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Scalar Multiplication on Koblitz Curves Using the Frobenius Endomorphism and Its Combination with Point Halving: Extensions and Mathematical Analysis

2006
*
Algorithmica
*

In this paper we prove

doi:10.1007/s00453-006-0105-9
fatcat:lnqnthkklnggbns5stz4wgyjay
*the*optimality*and*other properties of*the*τ -adic nonadjacent form: this expansion has been introduced in order to efficiently compute*scalar**multiplications**on**Koblitz**curves*. ... We also refine*and*extend results about double expansions of*scalars*introduced by Avanzi, Ciet*and*Sica in order to further improve*scalar**multiplications*. ...*The*present paper deals*with*properties of integer expansions which are associated to*Koblitz**curves*, as well as*with*techniques*combining**point**halving**and**Frobenius*expansions introduced in [2] . ...##
###
Minimality of the Hamming Weight of the τ-NAF for Koblitz Curves and Improved Combination with Point Halving
[chapter]

2006
*
Lecture Notes in Computer Science
*

At PKC 2004, Avanzi, Ciet,

doi:10.1007/11693383_23
fatcat:qwbemakqyfhxbl7i5sfm4n37ay
*and*Sica*combined**Frobenius*operations*with**one**point**halving*to compute*scalar**multiplications**on**Koblitz**curves**using**on*average 14% less group additions than*with**the*usual ... In order to efficiently perform*scalar**multiplications**on*elliptic*Koblitz**curves*, expansions of*the**scalar*to a complex base associated*with**the**Frobenius**endomorphism*are commonly*used*. ...*The*authors wish to express their gratitude to*the*anonymous reviewers for their remarks*and*suggestions. ...##
###
Fast elliptic scalar multiplication using new double-base chain and point halving

2006
*
Applied Mathematics and Computation
*

Based

doi:10.1016/j.amc.2006.05.111
fatcat:dh33u772jbgghdrbyd3ssuweii
*on**the*double-base chain representation of*scalar**using*powers of 2*and*3, we propose a new representation*with*powers of ½*and*3 instead. ... Thus*the*efficient*point**halving*operation can be incorporated in*the*new double-base chain to achieve fast*scalar**multiplication*. ... Acknowledgement*The*work presented in this paper was fully supported by a grant from*the*Research Grants ...##
###
Fast Point Multiplication Algorithms for Binary Elliptic Curves with and without Precomputation
[chapter]

2014
*
Lecture Notes in Computer Science
*

*Using*a left-to-right double-

*and*-add

*and*a right-to-left

*halve*-

*and*-add Montgomery ladder over a GLS

*curve*, we present some of

*the*fastest timings yet reported in

*the*literature for

*point*

*multiplication*... In addition, we

*combine*these two procedures to compute a multi-core protected

*scalar*

*multiplication*. Furthermore, we designed a novel regular τ -adic

*scalar*expansion for

*Koblitz*

*curves*. ...

*The*new variant presented in this work does not require

*point*doublings, but instead,

*it*

*uses*

*the*efficient

*point*

*halving*operation available

*on*binary elliptic

*curves*. ...

##
###
Analyzing the Galbraith-Lin-Scott Point Multiplication Method for Elliptic Curves over Binary Fields

2009
*
IEEE transactions on computers
*

*The*

*analysis*differs from

*the*q prime case because of several factors, including

*the*availability of

*the*

*point*

*halving*strategy for elliptic

*curves*over binary fields. ...

*point*

*multiplication*method for these

*curves*is significantly faster than

*point*

*multiplication*for general elliptic

*curves*over prime fields. ...

*The*GLV

*and*

*point*

*halving*techniques can be

*combined*,

*with*

*scalar*recoding performed as follows. Assume r ≈ 2 2ℓ

*and*let k ′ = 2 ℓ k mod r. ...

##
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Efficient implementation of elliptic curve cryptography in wireless sensors

2010
*
Advances in Mathematics of Communications
*

Our implementation of field

doi:10.3934/amc.2010.4.169
fatcat:6wdwdw6uujgtlh4t6ksj2oitmm
*multiplication**and*modular reduction algorithms focuses*on**the*reduction of memory accesses*and*appears as*the*fastest result for this platform. ... Finite field arithmetic was implemented in C*and*Assembly*and*elliptic*curve*arithmetic was implemented in*Koblitz**and*generic binary*curves*. ... Acknowledgements We would like to thank*the*referees for their valuable comments*and*suggestions. Diego F. Aranha is supported by FAPESP, grant no. 2007/06950-0. ...##
###
A Survey on Hardware Implementations of Elliptic Curve Cryptosystems
[article]

2017
*
arXiv
*
pre-print

We first discuss different elliptic

arXiv:1710.08336v1
fatcat:g3gpz5lzgvc27fboa5tv4kdhze
*curves*,*point**multiplication*algorithms*and*underling finite field operations over binary fields F2m*and*prime fields Fp which are*used*in*the*literature for hardware ... Therefore, in these categories to have a better presentation*and*comparison,*the*implementations are presented*and*distinguished based*on*type of finite fields. ... In*Koblitz**curves*, PD operation can be replaced efficiently by*Frobenius**endomorphism*[11] . ...##
###
Recent progress on the elliptic curve discrete logarithm problem

2015
*
Designs, Codes and Cryptography
*

*The*elliptic

*curve*discrete logarithm problem (ECDLP) is

*the*following computational problem: Given

*points*P, Q ∈ E(Fq) to find an integer a, if

*it*exists, such that Q = aP . ... We survey recent work

*on*

*the*elliptic

*curve*discrete logarithm problem. In particular we review index calculus algorithms

*using*summation polynomials,

*and*claims about their complexity. ...

*The*second author also thanks Maike Massierer, Pierre-Jean Spaenlehauer

*and*Vanessa Vitse for various discussions

*on*

*the*topic. ...

##
###
Twenty Female Mathematicians
[article]

2021
*
arXiv
*
pre-print

*The*work may be

*useful*as a historical resource, but there is very little biography or history,

*and*

*the*primary focus is

*on*

*the*

*mathematics*. ... In fact,

*the*main

*use*will probably be for a student who is coming to a new area of

*mathematics*for

*the*first time

*and*needs an overview of some of

*the*key results

*and*references viewed through

*the*work ...

*It*is an interesting exercise to consider how

*one*might take two

*points*

*on*an elliptic

*curve*

*and*obtain a new

*point*. This leads to

*the*group law for elliptic

*curves*. ...

##
###
Subgroup Cryptosystems
[chapter]

*
Encyclopedia of Cryptography and Security
*

*The*

*Frobenius*

*endomorphism*in

*the*quadratic

*extension*is

*used*for free inversion. ... A similar effect occurs

*with*

*Koblitz*

*curves*or other elliptic

*curves*where

*the*coefficients that define

*the*

*curve*are chosen from a smaller field than

*the*

*points*

*on*

*the*

*curve*(cf. Chapter 5). ...

##
###
High-speed cryptography and cryptanalysis
[article]

2011

*The*time

*and*effort they spent

*on*teaching, guiding

*and*supporting me is probably best expressed by

*using*

*the*German words for Ph.D. supervisor

*and*saying that they are really a "Doktormutter"

*and*"Doktorvater ... He supervised my Diplomarbeit, taught me a lot about elliptic

*curves*

*and*pairings during our joint time in Aachen

*and*Eindhoven, introduced me to my supervisors Tanja Lange

*and*Daniel J. ... All software described in this thesis is in

*the*public domain.

*It*is available for download at http://cryptojedi.org/users/peter/thesis/. ...

##
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On the Cryptanalysis of Public-Key Cryptography

2011

For instance, for socalled

doi:10.5075/epfl-thesis-5291
fatcat:5uxpphg2r5h2fmszp6xtaq6hqe
*Koblitz**curves*[125] over binary*extension*fields (which are not covered by our definition in Section 2.4),*the**Frobenius*automorphism of*the*finite field can be*used*to define ... A common optimization is to*halve**the*search space by identifying a*point**with**its*inverse [73, 86, 204] . ... B1 GMP-ECM [ Elliptic*Curve*Constant*Scalar**Multiplication*Most of*the*addition/subtraction chain based approaches to compute*the*ECSM*used*in practice*use**the*w-bit windowing technique, for some ...##
###
Parallel cryptanalysis
[article]

2012

Bernstein

doi:10.6100/ir731259
fatcat:vl52gfmjrbfr3myv5xcq4aiu6q
*and*Tanja Lange as well as my supervisors in Taiwan, Chen-Mou Cheng*and*Bo -YinYang, for*the*opportunity to enjoy my PhD studies commuting between*the*Netherlands*and*Taiwan. ... Furthermore, I would like to thank them for their support throughout my studies*and**the*writing of my thesis. ...*The*given elliptic*curve*is*the**Koblitz**curve*E : y 2 + xy = x 3 + 1 over*the*finite field F 2 131 ;*the*two given*points*P*and*Q have order , where is a 129-bit prime. ...