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

Let $E$ denote the set of twenty-seven integers between 0 and 26.

Let $A$ denote the set whose elements are the twenty-six letters of the alphabet and a separator between two words, denoted ``$\star$'' and considered as a character.

To encode the elements of $A$, we proceed as follows:

\begin{itemize}
  \item First: We associate to each of the letters of the alphabet, arranged in alphabetical order, a natural integer between 0 and 25, arranged in increasing order. We thus have $a \rightarrow 0 , b \rightarrow 1 , \ldots z \rightarrow 25$. We associate to the separator ``$\star$'' the number 26.
  \item Second: to each element $x$ of $E$, the function $g$ associates the remainder of the Euclidean division of $4 x + 3$ by 27. Note that for every $x$ in $E$, $g ( x )$ belongs to $E$.
  \item Third: The initial character is then replaced by the character of rank $g ( x )$.
\end{itemize}

Example: $s \rightarrow 18 , \quad g ( 18 ) = 21$ and $21 \rightarrow v$. So the letter $s$ is replaced during encoding by the letter $v$.

1. Find all integers $x$ of $E$ such that $g ( x ) = x$, that is, invariant under $g$.

Deduce the invariant characters in this encoding.

2. Prove that, for every natural number $x$ belonging to $E$ and every natural number $y$ belonging to $E$, if $y \equiv 4 x + 3$ modulo 27 then $x \equiv 7 y + 6$ modulo 27.

Deduce that two distinct characters are encoded by two distinct characters.

3. Propose a decoding method.

4. Decode the word ``vfv''.