I have been given that N=143 and the encoder E=7. An encrypted message 48 was received. I have to find the decoder and use it to compute the original message.
This is how I did it but i'm not sure if I did it right: I let $M$ = the original message, then $$M^7 \equiv 48mod143$$ and we want some decoder $D$ such that $$48^D \equiv Mmod143$$ Then i did $$7D = 1(mod(p-1)(q-1))$$ $$7D \equiv 1(mod143)$$ $$7(41) \equiv 1(mod43)$$ I found 41 by trial and error. Using this i found that $$48^41 \equiv 81(mod143)$$ Does this mean the original message is 81?
As you say, you want $7D \equiv 1 \pmod {(p-1)(q-1)}$, but it is $pq=143$, so you want $7D \equiv 1 \pmod {120}$, not $\pmod {143}$