a$.$ Let $x^{\text{'}}$ be the bitwise complement of $x.$ Prove that if the complement of the plain$-$

text block is taken and the complement of an encryption key is taken, then the

result of DES encryption with these values is the complement of the original ci$-$

phertext. That is$,$

$IfY,=E(K,x)$

Then $Y^{\text{'}},=,E(K^{\text{'}},x^{\text{'}})$

Hint: Begin by showing that for any two bit strings of equal length, A and $B,$

$(Ao+B{)}^{\text{'}}=A^{\text{'}}o+B.$

b$.$ It has been said that a brute$-$force attack on DES requires searching a key space of

$2^{56}$ keys. Does the result of part $($a$)$ change that?