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# Authenticating Communications Using Digital Signatures

In Chapter 3, you examined public key cryptography in detail. Public key cryptography involves generating two mathematically related keys, one of which can be used to encrypt a value and the other of which can be used to decrypt a value previously encrypted with the other. One important point to note is that it technically doesn't matter which key you use to perform the encryption, as long as the other one is available to perform the decryption. The RSA algorithm defines a public key that is used to encrypt, and a private key that is used to decrypt. However, the algorithm works if you reverse the keys — if you encrypt something with the private key, it can be decrypted with — and only with — the public key.

At first glance, this doesn't sound very useful. The public key, after all, is public. It's freely shared with anybody and everybody. Therefore, if a value is encrypted with the private key, it can be decrypted by anybody and everybody as well. However, the nature of public/private keypairs is such that it's also impossible — or, to be technically precise, mathematically infeasible — for anybody except the holder of the private key to generate something that can be decrypted using the public key. After all, the encryptor must find a number c such that ce%n = m for some arbitrary m. By definition, c = md satisfies this condition and it is believed to be computationally infeasible to find another such number c

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