<P> Public key cryptography systems often rely on cryptographic algorithms based on mathematical problems that currently admit no efficient solution, particularly those inherent in certain integer factorization, discrete logarithm, and elliptic curve relationships . Public key algorithms, unlike symmetric key algorithms, do not require a secure channel for the initial exchange of one or more secret keys between the parties . </P> <P> Because of the computational complexity of asymmetric encryption, it is usually used only for small blocks of data, typically the transfer of a symmetric encryption key (e.g. a session key). This symmetric key is then used to encrypt the rest of the potentially long message sequence . The symmetric encryption / decryption is based on simpler algorithms and is much faster . </P> <P> In a public key signature system, a person can combine a message with a private key to create a short digital signature on the message . Anyone with the corresponding public key can combine a message, a putative digital signature on it, and the known public key to verify whether the signature was valid, i.e. made by the owner of the corresponding private key . Changing the message, even replacing a single letter, will cause verification to fail . In a secure signature system, it is computationally infeasible for anyone who does not know the private key to deduce it from the public key or any number of signatures, or to find a valid signature on any message for which a signature has not hitherto been seen . Thus the authenticity of a message can be demonstrated by the signature, provided the owner of the private key keeps the private key secret . </P> <P> Public key algorithms are fundamental security ingredients in cryptosystems, applications and protocols . They underpin various Internet standards, such as Transport Layer Security (TLS), S / MIME, PGP, and GPG . Some public key algorithms provide key distribution and secrecy (e.g., Diffie--Hellman key exchange), some provide digital signatures (e.g., Digital Signature Algorithm), and some provide both (e.g., RSA). </P>

The combination of a public key and a private key is known as a