Some Observations

• Encryption systems used in practice employ keys d and e of 56, 128, or even 180 bits.
• To ``crack'' such encryption systems requires weeks or months of time on the fastest supercomputers.
• Public key cryptography becomes more and more important with the advent of electronic commerce.
• Public key cryptography can also be used for authentication purposes, i.e., for verifying the identity of a sender.
• This is often called digital signatures.
• In an authentication system, both parties, say A and B, have public and private keys e_A,d_A and e_B,d_B, respectively.
• To authenticate his signature S, B would first encrypt S with his private key d_B and then encrypt it again with A's public key e_A.
• A can decrypt this doubly encrypted signature by first using his private key d_A and then B's public key e_B.
• Furthermore, only A can decrypt the signature.
• And, B cannot refute later that this is his signature as only he knows the private key d_B.