Mathematical Tools for Cryptography

INTRODUCTION

In present world, with increasing usage of computer networks and internet, the importance of network, computer and information security is must. For the purpose of securing the data or, information which has to be protected from unauthorized access. Hence, data security has become a critical and important issue. One of the widely used approaches for information security is Cryptography. Cryptography, the mathematic of encryption, plays a vital role in many fields. The fundamental aim of cryptography is to enable two people, to communicate over an insecure channel in such a way that an opponent cannot understand what is being said. Encryption is the process of obscuring information to make it unreadable without special knowledge. A cipher is an algorithm for performing encryption (and on other hand, decryption) a series of well-defined steps that can be followed as a procedure. The original information is known as plain text, and the encrypted form as cipher text. The cipher text message contains all the information of the plaintext message, but is not in a format readable by a human or computer without the proper mechanism to decrypt it. Ciphers are usually parameterized by a piece of auxiliary information, called a key. The encrypting procedure is varied depending on the type of key which can changes the detailed operation of the algorithm. Without the key, the plaintext cannot be encrypted, or more importantly, to decryption varied depending on the key which changes the detailed operation of the algorithm.

This is based on theorems as mentioned below. 1. The given plane text in terms of Gi,o i=1,2,3,……., under Laplace of Gt2sinh 2r (that is by writing them as a coefficients of t2sinh 2r and then taking the Laplace transform) can be converted to cipher text Gi,1 = 22i+1(2i+2)(2i+3)Gi,0, mod 26 = qi,1-26ki,1, for i=0,1,2,3,……Z Where qi,1=22i+1(2i+2)(2i+3)Gi,0 i=0,1,2,3……. And key is Ki,1 = for i=0,1,2,3,…….. 2. The given plan text in terms of Gi,0 i=0,1,2,3,……. Under Laplace transform of Gi,0t2 sinh2t, successively j times ( that is writing them as a coefficients of t2 sinh 2t and then taking the Laplace transform =qi,j-26ki,j i,j=0,1,2,3,……. Where q i ,j = Gi,j-122i+1(2i+2)(2i+3) i , j =0,1,2,3…… and a key is k i , j = i , j =0,1,2,3,……….

1. The given cipher text in terms of Gi,1 i=0,1,2,3,4,…….with a given key k i , 0 , i = 0,1,2,3,…..which can be converted to plain text G i, 0 under the inverse Laplace transform of Where i = 01,2,3,….. and q i.0 = 26ki,0 + G i 1 i= 0,1,2,3,….. 2. The given cipher text in terms of Gi , j i , j =1,2,3,4,…..with a given key k i, j-1 i ,j =1,2,3,4,……can be converted to plain text Gi, j-1 under the inverse Laplace transform of Where i, j = 1,2,3,4,…… and qi, j =26ki,j + Gi,j i , j =1,2,3,4,……… institutions are adopting e-services and improving their Internet services. However, the e-service requirements are also opening up new opportunity to commit financial fraud. Internet banking fraud is one of the most serious electronic crimes and mostly committed by unauthorized users. The new method of key generation scheme developed in this paper may be used for a fraud prevention mechanism For computer network security random number generation is a prime important task and also it is very essential in constructing keys for cryptographic algorithm. Method used in this review paper can be useful for random number generation

[1] Alexander Stanoyevitch (2002). Introduction to cryptography with mathematical foundations and computer implementations, CRC Press. [2] Barr T.H. (2002). Invitation to Cryptography, Prentice Hall. [3] Blakley G.R. (1999). Twenty years of Cryptography in the open literature, Security and Privacy 1999, Proceedings of the IEEE Symposium. [4] Dhanorkar G.A. and Hiwarekar A.P. (2011). A generalized Hill cipher using matrix transformation, International J. of Math. Sci. & Engg. Appls,Vol.5 No. IV, pp. 19-23. [5] Eric C., Ronald K., James W.C. (2009). Netwark Security Bible Second edn.,Wiley India pub. [6] Erwin Kreyszing (1999). Advanced Engineering Mathematics, John Wiley and Sons Inc.

[1] G.Naga Lakshmi, B.Ravi Kumar and A.ChandraSekhar, A cryptographic scheme of Laplace transforms, International Journal of Mathematical Archive-2, 2515-2519, (2011). [2] Grewal B.S., Higher Engineering Mathematics, Khanna Pub., Delhi, (2005). [3] Gupta P., Mishra P. R., Cryptanalysis of ―A New Method of Cryptography Using Laplace Transform, Proceedings of the Third International Conference on Soft Computing for Problem Solving Advances

[4] Hiwarekar A.P., A new method of cryptography using Laplace transform, International Journal of Mathematical Archive, 3(3), 1193-1197, (2012). [5] Hiwarekar A.P., A new method of cryptography using Laplace transform of hyperbolic functions, International Journal of Mathematical Archive, 4(2), 208-213, (2013). [6] Johannes A. Buchmann, Introduction to Cryptography, Fourth Edn., Indian Reprint , Springer, (2009). [7] Lokenath Debnath, Dambaru Bhatta (2010). Integrl Transforms and Their Applications, Chapman and Hall/CRC, First Indian edn. [8] Overbey J., Traves W.and Wojdylo J. (2005). On the Keyspace of the Hill Cipher, Cryptologia, 29, pp. 59-72. [9] Ramana B.V. (2007). Higher Engineering Mathematics, Tata McGraw-Hills.

Assistant Professor, Department of Mathematics, Government First Grade College, Tumakuru