This question relates to hash functions for block ciphers $($sec $11\mathrm{.}3\mathrm{.}2)$

Block size $=4$ bits

Hash size $=4$ bits

Encryption function: Divide the key into two halves: LK and RK; Divide the plaintext into two halves: $LT$ and $RT$; Then ciphertext$=LC||RC$ where $LC=LK$ XOR $RT$; and $RC=$ RK XOR LT; where LC$,$ RC$,$ LT$,$ and RT are each $2$ bits; Plaintext and ciphertext are each $4$ bits.

$g(H)=$ a $4-$bit string that is equal to the complement of bits in H ; For example, if $H=A($Hexa$)=1010($binary$)$; then $g(H)=0101$

HO $=$ Initial hash $=$ C $($in Hexa$)$

Given message M: D$9($in Hexa$)$;

a$.[$Points $6]$ Determine the hash $($in hexadecimal$)$ of the message $M$ using Martyas$-$Meyer$-$Oseas hash function $($Fig$.11\mathrm{.}6).$

b$.[$Points $6]$ Determine the hash $($in hexadecimal$)$ of the message M using Davis$-$Meyer hash function $($Fig$.11\mathrm{.}7)$

c$.[$Points $6]$ Determine the hash $($in hexadecimal$)$ of the message M using MigayuchiPreneel hash function $($Fig$.11\mathrm{.}7)$

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