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Software Implementation of Koblitz Curves over Quadratic Fields
[chapter]

2016
*
Lecture Notes in Computer Science
*

We present a careful

doi:10.1007/978-3-662-53140-2_13
fatcat:2z2xlykue5hwhjxckcxkdhozjm
*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 ...##
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A Comparison of Practical Public-Key Cryptosystems based on Integer Factorization and Discrete Logarithms
[chapter]

1991
*
Lecture Notes in Computer Science
*

Elliptic

doi:10.1007/3-540-38424-3_40
fatcat:we4bib4mwzagpjco34alhoqfhi
*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. ...##
###
Fast Point Multiplication Algorithms for Binary Elliptic Curves with and without Precomputation
[chapter]

2014
*
Lecture Notes in Computer Science
*

[14] , constructed efficient endomorphisms for a class

doi:10.1007/978-3-319-13051-4_20
fatcat:fha5m43lenduxfs554h3sqy2ca
*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*...##
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Software Implementation of Binary Elliptic Curves: Impact of the Carry-Less Multiplier on Scalar Multiplication
[chapter]

2011
*
Lecture Notes in Computer Science
*

algorithms for binary

doi:10.1007/978-3-642-23951-9_8
fatcat:fgygdub5t5gkfjqqacobpq7xuu
*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. ...##
###
Design of elliptic curve cryptoprocessors over GF(2^163) using the Gaussian normal basis

2014
*
Ingeniería e Investigación
*

This paper presents an efficient hardware

doi:10.15446/ing.investig.v34n2.40542
fatcat:dmv4pxiwgjdkrouy6zmu6j4ihy
*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 wNAF method for*Koblitz**curves*. ... Return ( 1 , )*Koblitz**Curves**Koblitz**curves*, or anomalous binary*curves*, are elliptic*curves*defined*over*GF(2 m ). ...##
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Constructing Tower Extensions of Finite Fields for Implementation of Pairing-Based Cryptography
[chapter]

2010
*
Lecture Notes in Computer Science
*

We also show that for some families

doi:10.1007/978-3-642-13797-6_13
fatcat:3xoftcdjdzdf5ajyjucpspzmaq
*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 ...##
###
Short Memory Scalar Multiplication on Koblitz Curves
[chapter]

2005
*
Lecture Notes in Computer Science
*

In the second setting, well-suited for

doi:10.1007/11545262_7
fatcat:d3hmuhhgnjgptdiorrd5phpuwq
*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 ...##
###
Generating Elliptic Curves of Prime Order
[chapter]

2001
*
Lecture Notes in Computer Science
*

A variation

doi:10.1007/3-540-44709-1_13
fatcat:ucckqbokzzh2zb2vhls7j6smtm
*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. ...##
###
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*. ...

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

2012
*
Science China Information Sciences
*

The algorithm is tested by selecting a secure

doi:10.1007/s11432-012-4552-4
fatcat:qg55nmpnm5fjtf65rybqoefp3a
*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, ...##
###
A Review of Techniques for Implementing Elliptic Curve Point Multiplication on Hardware

2020
*
Journal of Sensor and Actuator Networks
*

This article presents a systematic review

doi:10.3390/jsan10010003
fatcat:bjzznwtk7zhhvnxcc5yyf27mxq
*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 . ...##
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Faster Implementation of Scalar Multiplication on Koblitz Curves
[chapter]

2012
*
Lecture Notes in Computer Science
*

We design a state-

doi:10.1007/978-3-642-33481-8_10
fatcat:cxt2n5qpwrcipglfn4nlmqmcwu
*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*. ...##
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Hardware architectures for algebra, cryptology, and number theory

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 ...

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

2016
*
Lecture Notes in Computer Science
*

We then extend the standard lattice attack on ECDSA

doi:10.1007/978-3-319-29485-8_14
fatcat:luuh3557azafvkch2dkhc2p3d4
*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 . ...##
###
Lambda Coordinates for Binary Elliptic Curves
[chapter]

2013
*
Lecture Notes in Computer Science
*

We also provide efficient elliptic

doi:10.1007/978-3-642-40349-1_18
fatcat:pcbkh2a4brd7deerbwlsvya4ve
*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] . ...
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