\textbf{Exercise 4 — Candidates who have followed the specialization course}

A person has developed the following encryption process:\\
— To each letter of the alphabet, we associate an integer $n$ as indicated below:

\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | c | c | c | c | c | }
\hline
A & B & C & D & E & F & G & H & I & J & K & L & M \\
\hline
0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\
\hline\hline
N & O & P & Q & R & S & T & U & V & W & X & Y & Z \\
\hline
13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 & 25 \\
\hline
\end{tabular}
\end{center}

— We choose two integers $a$ and $b$ between 0 and 25.\\
— Every integer $n$ between 0 and 25 is encoded by the remainder of the Euclidean division of $an + b$ by 26.\\
The following table gives the frequencies $f$ in percentage of letters used in a text written in French.

\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | c | c | c | c | c | c | }
\hline
Letter & A & B & C & D & E & F & G & H & I & J & K & L & M \\
\hline
Frequency & 9.42 & 1.02 & 2.64 & 3.38 & 15.87 & 0.94 & 1.04 & 0.77 & 8.41 & 0.89 & 0.00 & 5.33 & 3.23 \\
\hline\hline
\end{tabular}
\end{center}

\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | c | c | c | c | c | c | }
\hline
Letter & N & O & P & Q & R & S & T & U & V & W & X & Y & Z \\
\hline
Frequency & 7.14 & 5.13 & 2.86 & 1.06 & 6.46 & 7.90 & 7.26 & 6.24 & 2.15 & 0.00 & 0.30 & 0.24 & 0.32 \\
\hline
\end{tabular}
\end{center}

\textbf{Part A}

A text written in French and sufficiently long has been encoded according to this process. Frequency analysis of the encoded text showed that it contains $15.9\%$ of O and $9.4\%$ of E.\\
We wish to determine the numbers $a$ and $b$ that allowed the encoding.

\begin{enumerate}
  \item Which letters were encoded by the letters O and E?
  \item Show that the integers $a$ and $b$ are solutions of the system
$$\left\{ \begin{array}{l} 4a + b \equiv 14 [26] \\ b \equiv 4 [26] \end{array} \right.$$
  \item Determine all pairs of integers $(a, b)$ that could have allowed the encoding of this text.
\end{enumerate}

\textbf{Part B}
\begin{enumerate}
  \item We choose $a = 22$ and $b = 4$.\\
a. Encode the letters K and X.\\
b. Is this encoding feasible?
  \item We choose $a = 9$ and $b = 4$.\\
a. Show that for all natural integers $n$ and $m$, we have:
$$m \equiv 9n + 4 [26] \Longleftrightarrow n \equiv 3m + 14 [26]$$
b. Decode the word AQ.
\end{enumerate}