Please explain the solution process in detail and let me know the answer at the end.

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:

$111111111111111111111111111111111111111111111111(48$ bits$)$

a$.$ Case $1($illustration$)$: Let the input be:

$\{\begin{array}{ccc}\text{:}[1000,& 0000,0000,0000,0000,0000,0000,& 0001]\end{array}(32$ bits$)$

The table below is used to expand it$.$

$110000000000000000000000000000000000000000000011(48$ bits$)$

The result is XORed with the key round key to get

$001111111111111111111111111111111111111111111100(48$ 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$,$

$100000000000000000000000000000000000000000000010(48$ bits$)$

xOR it with the round key to get,

$011111111111111111111111111111111111111111111101(48$ bits$)$

and then use the substitution above to get

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 $2$ 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 don't forget

to count the number of flipped bits.