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{{short description|Algorithm for public-key cryptography}}
{{about|a cryptosystem|the company|RSA Security}}
{{Infobox block cipher
| name          = RSA
| designers     = [[Ron Rivest]],<ref name="rsa"/> [[Adi Shamir]], and [[Leonard Adleman]]
| publish date  = 1977
| type          = Public-Key
| derived from  =
| derived to    =
| related to    =
| certification = [[PKCS1|PKCS#1]], [[ANSI X9.31]]
| key size      = variable but typically 2,048 to 4,096 bits
| block size    =
| structure     =
| rounds        = 1
| cryptanalysis = [[General number field sieve]] for classical computers;<br />[[Shor's algorithm]] for quantum computers.<br />An [[RSA-250|829-bit key]] has been broken.
}}

'''RSA''' ('''Rivest–Shamir–Adleman''') is a [[public-key cryptography|public-key cryptosystem]], one of the oldest widely used for secure data transmission. The [[initialism]] "RSA" comes from the surnames of [[Ron Rivest]], [[Adi Shamir]] and [[Leonard Adleman]], who publicly described the algorithm in 1977. An equivalent system was developed secretly in 1973 at [[Government Communications Headquarters]] (GCHQ), the British [[signals intelligence]] agency, by the English mathematician [[Clifford Cocks]]. That system was [[classified information|declassified]] in 1997.<ref>{{cite web |url=http://www.bristol.ac.uk/graduation/honorary-degrees/hondeg08/cocks.html |first=Nigel |last=Smart |title=Dr Clifford Cocks CB |access-date=August 14, 2011 |date=February 19, 2008 |publisher=[[Bristol University]]}}</ref>

In a public-key [[cryptosystem]], the [[encryption key]] is public and distinct from the [[decryption key]], which is kept secret (private).
An RSA user creates and publishes a public key based on two large [[prime number]]s, along with an auxiliary value. The prime numbers are kept secret. Messages can be encrypted by anyone, via the public key, but can only be decrypted by someone who knows the private key.<ref name="rsa">{{cite journal
 | last1 = Rivest | first1 = R.
 | last2 = Shamir | first2 = A.
 | last3 = Adleman | first3 = L.
 | s2cid = 2873616
 | 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
 | archive-url = https://web.archive.org/web/20230127011251/http://people.csail.mit.edu/rivest/Rsapaper.pdf
 | archive-date = 2023-01-27
 | url-status = dead
 }}</ref>

The security of RSA relies on the practical difficulty of [[factorization|factoring]] the product of two large [[prime number]]s, the "[[factoring problem]]". Breaking RSA encryption is known as the [[RSA problem]]. Whether it is as difficult as the factoring problem is an open question.<ref>{{Cite journal|last=Castelvecchi|first=Davide|date=2020-10-30|title=Quantum-computing pioneer warns of complacency over Internet security|url=https://www.nature.com/articles/d41586-020-03068-9|journal=Nature|language=en|volume=587|issue=7833|pages=189|doi=10.1038/d41586-020-03068-9|pmid=33139910 |bibcode=2020Natur.587..189C |s2cid=226243008 }} 2020 interview of [[Peter Shor]].</ref> There are no published methods to defeat the system if a large enough key is used.

RSA is a relatively slow algorithm. Because of this, it is not commonly used to directly encrypt user data. More often, RSA is used to transmit shared keys for [[symmetric-key algorithm|symmetric-key]] cryptography, which are then used for bulk encryption–decryption.

==History==
[[File:Adi Shamir 2009 crop.jpg|thumb|upright=0.7|[[Adi Shamir]], co-inventor of RSA (the others are [[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 |first1 = W. |last1 = Diffie |first2 = M. E. |last2 = Hellman |citeseerx = 10.1.1.37.9720}}</ref> Moreover, like [[Diffie–Hellman key exchange|Diffie-Hellman]], RSA is based on [[modular exponentiation]].

[[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 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 |last = Rivest |first = Ronald}}</ref> In April 1977, they spent [[Passover]] at the house of a student and drank a good deal of 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 |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{{snd}} 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 a similar 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. |author-link=Clifford Cocks |date=20 November 1973 |website=www.gchq.gov.uk |access-date=2017-05-30 |archive-url=https://web.archive.org/web/20180928121748/https://www.gchq.gov.uk/sites/default/files/document_files/Cliff%20Cocks%20paper%2019731120.pdf |archive-date=28 September 2018 |url-status=dead}}</ref> However, given the relatively expensive computers needed to implement it at the time, it was considered to be mostly a curiosity and, as far as is publicly known, was never deployed. His ideas and concepts were not revealed until 1997 due to its top-secret classification.

Kid-RSA (KRSA) is a simplified, insecure 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.
[https://ww2.amstat.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.
[https://web.archive.org/web/20220319203917/https://www.cs.uri.edu/cryptography/publickeykidkrypto.htm "Public key Cryptography"].
</ref><ref>
Neal Koblitz.
[https://sites.math.washington.edu/~koblitz/crlogia.html "Cryptography As a Teaching Tool"].
Cryptologia, Vol. 21, No. 4 (1997).
</ref>

==Patent==

A [[patent]] describing the RSA algorithm was granted to [[Massachusetts Institute of Technology|MIT]] on 20 September 1983: {{US patent|4405829}} "Cryptographic communications system and method". From [[Derwent World Patent Index|DWPI]]'s abstract of the patent:
{{Blockquote|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).}}

A detailed description of the algorithm was published in August 1977, in [[Scientific American]]'s [[List of Martin Gardner Mathematical Games columns|Mathematical Games]] column.<ref name="SIAM" /> This preceded the patent's filing date of December 1977. Consequently, the patent had no legal standing outside the [[United States]]. Had Cocks' work been publicly known, a patent in the United States would not have been legal either.

When the patent was issued, [[term of patent|terms of patent]] were 17 years. The patent was about to expire on 21 September 2000, but [[RSA Security]] released the algorithm to the public domain on 6 September 2000.<ref>{{cite web |url=http://www.rsa.com/press_release.aspx?id=261 |title=RSA Security Releases RSA Encryption Algorithm into Public Domain |access-date=2010-03-03 |url-status=dead |archive-url=https://web.archive.org/web/20070621021111/http://www.rsa.com/press_release.aspx?id=261 |archive-date=June 21, 2007 }}</ref>

==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 for all integers {{mvar|m}} ({{math|0 &le; ''m'' &lt; ''n''}}), both <math>(m^e)^d</math> and <math>m</math> have the same [[Euclidean division|remainder]] when divided by <math>n</math> (they are [[Modular arithmetic#Congruence|congruent modulo]] <math>n</math>):<math display="block">(m^e)^d \equiv m \pmod{n}.</math>However, when given only {{mvar|e}} and {{mvar|n}}, it is extremely difficult to find {{mvar|d}}.

The integers {{mvar|n}} and {{mvar|e}} comprise the public key, {{mvar|d}} represents the private key, and {{mvar|m}} represents the message. The [[modular exponentiation]] to {{mvar|e}} and {{mvar|d}} corresponds to encryption and decryption, respectively.

In addition, because the two exponents [[Exponentiation#Identities and properties|can be swapped]], the private and public key can also be swapped, allowing for message [[Digital signature|signing and verification]] using the same algorithm.

===Key generation===
The keys for the RSA algorithm are generated in the following way:
# Choose two large [[prime number]]s {{mvar|p}} and {{mvar|q}}.
#* To make factoring harder, {{mvar|p}} and {{mvar|q}} should be chosen at random, be both large and have a large difference.<ref name="rsa"/> For choosing them the standard method is to choose random integers and use a [[primality test]] until two primes are found.
#* {{mvar|p}} and {{mvar|q}} are kept secret.
# Compute {{math|1=''n'' = ''pq''}}.
#* {{mvar|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]].
#* {{mvar|n}} is released as part of the public key.
# Compute {{math|''&lambda;''(''n'')}}, where {{mvar|&lambda;}} is [[Carmichael's totient function]]. Since {{math|1=''n'' = ''pq'', ''&lambda;''(''n'') = [[least common multiple|lcm]](''&lambda;''(''p''),&nbsp;''&lambda;''(''q''))}}, and since {{mvar|p}} and {{mvar|q}} are prime, {{math|1=''&lambda;''(''p'') = ''[[Euler totient function|&phi;]]''(''p'') = ''p'' − 1}}, and likewise {{math|1=''&lambda;''(''q'') = ''q'' − 1}}. Hence {{math|1=''&lambda;''(''n'') = lcm(''p'' − 1, ''q'' − 1)}}.
#* The {{math|1=lcm}} may be calculated through the [[Euclidean algorithm]], since {{math|1=lcm(''a'',&nbsp;''b'') = {{sfrac|{{abs|''ab''}}|gcd(''a'',&nbsp;''b'')}}}}.
#* {{math|''&lambda;''(''n'')}} is kept secret.
# Choose an integer {{mvar|e}} such that {{math|1 &lt; ''e'' &lt; ''&lambda;''(''n'')}} and {{math|1=[[greatest common divisor|gcd]](''e'', ''&lambda;''(''n'')) = 1}}; that is, {{mvar|e}} and {{math|''&lambda;''(''n'')}} are [[coprime]].
#* {{mvar|e}} having a short [[bit-length]] and small [[Hamming weight]] results in more efficient encryption{{snd}} the most commonly chosen value for {{mvar|e}} is {{math|1=2<sup>16</sup> + 1 = {{val|65,537}}}}. The smallest (and fastest) possible value for {{mvar|e}} is 3, but such a small value for {{mvar|e}} has been shown to be less secure in some settings.<ref name="Boneh99">
{{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>
#* {{mvar|e}} is released as part of the public key.
# Determine {{mvar|d}} as {{math|1=''d'' ≡ ''e''<sup>−1</sup> (mod ''&lambda;''(''n''))}}; that is, {{mvar|d}} is the [[modular multiplicative inverse]] of {{mvar|e}} modulo {{math|''&lambda;''(''n'')}}.
#* This means: solve for {{mvar|d}} the equation {{math|1=''de'' ≡ 1 (mod ''&lambda;''(''n''))}}; {{mvar|d}} can be computed efficiently by using the [[extended Euclidean algorithm]], since, thanks to {{mvar|e}} and {{math|''&lambda;''(''n'')}} being coprime, said equation is a form of [[Bézout's identity]], where {{mvar|d}} is one of the coefficients.
#* {{mvar|d}} is kept secret as the ''private key exponent''.
The ''public key'' consists of the modulus {{mvar|n}} and the public (or encryption) exponent {{mvar|e}}. The ''private key'' consists of the private (or decryption) exponent {{mvar|d}}, which must be kept secret. {{mvar|p}}, {{mvar|q}}, and {{math|''&lambda;''(''n'')}} must also be kept secret because they can be used to calculate {{mvar|d}}. In fact, they can all be discarded after {{mvar|d}} has been computed.<ref>Applied Cryptography, John Wiley & Sons, New York, 1996. [[Bruce Schneier]], p. 467.</ref>

{{anchor|OriginalWithPhiN}}In the original RSA paper,<ref name=rsa /> the [[Euler totient function]] {{math|1=''&phi;''(''n'') = (''p'' − 1)(''q'' − 1)}} is used instead of {{math|''&lambda;''(''n'')}} for calculating the private exponent {{mvar|d}}. Since {{math|''&phi;''(''n'')}} is always divisible by {{math|''&lambda;''(''n'')}}, the algorithm works as well. The possibility of using [[Euler totient function]] results also from [[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 {{mvar|d}} satisfying {{math|1=''d''⋅''e'' ≡ 1 (mod ''&phi;''(''n''))}} also satisfies {{math|1=''d''⋅''e'' ≡ 1 (mod ''&lambda;''(''n''))}}. However, computing {{mvar|d}} modulo {{math|''&phi;''(''n'')}} will sometimes yield a result that is larger than necessary (i.e. {{math|1=''d'' &gt; ''&lambda;''(''n'')}}). Most of the implementations of RSA will accept exponents generated using either method (if they use the private exponent {{mvar|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 such as [https://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.186-4.pdf#page=63 FIPS&nbsp;186-4] (Section B.3.1) may require that {{math|''d'' &lt; ''&lambda;''(''n'')}}. Any "oversized" private exponents not meeting this criterion may always be reduced modulo {{math|''&lambda;''(''n'')}} to obtain a smaller equivalent exponent.

{{anchor|CryptoStrengthOfPQ}}Since any common factors of {{math|(''p'' − 1)}} and {{math|(''q'' − 1)}} are present in the factorisation of {{math|''n'' − 1}} = {{math|''pq'' − 1}} = {{math|(''p'' − 1)(''q'' − 1) + (''p'' − 1) + (''q'' − 1)}},<ref>{{cite web|title=Further Attacks on Server-Aided RSA Cryptosystems |first1=James |last1=McKee |first2=Richard |last2=Pinch |citeseerx=10.1.1.33.1333 |year=1998|url=https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=64294c404088b69a519614510b8d12b4809a6b10}}</ref>{{self published inline|date=December 2023|reason=This paper is not from a journal and may not have been peer-reviewed}} it is recommended that {{math|(''p'' − 1)}} and {{math|(''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>{{Failed verification|date=April 2022}}<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>{{Failed verification|date=April 2022}}

Note: The authors of the original RSA paper carry out the key generation by choosing {{mvar|d}} and then computing {{mvar|e}} as the [[modular multiplicative inverse]] of {{mvar|d}} modulo {{math|''&phi;''(''n'')}}, whereas most current implementations of RSA, such as those following [[PKCS1|PKCS#1]], do the reverse (choose {{mvar|e}} and compute {{mvar|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 [[Padding (cryptography)|padded]] plaintext), such that {{math|0 &le; ''m'' &lt; ''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 display="block">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. Note that at least nine values of {{mvar|m}} will yield a ciphertext {{mvar|c}} equal to
{{mvar|m}},{{efn|Namely, the values of {{mvar|m}} which are equal to −1, 0, or 1 modulo {{mvar|p}} while also equal to −1, 0, or 1 modulo {{mvar|q}}. There will be more values of {{mvar|m}} having {{math|1=''c'' = ''m''}} if {{math|''p''&nbsp;−&nbsp;1}} or {{math|''q''&nbsp;−&nbsp;1}} has other divisors in common with {{math|''e''&nbsp;−&nbsp;1}} besides 2 because this gives more values of {{mvar|m}} such that <math>m^{e-1} \bmod p = 1</math> or <math>m^{e-1} \bmod q = 1</math> respectively.}} but this is very unlikely to occur in practice.

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

<math display="block">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:{{efn|The parameters used here are artificially small, but one can also [[b:Cryptography/Generate a keypair using OpenSSL|OpenSSL can also be used to generate and examine a real keypair]].}}

# Choose two distinct prime numbers, such as
#: <math>p = 61</math> and <math>q = 53</math>.
# Compute {{math|1=''n'' = ''pq''}} giving
#: <math>n = 61\times 53 = 3233.</math>
# Compute the [[Carmichael's totient function]] of the product as {{math|1=''&lambda;''(''n'') = [[least common multiple|lcm]](''p'' − 1, ''q'' − 1)}} giving
#: <math>\lambda(3233) = \operatorname{lcm}(60, 52) = 780.</math>
# Choose any number {{math|2 &lt; ''e'' &lt; 780}} that is [[coprime]] to 780. Choosing a prime number for {{mvar|e}} leaves us only to check that {{mvar|e}} is not a divisor of 780.
#: Let <math>e = 17</math>.
# Compute {{mvar|d}}, the [[modular multiplicative inverse]] of {{math|''e'' (mod ''&lambda;''(''n''))}}, yielding<br /><math display="block">d = 413,</math> as <math>1 = (17 \times 413) \bmod 780.</math>

The '''public key''' is {{math|1=(''n'' = 3233, ''e'' = 17)}}. For a padded [[plaintext]] message {{mvar|m}}, the encryption function is
<math display="block">\begin{align}
 c(m) &= m^{e} \bmod n \\
      &= m^{17} \bmod 3233.
\end{align}</math>

The '''private key''' is {{math|1=(''n'' = 3233, ''d'' = 413)}}. For an encrypted [[ciphertext]] {{mvar|c}}, the decryption function is
<math display="block">\begin{align}
 m(c) &= c^{d} \bmod n \\
      &= c^{413} \bmod 3233.
\end{align}</math>

For instance, in order to encrypt {{math|1=''m'' = 65}}, one calculates
<math display="block">c = 65^{17} \bmod 3233 = 2790.</math>

To decrypt {{math|1=''c'' = 2790}}, one calculates
<math display="block">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 {{math|1=''n'' = 3233}} (obtained from the freely available public key) back to the primes {{mvar|p}} and {{mvar|q}}. {{mvar|e}}, also from the public key, is then inverted to get {{mvar|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 {{mvar|d<sub>''p''</sub>}}, {{mvar|d<sub>''q''</sub>}} and {{mvar|q<sub>inv</sub>}}, which are part of the private key are computed as follows:
<math display="block">\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 {{mvar|d<sub>''p''</sub>}}, {{mvar|d<sub>''q''</sub>}} and {{mvar|q<sub>inv</sub>}} are used for efficient decryption (encryption is efficient by choice of a suitable {{mvar|d}} and {{mvar|e}} pair):
<math display="block">\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>

===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 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 {{mvar|d}} (modulo {{mvar|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 {{mvar|e}} (modulo {{mvar|n}}) (as he does when encrypting a message), and compares the resulting hash value with the message's 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 being sent.

This works because of [[exponentiation]] rules:
<math display="block">h = \operatorname{hash}(m),</math>
<math display="block">(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.

==Proofs of correctness==

===Proof using Fermat's little theorem===
The proof of the correctness of RSA is based on [[Fermat's little theorem]], stating that {{math| ''a''<sup>''p'' − 1</sup> ≡ 1 (mod ''p'')}} for any integer {{mvar|a}} and prime {{mvar|p}}, not dividing {{mvar|a}}.{{refn|group=note|We cannot trivially break RSA by applying the theorem (mod ''pq'') because {{math|''pq''}} is not prime.}}

We want to show that
<math display="block">(m^e)^d \equiv m \pmod{pq}</math>
for every integer {{mvar|m}} when {{mvar|p}} and {{mvar|q}} are distinct prime numbers and {{mvar|e}} and {{mvar|d}} are positive integers satisfying {{math|1=''ed'' ≡ 1 (mod ''&lambda;''(''pq''))}}.

Since {{math|1=''&lambda;''(''pq'') = [[least common multiple|lcm]](''p'' − 1, ''q'' − 1)}} is, by construction, divisible by both {{math|''p'' − 1}} and {{math|''q'' − 1}}, we can write
<math display="block">ed - 1 = h(p - 1) = k(q - 1)</math>
for some nonnegative integers {{mvar|h}} and {{mvar|k}}.{{refn|group=note|In particular, the statement above holds for any {{mvar|e}} and {{mvar|d}} that satisfy {{math|1=''ed'' ≡ 1 (mod (''p'' − 1)(''q'' − 1))}}, since {{math|1=(''p'' − 1)(''q'' − 1)}} is divisible by {{math|1=''&lambda;''(''pq'')}}, and thus trivially also by {{math|''p'' − 1}} and {{math|''q'' − 1}}. However, in modern implementations of RSA, it is common to use a reduced private exponent {{mvar|d}} that only satisfies the weaker, but sufficient condition {{math|1=''ed'' ≡ 1 (mod ''&lambda;''(''pq''))}}.}}

To check whether two numbers, such as {{mvar|m''<sup>ed</sup>''}} and {{mvar|m}}, are congruent {{math|mod&nbsp;''pq''}}, it suffices (and in fact is equivalent) to check that they are congruent {{math|mod&nbsp;''p''}} and {{math|mod&nbsp;''q''}} separately.{{refn|group=note|This is part of the [[Chinese remainder theorem]], although it is not the significant part of that theorem.}}

To show {{math|''m<sup>ed</sup>'' ≡ ''m'' (mod ''p'')}}, we consider two cases:
# If {{math|''m'' ≡ 0 (mod ''p'')}}, {{mvar|m}} is a multiple of {{mvar|p}}. Thus ''m<sup>ed</sup>'' is a multiple of {{mvar|p}}. So {{math|''m<sup>ed</sup>'' ≡ 0 ≡ ''m'' (mod ''p'')}}. 
# If {{math|''m'' <math>\not\equiv</math> 0 (mod ''p'')}}, 
#: <math>m^{ed} = m^{ed - 1} m = m^{h(p - 1)} m = (m^{p - 1})^h m \equiv 1^h m \equiv m \pmod{p},</math>
#: where we used [[Fermat's little theorem]] to replace {{math|''m''<sup>''p''−1</sup> mod ''p''}} with 1.

The verification that {{math|''m<sup>ed</sup>'' ≡ ''m'' (mod ''q'')}} proceeds in a completely analogous way:
# If {{math|''m'' ≡ 0 (mod ''q'')}}, ''m<sup>ed</sup>'' is a multiple of {{mvar|q}}. So {{math|''m<sup>ed</sup>'' ≡ 0 ≡ ''m'' (mod ''q'')}}. 
# If {{math|''m'' <math>\not\equiv</math> 0 (mod ''q'')}},
#: <math>m^{ed} = m^{ed - 1} m = m^{k(q - 1)} m = (m^{q - 1})^k m \equiv 1^k m \equiv m \pmod{q}.</math>

This completes the proof that, for any integer {{mvar|m}}, and integers {{mvar|e}}, {{mvar|d}} such that {{math|1=''ed'' ≡ 1 (mod ''&lambda;''(''pq''))}},
<math display="block">(m^e)^d \equiv m \pmod{pq}.</math>

====Notes====
{{reflist|group=note}}

===Proof using Euler's theorem===
Although the original paper of Rivest, Shamir, and Adleman used Fermat's little theorem to explain why RSA works, it is common to find proofs that rely instead on [[Euler's theorem]].

We want to show that {{math|''m<sup>ed</sup>'' ≡ ''m'' (mod ''n'')}}, where {{math|1=''n'' = ''pq''}} is a product of two different prime numbers, and {{mvar|e}} and {{mvar|d}} are positive integers satisfying {{math|''ed'' ≡ 1 (mod ''&phi;''(''n''))}}. Since {{mvar|e}} and {{mvar|d}} are positive, we can write {{math|1=''ed'' = 1 + ''h&phi;''(''n'')}} for some non-negative integer {{mvar|h}}. ''Assuming'' that {{mvar|m}} is relatively prime to {{mvar|n}}, we have
<math display="block">m^{ed} = m^{1 + h\varphi(n)} = m (m^{\varphi(n)})^h \equiv m (1)^h \equiv m \pmod{n},</math>

where the second-last congruence follows from [[Euler's theorem]].

More generally, for any {{mvar|e}} and {{mvar|d}} satisfying {{math|''ed'' ≡ 1 (mod ''&lambda;''(''n''))}}, the same conclusion follows from [[Carmichael function#Carmichael's theorem|Carmichael's generalization of Euler's theorem]], which states that {{math|''m''<sup>''&lambda;''(n)</sup> ≡ 1 (mod ''n'')}} for all {{mvar|m}} relatively prime to {{mvar|n}}.

When {{mvar|m}} is not relatively prime to {{mvar|n}}, the argument just given is invalid. This is highly improbable (only a proportion of {{math|1/''p'' + 1/''q'' − 1/(''pq'')}} numbers have this property), but even in this case, the desired congruence is still true. Either {{math|''m'' ≡ 0 (mod ''p'')}} or {{math|''m'' ≡ 0 (mod ''q'')}}, and these cases can be treated using the previous proof.

==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., {{math|1=''e'' = 3}}) and small values of the {{mvar|m}} (i.e., {{math|''m'' &lt; ''n''<sup>1/''e''</sup>}}), the result of {{math|''m''<sup>''e''</sup>}} is strictly less than the modulus {{mvar|n}}. In this case, ciphertexts can be decrypted easily by taking the {{mvar|e}}th root of the ciphertext over the integers.
* If the same clear-text message is sent to {{mvar|e}} or more recipients in an encrypted way, and the receivers share the same exponent {{mvar|e}}, but different {{mvar|p}}, {{mvar|q}}, and therefore {{mvar|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 clear texts 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]] (see [[Coppersmith's attack]]).<ref>{{cite journal |first=Don |last=Coppersmith |s2cid=15726802 |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>
* 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 whether 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. RSA without padding is not semantically secure.<ref>{{Cite book|last1=Goldwasser|first1=Shafi|last2=Micali|first2=Silvio|title=Proceedings of the fourteenth annual ACM symposium on Theory of computing - STOC '82 |chapter=Probabilistic encryption & how to play mental poker keeping secret all partial information |date=1982-05-05|chapter-url=https://doi.org/10.1145/800070.802212|location=New York, NY, USA|publisher=Association for Computing Machinery|pages=365–377|doi=10.1145/800070.802212|isbn=978-0-89791-070-5|s2cid=10316867 |author-link1=Shafi Goldwasser |author-link2=Silvio Micali }}</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, {{math|''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 {{math|''c'' ≡ ''m''<sup>''e''</sup> (mod ''n'')}} may ask the holder of the private key {{mvar|d}} to decrypt an unsuspicious-looking ciphertext {{math|''c''′ ≡ ''cr''<sup>''e''</sup> (mod ''n'')}} for some value {{mvar|r}} chosen by the attacker. Because of the multiplicative property, {{mvar|c}}' is the encryption of {{math|''mr'' (mod ''n'')}}. Hence, if the attacker is successful with the attack, they will learn {{math|''mr'' (mod ''n''),}} from which they can derive the message {{mvar|m}} by multiplying {{math|''mr''}} with the modular inverse of {{mvar|r}} modulo {{mvar|n}}.{{citation needed|reason=Someone had to have noticed this and published first, they should be cited|date=February 2015}}
* Given the private exponent {{mvar|d}}, one can efficiently factor the modulus {{math|1=''n'' = ''pq''}}. And given factorization of the modulus {{math|1=''n'' = ''pq''}}, one can obtain any private key ({{mvar|d}}',&nbsp;{{mvar|n}}) generated against a public key ({{mvar|e}}',&nbsp;{{mvar|n}}).<ref name="Boneh99"/>

===Padding schemes===
To avoid these problems, practical RSA implementations typically embed some form of structured, randomized [[Padding (cryptography)|padding]] into the value {{mvar|m}} before encrypting it. This padding ensures that {{mvar|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 {{mvar|m}} with some number of additional bits, the size of the un-padded message {{mvar|M}} must be somewhat smaller. RSA padding schemes must be carefully designed so as to prevent sophisticated attacks that 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.<ref>{{Cite book |last1=Coron |first1=Jean-Sébastien |last2=Joye |first2=Marc |last3=Naccache |first3=David |last4=Paillier |first4=Pascal |title=Advances in Cryptology — EUROCRYPT 2000 |chapter=New Attacks on PKCS#1 v1.5 Encryption |date=2000 |editor-last=Preneel |editor-first=Bart |series=Lecture Notes in Computer Science |volume=1807 |language=en |location=Berlin, Heidelberg |publisher=Springer| pages=369–381 |doi=10.1007/3-540-45539-6_25 |isbn=978-3-540-45539-4 |doi-access=free}}</ref> 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&nbsp;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 USA patents on PSS were granted ({{US patent|6266771}} and {{US patent|7036014}}); 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 (such as [[OpenSSL]], [[Java (programming language)|Java]] and [[.NET Framework|.NET]]) use for decryption and signing the following optimization based on the [[Chinese remainder theorem]].<ref>{{cite web |title=OpenSSL bn_s390x.c |url=https://github.com/openssl/openssl/blob/422a13fb5cd668cdc4c1eebce8accb4d25c3d8eb/crypto/bn/bn_s390x.c#L70 |website=Github |access-date=2 August 2024}}</ref>{{fact|date=December 2023}} The following values are precomputed and stored as part of the private key:
* <math>p</math> and <math>q</math>{{snd}} the primes from the key generation,
* <math>d_P = d \pmod{p - 1},</math>
* <math>d_Q = d \pmod{q - 1},</math>
* <math>q_\text{inv} = q^{-1} \pmod{p}.</math>

These values allow the recipient to compute the exponentiation {{math|1=''m'' = ''c''<sup>''d''</sup> (mod ''pq'')}} more efficiently as follows:
{{indent|5}}<math>m_1 = c^{d_P} \pmod{p}</math>,
{{indent|5}}<math>m_2 = c^{d_Q} \pmod{q}</math>,
{{indent|5}}<math>h = q_\text{inv}(m_1 - m_2) \pmod{p}</math>,{{efn|If <math>m_1 < m_2</math>, then some{{clarify|date=June 2020}} libraries compute {{mvar|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>.}}
{{indent|5}}<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 the 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 [[computational hardness assumption|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 |date=29 March 2013 |pages=167 |publisher=Trafford |isbn=978-1466985742}}</ref>

The [[RSA problem]] is defined as the task of taking {{mvar|e}}th roots modulo a composite {{mvar|n}}: recovering a value {{mvar|m}} such that {{math|''c'' ≡ ''m''<sup>''e''</sup> (mod ''n'')}}, where {{math|(''n'', ''e'')}} is an RSA public key, and {{mvar|c}} is an RSA ciphertext. Currently the most promising approach to solving the RSA problem is to factor the modulus {{mvar|n}}. With the ability to recover prime factors, an attacker can compute the secret exponent {{mvar|d}} from a public key {{math|(''n'', ''e'')}}, then decrypt {{mvar|c}} using the standard procedure. To accomplish this, an attacker factors {{mvar|n}} into {{mvar|p}} and {{mvar|q}}, and computes {{math|lcm(''p'' − 1, ''q'' − 1)}} that allows the determination of {{mvar|d}} from {{mvar|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.

The first RSA-512 factorization in 1999 used hundreds of computers and required the equivalent of 8,400 MIPS years, over an elapsed time of about seven months.<ref>{{Cite web |url=https://www.iacr.org/archive/eurocrypt2000/1807/18070001-new.pdf |title=Factorization of a 512-bit RSA Modulus |date=2000 |first=Arjen |last=Lenstra |collaboration=Group |publisher=Eurocrypt }}</ref> By 2009, Benjamin Moody could factor an 512-bit RSA 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 5&nbsp;gigabytes of disk storage was required and about 2.5&nbsp;gigabytes of RAM for the sieving process.

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 {{mvar|d}} from {{mvar|n}} and {{mvar|e}} is as hard as factoring {{mvar|n}} into {{mvar|p}} and {{mvar|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 as hard as factoring.

{{As of|2020}}, the largest publicly known factored [[RSA number]] had 829&nbsp;bits (250 decimal digits, [[RSA-250]]).<ref>{{cite web |title=Factorization of RSA-250 |date=2020-02-28 |first=Paul |last=Zimmermann |publisher=Cado-nfs-discuss |url=https://lists.gforge.inria.fr/pipermail/cado-nfs-discuss/2020-February/001166.html |access-date=2020-07-12 |archive-date=2020-02-28 |archive-url=https://web.archive.org/web/20200228234716/https://lists.gforge.inria.fr/pipermail/cado-nfs-discuss/2020-February/001166.html |url-status=dead }}</ref> Its factorization, by a state-of-the-art distributed implementation, took about 2,700 CPU-years. In practice, RSA keys are typically 1024 to 4096 bits long. In 2003, [[RSA Security]] estimated that 1024-bit keys were likely to become crackable by 2010.<ref name="twirl">{{cite web |url=http://emc.com/emc-plus/rsa-labs/historical/twirl-and-rsa-key-size.htm|title=TWIRL and RSA Key Size |publisher=[[RSA Security|RSA Laboratories]] |archive-url=https://web.archive.org/web/20170417095741/https://www.emc.com/emc-plus/rsa-labs/historical/twirl-and-rsa-key-size.htm |archive-date=2017-04-17 |url-status=dead |access-date=2017-11-24 |first=Burt |last=Kaliski |date=May 6, 2003 |df=ymd-all}}</ref> As of 2020, it is not known whether such keys can be cracked, but minimum recommendations have moved to at least 2048&nbsp;bits.<ref name="keymanagement">{{cite web |url=http://nvlpubs.nist.gov/nistpubs/SpecialPublications/NIST.SP.800-57Pt3r1.pdf |title=NIST Special Publication 800-57 Part 3 Revision 1: Recommendation for Key Management: Application-Specific Key Management Guidance |date=2015-01-22 |page=12 |access-date=2017-11-24 |publisher=[[National Institute of Standards and Technology]] |doi=10.6028/NIST.SP.800-57pt3r1 |first1=Elaine |last1=Barker |first2=Quynh |last2=Dang}}</ref> It is generally presumed that RSA is secure if {{mvar|n}} is sufficiently large, outside of quantum computing.

If {{mvar|n}} is 300&nbsp;[[bit]]s or shorter, it can be factored in a few hours on a [[personal computer]], using software already freely available. Keys of 512&nbsp;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.<ref name="twirl" />

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 {{mvar|p}} and {{mvar|q}} is usually done by testing random numbers of the correct size with probabilistic [[primality test]]s that quickly eliminate virtually all of the nonprimes.

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

It is important that the private exponent {{mvar|d}} be large enough. Michael J. Wiener showed that if {{mvar|p}} is between {{mvar|q}} and {{math|2''q''}} (which is quite typical) and {{math|''d'' &lt; ''n''<sup>1/4</sup>/3}}, then {{mvar|d}} can be computed efficiently from {{mvar|n}} and&nbsp;{{mvar|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 | s2cid=7120331 |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 {{math|1=''e'' = 3}}, provided that the proper padding is used. [[Coppersmith's attack]] has many applications in attacking RSA specifically if the public exponent {{mvar|e}} is small and if the encrypted message is short and not padded. [[65537]] is a commonly used value for&nbsp;{{mvar|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&nbsp;800-78 Rev.&nbsp;1 of August 2007) does not allow public exponents {{mvar|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]] known as RSALib. A large number of [[smart card]]s and [[trusted platform module]]s (TPM) 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 {{mvar|p}} and {{mvar|q}}. An analysis comparing millions of public keys gathered from the Internet was carried out in early 2012 by [[Arjen Klaas Lenstra|Arjen K.&nbsp;Lenstra]], James P.&nbsp;Hughes, Maxime Augier, Joppe W.&nbsp;Bos, Thorsten Kleinjung and Christophe Wachter. They were able to factor 0.2% of the keys using only Euclid's algorithm.<ref>{{cite web|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 web |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>{{self published inline|date=December 2023|reason=This paper is not from a journal and may not have been peer-reviewed}}

They exploited a weakness unique to cryptosystems based on integer factorization. If {{math|1=''n'' = ''pq''}} is one public key, and {{math|1=''n''′ = ''p''′''q''′}} is another, then if by chance {{math|1=''p'' = ''p''′}} (but {{mvar|q}} is not equal to {{mvar|q}}'), then a simple computation of {{math|1=gcd(''n'', ''n''′) = ''p''}} factors both {{mvar|n}} and {{mvar|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 {{mvar|q}} given {{mvar|p}}, instead of choosing {{mvar|p}} and {{mvar|q}} independently.

[[Nadia Heninger]] was part of a group that did a similar experiment. They used an idea of [[Daniel J.&nbsp;Bernstein]] to compute the GCD of each RSA key {{mvar|n}} against the product of all the other keys {{mvar|n}}' they had found (a 729-million-digit number), instead of computing each {{math|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 more than 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 is 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 Carl 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, Eve can deduce the decryption key {{mvar|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 name="Boneh03">{{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 {{math|''c''<sup>''d''</sup> (mod ''n'')}}, Alice first chooses a secret random value {{mvar|r}} and computes {{math|(''r''<sup>''e''</sup>''c'')<sup>''d''</sup> (mod ''n'')}}. The result of this computation, after applying [[Euler's theorem]], is {{math|''rc''<sup>''d''</sup> (mod ''n'')}}, and so the effect of {{mvar|r}} can be removed by multiplying by its inverse. A new value of {{mvar|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&nbsp;#1&nbsp;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&nbsp;#1 scheme, Bleichenbacher was able to mount a practical attack against RSA implementations of the [[Secure Sockets 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&nbsp;#1 that are not vulnerable to these attacks.

A variant of this attack, dubbed "BERserk", came back in 2014.<ref>{{cite web |title='BERserk' Bug Uncovered In Mozilla NSS Crypto Library Impacts Firefox, Chrome |date=25 September 2014 |url=https://www.darkreading.com/attacks-breaches/-berserk-bug-uncovered-in-mozilla-nss-crypto-library-impacts-firefox-chrome |access-date=4 January 2022}}</ref><ref>{{Cite web|url=https://www.mozilla.org/en-US/security/advisories/mfsa2014-73/|title=RSA Signature Forgery in NSS|website=Mozilla}}</ref> It impacted the Mozilla NSS Crypto Library, which was used notably by Firefox and Chrome.

===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&nbsp;bits of an RSA key in 10 iterations.

A power-fault attack on RSA implementations was described in 2010.<ref>{{cite book |last1=Pellegrini |first1=Andrea |last2=Bertacco |first2=Valeria |last3=Austin |first3=Todd |chapter=Fault-based attack of RSA authentication |title=2010 Design, Automation & Test in Europe Conference & Exhibition (DATE 2010) |date=March 2010 |pages=855–860 |doi=10.1109/DATE.2010.5456933 |isbn=978-3-9810801-6-2 |access-date=21 November 2024 |chapter-url=https://ieeexplore.ieee.org/document/5456933}}</ref> The author recovered the key by varying the CPU power voltage outside limits; this caused multiple power faults on the server.

===Tricky implementation===
There are many details to keep in mind in order to implement RSA securely (strong [[Pseudorandom number generator|PRNG]], acceptable public exponent, etc.). This makes the implementation challenging, to the point the book Practical Cryptography With Go suggests avoiding RSA if possible.<ref>{{cite web |last1=Isom |first1=Kyle |title=Practical Cryptography With Go |url=https://leanpub.com/gocrypto/read#leanpub-auto-rsa |access-date=4 January 2022}}</ref>

== Implementations ==
Some cryptography libraries that provide support for RSA include:

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

==See also==
{{Portal|Mathematics}}
* [[Acoustic cryptanalysis]]
* [[Computational complexity theory]]
* [[Diffie–Hellman key exchange]]
* [[Digital Signature Algorithm]]
* [[Elliptic-curve cryptography]]
* [[Key exchange]]
* [[Key management]]
* [[Key size]]
* [[Public-key cryptography]]
* [[Rabin cryptosystem]]
* [[Trapdoor function]]

==Notes==
{{notelist}}

==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. |author-link1=Thomas H. Cormen
 |last2=Leiserson |first2=Charles E. |author-link2=Charles E. Leiserson
 |last3=Rivest |first3=Ronald L. |author-link3=Ronald L. Rivest
 |last4=Stein |first4=Clifford |author-link4=Clifford Stein
 | title = Introduction to Algorithms
 | edition =  2nd
 | publisher =  MIT Press and McGraw-Hill | year = 2001
 | isbn = 978-0-262-03293-3 | pages = [https://archive.org/details/introductiontoal00corm_691/page/n903 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}}'''.
* [https://datatracker.ietf.org/doc/html/rfc8017 RFC 8017: PKCS #1: RSA Cryptography Specifications Version 2.2]
* {{YouTube|vgTtHV04xRI|Explanation of RSA using colored lamps}}
* [https://www.di-mgt.com.au/rsa_alg.html Thorough walk through of RSA]
* [https://www.muppetlabs.com/~breadbox/txt/rsa.html Prime Number Hide-And-Seek: How the RSA Cipher Works]
* [https://eprint.iacr.org/2006/351 Onur Aciicmez, Cetin Kaya Koc, Jean-Pierre Seifert: ''On the Power of Simple Branch Prediction Analysis'']

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