CONSTRUCTION OF “ SMART ENCRYPTION AND SMART DECRYPTION “

PROBLEM – 1 : Let a , b , P be three distinct given positive integers with P is an Odd Prime and a*b < P

How to construct an Encryption and Decryption using a , b , P

Define M : = a* b – P

d : = P*M + b

e : = P2 * M + a

using this M , d , e , we are going to construct a new positive integer N such that g.c.d ( d , N ) = 1 and

g. c . d ( e , N ) = 1

Now let us compute :

e *d - P = ( P2 * M + a ) ( P *M + b ) – P

= ( P3 * M2 + P2 * M * b + a * P *M + a *b ) – P

= P3 * M2 + P2 * M *b + a *P * M + M

NOW , WE DEFINE : N = P3 *M + P2 * b +P *a + 1

It is important to note that g.c.d ( d , N ) = 1 and g.c.d ( e , N ) = 1

[ 0 , e – 1 ] = { 0 ,1 , 2 , 3 , . . . , e – 1 }

[ 0 , d – 1 ] = {0 ,1 ,2 , 3 , . . . , d – 1 }

f : [ 0 ,e -1 ] x [ 0 , d – 1 ] ……….> [0 ,e – 1 ] x [ 0 , d – 1 ]

f ( x , y ) = ( N*x + a* ( Mod e ) , N*y + b* ( Mod d ) )

x , a* lies in [ 0 , e – 1 ] and y , b* lies in [ 0 , d – 1 ]

This f is a bijective function for our encryption