Exponention cipher - prove unique mapping from plain text to cipher text

Question

At the heart of RSA, is the exponention cipher: C=M^e mod P (where C=ciphertext, M=Plaintext e=exponent and P=modulus.)

How do you prove that two different plaintexts don't map to same ciphertext?

Answer 1

By showing that $C^d \mod P = M$ where $d$ is your secret deciphering exponent. Since (if set up correctly) you get an inverse function from ciphertext back to plaintext, the original mapping (plain to cipher) must have been one-to-one.