توصيفگر ها :
خم بيضوي , خم بيضوي مرتبه اول , خم وايرشتراس , قانون جمع خم بيضوي , فرمول جمع كامل , نقاط استثنا فرمول جمع , بخشياب
چكيده انگليسي :
Complete addition formulas for prime order elliptic curves RAZIEH NIKBAKHT r nikbakht@math iut ac ir September 19 2020 Master of Science Thesis in Farsi Departement of Mathematical Sciences Isfahan University of Technology Isfahan 84156 8311 IranSupervisor Dr Reza Rezaeian Farashahi farashahi@iut ac irAdvisor Dr Mostafa Einollahzadeh Samadi meinollahzadeh@iut ac ir2000 MSC 94A60 14H52 14H51 14G50 11G05 11T71 68P25Keywords Elliptic curve Prime order elliptic curves Weierstrass curve Elliptic curve addition law Com plete addition formulas Exceptional point DivisorAbstract This M Sc thesis is based on the following papers Complete addition formulas for prime order elliptic RENES J COSTELLO C BATINA L curves In Annual International Conference on the Theory and Applications of Cryptographic Techniques Springer Berlin Heidelberg 2016 403 428 LENSTRA H W BOSMA WIEB Complete system of two addition laws for elliptic curves J Numb Theory53 1995 229 240Elliptic Curves Cryptography ECC is one of the most efficient public key cryptosystem The security of ECC isbased on the difficulty of solving discrete logarithm problem in the group of points on an elliptic curve over a finitefield The main applications of elliptic curves cryptography are in Diffie Hellmann key exchange algorithm ECDH and Elliptic Curve Digital Signature Algorithm ECDSA that are widely used in commercial standards such as IPsecor Transport Layer Security TLS protocols Every elliptic curve can be represented by the Weierstrass equation The set of Weierstrass elliptic curve points underthe chord and tangent process form an Abelian group The addition law is said to be complete if the projectiveaddition formulas compute the sum of every two points on the curve In other words the addition formulas do nothave exceptional pairs of two points where their sum are not well computed To implement secure ECC completeaddition formulas are required to prevent side channel attacks based on exceptional cases In 1995 Bosma and Lenstra showed that every addition law on any elliptic curve has at least one exceptional pair ofinputs over the algebraic closure More precisely they proved that a complete set of addition formulas are given byat least two addition laws Fortunately in ECC elliptic curves over finite fields are used In this case it is possible tohave a complete single addition law that is well defined for all pairs of input points with coordinates on a finite field In 2016 Renes Costello and Batina proved that every odd order elliptic curve in short Weierstrass form defined overa finite field with characteristic bigger than 3 has a complete addition law Their result is applied for all prime orderelliptic curves presented by the National Institute for Standards and Technology NIST and many other standards In this thesis we study optimized addition formulas proposed by Renes et al that are complete on every odd orderelliptic curve given in Weierstrass form defined over a finite field These complete addition laws are interesting forECC first for their efficient and simplified implementation process and second for providing the security against sidechannel exceptional point attacks
