Editing Random number generator attack (section)

===RSA public key factoring===
An analysis comparing millions of [[RSA (algorithm)|RSA]] public keys gathered from the Internet was announced in 2012 by Lenstra, Hughes, Augier, Bos, Kleinjung, and Wachter. They were able to factor 0.2% of the keys using only [[Euclid's algorithm]].<ref>
{{cite news
|last=Markoff
|first=John
|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
|authorlink=John Markoff}}</ref><ref>
{{cite journal
|last=Lenstra
|first=Arjen
|author2=Hughes, James P. |author3=Augier, Maxime |author4=Bos, Joppe Willem |author5=Kleinjung, Thorsten |author6= Wachter, Christophe 
|title=Ron was wrong, Whit is right
|year=2012
|page=17
|url=http://eprint.iacr.org/2012/064.pdf
|location=Santa Barbara: IACR}}</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''′}}, then a simple computation of {{nowrap|1=gcd(''n'',''n''′) = ''p''}} factors both ''n'' and ''n''′, totally compromising both keys. [[Nadia Heninger]], part of a group that did a similar experiment, said that the bad keys occurred almost entirely in [[embedded system|embedded application]]s, and 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.<ref>{{cite web |last1=Heninger |first1=Nadia |title=New research: There's no need to panic over factorable keys–just mind your Ps and Qs |url=https://freedom-to-tinker.com/2012/02/15/new-research-theres-no-need-panic-over-factorable-keys-just-mind-your-ps-and-qs/ |website=Freedom to Tinker |date=15 February 2012 |access-date=27 November 2020 |archive-url=https://web.archive.org/web/20161224112657/https://freedom-to-tinker.com/2012/02/15/new-research-theres-no-need-panic-over-factorable-keys-just-mind-your-ps-and-qs/ |archive-date=2016-12-24}}</ref>