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You could also send an email to the <a href="https://meta.wikimedia.org/wiki/Special:MyLanguage/Stewards" class="extiw" title="m:Special:MyLanguage/Stewards">stewards</a> <a href="https://meta.wikimedia.org/wiki/Special:MyLanguage/VRT" class="extiw" title="m:Special:MyLanguage/VRT">VRT</a> queue at <kbd>stewards@wikimedia.org</kbd> including all above details. </p> </div> <p>Other useful links: <a href="https://meta.wikimedia.org/wiki/Global_blocks" class="extiw" title="m:Global blocks">Global blocks</a> · <a href="/wiki/Help:I_have_been_blocked" title="Help:I have been blocked">Help:I have been blocked</a> </p> </div></li></ul><hr /> <div id="viewsourcetext">You can view and copy the source of this page:</div><textarea readonly="" accesskey="," id="wpTextbox1" cols="80" rows="25" style="" class="mw-editfont-monospace" lang="en" dir="ltr" name="wpTextbox1">===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"/> </textarea><div class="templatesUsed"></div><p id="mw-returnto">Return to <a href="/wiki/RSA_(cryptosystem)" title="RSA (cryptosystem)">RSA (cryptosystem)</a>.</p> <!--esi <esi:include src="/esitest-fa8a495983347898/content" /> --><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/wiki/RSA_(cryptosystem)">https://en.wikipedia.org/wiki/RSA_(cryptosystem)</a>"</div></div> <div id="catlinks" class="catlinks catlinks-allhidden" data-mw="interface"></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> <li id="footer-places-contact"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-mobileview"><a href="//en.m.wikipedia.org/w/index.php?title=RSA_(cryptosystem)&action=edit&section=15&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-5c59558b9d-7jthg","wgBackendResponseTime":257,"wgPageParseReport":{"limitreport":{"cputime":"0.062","walltime":"0.090","ppvisitednodes":{"value":418,"limit":1000000},"postexpandincludesize":{"value":17740,"limit":2097152},"templateargumentsize":{"value":6556,"limit":2097152},"expansiondepth":{"value":9,"limit":100},"expensivefunctioncount":{"value":0,"limit":500},"unstrip-depth":{"value":0,"limit":20},"unstrip-size":{"value":469,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 59.007 1 -total"," 99.86% 58.925 2 Template:Blocked_text"," 41.69% 24.598 1 Template:Colocationwebhost"," 40.00% 23.605 2 Template:Replace"," 37.87% 22.345 1 Template:Hidden"," 16.11% 9.508 1 Template:Hidden_begin"," 15.56% 9.182 1 Template:Tlx"," 2.34% 1.382 1 MediaWiki:Wikimedia-globalblocking-blockedtext-mistake"," 2.28% 1.343 1 Template:Hidden_end"," 1.97% 1.164 1 MediaWiki:Wikimedia-globalblocking-blockedtext-mistake-email-steward"]},"scribunto":{"limitreport-timeusage":{"value":"0.012","limit":"10.000"},"limitreport-memusage":{"value":1043323,"limit":52428800}},"cachereport":{"origin":"mw-web.codfw.main-5c59558b9d-7jthg","timestamp":"20241129104426","ttl":2592000,"transientcontent":false}}});});</script> </body> </html>