Qu Question Eight: (10 marks) 1) If p is a shared secret key, and odd number, then when choose at random large q, and small r to encrypt a bit m, by using the following rule C=pq+2r+m Where 2r+m much smaller than p, Ciphertext is close to a multiple of p, and m = LSB of distance to nearest multiple of p. With to decrypt C, by m=( C mod p) mod 2 show that encryption above is homomorphic? 2) The Grigorchuk's group is the subgroup of A(T) generated by the automorphisms a, b, c, d defined as follows: 0 alb.by......) (1 bigba.....) 16.1- .....if b, .b(b.by......) (6.b......)) if bi (b. 1 - by......) if by cfb.by....b) (b.d(by.....6,)) if b, h. 1 - bs......) if by 0 0 . 1) Show that is non-commutative group? 2) If X=(b1,b2,63,babs,be)=(1,1,1,0,1,0), find alb(c(X)))? b(c(x)? Qu Question Eight: (10 marks) 1) If p is a shared secret key, and odd number, then when choose at random large q, and small r to encrypt a bit m, by using the following rule C=pq+2r+m Where 2r+m much smaller than p, Ciphertext is close to a multiple of p, and m = LSB of distance to nearest multiple of p. With to decrypt C, by m=( C mod p) mod 2 show that encryption above is homomorphic? 2) The Grigorchuk's group is the subgroup of A(T) generated by the automorphisms a, b, c, d defined as follows: 0 alb.by......) (1 bigba.....) 16.1- .....if b, .b(b.by......) (6.b......)) if bi (b. 1 - by......) if by cfb.by....b) (b.d(by.....6,)) if b, h. 1 - bs......) if by 0 0 . 1) Show that is non-commutative group? 2) If X=(b1,b2,63,babs,be)=(1,1,1,0,1,0), find alb(c(X)))? b(c(x)