**History**

During the early history of cryptography, two parties would rely upon a key using a secure, but non-cryptographic, method. For example, a face-to-face meeting or an exchange, via a trusted courier, could be used. This key, which both parties kept absolutely secret, could then be used to exchange encrypted messages. A number of significant practical difficulties arise with this approach to distributing keys. Public-key cryptography addresses these drawbacks so that users can communicate securely over a public channel without having to agree upon a shared key beforehand.

In 1874, a book by William Stanley Jevons described the relationship of one-way functions to cryptography, and went on to discuss specifically the factorization problem used to create the trapdoor function in the RSA system. In July 1996, one observer commented on the Jevons book in this way:

In his book *The Principles of Science: A Treatise on Logic and Scientific Method*, written and published in the 1890s, William S. Jevons observed that there are many situations where the "direct" operation is relatively easy, but the "inverse" operation is significantly more difficult. One example mentioned briefly is that enciphering (encryption) is easy while deciphering (decryption) is not. In the same section of Chapter 7: Introduction titled "Induction an Inverse Operation", much more attention is devoted to the principle that multiplication of integers is easy, but finding the (prime) factors of the product is much harder. Thus, Jevons anticipated a key feature of the RSA Algorithm for public key cryptography, although he certainly did not invent the concept of public key cryptography.

In 1997, it was publicly disclosed that asymmetric key algorithms were secretly developed by James H. Ellis, Clifford Cocks, and Malcolm Williamson at the Government Communications Headquarters (GCHQ) in the UK in 1973. These researchers independently developed Diffie–Hellman key exchange, and a special case of RSA. The GCHQ cryptographers referred to the technique as "non-secret encryption". This work was named an IEEE Milestone in 2010.

An asymmetric-key cryptosystem was published in 1976 by Whitfield Diffie and Martin Hellman who, influenced by Ralph Merkle's work on public-key distribution, disclosed a method of public-key agreement. This method of key exchange, which uses exponentiation in a finite field, came to be known as Diffie–Hellman key exchange. This was the first published practical method for establishing a shared secret-key over an authenticated (but not private) communications channel without using a prior shared secret. Merkle's "public-key-agreement technique" became known as Merkle's Puzzles, and was invented in 1974 and published in 1978.

A generalization of Cocks's scheme was independently invented in 1977 by Ron Rivest, Adi Shamir and Leonard Adleman, all then at MIT. The latter authors published their work in 1978, and the algorithm appropriately came to be known as RSA. RSA uses exponentiation modulo, a product of two very large primes, to encrypt and decrypt, performing both public key encryption and public key digital signature. Its security is connected to the (presumed) extreme difficulty of factoring large integers, a problem for which there is no known efficient (i.e. practicably fast) general technique. In 1979, Michael O. Rabin published a related cryptosystem that is provably secure, at least as long as the factorization of the public key remains difficult - it remains an assumption that RSA also enjoys this security.

Since the 1970s, a large number and variety of encryption, digital signature, key agreement, and other techniques have been developed in the field of public-key cryptography. The ElGamal cryptosystem, invented by Taher ElGamal. relies on the similar and related high level of difficulty of the discrete logarithm problem, as does the closely related DSA, which was developed at the US National Security Agency (NSA) and published by NIST as a proposed standard. The introduction of elliptic curve cryptography by Neal Koblitz and Victor Miller, independently and simultaneously in the mid-1980s, has yielded new public-key algorithms based on the discrete logarithm problem. Although mathematically more complex, elliptic curves provide smaller key sizes and faster operations for approximately equivalent estimated security.

Read more about this topic: Public-key Cryptography

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