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Four-Dimensional Gallant-Lambert-Vanstone Scalar Multiplication [chapter]

Patrick Longa, Francesco Sica
2012 Lecture Notes in Computer Science  
The GLV method of Gallant, Lambert and Vanstone (CRYPTO 2001) computes any multiple kP of a point P of prime order n lying on an elliptic curve with a low-degree endomorphism Φ (called GLV curve) over  ...  We show in this work how to merge the two approaches in order to get, for twists of any GLV curve over F p 2 , a four-dimensional decomposition together with fast endomorphisms Φ, Ψ over F p 2 acting on  ...  Introduction The Gallant-Lambert-Vanstone (GLV) method is a generic approach to speed up the computation of scalar multiplication on some elliptic curves defined over fields of large prime characteristic  ... 
doi:10.1007/978-3-642-34961-4_43 fatcat:xevrjchidrc63dof2tw4aa4jde

Four-Dimensional Gallant–Lambert–Vanstone Scalar Multiplication

Patrick Longa, Francesco Sica
2013 Journal of Cryptology  
The GLV method of Gallant, Lambert and Vanstone (CRYPTO 2001) computes any multiple kP of a point P of prime order n lying on an elliptic curve with a low-degree endomorphism Φ (called GLV curve) over  ...  We show in this work how to merge the two approaches in order to get, for twists of any GLV curve over F p 2 , a four-dimensional decomposition together with fast endomorphisms Φ, Ψ over F p 2 acting on  ...  Introduction The Gallant-Lambert-Vanstone (GLV) method is a generic approach to speed up the computation of scalar multiplication on some elliptic curves defined over fields of large prime characteristic  ... 
doi:10.1007/s00145-012-9144-3 fatcat:v6z6u3oktvbopnaql2va65f6ky

Four-Dimensional Gallant-Lambert-Vanstone Scalar Multiplication [article]

Peter Birkner, Patrick Longa, Francesco Sica
2011 arXiv   pre-print
The GLV method of Gallant, Lambert and Vanstone (CRYPTO 2001) computes any multiple $kP$ of a point $P$ of prime order $n$ lying on an elliptic curve with a low-degree endomorphism $\Phi$ (called GLV curve  ...  We show in this work how to merge the two approaches in order to get, for twists of any GLV curve over $\mathbb{F}_{p^2}$, a four-dimensional decomposition together with fast endomorphisms $\Phi, \Psi$  ...  We have shown how to generalize the Gallant-Lambert-Vanstone scalar multiplication method by combining it with the Galbraith-Lin-Scott ideas, to perform a proven almost fourfold speedup on GLV curves over  ... 
arXiv:1106.5149v4 fatcat:uekdpnpl5ngdhmd5dr7msri7ti

Easy scalar decompositions for efficient scalar multiplication on elliptic curves and genus 2 Jacobians [article]

Benjamin Smith
2013 arXiv   pre-print
The first step in elliptic curve scalar multiplication algorithms based on scalar decompositions using efficient endomorphisms-including Gallant-Lambert-Vanstone (GLV) and Galbraith-Lin-Scott (GLS) multiplication  ...  The shorter the basis vectors, the shorter the decomposed scalar coefficients, and the faster the resulting scalar multiplication.  ...  A spectacular (and easy) example of this phenomemon is scalar multiplication with endomorphism decompositions, originally proposed by Gallant, Lambert, and Vanstone [10] .  ... 
arXiv:1310.5250v1 fatcat:rbtyzwrvgncbdaxdlpjeopsvaq

Analysis of the Gallant-Lambert-Vanstone Method Based on Efficient Endomorphisms: Elliptic and Hyperelliptic Curves [chapter]

Francesco Sica, Mathieu Ciet, Jean-Jacques Quisquater
2003 Lecture Notes in Computer Science  
In this work we analyse the GLV method of Gallant, Lambert and Vanstone (CRYPTO 2001) which uses a fast endomorphism Φ with minimal polynomial X 2 + rX + s to compute any multiple kP of a point P of order  ...  The first one is a straightforward generalisation of the Gallant-Lambert-Vanstone arguments, which involve only lattice theory, to a higher dimensional setting (namely d ≤ 2g instead of 2 in the case of  ...  Gallant, Lambert and Vanstone then claim without proof that in fact M ≤ k √ n, for some constant k. 1 We overcome this omission in the next section.  ... 
doi:10.1007/3-540-36492-7_3 fatcat:e3u6kspg5zechpqr333nnwobby

Fast ECDH Key Exchange Using Twisted Edwards Curves with an Efficiently Computable Endomorphism

Johann Grobschadl, Zhe Liu, Zhi Hu, Chunhua Su, Lu Zhou
2019 2019 International Workshop on Secure Internet of Things (SIOT)  
Our software uses a special class of elliptic curves, namely twisted Edwards curves with an efficiently computable endomorphism similar to that of the socalled Gallant-Lambert-Vanstone (GLV) curves.  ...  A variable-base scalar multiplication on curves over the 159 and 207-bit field takes about 2.63 and 4.84 million clock cycles, respectively, on an MSP430F1611 processor.  ...  Gallant-Lambert-Vanstone (GLV) Curves At Crypto 2001, Gallant et al [13] presented an ingenious method to accelerate scalar multiplication on elliptic curves equipped with an endomorphism φ whose characteristic  ... 
doi:10.1109/siot48044.2019.9637091 fatcat:r33vqmchc5e2nb6mg7st2tehgu

Implementing the 4-dimensional GLV method on GLS elliptic curves with j-invariant 0

Zhi Hu, Patrick Longa, Maozhi Xu
2011 Designs, Codes and Cryptography  
The Gallant-Lambert-Vanstone (GLV) method is a very efcient technique for accelerating point multiplication on elliptic curves with eciently computable endomorphisms. Galbraith, Lin and Scott (J.  ...  We show how to get the 4-dimensional GLV decomposition with proper decomposed coecients, and thus reduce the number of doublings for point multiplication on these curves to only a quarter.  ...  Introduction The fundamental operation in elliptic curve cryptography is point multiplication. In 2001, Gallant, Lambert, and Vanstone [10] described a new method (a.k.a.  ... 
doi:10.1007/s10623-011-9558-1 fatcat:mr7vyfxvzfdcnfl6vhunrmj5ju

Efficient four-dimensional GLV curve with high security [article]

Olivier Bernard, Renaud Dubois, Simon Masson
2018 IACR Cryptology ePrint Archive  
We apply Smith's construction [9] to generate four-dimensional GLV curves with fast arithmetic in the group law as well as in the base eld.  ...  As Costello and Longa did in [5] for a 128-bit security level, we obtained an interesting curve for fast GLV scalar multiplication, providing a high level of security (254 bits).  ...  Introduction In , Gallant, Lambert and Vanstone introduce in [6] a new method named GLV 1 , to compute the scalar multiplication on certain elliptic curves.  ... 
dblp:journals/iacr/BernardDM18 fatcat:kycgpb5vbnenhpgdlzx65v53ii

Efficient Implementations of Four-Dimensional GLV-GLS Scalar Multiplication on 8-Bit, 16-Bit, and 32-Bit Microcontrollers

Jihoon Kwon, Seog Seo, Seokhie Hong
2018 Applied Sciences  
In this paper, we present the first constant-time implementations of four-dimensional Gallant-Lambert-Vanstone and Galbraith-Lin-Scott (GLV-GLS) scalar multiplication using curve Ted127-glv4 on 8-bit AVR  ...  In Asiacrypt 2012, Longa and Sica introduced the four-dimensional GLV-GLS scalar multiplication, and they reported the implementation results on Intel processors.  ...  Figure 1 . 1 The implementation hierarchy of four-dimensional Gallant-Lambert-Vanstone and Galbraith-Lin-Scott (GLV-GLS) scalar multiplication.  ... 
doi:10.3390/app8060900 fatcat:46julxtp4re2hk2potlxkgbahy

Four-Dimensional GLV via the Weil Restriction [chapter]

Aurore Guillevic, Sorina Ionica
2013 Lecture Notes in Computer Science  
The Gallant-Lambert-Vanstone (GLV) algorithm uses efficiently computable endomorphisms to accelerate the computation of scalar multiplication of points on an abelian variety.  ...  This leads to a four-dimensional GLV method on Freeman and Satoh's Jacobians and on two new families of elliptic curves defined over F p 2 .  ...  Secondly, in 2001, Gallant, Lambert and Vanstone [11] introduced a method which uses endomorphisms on the elliptic curve to decompose the scalar multiplication in a 2-dimensional multi-multiplication  ... 
doi:10.1007/978-3-642-42033-7_5 fatcat:jwogx3nulrdllclugfaizkejxa

Four $$\mathbb {Q}$$ : Four-Dimensional Decompositions on a $$\mathbb {Q}$$ -curve over the Mersenne Prime [chapter]

Craig Costello, Patrick Longa
2015 Lecture Notes in Computer Science  
At the highest arithmetic level, cryptographic scalar multiplications on FourQ can use a four-dimensional Gallant-Lambert-Vanstone decomposition to minimize the total number of elliptic curve group operations  ...  We show that this powerful combination facilitates scalar multiplications that are significantly faster than all prior works.  ...  It uses the endomorphisms ψ and φ to accelerate scalar multiplications via four-dimensional Gallant-Lambert-Vanstone (GLV)-style [22] decompositions.  ... 
doi:10.1007/978-3-662-48797-6_10 fatcat:klbaptuq3rhfrmd6ymc424tgmq

Efficient and Secure Algorithms for GLV-Based Scalar Multiplication and Their Implementation on GLV-GLS Curves [chapter]

Armando Faz-Hernández, Patrick Longa, Ana H. Sánchez
2014 Lecture Notes in Computer Science  
We propose efficient algorithms and formulas that improve the performance of sidechannel protected scalar multiplication exploiting the Gallant-Lambert-Vanstone (CRYPTO 2001) and Galbraith-Lin-Scott (EUROCRYPT  ...  Finally, we showcase the efficiency of the proposed techniques by implementing a state-of-the-art GLV-GLS curve in twisted Edwards form defined over F p 2 , which supports a four dimensional decomposition  ...  The Gallant-Lambert-Vanstone (GLV) method computes the scalar multiplication kP as k 1 P + k 2 φ(P ) [15] .  ... 
doi:10.1007/978-3-319-04852-9_1 fatcat:mvber7wazrcihetm33kw6jrowy

Lambda Coordinates for Binary Elliptic Curves [chapter]

Thomaz Oliveira, Julio López, Diego F. Aranha, Francisco Rodríguez-Henríquez
2013 Lecture Notes in Computer Science  
a protected single-core scalar multiplication in 114,800 cycles.  ...  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.  ...  This record was recently broken in [34, 35, 12] , where the authors merged the GLS (Galbraith, Lin and Scott) and GLV (Gallant, Lambert and Vanstone) methods to achieve a 4-dimensional decomposition of  ... 
doi:10.1007/978-3-642-40349-1_18 fatcat:pcbkh2a4brd7deerbwlsvya4ve

Families of fast elliptic curves from Q-curves [article]

Benjamin Smith
2013 arXiv   pre-print
We construct new families of elliptic curves over _p^2 with efficiently computable endomorphisms, which can be used to accelerate elliptic curve-based cryptosystems in the same way as Gallant-Lambert-Vanstone  ...  Our construction is based on reducing -curves-curves over quadratic number fields without complex multiplication, but with isogenies to their Galois conjugates-modulo inert primes.  ...  The classic Gallant-Lambert-Vanstone (GLV) construction [13] .  ... 
arXiv:1305.5400v1 fatcat:xmruklhivbdppfn22ocwm4nwsm

Families of Fast Elliptic Curves from ℚ-curves [chapter]

Benjamin Smith
2013 Lecture Notes in Computer Science  
We construct new families of elliptic curves over F p 2 with efficiently computable endomorphisms, which can be used to accelerate elliptic curvebased cryptosystems in the same way as Gallant-Lambert-Vanstone  ...  Our construction is based on reducing Q-curves-curves over quadratic number fields without complex multiplication, but with isogenies to their Galois conjugates-modulo inert primes.  ...  The classic Gallant-Lambert-Vanstone (GLV) construction [13] .  ... 
doi:10.1007/978-3-642-42033-7_4 fatcat:7crodbpsjfej3ej2vjhtwwg4ei
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