Modified Hill Cipher Using Vandermonde Matrix and Finite Field

P. L. Sharma and M. Rehan

Department of Mathematics, Himachal Pradesh University, Summer Hill, Shimla - 171005, India.

*Corresponding Author E-mail: plsharma1964@gmail.com, mansihverma16@gmail.com

ABSTRACT:

Cryptography is the science which provides confidentiality, authenticity and integrity to the users over the prevailing insecure communication channels. Encryption is the transformation of data into some unreadable form. The purpose is to ensure privacy by keeping the information hidden from anyone for whom it is not intended, even those who can see the encrypted data. Decryption is the reverse of encryption; it is the transformation of encrypted data back into some intelligible form. Encryption and decryption require the use of some secret information, usually referred to as a key. Hill cipher in cryptography is a symmetric key substitution algorithm, which is vulnerable to known plaintext attack. We modify the existing Hill cipher by using Vandermonde matrix and introducing the elements of finite field which provides more security to the said cipher.

KEY WORDS: Plain Text; Symmetric Key; Hill Cipher; Finite Field; Vandermonde Matrix.

MSC: 11T71, 94A60, 68P25.

1. INTRODUCTION:

The security of information to maintain its confidentiality has become a major issue in the present era of electronic transmission. Cryptography is a cornerstone of the modern electronic security technologies which is used to protect valuable information resources on intranets, extranets, and the internet [1, 3]. It is the study of techniques that provides confidentiality, integrity, availability and authenticity of information sent through insecure channels. Various techniques from different areas of mathematics like number theory [9], matrix analysis, finite fields [2], logical operators [4] etc. are used in building ciphers.

The Hill cipher in cryptography is a classical symmetric cipher which is based on matrix transformation. It was invented by Lester S. Hill in 1929 [5]. He extended this work in [6]. Hill cipher has its own advantages like frequency analysis, high speed, high throughput and the simplicity as it uses matrix multiplication and inversion for encryption and decryption. However it succumbs to the known plaintext attack [7, 8]. In Hill cipher for decryption to be possible, the key matrix should be invertible. According to Overbay [10] the key space of Hill cipher is the group of matrices that are invertible over where is ring of integers modulo [9].

Several researchers have contributed to improve the security of Hill Cipher using different techniques. Saeednia [11] made Hill cipher secure using the dynamic key matrix obtained by random permutations of columns and rows of the master key matrix. Chefranov [12] proposed a modification to [11] similar to Hill cipher permutation method but it uses a pseudo-random permutation generator. The number of dynamic keys is same as taken in [11]. Ismail et al. [13] repaired Hill cipher by introducing an initial vector that multiplies each row of current key matrix to form a different key for each block encryption. Adi et al. [14] modified the Hill cipher based on circulant matrices. Sharma et al. [15] used finite field and logical operator to make the cipher more secure. Vandermonde matrices have important role in many branches of applied mathematics such as Combinatorics, coding theory and cryptography. Due to wide range of applications of Vandermonde matrices in different area of mathematical sciences as well as other sciences, they have attained much importance. In the present paper, we have used Vandermonde matrix and elements of finite field for the encryption and decryption of plaintext. The illustration for the proposed algorithm is also given.

4. CONCLUSION:

The proposed algorithm is based on Vandermonde matrix and the elements of finite fields, which is an extension of the original Hill cipher. In this cryptosystem, prime Vandermonde matrix and a non-singular matrix is used such that determinant of the coefficient matrix is zero which generate infinite solutions. Hence, it becomes difficult for the hacker to break the cipher. Also, the second secret key is different for different block data which gives difficulty for adversary to break the cryptosystem. Therefore, there are least possibilities of Brute force attack. Also, the cipher text cannot be broken with the known plain text attack as there is no direct relation between plain text and cipher text.

5. REFERENCES:

[1] Stallings W., “Cryptography and Network Security” Fourth Edition, Pearson, 2006.

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[3] Schneier B., “Applied Cryptography: Protocols, Algorithms, and Source Code in C”, Second Edition, JohnWiley& Sons, 2007.

[4] Lakshami G. N., Kumar B. R., Suneetha C. and Chandra Shekhar A., A Cryptographic Scheme of Finite Fields Using Logical Operators, International Journal of Computer Applications, Vol. 31, No. 4, p.1-4, 2011.

[5] Hill L.S., Cryptography in an Algebraic Alphabet, American Mathematical Monthly, Vol.36, p. 306-312, 1929.

[6] Hill L. S., Concerning Certain Linear Transformation Apparatus of Cryptography, American Mathematical Monthly, Vol.38, p. 135-154, 1931.

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[9] Koblitz N.,“A Course in Number Theory and Cryptography”, Springer Verlag, New York, 1994.

[10] Overbey J., Traves W. and Wojdylo J., On The Key Space of The Hill Cipher, Cryptologia, Vol. 29, No.1, p. 59-72, 2005.

[11] Saeednia’s S., How to Make The Hill Cipher Secure, Cryptologia, Vol.24, p. 353-360, 2000.

[12] Chefranov A.G., Secure Hill Cipher Modification SHC-M, Proceedings of the First Internationl Conference on Security of Information and Networks, Trafford Publishing, Canada, p. 34-37, 2007.

[13] Ismail I.A., Amin M. and Diab H., How to Repair Hill Cipher, Journal of Zhejiang University-Science A, Vol.7, No. 12, p. 2022-2030, 2006.

[14] Adi N.R.K., Vishnuvardhan B., Madhuviswanath V. and Krishna A.V.N., A Modified Hill Cipher Based on Circulant Matrices, Procedia Technology (Elsevier), Vol. 4, p. 114-118, 2012.

[15] Sharma P.L. and Rehan M., On the Security of Hill Cipher Using Finite Field, International Journal of Computer Applications, International Journal of Computer Applications, Vol. 71, No.4, p. 30-33, 2013.

Received on 15.01.2014 Accepted on 02.02.2014

Int. J. Tech. 4(1): Jan.-June. 2014; Page 252-256