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Three Private Key Encryption, Proof - Coggle Diagram

- - - - \(N = 4301\) is one of our "public keys" that will be put out for person B to find.
        
        Person A then chooses \(e\) which must be relatively prime to \((p-1)(q-1)(r-1)= (10)(22)(16) = 3520\) so for our case, person A will choose \(e = 7\). \(e\) becomes our second "public key".
        
        Now B has enough information to encrypt a message to A.
        Our message will be \(M = 13\).
        
        B will now calculate the value of the code, \(C\),
        \(C = M^e(\mod N)= 13^7(\mod 4301) = \\ 62748517(\mod 4301) = 1228\)
        So 1228 is the message that B sends to A to decode.
        
        Now A wants to decode 1228. To do this, they need to find a number \(d\) such that:
        \(ed = 1(\mod((p-1)(q-1)(r-1)))\).
        
        In this case, \(7d = 1\mod (3520)\), one solution is \(d = 503\) since \(7*503 = 3521 = 1(3520) +1 = 1\mod 3520\).
        
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        For decoding the message, A must calculate \(C^d(\mod N) = 1228^{503}(\mod 4301)\), which would break our calculators.
        
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