DES has a somewhat surprising property related to bitwise complements of its inp

Question

DES has a somewhat surprising property related to bitwise complements of its inputs and outputs. We investigate this property in this problem. We denote the bitwise complement of a number A (that is, all bits are flipped) by A'. Let denote bitwise XOR. We want to show that if y = DES_k (X) then y' = DES_k' (x') This states that if we complement the plaintext and the key, then the ciphertext output will also be the complement of the original ciphertext. Your task is to prove this property. Try to prove this property using the following steps: 1. Show that for any bit strings A, B of equal length, A' B' = A B and A' B = (A B)'. (These two operations are needed for some of the following Steps.) 2. Show that PC - 1 (k') = (PC - 1(k))'. 3. Show that LS_i (C_i - 1) = (LS_i (C_i - 1))' 4. Using the two results from above, show that if k_i are the keys generated from K. then k'_i are the keys generated from k', where i = 1, 2, ..., 16. 5. Show that IP(x') = (IP(x))' 6. Show that E(R'_i) = (E(R_i))'. 7. Using all previous results, show that if R_i -1, L_i - 1, k_i generate then R'_i - 1, L'_i - 1, k'_i generate R'_i

Explanation / Answer

Solution:

1) we know that

a) A xor B = AB'+A'B

now consider left side

A' xor B' = A' (B')' + (A')' B' = A'B + AB' which is equivalent to right side

b) now in second case consider left side

A' xor B = A' (B)' + (A')' B = A'B' + AB

and now consider right hand side of second part in question 1

(A xor B)' = (AB'+A'B)' which is complement of xor gate so

(AB'+A'B)' = A'B' + AB ... which proves the second part in question 1

2) PC - 1 (k') =   (PC - 1 (k))'

here in this part we consider,

the complement of key gives the output to be complemented

which is proved by the complement property in the second part of the first question

3) LSi ( Cii-1) = ( LSi ( Ci-1))i

here in this part we consider,

the complement of key gives the output to be complemented

which is proved by the complement property in the second part of the first question

4) the keys Ki generated from the K will equivalent be like Kii from Ki

in the above two parts 2 and 3 we showed that the complement of the key at the input makes output to be complemented likewise in this case the complement of Ki Values would happen to be for various values of i= 1, 2, 3, ...

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