In this problem, consider a substitution permutation network composed of 8n bits, with n = 2^, s a positive integer. The bits are labelled by a pair (j, k), with j a number from 0 to 7 and k an s-bit binary number kokyk...k-1. That is, ko is the most significant bit of k and k.-1 is the least significant bit, so for instance, if k= 4, its binary representation is 100 and ko = 1, k = 0, and k = 0. In the substitution permutation network, the bits are mixed with a key via XOR, then divided up into groups of 8 and passed through S-boxes. Bits with the same k label go into the same S-box. The S-boxes take 8-bit inputs and produce 8-bit outputs. Then the bits are passed to one of the following transformations, which relabel bit (j, k) as bit (j',k) (for all (j. k)), completing one round. The same sequence of steps is repeated for many rounds. Only one of the transformations below is a possible candidate to produce a substitution permutation network with an avalanche effect. Identify which one and for each of the candidates, describe your reasoning for why it is or is not the correct choice. (10 points for each candidate.) 

a) (j,k)  (j', k') with kk for all k, j'=j+1 if j is even, and j'=j-1 if j is odd. 

b) (j,k)  (j',k) with j'=j+ 4 mod 8, k = 1k for i odd, and kk, for i even. (That is, the odd numbered bits of k are flipped.) 

e) (j.k)  (j', k') with j' = 3j mod 8 and k' = (3k+j) mod n. 

d) (j. k)  (j', k') with j'=4j mod 8 and k' = (4k+j) mod n.