The S-boxes of the Feistel cipher are designed to support the avalanche criterion. In particular, by flipping one bit of the input of an S-box, on average, at least two bits of the output will flip. This exercise wants you to show this for particular values. To help you, we first illustrate this for one input; you then must show this for another input. In both cases, assume that all round keys are: 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 (48 bits)

a. Case 1 (illustration): Let the input be: 1000 0000 0000 0000 0000 0000 0000 0001 (32 bits) The table below is used to expand it.

110000 000000 000000 000000 000000 000000 000000 000011 (48 bits) The result is XORed with the key round key to get 001111 111111 111111 111111 111111 111111 111111 111100 (48 bits) Then we use the substitution below

to get: 0001 1101 1101 1101 1101 1101 1101 0101 (*) This is the 32-bit output. Next flip the first bit of the input, to get 0000 0000 0000 0000 0000 0000 0000 0001 expand it to, 100000 000000 000000 000000 000000 000000 000000 000010 (48 bits) XOR it with the round key to get, 011111 111111 111111 111111 111111 111111 111111 111101 (48 bits) and then use the substitution above to get 1000 1101 1101 1101 1101 1101 1101 0110 (**) This is the output. Notice that one flipped bit in the input of the first S-Box has caused 2 bits to flipped in the output of the first substitution and anther 3 bits in the output of the last substitution. b. Case 2. Now do this for input (same round key): 1100 0000 0000 0000 0000 0000 0000 0001 and then flip the third bit to see how many bits get flipped in the output. Show all your work, and dont forget to count the number of flipped bits.