Modified Diffie-Hellman Protocol By Extend The Theory of The Congruence

This paper is introduced a modification to Diffie- Hellman protocol to be applicable on the decimal numbers, which they are the numbers between zero and one. For this purpose we extend the theory of the congruence. The new congruence is over the set of the real numbers and it is called the “real congruence" or the “real modulus". We will refer to the existing congruence by the “integer congruence" or the “integer modulus". This extension will define new terms and redefine the existing terms. As the properties and the theorems of the integer modulus are extended as well. Modified Diffie-Hellman key exchange protocol is produced a sharing, secure and decimal secret key for the the cryptosystems that depend on decimal numbers.

Authors:

• Rand Alfaris
• Mohamed Rushdan MD Said
• Mohamed Othman
• Fudziah Ismail

Keywords:

• Extended theory of the congruence
• modified Diffie- Hellman protocol.

References:

[1] Burton, D. M. The Theory of Congruences. In Elementary Number
Theory, 4th ed. Boston, MA: Allyn and Bacon, pp. 80-105, 1989.
[2] Conway, J. H. and Guy, R. K. Arithmetic Modulo. In The Book of
Numbers. New York: Springer-Verlag, pp. 130-132, 1996.
[3] Cormen, T. H., Leiserson, C. E., Rivest, R. and Stein, C. Greatest common
divisor. In Introduction to Algorithms, MIT Press and McGraw-Hill, 2nd
edn., pp. 856-862, 2001.
[4] Diffie, W. and Hellman, M. New Directions in Cryptography. IEEE
Transactions on Information Theory IT-22: pp. 472-492, 1976.
[5] Hejhal, D. A., Friedman, J., Gutzwiller, M. C. and Odlyzko, A. M.
Emerging Applications of Number Theory. New York: Springer. 2nd edn.,
1999.
[6] Hrbacek, K. and Jech, T. Finite, Countable, and Uncountable Sets. In
Introduction to Set Theory. New York, 3rd edn., pp. 65-93. 1999.
[7] Gilbert, J. and Gilbert, L. Elements of Modern Algebra. 6th ed. Thomson,
brooks/cole, pp. 57-117, 2005.
[8] Nagell, T. Theory of Congruences. In Introduction to Number Theory.
New York: Wiley, pp. 68-131, 1951.
[9] Niven, I. and Zuckerman, H. S. An Introduction to the Theory of Numbers.
New York: John Wiley and Sons. 4th edn., 1980.
[10] Rand Alfaris. Modified Diffie-Hellman Protocol. In A New Decimal
Cryptosystem Based on Decimal Numbers. Ph.D. Thesis, University Putra
Malaysia, pp. 55-81, 2008.
[11] Rand Alfaris, Muhamad Rezal Kamel Ariffin and Mohamed Rushdan
Md Said. Rounding Theorem the Possibility of Applying Cryptosystems
on Decimal Numbers. Journal of Mathematics and Statistics, Vol. 4(1),
pp. 15-20, 2008.
[12] S'eroul, R. Congruences. In Programming for Mathematicians. Berlin:
Springer-Verlag, pp. 11-12, 2000.