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{{short description|Algorithm for public-key cryptography}}
{{Infobox block cipher
| name          = RSA
| designers     = [[Ron Rivest]], [[Adi Shamir]], and [[Leonard Adleman]]
| publish date  = 1977
| type          = Public-Key
| derived from  =
| derived to    =
| related to    =
| certification = [[PKCS1|PKCS#1]], [[ANSI X9.31]], [[P1363|IEEE 1363]]
| key size      =1,024 to 4,096 bit typical
| block size    =
| structure     =
| rounds        = 1
| cryptanalysis = [[General number field sieve]] for classical computers;<br />[[Shor's algorithm]] for quantum computers.<br />A [[RSA-768|768-bit key]] has been broken.
}}

'''RSA''' ('''Rivest–Shamir–Adleman''') is one of the first [[public-key cryptography|public-key cryptosystems]] and is widely used for secure data transmission. In such a [[cryptosystem]], the [[encryption key]] is public and it is different from the [[decryption key]] which is kept secret (private). In RSA, this asymmetry is based on the practical difficulty of the [[factorization]] of the product of two large [[prime number]]s, the "[[factoring problem]]". The [[acronym]] RSA is made of the initial letters of the surnames of [[Ron Rivest]], [[Adi Shamir]], and [[Leonard Adleman]], who first publicly described the algorithm in 1977. [[Clifford Cocks]], an English mathematician working for the British intelligence agency [[Government Communications Headquarters]] (GCHQ), had developed an equivalent system in 1973, but this was not [[classified information|declassified]] until 1997.<ref>{{cite web |url=http://www.bristol.ac.uk/pace/graduation/honorary-degrees/hondeg08/cocks.html |first=Nigel |last=Smart |title=Dr Clifford Cocks CB |accessdate=August 14, 2011 |date=February 19, 2008 |publisher=[[Bristol University]]}}</ref>

A user of RSA creates and then publishes a public key based on two large [[prime number]]s, along with an auxiliary value. The prime numbers must be kept secret. Anyone can use the public key to encrypt a message,  but only someone with knowledge of the prime numbers can decode the message.<ref name="rsa">{{cite journal
 | last1 = Rivest | first1 = R.
 | last2 = Shamir | first2 = A.
 | last3 = Adleman | first3 = L.
 | url = http://people.csail.mit.edu/rivest/Rsapaper.pdf
 | title = A Method for Obtaining Digital Signatures and Public-Key Cryptosystems
 | journal = [[Communications of the ACM]]
 | volume = 21  | issue = 2 | pages = 120–126 | date = February 1978
 | doi = 10.1145/359340.359342| citeseerx = 10.1.1.607.2677
 }}</ref>
Breaking RSA [[encryption]] is known as the [[RSA problem]]. Whether it is as difficult as the factoring problem remains an open question.  There are currently no published methods to defeat the system if a large enough key is used.

RSA is a relatively slow algorithm, and because of this, it is less commonly used to directly encrypt user data. More often, RSA passes encrypted shared keys for [[symmetric-key algorithm|symmetric key]] cryptography which in turn can perform bulk encryption-decryption operations at much higher speed.

==History==
[[File:Adi Shamir 2009 crop.jpg|thumb|175px|[[Adi Shamir]], one of the three inventors of RSA (two others being [[Ron Rivest]] and [[Leonard Adleman]])]]
The idea of an asymmetric public-private key cryptosystem is attributed to [[Whitfield Diffie]] and [[Martin Hellman]], who published this concept in 1976. They also introduced digital signatures and attempted to apply number theory. Their formulation used a shared-secret-key created from exponentiation of some number, modulo a prime number. However, they left open the problem of realizing a one-way function, possibly because the difficulty of factoring was not well-studied at the time.<ref>{{Cite journal|title = New directions in cryptography|journal = IEEE Transactions on Information Theory|date = November 1976|issn = 0018-9448|pages = 644–654|volume = 22|issue = 6|doi = 10.1109/TIT.1976.1055638|first = W.|last = Diffie|first2 = M.E.|last2 = Hellman|citeseerx = 10.1.1.37.9720}}</ref>

[[Ron Rivest]], [[Adi Shamir]], and [[Leonard Adleman]] at the [[Massachusetts Institute of Technology]], made several attempts over the course of a year to create a one-way function that was hard to invert. Rivest and Shamir, as computer scientists, proposed many potential functions, while Adleman, as a mathematician, was responsible for finding their weaknesses. They tried many approaches including "[[Knapsack problem|knapsack]]-based" and "permutation polynomials". For a time, they thought what they wanted to achieve was impossible due to contradictory requirements.<ref>{{Cite web|url = https://people.csail.mit.edu/rivest/pubs/ARS03.rivest-slides.pdf|title = The Early Days of RSA -- History and Lessons|date = |accessdate = |website = |publisher = |last = Rivest|first = Ronald}}</ref> In April 1977, they spent [[Passover]] at the house of a student and drank a good deal of [[Manischewitz]] wine before returning to their homes at around midnight.<ref>{{Cite web|url = http://www.math.uchicago.edu/~may/VIGRE/VIGRE2007/REUPapers/FINALAPP/Calderbank.pdf|title = The RSA Cryptosystem: History, Algorithm, Primes|date = 2007-08-20|accessdate = |website = |publisher = |last = Calderbank|first = Michael}}</ref> Rivest, unable to sleep, lay on the couch with a math textbook and started thinking about their one-way function. He spent the rest of the night formalizing his idea, and he had much of the paper ready by daybreak. The algorithm is now known as RSA – the initials of their surnames in same order as their paper.<ref name="SIAM">{{cite journal |url=http://www.msri.org/people/members/sara/articles/rsa.pdf |journal=SIAM News |volume=36 |issue=5 |date=June 2003 |title=Still Guarding Secrets after Years of Attacks, RSA Earns Accolades for its Founders |first=Sara |last=Robinson }}</ref>

[[Clifford Cocks]], an English [[mathematician]] working for the [[United Kingdom|British]] intelligence agency [[Government Communications Headquarters]] (GCHQ), described an equivalent system in an internal document in 1973.<ref>{{Cite web|url=https://www.gchq.gov.uk/sites/default/files/document_files/Cliff%20Cocks%20paper%2019731120.pdf|title=A Note on Non-Secret Encryption|last=Cocks|first=C.C.|authorlink=Clifford Cocks|date=20 November 1973|website=www.gchq.gov.uk|access-date=2017-05-30}}</ref> However, given the relatively expensive computers needed to implement it at the time, RSA was considered to be mostly a curiosity and, as far as is publicly known, was never deployed. His discovery, however, was not revealed until 1997 due to its top-secret classification.

Kid-RSA (KRSA) is a simplified public-key cipher published in 1997, designed for educational purposes. Some people feel that learning Kid-RSA gives insight into RSA and other public-key ciphers, analogous to [[Data Encryption Standard#Simplified DES|simplified DES]].<ref>
Jim Sauerberg
[http://www.mathaware.org/mam/06/Sauerberg_PKC-essay.html "From Private to Public Key Ciphers in Three Easy Steps"].
</ref><ref>
Margaret Cozzens and Steven J. Miller.
[https://books.google.com/books?id=GbKyAAAAQBAJ "The Mathematics of Encryption: An Elementary Introduction"].
p. 180.
</ref><ref>
Alasdair McAndrew.
[https://books.google.com/books?id=9lTRBQAAQBAJ "Introduction to Cryptography with Open-Source Software"].
p. 12.
</ref><ref>
Surender R. Chiluka.
[http://www.cs.uri.edu/cryptography/publickeykidkrypto.htm "Public key Cryptography"].
</ref><ref>
Neal Koblitz.
[http://www.math.washington.edu/~koblitz/crlogia.html "Cryptography As a Teaching Tool"].
Cryptologia, Vol. 21, No. 4 (1997).
</ref>

==Patent==
[[Massachusetts Institute of Technology|MIT]] was granted {{US patent|4405829}} for a "Cryptographic communications system and method" that used the algorithm, on September 20, 1983. Though the patent was going to expire on September 21, 2000 (the [[term of patent]] was 17 years at the time), the algorithm was released to the public domain by [[RSA Security]] on September 6, 2000, two weeks earlier.<ref>{{cite web |url=http://www.rsa.com/press_release.aspx?id=261 |title=RSA Security Releases RSA Encryption Algorithm into Public Domain |accessdate=2010-03-03 |url-status=dead |archiveurl=https://web.archive.org/web/20070621021111/http://www.rsa.com/press_release.aspx?id=261 |archivedate=June 21, 2007 }}</ref> Since a detailed description of the algorithm had been published in the [[List of Martin Gardner Mathematical Games columns|Mathematical Games]]  column in the August 1977 issue of [[Scientific American]],<ref name="SIAM" /> prior to the December 1977 [[filing date]] of the [[patent application]], regulations in much of the rest of the world precluded [[patent]]s elsewhere and only the [[United States|US]] patent was granted. Had Cocks's work been publicly known, a patent in the United States would not have been legal either.

From the [[Derwent World Patent Index|DWPI]]'s abstract of the patent,
{{quote|The system includes a communications channel coupled to at least one terminal having an encoding device and to at least one terminal having a decoding device. A message-to-be-transferred is enciphered to ciphertext at the encoding terminal by encoding the message as a number M in a predetermined set. That number is then raised to a first predetermined power (associated with the intended receiver) and finally computed. The remainder or residue, C, is... computed when the exponentiated number is divided by the product of two predetermined prime numbers (associated with the intended receiver).}}

==Operation==
The RSA algorithm involves four steps: [[Key (cryptography)|key]] generation, key distribution, encryption and decryption.

A basic principle behind RSA is the observation that it is practical to find three very large positive integers {{mvar|e}}, {{mvar|d}} and {{mvar|n}} such that with [[modular exponentiation]] for all integers {{mvar|m}} (with {{math|0 ≤ ''m'' < ''n''}}):

:<math>(m^e)^d \equiv m \pmod{n}</math>

and that even knowing {{mvar|e}} and {{mvar|n}} or even {{mvar|m}} it can be extremely difficult to find {{mvar|d}}. The [[triple bar]] (≡) here denotes [[Modular arithmetic|modular congruence]].

In addition, for some operations it is convenient that the order of the two exponentiations can be changed and that this relation also implies:

:<math>(m^d)^e \equiv m \pmod{n}</math>

RSA involves a ''public key'' and a ''[[private key]].''  The public key can be known by everyone, and it is used for encrypting messages. The intention is that messages encrypted with the public key can only be decrypted in a reasonable amount of time by using the private key. The public key is represented by the integers {{mvar|n}} and {{mvar|e}}; and, the private key, by the integer {{mvar|d}} (although {{mvar|n}} is also used during the decryption process. Thus, it might be considered to be a part of the private key, too). {{mvar|m}} represents the message (previously prepared with a certain technique explained below).

===Key generation===
The keys for the RSA algorithm are generated in the following way:
# Choose two distinct [[prime number]]s ''p'' and ''q''.
#* For security purposes, the integers ''p'' and ''q'' should be chosen at random, and should be similar in magnitude but differ in length by a few digits to make factoring harder.<ref name="rsa"/> Prime integers can be efficiently found using a [[primality test]].
#* ''p'' and ''q'' are kept secret.
# Compute {{nowrap|1=''n'' = ''pq''}}.
#* ''n'' is used as the [[Modular arithmetic|modulus]] for both the public and private keys.  Its length, usually expressed in bits, is the [[key length]].
#* ''n'' is released as part of the public key.
# Compute ''λ''(''n''), where ''λ'' is [[Carmichael's totient function]]. Since ''n'' = ''pq'', ''λ''(''n'') = [[least common multiple|lcm]](''λ''(''p''),''λ''(''q'')), and since ''p'' and ''q'' are prime, ''λ''(''p'') = ''[[Euler totient function|φ]]''(''p'') = ''p'' − 1 and likewise ''λ''(''q'') = ''q'' − 1. Hence ''λ''(''n'') = lcm(''p'' − 1, ''q'' − 1). 
#* ''λ''(''n'') is kept secret.
# Choose an integer ''e'' such that {{nowrap|1 < ''e'' < ''λ''(''n'')}} and {{nowrap|1=[[greatest common divisor|gcd]](''e'', ''λ''(''n'')) = 1}}; that is, ''e'' and ''λ''(''n'') are [[coprime]].
#* ''e'' having a short [[bit-length]] and small [[Hamming weight]] results in more efficient encryption {{snd}} the most commonly chosen value for ''e'' is {{nowrap|1=2<sup>16</sup> + 1 = 65,537}}. The smallest (and fastest) possible value for ''e'' is 3, but such a small value for ''e'' has been shown to be less secure in some settings.<ref name="Boneh">
{{cite journal
 | url = http://crypto.stanford.edu/~dabo/abstracts/RSAattack-survey.html
 | last = Boneh | first = Dan
 | title = Twenty Years of attacks on the RSA Cryptosystem
 | journal = [[Notices of the American Mathematical Society]]
 | volume = 46 | issue = 2
 | pages = 203–213 | year = 1999
}}</ref>
#*''e'' is released as part of the public key.
# Determine ''d'' as {{nowrap|''d'' ≡ ''e''<sup>−1</sup> (mod ''λ''(''n''))}}; that is, ''d'' is the [[modular multiplicative inverse]] of ''e'' modulo ''λ''(''n'').
#* This means: solve for ''d'' the equation {{nowrap|''d''⋅''e'' ≡ 1 (mod ''λ''(''n''))}}. ''d'' can be computed efficiently by using the [[Extended Euclidean algorithm]].
#* ''d'' is kept secret as the ''private key exponent''.
The ''public key'' consists of the modulus ''n'' and the public (or encryption) exponent ''e''. The ''private key'' consists of the private (or decryption) exponent ''d'', which must be kept secret. ''p'', ''q'', and ''λ''(''n'') must also be kept secret because they can be used to calculate ''d''. In fact, they can all be discarded after ''d'' has been computed.<ref>Applied Cryptography, John Wiley & Sons, New York, 1996. [[Bruce Schneier]], p.467</ref>

In the original RSA paper,<ref name=rsa /> the [[Euler totient function]] {{nowrap|1=''φ''(''n'') = (''p'' − 1)(''q'' − 1)}} is used instead of ''λ''(''n'') for calculating the private exponent ''d''. Since ''φ''(''n'') is always divisible by ''λ''(''n'') the algorithm works as well.  That the [[Euler totient function]] can be used can also be seen as a consequence of the [[Lagrange's theorem (group theory)|Lagrange's theorem]] applied to the [[multiplicative group of integers modulo n|multiplicative group of integers modulo pq]]. Thus any ''d'' satisfying {{nowrap|''d''⋅''e'' ≡ 1 (mod ''φ''(''n''))}} also satisfies {{nowrap|''d''⋅''e'' ≡ 1 (mod ''λ''(''n''))}}. However, computing ''d'' modulo ''φ''(''n'') will sometimes yield a result that is larger than necessary (i.e. {{nowrap|''d'' > ''λ''(''n'')}}). Most of the implementations of RSA will accept exponents generated using either method (if they use the private exponent ''d'' at all, rather than using the optimized decryption method [[#Using the Chinese remainder algorithm|based on the Chinese remainder theorem]] described below), but some standards like [http://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.186-4.pdf#page=62 FIPS 186-4] may require that {{nowrap|''d'' < ''λ''(''n'')}}.  Any "oversized" private exponents not meeting that criterion may always be reduced modulo ''λ''(''n'') to obtain a smaller equivalent exponent.

Since any common factors of {{nowrap|(''p'' − 1)}} and {{nowrap|(''q'' − 1)}} are present in the factorisation of {{nowrap|''n'' − 1}} = {{nowrap|''pq'' − 1}} = {{nowrap|(''p'' − 1)(''q'' − 1) + (''p'' − 1) + (''q'' − 1)}},<ref>{{cite journal|title=Further Attacks on Server-Aided RSA Cryptosystems|first1=James|last1=McKee|first2=Richard|last2=Pinch|citeseerx=10.1.1.33.1333|year=1998}}</ref> it is recommended that {{nowrap|(''p'' − 1)}} and {{nowrap|(''q'' − 1)}} have only very small common factors, if any besides the necessary 2.<ref name="rsa" /><ref>A Course in Number Theory and Cryptography, Graduate Texts in Math. No. 114, Springer-Verlag, New York, 1987. [[Neal Koblitz]], Second edition, 1994. p. 94</ref><ref>{{cite mailing list|title=common factors in (''p'' − 1) and (''q'' − 1)|first=Viktor|last=Dukhovni|mailing-list=openssl-dev|url=https://mta.openssl.org/pipermail/openssl-dev/2015-July/002266.html|date=July 31, 2015}}</ref><ref>{{cite mailing list|title=common factors in (''p'' − 1) and (''q'' − 1)|first=Viktor|last=Dukhovni|mailing-list=openssl-dev|url=https://mta.openssl.org/pipermail/openssl-dev/2015-August/002277.html|date=August 1, 2015}}</ref>

Note: The authors of the original RSA paper carry out the key generation by choosing ''d'' and then computing ''e'' as the [[modular multiplicative inverse]] of ''d'' modulo ''φ''(''n''), whereas most current implementations of RSA, such as those following [[PKCS1|PKCS#1]], do the reverse (choose ''e'' and compute ''d''). Since the chosen key can be small whereas the computed key normally is not, the RSA paper's algorithm optimizes decryption compared to encryption, while the modern algorithm optimizes encryption instead.<ref name="rsa" /><ref>{{Cite IETF|rfc=3447|title=Public-Key Cryptography Standards (PKCS) #1: RSA Cryptography Specifications Version 2.1|last=Johnson|first=J.|last2=Kaliski|first2=B.|date=February 2003|publisher=Network Working Group|access-date=9 March 2016}}</ref>

===Key distribution===
Suppose that [[Alice and Bob|Bob]] wants to send information to [[Alice and Bob|Alice]]. If they decide to use RSA, Bob must know Alice's public key to encrypt the message and Alice must use her private key to decrypt the message.
To enable Bob to send his encrypted messages, Alice transmits her public key {{math|(''n'', ''e'')}} to Bob via a reliable, but not necessarily secret, route. Alice's private key {{math|(''d'')}} is never distributed.

===Encryption===
After Bob obtains Alice's public key, he can send a message {{mvar|M}} to Alice.

To do it, he first turns {{mvar|M}} (strictly speaking, the un-padded plaintext) into an integer {{mvar|m}} (strictly speaking, the padded plaintext), such that {{math|0 ≤ ''m'' < ''n''}} by using an agreed-upon reversible protocol known as a [[#Padding schemes|padding scheme]]. He then computes the ciphertext {{mvar|c}}, using Alice's public key {{mvar|e}}, corresponding to

: <math> c \equiv m^e \pmod{n} </math>

This can be done reasonably quickly, even for very large numbers, using [[modular exponentiation]]. Bob then transmits {{mvar|c}} to Alice.
<!-- What is this? I think it should be removed -->

===Decryption===
Alice can recover {{mvar|m}} from {{mvar|c}} by using her private key exponent {{mvar|d}} by computing

:<math>c^d \equiv (m^e)^d \equiv m \pmod{n}</math>

Given {{mvar|m}}, she can recover the original message {{mvar|M}} by reversing the padding scheme.

===Example===
Here is an example of RSA encryption and decryption. The parameters used here are artificially small, but one can also [[b:Cryptography/Generate a keypair using OpenSSL|use OpenSSL to generate and examine a real keypair]].

# Choose two distinct prime numbers, such as
#: <math>p = 61</math> and <math>q = 53</math>
# Compute {{nowrap|1=''n'' = ''pq''}} giving
#: <math>n = 61\times 53 = 3233</math>
# Compute the [[Carmichael's totient function]] of the product as {{nowrap|1=''λ''(''n'') = [[least common multiple|lcm]](''p'' − 1, ''q'' − 1)}} giving
#: <math>\lambda(3233) = \operatorname{lcm}(60, 52) = 780</math>
# Choose any number {{nowrap|1 < ''e'' < 780}} that is [[coprime]] to 780. Choosing a prime number for ''e'' leaves us only to check that ''e'' is not a divisor of 780.
#: Let <math>e = 17</math>
# Compute ''d'', the [[modular multiplicative inverse]] of {{nowrap|''e'' (mod ''λ''(''n''))}} yielding,
#: <math>d = 413</math>
#: Worked example for the modular multiplicative inverse:
#: <math>d \times e = 1 \bmod \lambda(n)</math>
#: <math>413 \times 17 = 1 \bmod 780 </math>

The '''public key''' is ({{nowrap|1=''n'' = 3233}}, {{nowrap|1=''e'' = 17}}). For a padded [[plaintext]] message ''m'', the encryption function is
:<math>c(m) = m^{17} \bmod 3233</math>

The '''private key''' is ({{nowrap|1=''n'' = 3233}}, {{nowrap|1=''d'' = 413}}). For an encrypted [[ciphertext]] ''c'', the decryption function is
:<math>m(c) = c^{413} \bmod 3233</math>

For instance, in order to encrypt {{nowrap|1=''m'' = 65}}, we calculate
:<math>c = 65^{17} \bmod 3233 = 2790 </math>

To decrypt {{nowrap|1=''c'' = 2790}}, we calculate
:<math>m = 2790^{413} \bmod 3233 = 65 </math>

Both of these calculations can be computed efficiently using the [[square-and-multiply algorithm]] for [[modular exponentiation]]. In real-life situations the primes selected would be much larger; in our example it would be trivial to factor ''n'', 3233 (obtained from the freely available public key) back to the primes ''p'' and ''q''. ''e'', also from the public key, is then inverted to get ''d'', thus acquiring the private key.

Practical implementations use the [[Chinese remainder theorem]] to speed up the calculation using modulus of factors (mod ''pq'' using mod ''p'' and mod ''q'').

The values ''d''<sub>''p''</sub>, ''d''<sub>''q''</sub> and ''q''<sub>inv</sub>, which are part of the private key are computed as follows:
: <math>\begin{align}
           d_p = {} &d \bmod (p-1) = 413 \bmod (61-1) = 53 \\
           d_q = {} &d \bmod (q-1) = 413 \bmod (53-1) = 49 \\
  q_\text{inv} = {} &q^{-1} \bmod p = 53^{-1} \bmod 61 = 38 \\
     \Rightarrow {} &(q_\text{inv} \times q) \bmod p = 38 \times 53 \bmod 61= 1
\end{align}</math>

Here is how ''d''<sub>''p''</sub>, ''d''<sub>''q''</sub> and ''q''<sub>inv</sub> are used for efficient decryption. (Encryption is efficient by choice of a suitable ''d'' and ''e'' pair)
: <math>\begin{align}
  m_1 &= c^{d_p} \bmod p = 2790^{53} \bmod 61 = 4 \\
  m_2 &= c^{d_q} \bmod q = 2790^{49} \bmod 53 = 12 \\
    h &= (q_\text{inv} \times (m_1 - m_2)) \bmod p = (38 \times -8) \bmod 61 = 1 \\
    m &= m_2 + h \times q = 12 + 1 \times 53 = 65
\end{align}</math>

===Code===
A working example in JavaScript using [https://github.com/peterolson/BigInteger.js BigInteger.js]. This code should not be used in production, as <code>bigInt.randBetween()</code> uses <code>Math.random()</code>, which is not a [[cryptographically secure pseudorandom number generator]].<ref>{{cite web|last1=Scholz|first1=Florian|last2=Shepherd|first2=Eric|title=Math.random()|url=https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Math/random|website=Mozilla Developer Network|accessdate=5 June 2016}}</ref>

<syntaxhighlight lang="javascript">
'use strict';

/**
 * RSA hash function reference implementation.
 * Uses BigInteger.js https://github.com/peterolson/BigInteger.js
 * Code originally based on https://github.com/kubrickology/Bitcoin-explained/blob/master/RSA.js
 */
const RSA = {};

/**
 * Generates a k-bit RSA public/private key pair
 * https://en.wikipedia.org/wiki/RSA_(cryptosystem)#Code
 *
 * @param   {keysize} int, bitlength of desired RSA modulus n (should be even)
 * @returns {array} Result of RSA generation (object with three bigInt members: n, e, d)
 */
RSA.generate = function(keysize) {
  /**
   * Generates a random k-bit prime greater than √2 × 2^(k-1)
   *
   * @param   {bits} int, bitlength of desired prime
   * @returns {bigInt} a random generated prime
   */
  function randomPrime(bits) {
    const min = bigInt(6074001000).shiftLeft(bits - 33);  // min ≈ √2 × 2^(bits - 1)
    const max = bigInt.one.shiftLeft(bits).minus(1);  // max = 2^(bits) - 1
    for (;;) {
      const p = bigInt.randBetween(min, max);  // WARNING: not a cryptographically secure RNG!
      if (p.isProbablePrime(256)) {
        return p;
      }
    }
  }

  // set up variables for key generation
  const e = bigInt(65537);  // use fixed public exponent
  let p;
  let q;
  let lambda;

  // generate p and q such that λ(n) = lcm(p − 1, q − 1) is coprime with e and |p-q| >= 2^(keysize/2 - 100)
  do {
    p = randomPrime(keysize / 2);
    q = randomPrime(keysize / 2);
    lambda = bigInt.lcm(p.minus(1), q.minus(1));
  } while (bigInt.gcd(e, lambda).notEquals(1) || p.minus(q).abs().shiftRight(
      keysize / 2 - 100).isZero());

  return {
    n: p.multiply(q),  // public key (part I)
    e: e,  // public key (part II)
    d: e.modInv(lambda),  // private key d = e^(-1) mod λ(n)
  };
};

/**
 * Encrypt
 *
 * @param   {m} int / bigInt: the 'message' to be encoded
 * @param   {n} int / bigInt: n value returned from RSA.generate() aka public key (part I)
 * @param   {e} int / bigInt: e value returned from RSA.generate() aka public key (part II)
 * @returns {bigInt} encrypted message
 */
RSA.encrypt = function(m, n, e) {
  return bigInt(m).modPow(e, n);
};

/**
 * Decrypt
 *
 * @param   {c} int / bigInt: the 'message' to be decoded (encoded with RSA.encrypt())
 * @param   {d} int / bigInt: d value returned from RSA.generate() aka private key
 * @param   {n} int / bigInt: n value returned from RSA.generate() aka public key (part I)
 * @returns {bigInt} decrypted message
 */
RSA.decrypt = function(c, d, n) {
  return bigInt(c).modPow(d, n);
};

</syntaxhighlight>

===Signing messages===
Suppose [[Alice and Bob|Alice]] uses [[Alice and Bob|Bob]]'s public key to send him an encrypted message.  In the message, she can claim to be Alice but Bob has no way of verifying that the message was actually from Alice since anyone can use Bob's public key to send him encrypted messages.  In order to verify the origin of a message, RSA can also be used to [[digital signature|sign]] a message.

Suppose Alice wishes to send a signed message to Bob. She can use her own private key to do so. She produces a [[cryptographic hash function|hash value]] of the message, raises it to the power of ''d'' (modulo ''n'') (as she does when decrypting a message), and attaches it as a "signature" to the message. When Bob receives the signed message, he uses the same hash algorithm in conjunction with Alice's public key. He raises the signature to the power of ''e'' (modulo ''n'') (as he does when encrypting a message), and compares the resulting hash value with the message's actual hash value. If the two agree, he knows that the author of the message was in possession of Alice's private key, and that the message has not been tampered with since.

This works because multiplication is commutative so
<math>
h=hash(m);
(h^e)^d = h^{ed}= h^{de}=(h^d)^e \equiv h \pmod{n}
</math>
Thus, the keys may be swapped without loss of generality, that is a private key of a key pair may be used either to:
# Decrypt a message only intended for the recipient, which may be encrypted by anyone having the public key (asymmetric encrypted transport).
# Encrypt a message which may be decrypted by anyone, but which can only be encrypted by one person; this provides a digital signature.

==Padding==

===Attacks against plain RSA===
There are a number of attacks against plain RSA as described below.
* When encrypting with low encryption exponents (e.g., {{nowrap|1=''e'' = 3}}) and small values of the ''m'', (i.e., {{nowrap|''m'' < ''n''<sup>1/''e''</sup>}}) the result of {{nowrap|''m''<sup>''e''</sup>}} is strictly less than the modulus ''n''.  In this case, ciphertexts can be easily decrypted by taking the ''e''th root of the ciphertext over the integers.
* If the same clear text message is sent to ''e'' or more recipients in an encrypted way, and the receivers share the same exponent ''e'', but different ''p'', ''q'', and therefore ''n'', then it is easy to decrypt the original clear text message via the [[Chinese remainder theorem]]. [[Johan Håstad]] noticed that this attack is possible even if the cleartexts are not equal, but the attacker knows a linear relation between them.<ref>{{cite book |first=Johan |last=Håstad |chapter=On using RSA with Low Exponent in a Public Key Network |title=Advances in Cryptology — CRYPTO '85 Proceedings |series=Lecture Notes in Computer Science |volume=218 |year=1986 |pages=403–408 |doi=10.1007/3-540-39799-X_29 |isbn=978-3-540-16463-0 }}</ref> This attack was later improved by [[Don Coppersmith]].<ref>{{cite journal |first=Don |last=Coppersmith |title=Small Solutions to Polynomial Equations, and Low Exponent RSA Vulnerabilities |journal=[[Journal of Cryptology]] |volume=10 |issue=4 |pages=233–260 |year=1997 |doi=10.1007/s001459900030 |url=https://www.di.ens.fr/~fouque/ens-rennes/coppersmith.pdf |citeseerx=10.1.1.298.4806 }}</ref>
{{See also|Coppersmith's Attack}}
* Because RSA encryption is a [[deterministic algorithm|deterministic encryption algorithm]] (i.e., has no random component) an attacker can successfully launch a [[chosen plaintext attack]] against the cryptosystem, by encrypting likely plaintexts under the public key and test if they are equal to the ciphertext. A cryptosystem is called [[semantically secure]] if an attacker cannot distinguish two encryptions from each other, even if the attacker knows (or has chosen) the corresponding plaintexts. As described above, RSA without padding is not semantically secure.<ref>[[Shafi Goldwasser|S. Goldwasser]] and [[Silvio Micali|S. Micali]], [http://portal.acm.org/citation.cfm?id=802212 Probabilistic encryption & how to play mental poker keeping secret all partial information], Annual ACM Symposium on Theory of Computing, 1982.</ref>
* RSA has the property that the product of two ciphertexts is equal to the encryption of the product of the respective plaintexts. That is {{nowrap|''m''<sub>1</sub><sup>''e''</sup>''m''<sub>2</sub><sup>''e''</sup> ≡ (''m''<sub>1</sub>''m''<sub>2</sub>)<sup>''e''</sup> (mod ''n'')}}. Because of this multiplicative property a [[chosen-ciphertext attack]] is possible. E.g., an attacker who wants to know the decryption of a ciphertext {{nowrap|''c'' ≡ ''m''<sup>''e''</sup> (mod ''n'')}} may ask the holder of the private key  ''d'' to decrypt an unsuspicious-looking ciphertext {{nowrap|''c''′ ≡ ''cr''<sup>''e''</sup> (mod ''n'')}} for some value ''r'' chosen by the attacker. Because of the multiplicative property ''c''′ is the encryption of {{nowrap|''mr'' (mod ''n'')}}. Hence, if the attacker is successful with the attack, he will learn {{nowrap|''mr'' (mod ''n'')}} from which he can derive the message ''m'' by multiplying ''mr'' with the modular inverse of ''r'' modulo ''n''.{{citation needed|reason=Someone had to have noticed this and published first, they should be cited|date=February 2015}}

===Padding schemes===
To avoid these problems, practical RSA implementations typically embed some form of structured, randomized [[Padding (cryptography)|padding]] into the value ''m'' before encrypting it. This padding ensures that ''m'' does not fall into the range of insecure plaintexts, and that a given message, once padded, will encrypt to one of a large number of different possible ciphertexts.

Standards such as [[PKCS1|PKCS#1]] have been carefully designed to securely pad messages prior to RSA encryption. Because these schemes pad the plaintext ''m'' with some number of additional bits, the size of the un-padded message ''M'' must be somewhat smaller. RSA padding schemes must be carefully designed so as to prevent sophisticated attacks which may be facilitated by a predictable message structure.  Early versions of the PKCS#1 standard (up to version 1.5) used a construction that appears to make RSA semantically secure. However, at [[International Cryptology Conference|Crypto]] 1998, Bleichenbacher showed that this version is vulnerable to a practical [[adaptive chosen ciphertext attack]]. Furthermore, at [[Eurocrypt]] 2000, Coron et al.{{citation needed|reason=a link to the paper or at least its title would be helpful|date=June 2019}} showed that for some types of messages, this padding does not provide a high enough level of security. Later versions of the standard include [[Optimal Asymmetric Encryption Padding]] (OAEP), which prevents these attacks. As such, OAEP should be used in any new application, and PKCS#1 v1.5 padding should be replaced wherever possible. The PKCS#1 standard also incorporates processing schemes designed to provide additional security for RSA signatures, e.g. the Probabilistic Signature Scheme for RSA ([[RSA-PSS]]).

Secure padding schemes such as RSA-PSS are as essential for the security of message signing as they are for message encryption. Two US patents on PSS were granted (USPTO 6266771 and USPTO 70360140); however, these patents expired on 24 July 2009 and 25 April 2010, respectively.  Use of PSS no longer seems to be encumbered by patents.{{Original research inline|date=August 2019}}  Note that using different RSA key-pairs for encryption and signing is potentially more secure.<ref>{{Cite web | url=https://www.di-mgt.com.au/rsa_alg.html#weaknesses | title=RSA Algorithm}}</ref>

==Security and practical considerations==

===Using the Chinese remainder algorithm===
For efficiency many popular crypto libraries (like [[OpenSSL]], [[Java (programming language)|Java]] and [[.NET]]) use the following optimization for decryption and signing based on the [[Chinese remainder theorem]]. The following values are precomputed and stored as part of the private key:
* <math>p</math> and <math>q</math>: the primes from the key generation,
* <math>d_P = d \pmod{p - 1}</math>,
* <math>d_Q = d \pmod{q - 1}</math> and
* <math>q_\text{inv} = q^{-1} \pmod{p}</math>.

These values allow the recipient to compute the exponentiation {{nowrap|1=''m'' = ''c''<sup>''d''</sup> (mod ''pq'')}} more efficiently as follows:
* <math>m_1 = c^{d_P} \pmod{p}</math>
* <math>m_2 = c^{d_Q} \pmod{q}</math>
* <math>h = q_\text{inv}(m_1 - m_2) \pmod{p}</math> (if <math>m_1 < m_2</math> then some libraries compute ''h'' as <math>q_\text{inv}\left[\left(m_1 + \left\lceil \frac{q}{p} \right\rceil p\right) - m_2\right] \pmod{p}</math>)
* <math>m = m_2 + hq\,</math>

This is more efficient than computing [[exponentiation by squaring]] even though two modular exponentiations have to be computed. The reason is that these two modular exponentiations both use a smaller exponent and a smaller modulus.

===Integer factorization and RSA problem===
{{See also|RSA Factoring Challenge|Integer factorization records|Shor's algorithm|}}
The security of the RSA cryptosystem is based on two mathematical problems: the problem of [[integer factorization|factoring large numbers]] and the [[RSA problem]]. Full decryption of an RSA ciphertext is thought to be infeasible on the assumption that both of these problems are hard, i.e., no efficient algorithm exists for solving them. Providing security against ''partial'' decryption may require the addition of a secure [[padding (cryptography)|padding scheme]].<ref>{{cite book|first=Edmond K.|last=Machie|url=https://books.google.com/books?id=AK5MySZbbuMC&pg=PA167|title=Network security traceback attack and react in the United States Department of Defense network|pages=167|isbn=1466985747}}</ref>

The [[RSA problem]] is defined as the task of taking ''e''th roots modulo a composite ''n'': recovering a value ''m'' such that {{nowrap|''c'' ≡ ''m''<sup>''e''</sup> (mod ''n'')}}, where {{nowrap|(''n'', ''e'')}} is an RSA public key and ''c'' is an RSA ciphertext.  Currently the most promising approach to solving the RSA problem is to factor the modulus ''n''.  With the ability to recover prime factors, an attacker can compute the secret exponent ''d'' from a public key {{nowrap|(''n'', ''e'')}}, then decrypt ''c'' using the standard procedure.  To accomplish this, an attacker factors ''n'' into ''p'' and ''q'', and computes {{nowrap|lcm(''p'' − 1, ''q'' − 1)}} which allows the determination of ''d'' from ''e''.  No polynomial-time method for factoring large integers on a classical computer has yet been found, but it has not been proven that none exists.  ''See [[integer factorization]] for a discussion of this problem''.

Multiple polynomial quadratic sieve (MPQS) can be used to factor the public modulus ''n''. The time taken to factor 128-bit and 256-bit ''n'' on a desktop computer {{nowrap|(Processor: Intel Dual-Core i7-4500U 1.80GHz)}} are respectively 2 seconds and 35 minutes.
{| class="wikitable"
|-
! Bits !! Time
|-
| 128 || Less than 2 seconds
|-
| 192 || 16 seconds
|-
| 256 || 35 minutes
|-
| 260 || 1 hour
|}{{original research inline|date=November 2018}}

A tool called YAFU can be used to optimize this process.<ref>{{Cite web | url=https://sourceforge.net/projects/yafu/ | title=Yafu}}</ref> It took about 5720s to factor ''320bit-N'' on the same computer.

{| class="wikitable"
|-
! Bits !! Time !! Memory used
|-
| 128 || 0.4886 seconds    || 0.1&nbsp;MiB
|-
| 192 || 3.9979 seconds    || 0.5&nbsp;MiB
|-
| 256 || 103.1746 seconds  || 3&nbsp;MiB
|-
| 300 || 1175.7826 seconds || 10.9&nbsp;MiB
|}

In 2009, Benjamin Moody factored an RSA-512 bit key in 73 days using only public software (GGNFS) and his desktop computer (a dual-core Athlon64 with a 1,900&nbsp;MHz cpu.). Just less than five gigabytes of disk storage was required and about 2.5 gigabytes of RAM for the sieving process. The first RSA-512 factorization in 1999 required the equivalent of 8,400 MIPS years, over an elapsed time of about seven months.<ref>{{Cite web | url=http://lukenotricks.blogspot.se/2009/08/solo-desktop-factorization-of-rsa-512.html | title=Solo Desktop Factorization of an RSA-512 Key| date=2009-08-25}}</ref>{{better source|date=November 2018}}

Rivest, Shamir, and Adleman noted<ref name="rsa" /> that Miller has shown that – assuming the truth of the [[Generalized Riemann hypothesis|Extended Riemann Hypothesis]] – finding ''d'' from ''n'' and ''e'' is as hard as factoring ''n'' into ''p'' and ''q'' (up to a polynomial time difference).<ref>{{cite conference|url=https://www.cs.cmu.edu/~glmiller/Publications/Papers/Mi75.pdf|first=Gary L.|last=Miller|title=Riemann's Hypothesis and Tests for Primality|book-title=Proceedings of Seventh Annual ACM Symposium on Theory of Computing|year=1975|pages=234–239}}</ref>  However, Rivest, Shamir, and Adleman noted, in section IX/D of their paper, that they had not found a proof that inverting RSA is equally as hard as factoring.

{{As of|2010}}, the largest factored [[RSA number]] was 768 bits long (232 decimal digits, see [[RSA-768]]). Its factorization, by a state-of-the-art distributed implementation, took around fifteen hundred CPU years (two years of real time, on many hundreds of computers). No larger RSA key is known publicly to have been factored. In practice, RSA keys are typically 1024 to 4096 bits long. Some experts believe that 1024-bit keys may become breakable in the near future or may already be breakable by a sufficiently well-funded attacker, though this is disputable. Few people see any way that 4096-bit keys could be broken in the foreseeable future. Therefore, it is generally presumed that RSA is secure if ''n'' is sufficiently large. If ''n'' is 300 [[bit]]s or shorter, it can be factored in a few hours in a [[personal computer]], using software already freely available. Keys of 512 bits have been shown to be practically breakable in 1999 when [[RSA-155]] was factored by using several hundred computers, and these are now factored in a few weeks using common hardware. Exploits using 512-bit code-signing certificates that may have been factored were reported in 2011.<ref>{{cite web|url=https://blog.fox-it.com/2011/11/21/rsa-512-certificates-abused-in-the-wild/|title=RSA-512 certificates abused in-the-wild|website=Fox-IT International blog|date=November 21, 2011|first=Michael|last=Sandee}}</ref> A theoretical hardware device named [[TWIRL]], described by Shamir and Tromer in 2003, called into question the security of 1024 bit keys. It is currently recommended that ''n'' be at least 2048 bits long.<ref>{{cite web|url=http://www.emc.com/emc-plus/rsa-labs/historical/has-the-rsa-algorithm-been-compromised.htm|archive-url=https://web.archive.org/web/20130923062442/http://www.emc.com/emc-plus/rsa-labs/historical/has-the-rsa-algorithm-been-compromised.htm|archive-date=2013-09-23|title=Has the RSA algorithm been compromised as a result of Bernstein's Paper?|first=Robert D.|last=Silverman|date=April 8, 2002|at=sec. "What key size should I be using?"}}</ref>

In 1994, [[Peter Shor]] showed that a [[quantum computer]] – if one could ever be practically created for the purpose – would be able to factor in [[polynomial time]], breaking RSA; see [[Shor's algorithm]].

===Faulty key generation===
{{more citations needed|section|date=October 2017}}
{{See also|Coppersmith's Attack|Wiener's Attack}}

Finding the large primes ''p'' and ''q'' is usually done by testing random numbers of the right size with probabilistic [[primality test]]s that quickly eliminate virtually all of the nonprimes.

The numbers ''p'' and ''q'' should not be "too close", lest the [[Fermat factorization]] for ''n'' be successful. If ''p'' − ''q'' is less than 2''n''<sup>1/4</sup> (''n'' = ''p'' * ''q'', which for even small 1024-bit values of ''n'' is {{val|3|e=77}}) solving for ''p'' and ''q'' is trivial. Furthermore, if either ''p'' − 1 or ''q'' − 1 has only small prime factors, ''n'' can be factored quickly by [[Pollard's p − 1 algorithm]], and such values of ''p'' or ''q'' should hence be discarded.

It is important that the private exponent ''d'' be large enough. [[Michael J. Wiener]] showed that if ''p'' is between ''q'' and 2''q'' (which is quite typical) and {{nowrap|''d'' < ''n''<sup>1/4</sup>/3}}, then ''d'' can be computed efficiently from ''n'' and&nbsp;''e''.<ref name="wiener">{{Cite journal | title=Cryptanalysis of short RSA secret exponents |
first1=Michael J. | last1=Wiener |
journal=IEEE Transactions on Information Theory |
volume=36 | issue=3 | pages=553–558 | date=May 1990 |
doi=10.1109/18.54902
|url=http://www.cits.rub.de/imperia/md/content/may/krypto2ss08/shortsecretexponents.pdf }}
</ref>

There is no known attack against small public exponents such as {{nowrap|1=''e'' = 3}}, provided that the proper padding is used. [[Coppersmith's Attack]] has many applications in attacking RSA specifically if the public exponent ''e'' is small and if the encrypted message is short and not padded. [[65537]] is a commonly used value for&nbsp;''e''; this value can be regarded as a compromise between avoiding potential small exponent attacks and still allowing efficient encryptions (or signature verification). The NIST Special Publication on Computer Security (SP 800-78 Rev 1 of August 2007) does not allow public exponents ''e'' smaller than 65537, but does not state a reason for this restriction.

In October 2017 a team of researchers from [[Masaryk University]] announced the [[ROCA vulnerability]], which affects RSA keys generated by an algorithm embodied in a library from [[Infineon]]. Large number of smart cards and TPMs were shown to be affected. Vulnerable RSA keys are easily identified using a test program the team released.<ref name=nemecsys>{{cite conference|url=https://crocs.fi.muni.cz/_media/public/papers/nemec_roca_ccs17_preprint.pdf|title=The Return of Coppersmith’s Attack: Practical Factorization of Widely Used RSA Moduli|first1=Matus|last1=Nemec|first2=Marek|last2=Sys|first3=Petr|last3=Svenda|first4=Dusan|last4=Klinec|first5=Vashek|last5=Matyas|date=November 2017|doi=10.1145/3133956.3133969|book-title=Proceedings of the 2017 ACM SIGSAC Conference on Computer and Communications Security|series=CCS '17}}</ref>

===Importance of strong random number generation===
A cryptographically strong [[random number generator]], which has been properly seeded with adequate entropy, must be used to generate the primes ''p'' and ''q''. An analysis comparing millions of public keys gathered from the Internet was carried out in early 2012 by [[Arjen Klaas Lenstra|Arjen K. Lenstra]], James P. Hughes, Maxime Augier, Joppe W. Bos, Thorsten Kleinjung and Christophe Wachter. They were able to factor 0.2% of the keys using only [[Euclid's algorithm]].<ref>{{cite news |title=Flaw Found in an Online Encryption Method |url=https://www.nytimes.com/2012/02/15/technology/researchers-find-flaw-in-an-online-encryption-method.html |newspaper=[[The New York Times]] |date=February 14, 2012 |first=John |last=Markoff }}</ref><ref>{{cite journal |title=Ron was wrong, Whit is right |url=http://eprint.iacr.org/2012/064.pdf |year=2012 |last1=Lenstra |first1=Arjen K. |last2=Hughes |first2=James P. |last3=Augier |first3=Maxime |last4=Bos |first4=Joppe W. |last5=Kleinjung |first5=Thorsten |last6=Wachter |first6=Christophe }}</ref>

They exploited a weakness unique to cryptosystems based on integer factorization. If {{nowrap|1=''n'' = ''pq''}} is one public key and {{nowrap|1=''n''′ = ''p''′''q''′}} is another, then if by chance {{nowrap|1=''p'' = ''p''′}} (but ''q'' is not equal to ''q''′), then a simple computation of {{nowrap|1=gcd(''n'',''n''′) = ''p''}} factors both ''n'' and ''n''′, totally compromising both keys. Lenstra et al. note that this problem can be minimized by using a strong random seed of bit-length twice the intended security level, or by employing a deterministic function to choose ''q'' given ''p'', instead of choosing ''p'' and ''q'' independently.

[[Nadia Heninger]] was part of a group that did a similar experiment. They used an idea of [[Daniel J. Bernstein]] to compute the GCD of each RSA key ''n'' against the product of all the other keys ''n''′ they had found (a 729 million digit number), instead of computing each gcd(''n'',''n''′) separately, thereby achieving a very significant speedup since after one large division the GCD problem is of normal size.

Heninger says in her blog that the bad keys occurred almost entirely in embedded applications, including "firewalls, routers, VPN devices, remote server administration devices, printers, projectors, and VOIP phones" from over 30 manufacturers. Heninger explains that the one-shared-prime problem uncovered by the two groups results from situations where the pseudorandom number generator is poorly seeded initially and then reseeded between the generation of the first and second primes. Using seeds of sufficiently high entropy obtained from key stroke timings or electronic diode noise or [[atmospheric noise]] from a radio receiver tuned between stations should solve the problem.<ref>{{Cite web | url=https://freedom-to-tinker.com/blog/nadiah/new-research-theres-no-need-panic-over-factorable-keys-just-mind-your-ps-and-qs | title=New research: There's no need to panic over factorable keys–just mind your Ps and Qs | date=February 15, 2012 | website=Freedom to Tinker | first=Nadia | last=Heninger}}</ref>

Strong random number generation is important throughout every phase of public key cryptography. For instance, if a weak generator is used for the symmetric keys that are being distributed by RSA, then an eavesdropper could bypass RSA and guess the symmetric keys directly.

===Timing attacks===
[[Paul Kocher|Kocher]] described a new attack on RSA in 1995: if the attacker Eve knows Alice's hardware in sufficient detail and is able to measure the decryption times for several known ciphertexts, she can deduce the decryption key ''d'' quickly.  This attack can also be applied against the RSA signature scheme.  In 2003, [[Dan Boneh|Boneh]] and [[David Brumley|Brumley]] demonstrated a more practical attack capable of recovering RSA factorizations over a network connection (e.g., from a [[Secure Sockets Layer]] (SSL)-enabled webserver)<ref>{{cite conference|url=http://crypto.stanford.edu/~dabo/papers/ssl-timing.pdf|title=Remote timing attacks are practical|first1=David|last1=Brumley|first2=Dan|last2=Boneh|year=2003|series=SSYM'03|book-title=Proceedings of the 12th Conference on USENIX Security Symposium}}</ref>  This attack takes advantage of information leaked by the [[Chinese remainder theorem]] optimization used by many RSA implementations.

One way to thwart these attacks is to ensure that the decryption operation takes a constant amount of time for every ciphertext.  However, this approach can significantly reduce performance.  Instead, most RSA implementations use an alternate technique known as [[blinding (cryptography)|cryptographic blinding]].  RSA blinding makes use of the multiplicative property of RSA.  Instead of computing {{nowrap|''c''<sup>''d''</sup> (mod ''n'')}}, Alice first chooses a secret random value ''r'' and computes {{nowrap|(''r''<sup>''e''</sup>''c'')<sup>''d''</sup> (mod ''n'')}}.  The result of this computation after applying [[Euler's Theorem]] is {{nowrap|''rc''<sup>''d''</sup> (mod ''n'')}} and so the effect of ''r'' can be removed by multiplying by its inverse.  A new value of ''r'' is chosen for each ciphertext.  With blinding applied, the decryption time is no longer correlated to the value of the input ciphertext and so the timing attack fails.

===Adaptive chosen ciphertext attacks===
In 1998, [[Daniel Bleichenbacher]] described the first practical [[adaptive chosen ciphertext attack]], against RSA-encrypted messages using the PKCS #1 v1 [[Padding (cryptography)|padding scheme]] (a padding scheme randomizes and adds structure to an RSA-encrypted message, so it is possible to determine whether a decrypted message is valid). Due to flaws with the PKCS #1 scheme, Bleichenbacher was able to mount a practical attack against RSA implementations of the [[Secure Socket Layer]] protocol, and to recover session keys. As a result of this work, cryptographers now recommend the use of provably secure padding schemes such as [[Optimal Asymmetric Encryption Padding]], and RSA Laboratories has released new versions of PKCS #1 that are not vulnerable to these attacks.

===Side-channel analysis attacks===
A side-channel attack using branch prediction analysis (BPA) has been described. Many processors use a [[branch predictor]] to determine whether a conditional branch in the instruction flow of a program is likely to be taken or not. Often these processors also implement [[simultaneous multithreading]] (SMT). Branch prediction analysis attacks use a spy process to discover (statistically) the private key when processed with these processors.

Simple Branch Prediction Analysis (SBPA) claims to improve BPA in a non-statistical way. In their paper, "On the Power of Simple Branch Prediction Analysis",<ref>{{Cite conference |citeseerx = 10.1.1.80.1438|title = On the power of simple branch prediction analysis|first1=Onur|last1=Acıiçmez|first2=Çetin Kaya|last2=Koç|first3=Jean-Pierre|last3=Seifert|pages = 312–320|year = 2007|book-title=Proceedings of the 2nd ACM Symposium on Information, Computer and Communications Security|series=ASIACCS '07|doi=10.1145/1229285.1266999}}</ref> the authors of SBPA ([[Onur Aciicmez]] and [[Cetin Kaya Koc]]) claim to have discovered 508 out of 512 bits of an RSA key in 10 iterations.

A power fault attack on RSA implementations has been described in 2010.<ref>{{cite journal|url=http://www.eecs.umich.edu/~valeria/research/publications/DATE10RSA.pdf|title=Fault-Based Attack of RSA Authentication|first1=Andrea|last1=Pellegrini|first2=Valeria|last2=Bertacco|first3=Todd|last3=Austin|year=2010}}</ref> The author recovered the key by varying the CPU power voltage outside limits; this caused multiple power faults on the server.

===Rainbow tables attacks===
The generated primes can be attacked by [[rainbow tables]] because the random numbers are fixed and finite sets.<ref>[http://www.cqvip.com/QK/72003x/201603/epub1000000177403.html 现实RSA算法的破解]</ref>

== Implementations ==
Below are some cryptography libraries that provide support for RSA:

* [[Botan (programming library)|Botan]]
* [[Bouncy Castle (cryptography)|Bouncy Castle]]
* [[cryptlib]]
* [[Crypto++]]
* [[Libgcrypt]]
* [[Nettle (cryptographic library)|Nettle]]
* [[OpenSSL]]
* [[WolfSSL#wolfCrypt|wolfCrypt]]

==See also==
{{Portal|Mathematics}}
* [[Acoustic cryptanalysis]]
* [[Computational complexity theory]]
* [[Cryptographic key length]]
* [[Diffie–Hellman key exchange]]
* [[Key exchange]]
* [[Key management]]
* [[Public-key cryptography]]
* [[Trapdoor function]]

==References==
{{Reflist|30em}}

==Further reading==
* {{cite book | last1 = Menezes | first1 = Alfred | last2 = van Oorschot | first2 = Paul C. | last3 = Vanstone | first3 = Scott A. | url = https://archive.org/details/handbookofapplie0000mene | title = Handbook of Applied Cryptography | publisher = CRC Press | date = October 1996 | isbn = 978-0-8493-8523-0 | url-access = registration }}
* {{cite book
 |last1=Cormen |first1=Thomas H. |authorlink1=Thomas H. Cormen
 |last2=Leiserson |first2=Charles E. |authorlink2=Charles E. Leiserson
 |last3=Rivest |first3=Ronald L. |authorlink3=Ronald L. Rivest
 |last4=Stein |first4=Clifford |authorlink4=Clifford Stein
 | title = Introduction to Algorithms
 | edition =  2nd
 | publisher =  MIT Press and McGraw-Hill | year = 2001
 | isbn = 978-0-262-03293-3 | pages = 881–887|title-link=Introduction to Algorithms }}

==External links==
* The Original RSA Patent as filed with the U.S. Patent Office by Rivest; Ronald L. (Belmont, MA), Shamir; Adi (Cambridge, MA), Adleman; Leonard M. (Arlington, MA), December 14, 1977, '''{{US patent|4405829}}'''.
* [http://www.emc.com/emc-plus/rsa-labs/standards-initiatives/pkcs-rsa-cryptography-standard.htm PKCS #1: RSA Cryptography Standard] ([[RSA Laboratories]] website)
** The ''[[PKCS]] #1'' [[standardization|standard]] ''"provides recommendations for the implementation of [[public-key cryptography]] based on the '''RSA''' algorithm, covering the following aspects: cryptographic [[Primitive type|primitives]]; [[encryption]] schemes; [[Digital signature|signature]] schemes with appendix; [[ASN.1]] syntax for representing keys and for identifying the schemes"''.
* {{YouTube|vgTtHV04xRI|Explanation of RSA using colored lamps}}
* [http://www.di-mgt.com.au/rsa_alg.html Thorough walk through of RSA]
* [http://www.muppetlabs.com/~breadbox/txt/rsa.html Prime Number Hide-And-Seek: How the RSA Cipher Works]
* [http://eprint.iacr.org/2006/351 Onur Aciicmez, Cetin Kaya Koc, Jean-Pierre Seifert: ''On the Power of Simple Branch Prediction Analysis'']
* [http://blog.cacert.org/2006/11/193.html A New Vulnerability In RSA Cryptography, CAcert NEWS Blog]
* [http://polarssl.org/source_code Example of an RSA implementation with PKCS#1 padding (GPL source code)]
* [http://www.cryptography.com/resources/whitepapers/TimingAttacks.pdf Kocher's article about timing attacks]
* [http://www.cryptool.org/images/ct1/presentations/RSA/RSA-Flash-en/player.html An animated explanation of RSA with its mathematical background by CrypTool]
* [http://inventwithpython.com/hacking ''Hacking Secret Ciphers with Python''], [http://inventwithpython.com/hacking/chapter24.html Chapter 24, Public Key Cryptography and the RSA Cipher]
* {{cite web|last=Grime|first=James|title=RSA Encryption|url=http://www.numberphile.com/videos/RSA.html|work=Numberphile|publisher=[[Brady Haran]]}}
* [http://www.gnudeveloper.com/groups/cyber_security/Cryptography_RSA_Key_Exchange_works_in_realtime_using_Keytool_openSSL%20.html How RSA Key used for Encryption in real world]

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