10084

9314

كارشناسي ارشد

رياضي كاربردي

اصفهان: دانشگاه صنعتي اصفهان، دانشكده علوم رياضي

1393

ده، 100ص.: مصور، جدول، نمودار

ص.ع. به فارسي و انگليسي

رضا رضائيان فراشاهي

غلامرضا اميدي

30/2/94

منصور معتمدي، علي رجالي

1396/09/27

كتابنامه

علوم رياضي

رياضي

ID9314

Factoring Integers by Elliptic Curves Using Edwards Curves Mohsen Keikhaie m keikhaie@math iut ac ir January 18 2015 Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 Iran Supervisor Dr Reza Rezaeian Farashahi farashahi@cc iut ac ir Advisor Dr Gholamreza Omidi romidi@cc iut ac ir 2010 MSC 11Y05 11G05 11G07 20K10 11F06 Keywords Factorization Elliptic curve method Curve selection Edwards coordinates Extended Edwardscoordinates Elliptic curves over local elds Torsion groups Structure of modular groups AbstractFactoring integers is one of the most analysed problems in number theory and cryptology Elliptic Curve Method ECM which is known by Lenstra method for factoring integers is one of the best exsitance method forfactorization problem This method introduced in 1987 by Hendrik Willem Lenstra Jr 45 For this method several forms of elliptic curves have been studied which can be Weierstras curves Suyama curves Montgomerycurves Edwards curves and extension families of Edwards curves The end case is one of the newest of ellipticcurves which is known Using the historical results of Euler and Gauss Edwards introduced a normal form for elliptic curves and statedthe group law in 35 These curves are de ned by the equation x2 y 2 c2 c2 x2 y 2 Edwards curves since tohave low cost for group law and memory arithmetic in cryptographic applications are drawing most attention ofcryptologists to their In recent years there has been a rapid development of Edwards curves and their applicationin elliptic curve cryptology In this article studied family of Edwards curves and these applications ECM ability to factoring the random integers that interest to number theorists this method is not as fast astrial division and Pollard s rho method for nding small prime divisers but it is the method of random choiceelliptic curves for nding medium size prime divisers ECM also ability to factoring the hard integers thatinterest to cryptologists those integers are factored by sieving methods which use ECM to nd medium sizeprime divisers of auxiliary integers ECM ability to nd large prime divisers the best record is in 2013 that nds a number 274 bit of the number 947 bit 7337 1 More data for all recordes of elliptic curve factorizationmethod can be found in http www loria fr zimmerma records ecmnet html Many analysis and studies have been done to develop this method particularly use of the Q torsion points and

رضا رضائيان فراشاهي

غلامرضا اميدي

منصور معتمدي، علي رجالي