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This article is about an algorithm for public-key

encryption. For the U.S. encryption and network

security company, see RSA Security. For otheruses, see RSA (disambiguation).

In cryptography, RSA (which standsforRivest, Shamirand Adleman who first publiclydescribed it) is an algorithm forpublic-keycryptography.[1] It is the first algorithm known to besuitable forsigning as well as encryption, andwas one of the first great advances in public keycryptography. RSA is widely used in electroniccommerce protocols, and is believed to besufficiently secure given sufficiently long keys andthe use of up-to-date implementations.

Contents

[hide]

1 History

2 Operation

o 2.1 Key generation

o 2.2 Encryption

o 2.3 Decryption

o 2.4 A worked exampleo 2.5 Using the Chinese remainder algorithm

o 2.6 Attacks against plain RSA

o 2.7 Padding schemes

o 2.8 Signing messages

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3 Security and practical considerations

o 3.1 Integer factorization and RSA problem

o 3.2 Key generation

o 3.3 Timing attackso 3.4 Adaptive chosen ciphertext attacks

o 3.5 Side-channel analysis attacks

4 Proofs of correctness

o 4.1 Concise proof using Euler's Theorem

o 4.2 Proof using Fermat's Little Theorem and

Chinese Remainder Theorem

5 See also 6 Notes

7 References

8 External links

[edit]History

Clifford Cocks, a British mathematician workingfor the UK intelligence agency GCHQ, describedan equivalent system in an internal document in1973, but given the relatively expensivecomputers needed to implement it at the time, itwas mostly considered a curiosity and, as far asis publicly known, was never deployed. Hisdiscovery, however, was not revealed until 1998due to its top-secret classification, and Rivest,Shamir, and Adleman devised RSAindependently of Cocks' work.

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Adi Shamir, one of the authors ofRSA: Rivest, ShamirandAdleman

The RSA algorithm was publicly described in1978 by Ron Rivest, Adi Shamir, and LeonardAdleman at MIT; the letters RSA are the initials oftheir surnames, listed in the same order as on thepaper.[2]

MIT was granted U.S. Patent 4,405,829 for a"Cryptographic communications system andmethod" that used the algorithm in 1983. Thepatent would have expired on September 21,2000 (the term of patent was 17 years at thetime), but the algorithm was released to the publicdomain by RSA Security on 6 September 2000,

two weeks earlier.[3] Since a paper describing thealgorithm had been published in August 1977,[2] prior to the December 1977 filing date ofthe patent application, regulations in much of therest of the world precluded patents elsewhere and

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only the US patent was granted. Had Cocks' workbeen publicly known, a patent in the US might nothave been possible.

From the DWPI's abstract of the patent,

The system includes a communications channelcoupled to at least one terminal having anencoding device and to at least one terminalhaving a decoding device. A message-to-be-transferred is enciphered to ciphertext at the

encoding terminal by encoding the message as anumber M in a predetermined set. That number isthen raised to a first predetermined power(associated with the intended receiver) and finallycomputed. The remainder or residue, C, is...computed when the exponentiated number isdivided by the product of two predetermined

prime numbers (associated with the intendedreceiver).

[edit]Operation

The RSA algorithm involves threesteps: key generation, encryption and decryption.

[edit]Key generationRSA involves a public key and a privatekey. The public key can be known to everyoneand is used for encrypting messages. Messagesencrypted with the public key can only be

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decrypted using the private key. The keys for theRSA algorithm are generated the following way:

1.

Choose two distinct primenumbersp and q. For security purposes, theintegersp and q should be chosen atrandom, and should be of similar bit-length.Prime integers can be efficiently foundusing a primality test.

2. Compute n =pq. n is used as the modulus for both thepublic and private keys

3. Compute (n) = (p 1)( q 1), where is Euler's totient function.4. Choose an integere such that 1 < e