Key exchange (also known as "key establishment") is any method in cryptography by which cryptographic keys are exchanged between users, allowing use of a cryptographic algorithm.
If Alice and Bob wish to exchange encrypted messages, each must be equipped to encrypt messages to be sent and decrypt messages received. The nature of the equipping they require depends on the encryption technique they might use. If they use a code, both will require a copy of the same codebook. If they use a cipher, they will need appropriate keys. If the cipher is a symmetric key cipher, both will need a copy of the same key. If an asymmetric key cipher with the public/private key property, both will need the other's public key.
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The key exchange problem is how to exchange whatever keys or other information are needed so that no one else can obtain a copy. Traditionally, this required trusted couriers, diplomatic bags, or some other secure channel. With the advent of public key / private key cipher algorithms, the encrypting key (aka public key) could be made public, since (at least for high quality algorithms) no one without the decrypting key (aka, the private key) could decrypt the message.
In principle, the only remaining problem was to be sure (or at least confident) that a public key actually belonged to its supposed owner. Because it is possible to 'spoof' another's identity in any of several ways, this is not a trivial or easily solved problem, particularly when the two users involved have never met and know nothing about each other.
In 1976, Whitfield Diffie and Martin Hellman published a cryptographic protocol, (Diffie–Hellman key exchange), which allows users to establish 'secure channels' on which to exchange keys, even if an Opponent is monitoring that communication channel. However, D–H key exchange did not address the problem of being sure of the actual identity of the person (or 'entity').
Public key infrastructures have been proposed as a way around this problem of identity authentication. In their most usual implementation, each user applies to a 'certificate authority' for a digital certificate which serves for other users as a non-tamperable authentication of identity. Several countries and other jurisdictions have passed legislation or issued regulations encouraging PKIs by giving (more or less) legal effect to these digital certificates. Several commercial firms, and a few government departments, have established such certificate authorities. VeriSign is the most prominent commercial firm. For those new to such things, these arrangements are best thought of as electronic notary endorsements that "this public key belongs to this user". As with notary endorsements, there can be mistakes or misunderstandings in such vouchings. There have been several high profile public failures by assorted certificate authorities.
At the other end of the conceptual range is the web of trust system, which avoids central Certificate Authorities entirely. Each user is responsible for vetting any certificate from another before using that certificate to communicate with, vet digital signatures from, ... the user claimed to be associated with the particular public key in a certificate. PGP (and GPG, an implementation of the OpenPGP Internet Standard) employ just such a web of trust mechanism. Together they are the most widely used high quality crypto system in the world.
Password-authenticated key agreement algorithms can perform a cryptographic key exchange utilizing knowledge of a user's password.
The problem of key exchange has not yet been solved. In particular, it has not yet been solved for the modern situation of two previously unknown users attempting to communicate electronically, as, for instance, in electronic commerce. Some of the existing work-around designs work, more or less, but are not fully satisfactory.