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| In [[number theory]], a [[prime number]] ''p'' is a '''Sophie Germain prime''' if 2''p'' + 1 is also prime. The number 2''p'' + 1 associated with a Sophie Germain prime is called a [[safe prime]]. For example, 29 is a Sophie Germain prime and 2 × 29 + 1 = 59 is its associated safe prime. Sophie Germain primes are named after French mathematician [[Sophie Germain]], who used them in her investigations of [[Fermat's Last Theorem]].<ref>Specifically, Germain proved that the first case of Fermat's Last Theorem, in which the exponent divides one of the bases, is true for every Sophie Germain prime, and she used similar arguments to prove the same for all other primes up to 100. For details see {{citation|title=Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory|volume=50|series=Graduate Texts in Mathematics|first=Harold M.|last=Edwards|authorlink=Harold Edwards (mathematician)|publisher=Springer|year=2000|isbn=9780387950020|pages=61–65}}.</ref> Sophie Germain primes and safe primes have applications in [[public key cryptography]] and [[primality testing]]. It has been conjectured that there are infinitely many Sophie Germain primes, but this remains unproven.
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| ==Individual numbers==
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| The first few Sophie Germain primes are:
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| :[[2 (number)|2]], [[3 (number)|3]], [[5 (number)|5]], [[11 (number)|11]], [[23 (number)|23]], [[29 (number)|29]], [[41 (number)|41]], [[53 (number)|53]], [[83 (number)|83]], [[89 (number)|89]], [[113 (number)|113]], [[131 (number)|131]], [[173 (number)|173]], [[179 (number)|179]], [[191 (number)|191]], [[233 (number)|233]], … {{OEIS2C|id=A005384}}.
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| Two distributed computing projects, [[PrimeGrid]] and [[Twin Prime Search]], include searches for large Sophie Germain primes.
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| The largest known Sophie Germain primes {{as of|2013|8|lc=on}} are:<ref>[http://primes.utm.edu/top20/page.php?id=2 The Top Twenty Sophie Germain Primes] — from the [[Prime Pages]]. Retrieved 5 August 2013.</ref>
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| {| class="wikitable"
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| ! Value !! Number of digits !! Time of discovery !! Discoverer
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| | 18543637900515 × 2<sup>666667</sup>−1 || align="right" | 200701 || April 2012 || Philipp Bliedung in a distributed [[PrimeGrid]] search using the programs TwinGen and [[Lucas–Lehmer–Riesel test|LLR]]<ref>{{cite web|title=PrimeGrid’s Sophie Germain Prime Search|url=http://www.primegrid.com/download/SGS_666667.pdf|publisher=PrimeGrid|format=PDF|accessdate=18 April 2012}}</ref>
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| |183027 × 2<sup>265440</sup>−1 || align="right" | 79911 || March 2010 || Tom Wu using LLR<ref>[http://primes.utm.edu/primes/page.php?id=92222 The Prime Database: 183027*2^265440-1]. From The [[Prime Pages]].</ref>
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| |648621027630345 × 2<sup>253824</sup>−1 and 620366307356565 × 2<sup>253824</sup>−1 || align="right" | 76424 || November 2009 || Zoltán Járai, Gábor Farkas, Tímea Csajbók, János Kasza and Antal Járai<ref>[http://primes.utm.edu/primes/page.php?id=90907 The Prime Database: 648621027630345*2^253824-1].</ref><ref>[http://primes.utm.edu/primes/page.php?id=90711 The Prime Database: 620366307356565*2^253824-1]</ref>
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| |607095 × 2<sup>176311</sup>−1 || align="right" | 53081 || September 2009 || Tom Wu<ref>[http://primes.utm.edu/primes/page.php?id=89999 The Prime Database: 607095*2^176311-1].</ref>
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| |48047305725 × 2<sup>172403</sup>−1 || align="right" | 51910 || January 2007 || David Underbakke using TwinGen and LLR<ref>[http://primes.utm.edu/primes/page.php?id=79261 The Prime Database: 48047305725*2^172403-1].</ref>
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| |137211941292195 × 2<sup>171960</sup>−1 || align="right" | 51780 || May 2006 || Járai et al<ref>[http://primes.utm.edu/primes/page.php?id=77705 The Prime Database: 137211941292195*2^171960-1].</ref>
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| |}
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| ==Infinitude and density==
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| {{unsolved|mathematics|Are there infinitely many Sophie Germain primes?}}
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| It is [[conjecture]]d that there are [[Infinity|infinitely]] many Sophie Germain primes, but this has not been [[mathematical proof|proven]].<ref name="shoup">{{citation|title=A Computational Introduction to Number Theory and Algebra|first=Victor|last=Shoup|authorlink=Victor Shoup|publisher=Cambridge University Press|year=2009|isbn=9780521516440|contribution=5.5.5 Sophie Germain primes|pages=123–124|url=http://books.google.com/books?id=pWFdMf5hb5oC&pg=PA123}}.</ref> Several other famous conjectures in number theory generalize this and the [[twin prime conjecture]]; they include the [[Bunyakovsky conjecture]], [[Schinzel's hypothesis H]], and the [[Bateman–Horn conjecture]].
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| A [[heuristic]] estimate for the [[number]] of Sophie Germain primes less than ''n'' is<ref name="shoup"/>
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| :<math>2C \frac{n}{(\ln n)^2} \approx 1.32032\frac{n}{(\ln n)^2}</math>
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| where
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| :<math>C=\prod_{p>2} \frac{p(p-2)}{(p-1)^2}\approx 0.660161</math> | |
| is the [[Twin prime|twin prime constant]]. For ''n'' = 10<sup>4</sup>, this estimate predicts 156 Sophie Germain primes, which has a 20% error compared to the exact value of 190. For ''n'' = 10<sup>7</sup>, the estimate predicts 50822, which is still 10% off from the exact value of 56032. The form of this estimate is due to [[G. H. Hardy]] and [[J. E. Littlewood]], who applied a similar estimate to [[twin prime]]s.<ref>{{citation|title=Fermat's Last Theorem for Amateurs|first=Paulo|last=Ribenboim|authorlink=Paulo Ribenboim|publisher=Springer|year=1999|isbn=9780387985084|page=141|url=http://books.google.com/books?id=XPrQmE5trIgC&pg=PA141}}.</ref>
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| A sequence {''p'', 2''p'' + 1, 2(2''p'' + 1) + 1, ...} in which all of the numbers are prime is called a [[Cunningham chain]] of the first kind. Every term of such a sequence except the last is a Sophie Germain prime, and every term except the first is a safe prime. Extending the conjecture that there exist infinitely many Sophie Germain primes, it has also been conjectured that arbitrarily long Cunningham chains exist,<ref>{{citation|title=Prime Numbers: The Most Mysterious Figures in Math|first=David|last=Wells|publisher=John Wiley & Sons|year=2011|isbn=9781118045718|page=35|url=http://books.google.com/books?id=1MTcYrbTdsUC&pg=PA35|quote=If the strong prime ''k''-tuples conjecture is true, then Cunningham chains can reach any length.}}</ref> although infinite chains are known to be impossible.<ref>{{citation|last=Löh|first=Günter|title=Long chains of nearly doubled primes|journal=[[Mathematics of Computation]]|year=1989|volume=53|issue=188|pages=751–759|doi=10.1090/S0025-5718-1989-0979939-8|mr=0979939}}.</ref>
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| ==Modular restrictions==
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| If ''p'' is a Sophie Germain prime greater than 3, then ''p'' must be congruent to 2 mod 3. For, if not, it would be congruent to 1 mod 3 and 2''p'' + 1 would be congruent to 3 mod 3, impossible for a prime number.<ref>{{citation|title=An Episodic History of Mathematics: Mathematical Culture Through Problem Solving|publisher=Mathematical Association of America|first=Steven G.|last=Krantz|year=2010|isbn=9780883857663|page=206|url=http://books.google.com/books?id=ulmAH-6IzNoC&pg=PA206}}.</ref> Similar restrictions hold for larger prime moduli, and are the basis for the choice of the "correction factor" 2''C'' in the Hardy–Littlewood estimate on the density of the Sophie Germain primes.<ref name="shoup"/>
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| If a Sophie Germain prime ''p'' is [[Congruence relation|congruent]] to 3 (mod 4), then its matching safe prime 2''p'' + 1 will be a divisor of the [[Mersenne number]] 2<sup>''p''</sup> − 1. Historically, this result of [[Leonhard Euler]] was the first known criterion for a Mersenne number with a prime index to be composite.<ref>{{citation
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| | last = Ribenboim | first = P. | authorlink = Paulo Ribenboim
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| | doi = 10.1007/BF03023623
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| | issue = 2
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| | journal = [[The Mathematical Intelligencer]]
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| | mr = 737682
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| | pages = 28–34
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| | title = 1093
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| | volume = 5
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| | year = 1983}}.</ref> It can be used to generate the largest Mersenne numbers (with prime indices) that are known to be composite.<ref>{{citation|title=Large Sophie Germain primes|first=Harvey|last=Dubner|authorlink=Harvey Dubner|journal=Mathematics of Computation|volume=65|year=1996|pages=393–396|doi=10.1090/S0025-5718-96-00670-9|mr=1320893}}.</ref>
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| ==Applications==
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| ===Cryptography===
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| A prime number ''p'' = 2''q'' + 1 is called a [[safe prime]] if ''q'' is prime. Thus, ''p'' = 2''q'' + 1 is a safe prime if and only if ''q'' is a Sophie Germain prime, so finding safe primes and finding Sophie Germain primes are equivalent in computational difficulty. The notion of a safe prime can be strengthened to a strong prime, for which both ''p'' − 1 and ''p'' + 1 have large prime factors. Safe and strong primes are useful as the factors of secret keys in the [[RSA (cryptosystem)|RSA cryptosystem]], because they prevent the system being broken by certain [[Integer factorization|factorization]] algorithms such as [[Pollard's rho algorithm]] that would apply to secret keys formed from non-strong primes.<ref>{{citation|title=Are 'strong' primes needed for RSA?|first1=Ronald L.|last1=Rivest|first2=Robert D.|last2=Silverman|date=November 22, 1999|url=https://people.csail.mit.edu/rivest/pubs/RS01.version-1999-11-22.pdf}}</ref>
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| Similar issues apply in other cryptosystems as well, including [[Diffie-Hellman key exchange]] and similar systems that depend on the security of the [[discrete log problem]] rather than on integer factorization.<ref>{{citation
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| | last = Cheon | first = Jung Hee
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| | contribution = Security analysis of the strong Diffie–Hellman problem
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| | doi = 10.1007/11761679_1
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| | pages = 1–11
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| | publisher = Springer-Verlag
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| | series = Lecture Notes in Computer Science
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| | title = 24th Annual International Conference on the Theory and Applications of Cryptographic Techniques (EUROCRYPT'06), St. Petersburg, Russia, May 28 – June 1, 2006, Proceedings
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| | volume = 4004
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| | year = 2006}}.</ref> For this reason, key generation protocols for these methods often rely on efficient algorithms for generating strong primes, which in turn rely on the conjecture that these primes have a sufficiently high density.<ref>{{citation
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| | last = Gordon | first = John A.
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| | contribution = Strong primes are easy to find
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| | doi = 10.1007/3-540-39757-4_19
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| | pages = 216–223
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| | publisher = Springer-Verlag
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| | series = Lecture Notes in Computer Science
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| | title = Proceedings of EUROCRYPT 84, A Workshop on the Theory and Application of Cryptographic Techniques, Paris, France, April 9–11, 1984
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| | volume = 209
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| | year = 1985}}.</ref>
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| In [[Sophie Germain Counter Mode]], it was proposed to use the arithmetic in the [[finite field]] of order equal to the Sophie Germain prime 2<sup>128</sup> + 12451, to counter weaknesses in [[Galois/Counter Mode]] using the binary finite field GF(2<sup>128</sup>). However, SGCM has been shown to be vulnerable to many of the same cryptographic attacks as GCM.<ref>{{citation
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| | last1 = Yap | first1 = Wun-She
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| | last2 = Yeo | first2 = Sze Ling
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| | last3 = Heng | first3 = Swee-Huay
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| | last4 = Henricksen | first4 = Matt
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| | doi = 10.1002/sec.798
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| | journal = Security and Communication Networks
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| | title = Security analysis of GCM for communication
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| | year = 2013}}.</ref>
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| ===Primality testing===
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| Sophie Germain primes play an important role in the [[AKS primality test]]: if they exist in the conjectured density, then they can be used as the primes over which the algorithm does its modular arithmetic. This would speed up its running time to O(''n''<sup>6</sup>) (where ''n'' denotes the number of digits of the input number) compared to a version of the algorithm that does not need this assumption and takes time O(''n''<sup>10.5</sup>).<ref>{{citation |first=Manindra |last=Agrawal |first2=Neeraj |last2=Kayal |first3=Nitin |last3=Saxena |url=http://www.cse.iitk.ac.in/users/manindra/algebra/primality_v6.pdf |title=PRIMES is in P |journal=[[Annals of Mathematics]] |volume=160 |year=2004 |issue=2 |pages=781–793 |doi=10.4007/annals.2004.160.781 |jstor=3597229 }}</ref>
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| ===Pseudorandom number generation===
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| Sophie Germain primes may be used in the generation of [[pseudo-random numbers]]. The decimal expansion of 1/''q'' will produce a [[Recurring decimal#Fractions with prime denominators|stream]] of ''q'' − 1 pseudo-random digits, if ''q'' is the safe prime of a Sophie Germain prime ''p'', with ''p'' congruent to 3, 9, or 11 (mod 20).<ref>{{citation
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| | last = Matthews | first = Robert A. J.
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| | issue = 9-10
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| | journal = Bulletin of the Institute of Mathematics and its Applications
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| | mr = 1192408
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| | pages = 147–148
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| | title = Maximally periodic reciprocals
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| | volume = 28
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| | year = 1992}}.</ref> Thus “suitable” prime numbers ''q'' are 7, 23, 47, 59, 167, 179, etc. (corresponding to ''p'' = 3, 11, 23, 29, 83, 89, etc.). The result is a stream of length ''q'' − 1 digits (including leading zeros). So, for example, using ''q'' = 23 generates the pseudo-random digits 0, 4, 3, 4, 7, 8, 2, 6, 0, 8, 6, 9, 5, 6, 5, 2, 1, 7, 3, 9, 1, 3. Note that these digits are not appropriate for cryptographic purposes, as the value of each can be derived from its predecessor in the digit-stream.
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| == In popular culture ==
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| Sophie Germain primes are mentioned in the stage play ''[[Proof (play)|Proof]]''<ref>{{citation|title=Drama in numbers: putting a passion for mathematics on stage|first=Ivars|last=Peterson|authorlink=Ivars Peterson|journal=[[Science News]]|date=Dec 21, 2002|url=http://www.thefreelibrary.com/Drama+in+numbers%3A+putting+a+passion+for+mathematics+on+stage.-a096417274|quote=[Jean E.] Taylor pointed out that the example of a Germain prime given in the preliminary text was missing the term "+ 1." "When I first went to see `Proof' and that moment came up in the play, I was happy to hear the `plus one' clearly spoken," Taylor says.}}</ref> and the [[Proof (2005 film)|subsequent film]].<ref>{{citation|url=http://www.ams.org/notices/200603/rev-ullman.pdf|title=Movie Review: Proof|first=Daniel|last=Ullman|journal=[[Notices of the AMS]]|volume=53|issue=3|year=2006|pages=340–342|quote=There are a couple of breaks from realism in ''Proof'' where characters speak in a way that is for the benefit of the audience rather than the way mathematicians would actually talk among themselves. When Hal remembers what a Germain prime is, he speaks to Catherine in a way that would be
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| patronizing to another mathematician.}}</ref>
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| == References ==
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| {{reflist|30em}}
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| {{Prime number classes}}
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| {{DEFAULTSORT:Sophie Germain Prime}}
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| [[Category:Classes of prime numbers]]
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