Given the following RSA generated public key: $P(3, 55)$. Which integer value should be chosen for $d$ to decrypt messages encrypted with $P$?
Check your answer with $M = 8$ and $C = 17$.
$P(3,55)$ is $p(e,n)$
So through trial and error i have got $d = 7$ but how do you calculate this the proper way?
To check this we can use the encryption formulas:
$p(M) = M^e \bmod n $
so $ p(8) = 8^3 \bmod 55 = 17 = c\\ s(c)= c^d \bmod n \\ s(17)=17^d \bmod 55 = M = 8$
so basically we have $17^d \bmod 55=8$
Now how do we find out $d$ from that equation other than trial and error, which was how I found $d=7$