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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  
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.  ...  We present a careful implementation of the base and quadratic field arithmetic required for computing the scalar multiplication operation in such curves.  ...  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

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.  ...  We have completed a preliminary base field and base point generating experiment for Koblitz ECs, which generated secure Koblitz ECs over 2 163 , 2 2000 , and obtained the same 5 Koblitz ECs recommended  ... 
doi:10.1007/s11432-012-4552-4 fatcat:qg55nmpnm5fjtf65rybqoefp3a

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  
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

Subject Index

2005 Journal of Discrete Algorithms  
Subject Index (1 + 1) EA On the analysis of a simple evolutionary algorithm on quadratic pseudo-boolean functions, 61 Window nonadjacent expansionEfficient algorithms for Koblitz curves over fields of  ...  characteristic three, 113 Algorithm Efficient algorithms for Koblitz curves over fields of characteristic three, 113; Decomposition al- gorithms for the tree edit distance problem, 448 Algorithm of  ... 
doi:10.1016/s1570-8667(05)00042-0 fatcat:phytlwupp5hmtddg3ffue5546u

Almost Primality of Group Orders of Elliptic Curves Defined over Small Finite Fields

Neal Koblitz
2001 Experimental Mathematics  
Curves defined over small fields often have special efficiency advantages.  ...  INTRODUCTION References Let E be an elliptic curve defined over the finite field ¥ q of q elements.  ... 
doi:10.1080/10586458.2001.10504675 fatcat:5hk2exwkmzepbjwttwubvefx4m

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  ...  A cryptographic pairing evaluates as an element of a finite extension field, and the evaluation itself involves a considerable amount of extension field arithmetic.  ...  Here we focus on the case of non-supersingular elliptic curves over prime fields, that is, q = p. In practice it is common to choose P as a point on the elliptic curve over the base field, E(F p ).  ... 
doi:10.1007/978-3-642-13797-6_13 fatcat:3xoftcdjdzdf5ajyjucpspzmaq

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  
form (2-NAF) halve-and-add algorithm and the wNAF method for Koblitz curves.  ...  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.  ...  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

Faster Batch Verification of Standard ECDSA Signatures Using Summation Polynomials [chapter]

Sabyasachi Karati, Abhijit Das
2014 Lecture Notes in Computer Science  
Our experiments on NIST prime and Koblitz curves demonstrate that our proposed algorithm increases the optimal batch size from seven to nine.  ...  We also mention how our algorithm can be adapted to Edwards curves.  ...  The security of Algorithm SP depends on the structures of the elliptic-curve groups over quadratic extensions of the base fields.  ... 
doi:10.1007/978-3-319-07536-5_26 fatcat:p36k6xt3zfbern2gugoyan7ugq

Page 981 of Mathematical Reviews Vol. , Issue 82c [page]

1982 Mathematical Reviews  
Morgantini (Padua) Szczepanik, Lucyna 82c:12018 Quadratic forms over quadratic extensions of some fields. (Polish summary) Uniw. Slaski w Katowicach Prace Nauk.-Prace Mat. No. \0 (1980), 23-30.  ...  In the third chapter the author first treats Gauss and Jacobi sums, L-series for algebraic varieties over finite fields, and Fermat and Artin-Schreier curves and their cohomology.  ... 

Halving for the 2-Sylow subgroup of genus 2 curves over binary fields

J.M. Miret, R. Moreno, J. Pujolàs, A. Rio
2009 Finite Fields and Their Applications  
We give a deterministic polynomial time algorithm to find the structure of the 2-Sylow subgroup of the Jacobian of a genus 2 curve over a finite field of characteristic 2.  ...  The stopping condition is triggered when certain polynomials fail to have roots in the base field, as previously shown by I. Kitamura, M. Katagi and T. Takagi.  ...  This implies that the rank of Jac [2] is 1 over the base field -and clearly 2 over a quadratic extension.  ... 
doi:10.1016/j.ffa.2009.05.007 fatcat:32bmymkvufgxrcvnexou4dod6a

Page 2494 of Mathematical Reviews Vol. , Issue 89E [page]

1989 Mathematical Reviews  
Summary: “In this paper, it is proved that the zeta functions of the real quadratic fields with disciminant d = n?  ...  Sen obtained formulas for the P”th norm residue symbol over an arbitrary p-adic field containing the P"th roots of unity, where P is an odd prime.  ... 

Defeating Simple Power Analysis on Koblitz Curves

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

A secure approach for embedding message text on an elliptic curve defined over prime fields, and building 'EC-RSA-ELGamal' Cryptographic System [article]

Ahmad Steef, M. N. Shamma, A. Alkhatib
2017 arXiv   pre-print
This paper presents a new probabilistic approach to embedding message text on an elliptic curve, by using the concept of the RSA Algorithm and its security, and such approach allows us discovering the  ...  Elliptic Curve over prime fields According to [3, 4] , EC over prime field p p F Z = and the group law are defined as the following: EC over prime field p F is a special kind of algebraic curves given  ...  The references [1,3], presented the most practical approach known for mapping message text onto a point on an elliptic curves which defined over finite fields, and this approach attributed to Koblitz as  ... 
arXiv:1707.04892v1 fatcat:w3vfdsdt4rg65lt7ylfhb2ol4u

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.  ...  Throughout, we avoid the imaginary quadratic field of exceptionally many units: We exclude D = 3.  ... 
doi:10.1007/3-540-44709-1_13 fatcat:ucckqbokzzh2zb2vhls7j6smtm
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