$(2)$ This problem introduces a hash function similar in spirit to SHA that operates on letters instead of binary data. It is called the toy tetragraph hash $($tth$)\mathrm{.}8$ Given a message consisting of a sequence of letters, tth produces a hash value consisting of four letters. First, tth divides the message into blocks of $16$ letters, ignoring spaces, punctuation, and capitalization. If the message length is not divisible by $16,$ it is padded out with nulls. A four$-$number running total is maintained that starts out with value $(0,0,0,0)$; this is input to a function, known as a compression function, for processing the first block. The compression function consists of two rounds. Round $1$: Get the next block of text and arrange it as a row$-$wise $4$ X $4$ block of text and convert it to numbers $($A $=0,$ B $=1),$ for example, the block ABCDEFGHIJKLMNOP, we have

A B C D

$0123$

E F G H $4567$

I J K L $891011$

M N O P $12131415$

Then, add each column mod $26$ and add the results to the running total, mod $26.$ In this example, the running total is $(24,2,6,10).$ Round $2$: Using the matrix from round $1,$ rotate the first row left by $1,$ second row left by $2,$ third row left by $3,$ and reverse the order of the fourth row. In our example,

B C D A

$1230$

G H E F $6745$

L I J K $118910$

P O N M $15141312$

Now, add each column mod $26$ and add the result to the running total. The new running total is $(5,7,9,11).$ This running total is now the input into the first round of the compression function for the next block of text. After the final block is processed, convert the final running total to letters. For example, if the message is ABCDEFGHIJKLMNOP, then the hash is FHJL$.$

a$.$ Draw figures of the overall tth logic and the compression function logic.

b$.$ Calculate the hash function for the $48-$letter message $$I leave twenty million dollars to my friendly cousin Bill.$$

c$.$ To demonstrate the weakness of tth$,$ find a $48-$letter block that produces the same hash as that just derived. Hint: Use lots of As$.$