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Software Implementation of Koblitz Curves over Quadratic Fields [chapter]

Thomaz Oliveira, Julio López, Francisco Rodríguez-Henríquez
2016 Lecture Notes in Computer Science  
We present a careful implementation of the base and quadratic field arithmetic required for computing the scalar multiplication operation in such curves.  ...  In this work, we retake an old idea that Koblitz presented in his landmark paper [21] , where he suggested the possibility of defining anomalous elliptic curves over the base field F4.  ...  Conclusion We have presented a comprehensive study of how to implement efficiently Koblitz elliptic curves defined over quaternary fields F 4 m , using vectorized instructions on the Intel micro-architectures  ... 
doi:10.1007/978-3-662-53140-2_13 fatcat:2z2xlykue5hwhjxckcxkdhozjm

A Comparison of Practical Public-Key Cryptosystems based on Integer Factorization and Discrete Logarithms [chapter]

Paul C. van Oorschot
1991 Lecture Notes in Computer Science  
Elliptic curve cryptosystems, introduced by Miller [241 and Koblitz [ 121, have also recently received much attention as cryptographic alternatives.  ...  The security of elliptic curve cryptosystems, which is generally equated to the difficulty of extracting elliptic curve logarithms (the elliptic curve analogue of the discrete logarithm problem), is also  ...  Implementation of elliptic curve cryptosystems over fields of characteristic two has recently been studied by Menezes and Vanstone (221 and Koblitz 1131.  ... 
doi:10.1007/3-540-38424-3_40 fatcat:we4bib4mwzagpjco34alhoqfhi

Fast Point Multiplication Algorithms for Binary Elliptic Curves with and without Precomputation [chapter]

Thomaz Oliveira, Diego F. Aranha, Julio López, Francisco Rodríguez-Henríquez
2014 Lecture Notes in Computer Science  
[14] , constructed efficient endomorphisms for a class of elliptic curves defined over the quadratic field F q 2 , where q is a prime number [13] .  ...  The authors applied that coordinate system into a binary GLS curve that admits a degree-2 endomorphism and a fast field arithmetic associated with the quadratic field extension of the binary field F 2  ...  On the other hand, since the elliptic curve of Eq. (2) is defined over a quadratic extension of the field F q , the generalized Gaudry-Hess-Smart (gGHS) attack [15, 19] to solve the ECDLP on the curve  ... 
doi:10.1007/978-3-319-13051-4_20 fatcat:fha5m43lenduxfs554h3sqy2ca

Software Implementation of Binary Elliptic Curves: Impact of the Carry-Less Multiplier on Scalar Multiplication [chapter]

Jonathan Taverne, Armando Faz-Hernández, Diego F. Aranha, Francisco Rodríguez-Henríquez, Darrel Hankerson, Julio López
2011 Lecture Notes in Computer Science  
algorithms for binary field arithmetic and scalar multiplication over elliptic curves.  ...  These improvements in field arithmetic are complemented by a study on serial and parallel approaches for Koblitz and random curves, where parallelization strategies are implemented and compared.  ...  this project was discussed, planned and a portion of the development phase of this work was done.  ... 
doi:10.1007/978-3-642-23951-9_8 fatcat:fgygdub5t5gkfjqqacobpq7xuu

Design of elliptic curve cryptoprocessors over GF(2^163) using the Gaussian normal basis

Paulo Cesar Realpe, Vladimir Trujillo-Olaya, Jaime Velasco-Medina
2014 Ingeniería e Investigación  
This paper presents an efficient hardware implementation of cryptoprocessors that perform the scalar multiplication kP over a finite field GF(2 163 ) using two digit-level multipliers.  ...  form (2-NAF) halve-and-add algorithm and the wNAF method for Koblitz curves.  ...  Return ( 1 ,  ) Koblitz Curves Koblitz curves, or anomalous binary curves, are elliptic curves defined over GF(2 m ).  ... 
doi:10.15446/ing.investig.v34n2.40542 fatcat:dmv4pxiwgjdkrouy6zmu6j4ihy

Constructing Tower Extensions of Finite Fields for Implementation of Pairing-Based Cryptography [chapter]

Naomi Benger, Michael Scott
2010 Lecture Notes in Computer Science  
We also show that for some families of pairing-friendly elliptic curves defined over Fp there are a large number of instances for which an efficient tower extension F p k is given immediately if the parameter  ...  Here we consider criteria that apply when choosing the best towering construction, and the associated choice of irreducible polynomials for the implementation of pairing-based cryptosystems.  ...  For example, F p 12 could be implemented as a quadratic extension, of a cubic extension, of a very efficiently implemented (and reusable) quadratic extension field F p 2 , as implemented by Devegili et  ... 
doi:10.1007/978-3-642-13797-6_13 fatcat:3xoftcdjdzdf5ajyjucpspzmaq

Short Memory Scalar Multiplication on Koblitz Curves [chapter]

Katsuyuki Okeya, Tsuyoshi Takagi, Camille Vuillaume
2005 Lecture Notes in Computer Science  
In the second setting, well-suited for software implementations, our technique reduces the memory consumption by 70%.  ...  Thus, with much smaller memory usage, the proposed method yields the same efficiency as the fastest scalar multiplication schemes on Koblitz curves.  ...  Koblitz Curves Koblitz curves belong to a special class of elliptic curves defined over binary fields, and additionally offer a very efficient arithmetic, with no significant security flaw compared to  ... 
doi:10.1007/11545262_7 fatcat:d3hmuhhgnjgptdiorrd5phpuwq

Generating Elliptic Curves of Prime Order [chapter]

Erkay Savaş, Thomas A. Schmidt, Çetin K. Koç
2001 Lecture Notes in Computer Science  
A variation of the Complex Multiplication (CM) method for generating elliptic curves of known order over finite fields is proposed.  ...  We give heuristics and timing statistics in the mildly restricted setting of prime curve order. These may be seen to corroborate earlier work of Koblitz in the class number one setting.  ...  Koblitz [8] used the Hardy-Littlewood heuristics to derive conjectures on the number of primes p for which the reduction of an elliptic curve defined over Q is an elliptic curve of prime order.  ... 
doi:10.1007/3-540-44709-1_13 fatcat:ucckqbokzzh2zb2vhls7j6smtm

Defeating Simple Power Analysis on Koblitz Curves

C. VUILLAUME
2006 IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences  
Koblitz curves belong to a special class of binary curves on which the scalar multiplication can be computed very efficiently.  ...  We show that in practical cases, the recoding can be performed from left to right, and can be easily stored or even randomly generated. key words: elliptic curve cryptosystems, Koblitz curves, smartcard  ...  curves defined over prime fields.  ... 
doi:10.1093/ietfec/e89-a.5.1362 fatcat:iigji4eecvbunhgoszzd4tft74

Evolutionary cryptography theory based generating method for a secure Koblitz elliptic curve and its improvement by a hidden Markov models

Chao Wang, HuanGuo Zhang, LiLi Liu
2012 Science China Information Sciences  
The algorithm is tested by selecting a secure Koblitz EC over the field F(2 2000 ), with experiments showing that both the base field and base point of the secure curve generated exceed the parameter range  ...  for Koblitz curves recommended by NIST.  ...  for both kinds of Koblitz ECs over 2 163 , 2 2000 , and find a secure EC base field exceeding 1900 bits (5 h), which is larger than the largest recommended by NIST. • For Koblitz ECs with a = 0, b = 1,  ... 
doi:10.1007/s11432-012-4552-4 fatcat:qg55nmpnm5fjtf65rybqoefp3a

A Review of Techniques for Implementing Elliptic Curve Point Multiplication on Hardware

Arielle Verri Lucca, Guilherme Augusto Mariano Sborz, Valderi Reis Quietinho Leithardt, Marko Beko, Cesar Albenes Zeferino, Wemerson Delcio Parreira
2020 Journal of Sensor and Actuator Networks  
This article presents a systematic review of literature on ECPM implementations on both Field-Programmable Gate Array (FPGA) and Application-Specific Integrated Circuit (ASIC).  ...  Lots of ECPM implementations have been applied on hardware targeting the acceleration of its calculus.  ...  curves over a binary field F 2 m ; and (ii) Koblitz elliptic curves over a binary field F 2 m .  ... 
doi:10.3390/jsan10010003 fatcat:bjzznwtk7zhhvnxcc5yyf27mxq

Faster Implementation of Scalar Multiplication on Koblitz Curves [chapter]

Diego F. Aranha, Armando Faz-Hernández, Julio López, Francisco Rodríguez-Henríquez
2012 Lecture Notes in Computer Science  
We design a state-of-the-art software implementation of field and elliptic curve arithmetic in standard Koblitz curves at the 128-bit security level.  ...  Since Koblitz curves are defined over the binary field F 2 , the Frobenius map and its inverse naturally extend to an automorphism of the curve denoted by τ .  ...  Conclusion In this work, we presented a software implementation of elliptic curve arithmetic in Koblitz curves defined over binary fields.  ... 
doi:10.1007/978-3-642-33481-8_10 fatcat:cxt2n5qpwrcipglfn4nlmqmcwu

Hardware architectures for algebra, cryptology, and number theory

Kris Gaj, Rainer Steinwandt
2011 Integration  
Over the past few years, the interaction of research in computer engineering with research in algebra and number theory has intensified.  ...  Below, we provide a quick bird's-eye view of the research fields of interest to this issue, and show how the five selected papers fit in this view, and enhance and enrich what was accomplished so far.  ...  His architecture takes advantage of multiple optimizations, such as a specific type of elliptic curve (Koblitz curve), fixed field size (GF(2 163 )), fixed polynomial (a NIST-recommended sparse pentanomial  ... 
doi:10.1016/j.vlsi.2011.04.002 fatcat:tbrvsxom65fdrogksrpp6ot6rm

Side-Channel Analysis of Weierstrass and Koblitz Curve ECDSA on Android Smartphones [chapter]

Pierre Belgarric, Pierre-Alain Fouque, Gilles Macario-Rat, Mehdi Tibouchi
2016 Lecture Notes in Computer Science  
We then extend the standard lattice attack on ECDSA over prime fields to binary Koblitz curves. This is the first time that such an attack is described on Koblitz curves.  ...  We show that, for elliptic curves over prime fields, one can recover the secret key very efficiently on smartphones using electromagnetic side-channel and well-known lattice reduction techniques.  ...  Koblitz curves are anomalous binary curves defined over F 2 and considered over the extension field F 2 m .  ... 
doi:10.1007/978-3-319-29485-8_14 fatcat:luuh3557azafvkch2dkhc2p3d4

Lambda Coordinates for Binary Elliptic Curves [chapter]

Thomaz Oliveira, Julio López, Diego F. Aranha, Francisco Rodríguez-Henríquez
2013 Lecture Notes in Computer Science  
We also provide efficient elliptic curve operations based on the new representation and timing results of our software implementation over the field F 2 254 .  ...  As a result, we improve speed records for protected/unprotected single/multi-core software implementations of random-point elliptic curve scalar multiplication at the 128-bit security level.  ...  of efficient software implementations of scalar multiplication in elliptic curves defined over binary fields [41, 5, 4] .  ... 
doi:10.1007/978-3-642-40349-1_18 fatcat:pcbkh2a4brd7deerbwlsvya4ve
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