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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?

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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.

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