In a certain network protocol, each datagram consists of a $512-$bit header and a $512-$bit payload.

An organisation decides to store hashed datagrams for analytic purposes. These hash values

need to be calculated on resource$-$limited devices that can only process data in $512-$bit chunks.

Thankfully, the devices have hardware implementations of the following cryptographic algorithms:

$$ E: a secure block cipher with $512-$bit keys, $512-$bit inputs, and $512-$bit outputs, and

$$ H: asecure internal hash function with $512-$bit inputs and $256-$bit outputs.

Since the internal hash function H cannot process entire datagrams, it is necessary to design a

hashing method that can compute datagram hashes based on the internally implemented

algorithms. The engineers come up with the following designs for hashing datagrams:

Design A: hash the header and payload separately, then XOR the results, i$.$e$.$

Hash$($datagram$)=$ H$($header$)$H$($payload$)$

Design B: hash the header and payload separately, then concatenate and hash the results, i$.$e$.$

Hash$($datagram$)=$ H$($ H$($header$)\backslash |$ H$($payload$))$

Design C: encipher the payload using E with the header as key, then XOR the result with the

header, i$.$e$.$

Hash$($datagram$)=$ H$($ Eheader$($payload$)$ header $)$

In these formulas, $$ denotes the bitwise exclusive or operation and $\backslash |$ denotes string

concatenation. The designed datagram hashing method is called Hash to distinguish it from the

internal hash function H$.$

Tasks

$($i$)[2$ Marks$]$ The organisation needs to develop a program that looks at the stored datagram

hash values and for any two hashed datagrams reliably distinguishes if the original

datagrams were identical or not. Identify the security requirement on datagram hashing in

this case, and briefly justify your answer.

$($ii$)[8$ Marks$]$ Consider the three candidate designs, i$.$e$.,$ Designs A$,$ B$,$ and C$,$ above. For

each design, discuss if it satisfies the security requirement identified in part $($i$)$ or not. In

each case provide a convincing argument or an attack to support your answer.