Three Dimensional Montgomery Ladder, Differential Point Tripling on Montgomery Curves and Point Quintupling on Weierstrass’ and Edwards Curves

Rao, Srinivasa Rao Subramanya

doi:10.1007/978-3-319-31517-1_5

Cited by 10 publications

(3 citation statements)

References 22 publications

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“…In [28], Subramanya Rao worked on Montgomery curves and found an efficient technique to find point tripling. Simply, we optimize an application of a single double to some point P, then perform a point addition.…”

Section: Intermediate Operationsmentioning

confidence: 99%

Efficient Elliptic Curve Operators for Jacobian Coordinates

2022

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The speed up of group operations on elliptic curves is proposed using a new type of projective coordinate representation. These operations are the most common computations in key exchange and encryption for both current and postquantum technology. The boost this improvement brings to computational efficiency impacts not only encryption efforts but also attacks. For maintaining security, the community needs to take note of this development as it may need to operate changes in the key size of various algorithms. Our proposed projective representation can be viewed as a warp on the Jacobian projective coordinates, or as a new operation replacing the addition in a Jacobian projective representation, basically yielding a new group with the same algebra elements and homomorphic to it. Efficient algorithms are introduced for computing the expression Pk+Q where P and Q are points on the curve and k is an integer. They exploit optimized versions for particular k values. Measurements of the numbers of basic computer instructions needed for operations based on the new representation show clear improvements. The experiments are based on benchmarks selected using standard NIST elliptic curves.

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Section: Intermediate Operationsmentioning

confidence: 99%

Efficient Elliptic Curve Operators for Jacobian Coordinates

2022

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“…At the first CHES workshop in 1999, López and Dahab [58] presented a Montgomery ladder based ECSM algorithm for general ECC over binary fields, which is faster than the binary NAF method on average and resists against SPA which started to be extensively explored about that time. The Montgomery ladder as a regular chain attracted ECC researchers [49,65,4,13,68]. In 2002, Montgomery ladder based ECSM algorithms for general elliptic curves over large characteristic fields have been appeared in the literatures [12,45,27,44].…”

Section: Introductionmentioning

confidence: 99%

Speeding up regular elliptic curve scalar multiplication without precomputation

Kim¹,

Choe²,

Kim³

et al. 2020

Advances in Mathematics of Communications

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This paper presents a series of Montgomery scalar multiplication algorithms on general short Weierstrass curves over fields with characteristic greater than 3, which need only 12 field multiplications per scalar bit using 8 ∼ 9 field registers, thus outperform the binary NAF method on average. Over binary fields, the Montgomery scalar multiplication algorithm which was presented at the first CHES workshop by López and Dahab has been a favorite of ECC implementors, due to its nice properties such as high efficiency (outperforming the binary NAF), natural SPA-resistance, generality (coping with all ordinary curves) and implementation easiness. Over odd characteristic fields, the new scalar multiplication algorithms are the first ones featuring all these properties. Building-blocks of our contribution are new efficient differential addition-and-doubling formulae and a novel conception of on-the-fly adaptive coordinates which varies in accordance with not only the base point but also the bits of the given scalar.

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“…As a starting point, note that calculating x(3P ) using a point doubling followed by a differential point addition takes 7M+4S+8A field operations. Subramanya Rao [258] showed an efficient formula for point tripling. Let A be the Montgomery curve parameter and given x(P ) = (X P : Z P ), such a formula calculates x(3P ) = (X 3P : Z 3P ) as…”

Section: An Optimized Point Tripling Formulamentioning

confidence: 99%

High-performance elliptic curve cryptography

Faz Hernández

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ResumoA criptografia baseada em curvas elípticas fornece métodos eficientes para a criptografia de chave pública. Pesquisa recente tem mostrado a superioridade das curvas de Montgomery e de Edwards sobre as curvas de Weierstrass pois elas precisam de menos operações aritméticas. O uso destas curvas modernas, porém, traz consigo diversos desafios na construção de algoritmos criptográficos deixando em aberto novos alvos de otimizações.Nosso objetivo principal é propor otimizações algorítmicas e técnicas de implementação para os algoritmos criptográficos baseados em curvas elípticas. Visando acelerar a execução destes algoritmos, nossa abordagem fundamenta-se na utilização extensões ao conjunto de instruções da arquitetura. Além daquelas específicas para a criptografia, nós usamos extensões que seguem o paradigma de cômputo paralelo SIMD (do inglês, Single Instruction, Multiple Data). Neste modelo, o processador executa a mesma operação sobre um conjunto de dados de forma paralela. Nós investigamos como aplicar o modelo SIMD na implementação de algoritmos de curvas elípticas.Como parte de nossas contribuições, projetamos algoritmos paralelos para a aritmética de corpos primos e de curvas elípticas. Projetamos também um algoritmo de multiplicação escalar para calcular P + kQ e uma fórmula otimizada para calcular 3P em curvas de Montgomery. Estes algoritmos encontraram aplicabilidade na criptografia baseada em isogenias. Usando extensões SIMD tais como SSE, AVX e AVX2, desenvolvemos implementações otimizadas dos seguintes algoritmos criptográficos: X25519, X448, SIDH, ECDH, ECDSA, EdDSA e qDSA. Testes de desempenho mostram que essas implementações são mais rápidas do que as implementações existentes no estado da arte.Nosso estudo confirma que o uso de extensões ao conjunto de instruções da arquitetura é uma ferramenta efetiva para otimizar implementações de algoritmos criptográficos baseados em curvas elípticas. Seja isto um incentivo não somente para aqueles que procuram acelerar os programas em geral, mas também para que os fabricantes de computadores incluam mais extensões avançadas para apoiar a demanda crescente da criptografia.

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Three Dimensional Montgomery Ladder, Differential Point Tripling on Montgomery Curves and Point Quintupling on Weierstrass’ and Edwards Curves

Cited by 10 publications

References 22 publications

Efficient Elliptic Curve Operators for Jacobian Coordinates

Efficient Elliptic Curve Operators for Jacobian Coordinates

Speeding up regular elliptic curve scalar multiplication without precomputation

High-performance elliptic curve cryptography

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