## A.2. Public-key Encryption

Public-key cryptography is a cryptographic approach, employed by many cryptographic algorithms and cryptosystems, whose distinguishing characteristic is the use of asymmetric key algorithms instead of or in addition to symmetric key algorithms. Using the techniques of public key-private key cryptography, many methods of protecting communications or authenticating messages formerly unknown have become practical. They do not require a secure initial exchange of one or more secret keys as is required when using symmetric key algorithms. It can also be used to create digital signatures.

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Public key cryptography is a fundamental and widely used technology around the world, and is the approach which underlies such Internet standards as Transport Layer Security (TLS) (successor to SSL), PGP and GPG.

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The distinguishing technique used in public key cryptography is the use of asymmetric key algorithms, where the key used to encrypt a message is not the same as the key used to decrypt it. Each user has a pair of cryptographic keys — a public key and a private key. The private key is kept secret, whilst the public key may be widely distributed. Messages are encrypted with the recipient's public key and can only be decrypted with the corresponding private key. The keys are related mathematically, but the private key cannot be feasibly (ie, in actual or projected practice) derived from the public key. It was the discovery of such algorithms which revolutionized the practice of cryptography beginning in the middle 1970s.

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In contrast, Symmetric-key algorithms, variations of which have been used for some thousands of years, use a single secret key shared by sender and receiver (which must also be kept private, thus accounting for the ambiguity of the common terminology) for both encryption and decryption. To use a symmetric encryption scheme, the sender and receiver must securely share a key in advance.

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Because symmetric key algorithms are nearly always much less computationally intensive, it is common to exchange a key using a key-exchange algorithm and transmit data using that key and a symmetric key algorithm. PGP, and the SSL/TLS family of schemes do this, for instance, and are called hybrid cryptosystems in consequence.

^{[13]}### A.2.1. Diffie-Hellman

Diffie–Hellman key exchange (D–H) is a cryptographic protocol that allows two parties that have no prior knowledge of each other to jointly establish a shared secret key over an insecure communications channel. This key can then be used to encrypt subsequent communications using a symmetric key cipher.

^{[14]}#### A.2.1.1. Diffie-Hellman History

The scheme was first published by Whitfield Diffie and Martin Hellman in 1976, although it later emerged that it had been separately invented a few years earlier within GCHQ, the British signals intelligence agency, by Malcolm J. Williamson but was kept classified. In 2002, Hellman suggested the algorithm be called Diffie–Hellman–Merkle key exchange in recognition of Ralph Merkle's contribution to the invention of public-key cryptography (Hellman, 2002).

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Although Diffie–Hellman key agreement itself is an anonymous (non-authenticated) key-agreement protocol, it provides the basis for a variety of authenticated protocols, and is used to provide perfect forward secrecy in Transport Layer Security's ephemeral modes (referred to as EDH or DHE depending on the cipher suite).

^{[16]}### A.2.2. RSA

In cryptography, RSA (which stands for Rivest, Shamir and Adleman who first publicly described it) is an algorithm for public-key cryptography. It is the first algorithm known to be suitable for signing as well as encryption, and was one of the first great advances in public key cryptography. RSA is widely used in electronic commerce protocols, and is believed to be secure given sufficiently long keys and the use of up-to-date implementations.

### A.2.3. DSA

### A.2.4. SSL/TLS

Transport Layer Security (TLS) and its predecessor, Secure Sockets Layer (SSL), are cryptographic protocols that provide security for communications over networks such as the Internet. TLS and SSL encrypt the segments of network connections at the Transport Layer end-to-end.

### A.2.5. Cramer-Shoup Cryptosystem

The Cramer–Shoup system is an asymmetric key encryption algorithm, and was the first efficient scheme proven to be secure against adaptive chosen ciphertext attack using standard cryptographic assumptions. Its security is based on the computational intractability (widely assumed, but not proved) of the decisional Diffie–Hellman assumption. Developed by Ronald Cramer and Victor Shoup in 1998, it is an extension of the ElGamal cryptosystem. In contrast to ElGamal, which is extremely malleable, Cramer–Shoup adds additional elements to ensure non-malleability even against a resourceful attacker. This non-malleability is achieved through the use of a collision-resistant hash function and additional computations, resulting in a ciphertext which is twice as large as in ElGamal.

^{[20]}### A.2.6. ElGamal Encryption

In cryptography, the ElGamal encryption system is an asymmetric key encryption algorithm for public-key cryptography which is based on the Diffie-Hellman key agreement. It was described by Taher ElGamal in 1985. ElGamal encryption is used in the free GNU Privacy Guard software, recent versions of PGP, and other cryptosystems.

^{[21]}^{[9] }"Public-key Encryption."

*Wikipedia.*14 November 2009 http://en.wikipedia.org/wiki/Public-key_cryptography

^{[10] }"Public-key Encryption."

*Wikipedia.*14 November 2009 http://en.wikipedia.org/wiki/Public-key_cryptography

^{[11] }"Public-key Encryption."

*Wikipedia.*14 November 2009 http://en.wikipedia.org/wiki/Public-key_cryptography

^{[12] }"Public-key Encryption."

*Wikipedia.*14 November 2009 http://en.wikipedia.org/wiki/Public-key_cryptography

^{[13] }"Public-key Encryption."

*Wikipedia.*14 November 2009 http://en.wikipedia.org/wiki/Public-key_cryptography

^{[18] }"DSA."

*Wikipedia.*24 February 2010 http://en.wikipedia.org/wiki/Digital_Signature_Algorithm

^{[19] }"TLS/SSL."

*Wikipedia.*24 February 2010 http://en.wikipedia.org/wiki/Transport_Layer_Security

^{[20] }"Cramer-Shoup cryptosystem."

*Wikipedia.*24 February 2010 http://en.wikipedia.org/wiki/Cramer–Shoup_cryptosystem

^{[21] }"ElGamal encryption"

*Wikipedia.*24 February 2010 http://en.wikipedia.org/wiki/ElGamal_encryption