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'''Public-key encryption''' is an [[encryption]] method in which both the message and the public key are public. There is a private key used known only to the individual. This differs from the traditional (so-called "private-key") encryption, in which the key is kept secret from the intended recipient. Public-key encryption is a form of ''[[symmetry|asymmetric]] encryption'', meaning that it is much easier to encrypt a message than to decrypt it, at least without knowing the encryption key; this is the reason that the key has to be made public.

'''Public-key encryption''' is an [[encryption]] method in which both the encryption key is public. There is a private key used known only to the individual. This differs from the traditional (so-called "private-key") encryption, in which the key is kept secret from the intended recipient. Public-key encryption is a form of ''[[symmetry|asymmetric]] encryption'', meaning that it is much easier to encrypt a message than to decrypt it, at least without knowing the decryption key.

Well-known public-key encryption algorithms include '''Diffie-Hellman''',

Well-known public-key encryption algorithms include '''Diffie-Hellman''' and '''RSA'''. In Diffie-Hellman, the hardness is based on the [[discrete logarithm]]; in RSA it is [[factoring]].

'''El-Gamal''', and '''RSA'''. These algorithms all work on the same basic

model, which is that of a [[cyclic group]] ''G'' distributing over an operation ''o'' which may or may not be [[commutative]]. In Diffie-Hellman and El-Gamal, the basic operation is the [[discrete logarithm]]; in RSA it is [[factoring]].

==Diffie-Hellman encryption==

==Diffie-Hellman encryption==

Diffie-Hellman encryption, also known as "Diffie-Hellman key exchange", relies on the fundamental [[hardness|difficulty]] of [[computing]] the discrete logarithm ''g^a'' of a number ''x'' in the [[Group (mathematics)|group]] ''G''. It was invented by [[Whitfield Diffie]] and [[Lillian Hellman]] in 1976. The protocol proceeds in three steps:

Diffie-Hellman encryption, also known as "Diffie-Hellman key exchange", relies on the fundamental [[hardness|difficulty]] of [[computing]] the discrete logarithm of a number in the [[Group (mathematics)|group]] ''G''. It was invented by [[Whitfield Diffie]] and Martin Hellman in 1976. The protocol proceeds in three steps:

# Alice and Bob secretly decide on a large prime number ''p'' and a group ''G'' in which to work, with [[generator]] ''g''.

# Alice and Bob decide on a large prime number ''p'' and a group ''G'' in which to work, with [[generator]] ''g''.

# Alice chooses a secret integer ''a'', for example 6. She sends ''a^g'' mod ''p'' to Bob.

# Alice chooses a secret integer ''a''. She sends ''g^a'' mod ''p'' to Bob.

# Bob chooses a different integer ''b'', for example 7, and sends ''b^g'' mod ''p'' to Alice.

# Bob chooses a different integer ''b'', and sends ''g^b'' mod ''p'' to Alice.

# Alice computes ''(a^g)^b'', while Bob computes ''(b^g)^a''. Whoever finishes first sends their result to the other person.

# Alice computes ''(g^b)^a'', while Bob computes ''(g^a)^b''.

At this point both parties (Alice and Bob) have the same (secret) information, and can use the shared secret as a key for sending [[encryption|encrypted]] messages back and forth by the usual methods.

At this point both parties (Alice and Bob) have the same (secret) information, and can use the shared secret as a key for sending [[encryption|encrypted]] messages back and forth by the usual methods.

==RSA encryption==

==RSA encryption==

"RSA" stands for "[[Ron Rivest|Rivest]]&ndash;[[Shamir]]&ndash;[[Adelson]]",<ref>[http://www.all-acronyms.com/?t=rsa&d=rivest%2Dshamir%2Dadelson ''AllAcronyms.com'']</ref><ref>[http://www.cogs.susx.ac.uk/courses/mct/Handouts/notes-6.pdf Course notes] from G6016 "Networks", at the [[University of Sussex]]</ref> the three [[MIT]] researchers who discovered the RSA algorithm in 1977, seven years before Taher El-Gamal developed his competing system.

"RSA" stands for "[[Ron Rivest|Rivest]]&ndash;[[Shamir]]&ndash;[[Adelman]]",<ref>[http://www.all-acronyms.com/?t=rsa&d=rivest%2Dshamir%2Dadelson ''AllAcronyms.com'']</ref><ref>[http://www.cogs.susx.ac.uk/courses/mct/Handouts/notes-6.pdf Course notes] from G6016 "Networks", at the [[University of Sussex]]</ref> the three [[MIT]] researchers who discovered the RSA algorithm in 1977.

RSA, unlike the previous two systems, is based on the fundamental difficulty of [[factoring]] large [[prime]]s. For example, although it is easy to factor 6=2&times;3, it is exponentially more difficult to factor 6000000=2000&times;3000. (Of course, the digital computers that implement RSA encryption use numbers that are bigger still; a "strong" key used by an organization such as the [[NSA]] might have 6000000=2000&times;3000 ''digits!'')

RSA is based on the fundamental difficulty of [[factoring]] large integers into [[prime]]s.  

The RSA algorithm was put to the test in 1991, when RSA Laboratories released the "RSA Factoring Challenge". The challenge consisted of a list of progressively larger numbers, which, when fully decrypted, read "The magic words are [[squeamish ossifrage]]."<ref>[http://citeseer.ist.psu.edu/1393.html "The Magic Words Are Squeamish Ossifrage"], by Atkins, Graff, Lenstra, and Leyl</ref> Although the challenge was withdrawn in 2007, the RSA algorithm is still widely considered acceptable for business purposes.

The RSA algorithm was put to the test in 1991, when RSA Laboratories released the "RSA Factoring Challenge". The challenge consisted of a list of progressively larger numbers, which, when fully decrypted, read "The magic words are [[squeamish ossifrage]]."<ref>[http://citeseer.ist.psu.edu/1393.html "The Magic Words Are Squeamish Ossifrage"], by Atkins, Graff, Lenstra, and Leyl</ref> Although the challenge was withdrawn in 2007, the RSA algorithm is still widely considered acceptable for business purposes.

==Example==

==Example==

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Fred needs to send documents to many people. They dont need to be secure , but its important that they know for certain that the messge actually came from Fred.  

Fred needs to send documents to many people. They don't need to be secure, but its important that they know for certain that the message actually came from Fred.  

* Fred '''encrypts''' the message using his ''private key''. (only fred knows this key)

* Fred '''encrypts''' the message using his ''private key''. (only fred knows this key)

* Fred sends the message  

* Fred sends the encrypted message  

* George gets the message (so do many other people , it's not important)

* George gets the message (so do many other people , it's not important)

* George '''decrypts''' the message using Freds ''Public key''.  

* George '''decrypts''' the message using Fred's ''Public key''.  

* Only a message coded with the private key can be decoded wth the public key so George knows it came from Fred.  

* Only a message coded with the private key can be decoded wth the public key so George knows it came from Fred.  

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