Hello! I need help with below questions. Thank You

(1) Let DES(x, K) represent the encryption of plaintext x with key K using the DES cryptosystem.

Suppose y=DES(x, K) and y=(DES(c(x),c(K)), where c(*) denotes the bitwise complement of its argument.

Prove that y=c(y) (i.e., if we complement the plaintext and key, then the ciphertext is also complemented).

Note that this can be proved using only the high-level description of DES

the actual structure of S-boxes and other components of the system may be irrelevant.

(2) With the ECB mode of DES, if there is an error in a block of the transmitted ciphertext,

only the corresponding plaintext block is affected. However, in the CBC mode,

this error propagates. For example, an error in the transmitted C1 obviously corrupts P1 and P2.

Are any blocks beyond P2 affected? Suppose that there is a bit error in the source version of P1.

Through how many ciphertext blocks is this error propagated?

(3) Suppose we have a computer that can do 2^45 DES decryptions per second.

What is the expected time (in hours ) to find the correct key by brute force search given that the DES key space is 2^56?