1. Today , We Shall see the most Important Finite Non abelian group from " Cryptography "
point of view.
1. Let P , Q R , S be any given distinct positive integers greater than or equal to " 2 "
2. DEFINE N = P.Q.R.S + 1 . THEN ( P , N ) = ( Q , N ) = ( R , N ) = ( S , N ) = 1
3. G = { ( a , b , c , d , e ) | a , b , c , d , e BELONGS TO { 0 ,1 , 2 , 3 , . . . N - 1 } }
Now in " G " WE DEFINE A BINARY OPERATION AS FOLLOWS :
4. ( a , b , c , d , e) * ( f , g , h , i , k ) = ( x , y , z , w , l
)
[ a ,b ,c,d,e ,f ,g ,h i , k belongs to { 0 ,1,2,3, . . . , N - 1 } ]
where x = a + f ( Mod N )
y = b + g ( Mod N )
z = c + h ( Mod N )
w = d + i ( Mod N )
l = f b + g c + h d + e + k ( Mod N )
Now " G " is Non abelian group of order = N.N.N.N.N
LET US WORK on this Non abelian group " G "
Then the map " T " from G to G
( a , b , c , d , e ) ------------> ( a , b , c , d ,e) * ( a , b , c , d ,e ) * .... * ( a , b ,c
,d ,e )
[ " P " times ] is a Permutation on " G " ( MULTIPLICATION BY * APPEARS " P
" TIMES )
Similarly , we can replace " P " BY Q , R , S [ N = P .Q .R.S + 1 ]