# Private Keys and Message Interception

1. Using RSA, let p = 29, q = 23 and e = 3. What is the complete private key?

2. Suppose that Eve runs a key server. Alice downloads a key from the key server which Eve claims is Bob's public key. Bob downloads a key from the key server which Eve claims is Alice's public key. Given that Alice and Bob both assume that they have the correct public keys for the other party, and assuming that Eve can intercept any messages passed between Alice and Bob, is there any way that Eve can read the encrypted communications between the two parties? If so, how could she do it, and would Bob or Alice know that Eve was reading their messages? How could Bob and Alice mitigate this situation?

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#### Solution Preview

1. Using RSA, let p = 29, q = 23 and e = 3. What is the complete private key?

Answer: public key=(e,n)=(e,p*q)=(3,29*23)=(3,667)

Private key=d=e-1 mod((p-1)*(q-1))=3-1 mod(28*22)=3-1 mod(616).

That is, de=3d=1 mod(616). So 3d=1+616h where h is the positive and least whole number which counts from 1.

So if we try h=2 and get d=411.

So the complete private key is (411, 667).

2. Suppose that Eve runs a key server. Alice downloads a key from the key server which Eve claims is Bob's public ...

#### Solution Summary

The solution gives detailed steps on calculating the private keys and discusses the interception of a message on the server.