Using an affine transformation to cryptanalyze an enciphered message

In the enciphered text the letters $L$ and $U$ and $Y$ have the largest number of occurrences, where $L$ occurs $5$ times and both letters $U$ and $Y$ occur $4$ times. $$

Choosing $L \rightarrow E$ and $U \rightarrow T$ $$

$\implies$ $$

$4 * a + b \equiv 11 \mod{26}$ $19 * a + b \equiv 20 \mod{26}$

$\implies 15 * a \equiv 9 \mod{26}$ $$

Using a repeated application of the Euclidean algorithm you can verify that $7$ is an inverse of $15$ modulo $26 \implies$ $a \equiv 63 \mod 26 \equiv 11 \mod{26} \implies b \equiv -33 \mod{26} \equiv -7 \mod{26} \equiv 19 \mod{26}.$ $$

$\implies 11 * P \equiv (C - 19) \mod{26}$ $$

You can use a repeated application of the Euclidean algorithm to verify that $19$ is an inverse of $11$ modulo $26 \implies$ $$

$\implies P \equiv 19 * (C - 19) \mod{26} \equiv (19* C - 23) \mod{26} \equiv (19 * C + 3) \mod{26}$ $$

$\therefore P \equiv (19 * C + 3) \mod{26} .$ $$

Now we attempt the tedious task of determining if we obtain an intelligible message.

Using each $C_{j},$ where $(1 <= j <= 34)$ in the enciphered message $USLEL \: JUTCC \: \: YRTPS \: \: URKLT \: \: YGGFV \: \: ELYUS \: \: LRYX$ we obtain: $$

$P \equiv 383 \mod{26} \equiv 19 \mod{26}$ which corresponds to the letter $T$ .

$P \equiv 345 \mod{26} \equiv 7 \mod{26}$ which corresponds to the letter $H$ .

$p \equiv 212 \mod{26} \equiv 4 \mod{26}$ which corresponds to the letter $E$ .

$P \equiv 79 \mod{26} \equiv 1 \mod{26}$ which corresponds to the letter $B$ . $$

$\hspace{260pt} E$ $$

$P \equiv 175 \mod{26} \equiv 18 \mod{26}$ which corresponds to the letter $S$ . $$

$\hspace{260pt} T$

$P \equiv 364 \mod{26} \equiv 0 \mod{26}$ which corresponds to the letter $A$ .

$P \equiv 41 \mod{26} \equiv 15 \mod{26}$ which corresponds to the letter $P$ .

$\hspace{260pt} P$

$P \equiv 549 \mod{26} \equiv 17 \mod{26}$ which corresponds to the letter $R$ .

$P \equiv 326 \mod{26} \equiv 14 \mod{26}$ which corresponds to the letter $O$ .

$\hspace{260pt} A$

$P \equiv 288 \mod{26} \equiv 2 \mod{26}$ which corresponds to the letter $C$ .

$\hspace{260pt} H$ $$

$\hspace{260pt} T$

$\hspace{260pt} O$

$P \equiv 193 \mod{26} \equiv 11 \mod{26}$ which corresponds to the letter $L$ .

$\hspace{260pt} E$ $$

$\hspace{260pt} A$

$\hspace{260pt} R$

$P \equiv 117 \mod{26} \equiv 13 \mod{26}$ which corresponds to the letter $N$ .

$\hspace{260pt} N$

$P \equiv 98 \mod{26} \equiv 20 \mod{26}$ which corresponds to the letter $U$ . $$

$P \equiv 402 \mod{26} \equiv 12 \mod{26}$ which corresponds to the letter $M$

$\hspace{260pt} B$

$\hspace{260pt} E$

$\hspace{260pt} R$

$\hspace{260pt} T$

$\hspace{260pt} H$

$\hspace{260pt} E$

$\hspace{260pt} O$

$\hspace{260pt} R$

$P \equiv 440 \mod{26} \equiv 24 \mod{26}$ which corresponds to the letter $Y$ . $$

$$

$THEBE \: STAPP \: \: ROACH \: \: TOLEA \: \: RNNUM \: \: BERTH \: \: EORY$ $$

$$

$THE \: MESSAGE \: IS:$ $$

$THE \: BEST \: APPROACH \: TO \: LEARN \: NUMBER \: \: THEORY$ $$

The string of integers is: $$

$\boxed{1974141819015151714027191411401713813613201214171974141724}$

$$ If this this didn't work we could try choosing $L \rightarrow E$ and $Y \rightarrow T$

The number of occurrences of the enciphered letters: $$

$L: 5$ $$

$U: 4$ $$

$Y: 4$ $$

$S: 3$ $$

$J: 3$ $$

$R: 3$ $$

$C: 2$ $$

$E: 2$ $$

$G: 2$ $$

$J: 1$ $$

$P: 1$ $$

$K: 1$ $$

$F: 1$ $$

$V: 1$ $$

$X: 1$