For the questions in this project, please ensure you answer in the following manner: include all work for solving any particular problem. Problems that allow you to use an outside calculator will state so; if you use an outside calculator, you must cite the calculator or website used. If the answer is non$-$computational, please write in complete sentences.

In this project, we will be exploring the Hill cipher. Ciphers are methods for encrypting or decrypt$-$ ing messages. All mathematical ciphers begin by mapping the letters of the alphabet to a corresponding set of numbers. There are multiple ways to create this map, depending on what you want to include in your cipher. For this project, we will use the cipher map below, where the first row maps to the second, and the third row maps to the fourth:

The $0$ element $($which$,$ in the alphabet is denoted by $-)$ will take the place of spaces in any phrases, and can

$-$ABCDEFGHIJKLMN

$01234567891011121314$ OPQRSTUVWXYZ.$/$

$1516171819202122232425262728$

also be used as placeholders in the message $($which you will see below$).$ In general, we wish to make sure that the size of our alphabet is a prime number; you will note that this alphabet includes $29$ elements $-$ the $26$ letters, a space, period, and forward slash. What goes into the alphabet is a choice that should be known to both the sender of the message and the recipient of the message.

The most widely known cipher is the Caesar cipher. In the Caesar cipher $($so named because Julius Caesar was known to use it in his private correspondence$),$ every letter of the alphabet is shifted by a predetermined amount. We will require one of the mathematical techniques used in this cipher, so let us explore this before returning to the Hill cipher.

Using the map above, we will next determine the shift value, S$,$ of our cipher. Since our map has $29$ elements, the shift value must between $1$ and $28.$ Then, for every element x in our alphabet, we apply the function f$($x$)=$ x $+$ S$.$ For example, if the shift value is $4,$ then f$(1)=1+4=5$; looking at our table, this has transformed the letter A to the letter E$.$ For the use of this cipher, all numbers should be two digit numbers $-$ so you will encode $1$ as $01,2$ as $02,$ and so on$,$ while using the Caesar cipher.

$1.$ Suppose I want to sent the message MATH using the Caesar cipher with a shift value of $2.($a$)$ Before the shift, what is the numeric code for MATH?