Examples of groups ZN: the multiplicative group mod N G=Z n the positive integers modulo n relatively prime to N the group operator is"x>, modular multiplication the integers modulo n are closed under multiplication this is so because if GCD(x, n)=l and GCD( n)=1 then gCd(xy, n)=1 · the identity is the inverse of x is from Euclids algorithm Ux+VN=1=GCD(X, N) So X=uX(N-) multiplication is associative multiplication is commutative(so the group is abelian)Examples of Groups Z* N : the multiplicative group mod N G = Z* N = the positive integers modulo N relatively prime to N the group operator is “*”, modular multiplication • the integers modulo N are closed under multiplication: this is so because if GCD(x, N) =1 and GCD(y,N) = 1 then GCD(xy,N) = 1 • the identity is 1 • the inverse of x is from Euclid’s algorithm: ux + vN = 1 = GCD(x,N) so x-1 = u (= x f(N)-1 ) • multiplication is associative • multiplication is commutative (so the group is abelian)