements is used for encryption and decryption.

4. Involutory Matrix Generation Method

As described in section 2, Hill cipher requires inverse of the key matrix while decryption. In fact that not all the matrices have an inverse and therefore they will not be eligible as key matrices in the Hill cipher scheme [7-10].

4.1. Algorithm for Involutory matrix generation:

(1) A random non-singular matrix A22of dimension m/2 x m/2 with integer elements is generated.

(2) Now Ajj = -A22 .

(3) A random number k (k > 1) is selected.

A21 = (I - A11)/k A12 = (I + A11)* k

The involutory key matrix K will be:

¿11 ¿12 ¿21 ¿22.

This Involutory key matrix has all integer elements. It can be of any size depending upon dimensions of plain image. The property of involutory matrix K is

K = K (6)

So the same matrix can be used for both encryption as well as decryption.

5. Robust Cryptosystem Algorithm

Rushdi A. Hamamreh and Mousa Farajallah in their research paper "Design of a robust cryptosystem algorithm for non-invertible matrices based on Hill cipher" [11] have proposed an efficient technique for safe transmission of cipher and key. This technique can also be used for non-invertible keys.

5.1. Encryption algorithm:

(1) Convert the plaintext char into numerical numbers.

(2) If the determinant matrix is zero then add identity matrix else does nothing

(3) Calculate the column vector

C = K x X (7)

(4) Calculate

C1 = fix(C / P) (8)

C2 = mod (C, P) (9)

(5) Convert the numerical numbers (C1,C2)into Char.

5.2. Decryption algorithm:

(1) Convert the two sequence of ciphertext into numerical numbers (F1,F2).

(2) If the determinant matrix is zero then add identity matrix else does nothing.

(3) Calculate the column vector

P = inv(K) x ((Y1 x 256) + Y2). (10)

(4) Convert the numerical numbers P into char.

6. Proposed Technique

In this paper, we propose a technique for encryption and decryption of biometric template efficiently. This cryptographic system is a combination of the Modified Hill cipher using involutory key matrix and the robust cryptosystem. The combined cryptosystem takes advantage of all the three techniques and it gives nearly uniform histogram suggesting the high quality of encryption. The encryption and decryption algorithms are mentioned below:

6.1. ¿Igorithm For Encryption:

1. The template image is converted to a square matrix P .

2. A non-singular involutory key matrix K is formed with same order as the size of the template image.

3. Now the image matrix P is encrypted using Modified Hill cipher technique with involutory key using binary interlacing and iteration as explained in section 3.

4. The cipher C created above is split into two parts C1 and C2 as explained in section 5.

6.2. ¿lgorithm For Decryption:

1. The two parts of cipher are joined together to form the encrypted image matrix C as

C = (CI * p) + C2 (11)

2. The cipher text so formed is decrypted using Modified Hill cipher using interlacing and iteration as explained in section 3.

3. The recovered matrix P is the decrypted image or template.

Modified Hill cipher using interlacing and iteration cannot be used for encrypting an image because of the loss of data during interlacing (binary conversion and rearrangement) of temporary cipher though it provides a robust encryption. So use of involutory key matrix eliminates the possibility of any decimal value and makes the Modified Hill cipher evenly applicable to images too. The encryption and decryption process is illustrated by flow diagram in Fig.1. Interlacing, decomposition, fix and mod are the functions for performing Modified Hill cipher and robust cryptosystem algorithm.

Fig. 1. Flow diagram of proposed technique for encryption and decryption

7. Results and Discussion

We use an image of Iris as biometric template. Encryption and decryption is performed by proposed algorithm and the Fig.2. (a) shows the plaintext biometric template to be encrypted. Fig.2. (b) shows the encrypted cipher image which is completely different from original image. Fig.2. (c) shows the histogram of plaintext biometric template. Fig.2. (d) shows histogram plot of encrypted image. It is clear that the elements in encrypted image are uniformly distributed.

8. Conclusion

In the proposed method, we have generated a involutory key matrix with integer elements. It solves the drawbacks of modified Hill cipher. The biometric template image is encrypted using this key matrix. Key matrix can be of any dimension as per the plain image. So the technique works for all images. The key is involutory, so there is no need of calculating inverse of key. Non-invertiblity of key matrix problem will not arise.

In this analysis, we have adopted an iterative process. In each round of the iteration, we have performed multiplication of the plaintext matrix with the key matrix and modulo operation with 256. In every round, the modified plaintext is represented in terms of binary bits and these binary bits are interlaced so that we get a plaintext matrix of the same size. This sort of interlacing of the binary bits of the plaintext is expected to cause a lot of confusion in the structure of the plaintext.

Also we are converting the cipher matrix into two different ciphers while encryption. In decryption side these two matrices are combined to generate the original cipher back. In this method the cipher is never visible in the time of transmission. It increases the security of biometric trait.

The histogram shown in this paper, clearly indicate that the cipher is strong and original image can never be generated by any attack.

In the light of the above analysis, we find that the cipher under consideration can be applied to a biometric template image of any size and the strength of the cipher is quite significant as interlacing is causing a lot of transposition in the elements of the plaintext.

References

[1] William Stailings, "Cryptography and Network Security Principles and Practices", Prentice Hall.2006.

[2] Forouzan Behrouz . A "Cryptography And Network Security", McGraw Hill 2008.

[3] G.R. Blakley, Twenty years of cryptography in the open literature, Security and Privacy 1999, Proceedings of the IEEE Symposium, 9-12,May 1999.

[4] W.-K. Chen, Scott Sutherland, "An Introduction of Cryptography", MSTP MATH WORKSHOP, 2005.

[5] Lester S. Hill, Cryptography in an Algebraic Alphabet, The American Mathematical Monthly, Vol. 36, No. 6. (Jun. - Jul., 1929), pp. 306-312.

[6] Sastry, V.U.K. and N. Ravi Shankar, "Modified Hill Cipher for a Large Block of Plain Text with Interlacing and Iteration", Journal of Computer Science 4 (1): 15-20, 2008.

[7] Bibhudendra Acharya, Girija Sankar Rath, Sarat Kumar Patra, and Saroj Kumar Panigrahy. "Novel Methods of Generating Self-Invertible Matrix for Hill Cipher Algorithm". International Journal of Security (CSC Journals). Vol. 1, Issue. (1), pp. 14-21, 2007.

[8] Bibhudendra Acharya Sarat Kumar Patra , Ganapati Panda "Invertible, Involutory and Permutation Matrix Generation Methods for Hill Cipher System", IEEE International Conference on Advanced Computer Control, Jan 2009.

[9] Bibhudendra Acharya, Sambit Kumar Shukla, Saroj Kumar Panigrahy, Sarat Kumar Patra and Ganapati Panda, "H-S-X Cryptosystem and its Application to Image Encryption" International Conference on Advances in Computing, Control, and TelecommunicationTechnologies, 2009, pp 720-724.

[10] Charlie Obimbo, Behzad Salami, "A Parallel Algorithm for determining the inverse of a matrix for use in block cipher encryption/decryption". J Supercomput. 200, 39: pp 113-130

[11] Rushdi A. Hamamreh and Mousa Farajallah , "Design of a Robust Cryptosystem Algorithm for Non-Invertible Matrices based on Hill Cipher", IJCSNS International Journal of Computer Science and Network Security, VOL.9 No.5, May 2009.

[12] Alper Kanak, "Biometrics For Computer Security And Cryptography", June 3rd, 2004.