An Application of Finite Field in Hill Cipher

P. L. Sharma and S. Sharma

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

*Corresponding Author E-mail: plsharma1964@gmail.com, shabz0887@gmail.com

ABSTRACT:

Lester S. Hill in 1929 introduced non-singular matrices to encrypt and decrypt the message in the symmetric key cryptography. Block ciphers designing and cryptographic hash functions have the use of matrices. Here, we introduce permutations, iterations and finite field in the Hill cipher to provide more security and make it free from vulnerable attacks.

AMS Classification: 11T71, 94A60.

KEY WORDS: Encryption; Decryption; Finite field; Hill Cipher.

1. INTRODUCTION:

Information security has become a very critical aspect of modern computing systems. Cryptography is the science which provides confidentiality, authenticity and integrity of information passing through insecure channels [1]. Although the ultimate goal of cryptography, and the mechanisms that make it up, is to hide information from unauthorized individuals, most algorithms can be broken and the information can be revealed if the attacker has enough time, desire, and resources. So a more realistic goal of cryptography is to make obtaining the information too work-intensive to be worth it to the attacker. Various researchers have used different techniques to build ciphers so that the systems become more secure and no hacker will be able to break those ciphers. Finite field has greater importance in cryptography as the elements of finite field used in transforming computer data [2].

The Hill Cipher is classical symmetric cipher invented by Lester S. Hill in 1929 [3] and extension of this work is in [4]. The main advantages of Hill cipher includes its frequency analysis, high speed, high throughput and the simplicity because it uses matrix operations but it succumbs to the known plaintext attack [5]. Hill cipher is modified by several authors. Saeednia [6] used the dynamic key matrix while Chefranov [7] used a pseudo-random permutation generator. Ismail et al. [8] gave an initial vector to form a different key for each block encryption. Adi et al. [9] modified the Hill cipher using circulant matrices. Shastry et al. [10, 11] used key on both sides of the pain text to modify Hill cipher. Sharma and Rehan [12] modified Hill cipher using elements of finite field and logical operator. In the present paper, our aim is to modify the Hill Cipher by multiplying K and one on the left side and another on right side of the plain text matrix in encryption as well as decryption and use the permutation on the binary bits. Also, we use the elements of finite field for the purpose of more security. The illustration of the proposed modification is also given.

3. CONCLUSION:

The proposed cipher used a key matrix and its inverse on both sides of the plaintext. In this cipher, iteration and permutation is used which makes more difficulty for the hacker to break it. Therefore, there are least possibilities of chosen plaintext attack and known plaintext attack. Also the cipher cannot be broken by Brute Force attack as the modulus is taken a prime number.

4. REFERENCES:

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

[2] Lidl R. and Niederreiter H., “Finite Fields”, Cambridge University Press, Cambridge, Second Edition, 1997.

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

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

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

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

[7]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.

[8] 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.

[9]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.

[10] Sastry V.U.K., Murthy D.S.R. and Bhavani S.D., A Block Cipher Involving a Key Applied on Both Sides of The Plain text, International Journal of Computer and Network Security, Vol.1, No.1, p. 27-30, 2009.

[11] Sastry V.U.K., Murthy D.S.R. and Bhavani S.D., A Block Cipher Having a Key on One Side of the Plain text Matrix and Its Inverse on the Other Side, International Journal of Computer and Network Security, Vol.2, No.5, p. 1793-8201, 2010.

[12] 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, 2013.

Received on 14.01.2014 Accepted on 02.02.2014

Int. J. Tech. 4(1): Jan.-June. 2014; Page 248-251