Solution

Suppose a message is RSA-encrypted with modulus $N=323$ and encrypting key $e=49$. Find the decrypting key $d$.


Recall that the relationship between the encrypting key $e$, the decrypting key $d$, and the modulus $N$ is given by

$$ed \equiv 1 \pmod{\phi(N)}$$

Noting that $\phi(323) = \phi(17 \cdot 19) = \phi(17) \cdot \phi(19) = 16 \cdot 18 = 288$, it remains to solve

$$49d \equiv 1 \pmod{288}$$

We can find $d$ from the above in the usual way, yielding

$$d \equiv 241 \pmod{288}$$