Symmetric-key cryptosystems use the（1）key for encryption and decryption of a message, though a message or group of messages may have a difent key than others. A significant disadvantage of symmetric ciphers is the key management necessary to use them securely. Each ditinct pair of communicating parties must, ideally, share a diferent key, and perhaps each ciphertext exchanged as well. The number of keys required increases as the square of the number of network members, which very quickly requires complex key management schemes to keep them all straight and secret. The dificulty of securely establishing a secret （2） between two communicating parties, when a secure channel doesn't already exist between them, also presents a chicken-and-egg problem which is a considerable practical obstacle for cryptography users in the real world.
Whitfield Diffe and Martin Hellman, authors of the frst paper on public-key cryptography.
In a groundbreaking 1976 paper, Whitfield Difle and Martin Hellman proposed the notion of public-key (also, more generally, called asymmetric key) cryptography in which two different but mathematically related keys are used- a public key and a private key. A public key system is so constructed that calculation of one key (he private key) is computationally infeasible（3）the other (the pubic key), even though they are necessarily related. Instead, both keys are generated secretly, as an interrelated pair. The historian David Kahn described public-key cryptography as "the most revolutionary new concept in the field since poly-alphabetic substitution emerged in the Renaissance".
In public-key cryptosystems, the（4）key may be feely distributed, while its paired private key must remain secret. The public key is typically used for encryption, while the private or secret key is used for decryption. Dife and Hllman showed that public-key cryptography was possible by presenting the Difit-Hellman key exchange protocol.
In 1978, Ronald Rivest, Adi Shamir, and Len Adleman invented（5）, another public-key system.
In 1997, it finally became publicly known that asymmetric key cryptography had been invented by James H. Ellis at GCHQ, a British itelligence organization, and that, in the early 1970s, both the Diffie-Hellman and RSA algorithms had been previously developed (by Malcolm J. Williamson and Clifford Cocks, respectively).