1,715 research outputs found

    Computing cardinalities of Q-curve reductions over finite fields

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    We present a specialized point-counting algorithm for a class of elliptic curves over F\_{p^2} that includes reductions of quadratic Q-curves modulo inert primes and, more generally, any elliptic curve over F\_{p^2} with a low-degree isogeny to its Galois conjugate curve. These curves have interesting cryptographic applications. Our algorithm is a variant of the Schoof--Elkies--Atkin (SEA) algorithm, but with a new, lower-degree endomorphism in place of Frobenius. While it has the same asymptotic asymptotic complexity as SEA, our algorithm is much faster in practice.Comment: To appear in the proceedings of ANTS-XII. Added acknowledgement of Drew Sutherlan

    Efficient software implementation of elliptic curves and bilinear pairings

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    Orientador: Júlio César Lopez HernándezTese (doutorado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: O advento da criptografia assimétrica ou de chave pública possibilitou a aplicação de criptografia em novos cenários, como assinaturas digitais e comércio eletrônico, tornando-a componente vital para o fornecimento de confidencialidade e autenticação em meios de comunicação. Dentre os métodos mais eficientes de criptografia assimétrica, a criptografia de curvas elípticas destaca-se pelos baixos requisitos de armazenamento para chaves e custo computacional para execução. A descoberta relativamente recente da criptografia baseada em emparelhamentos bilineares sobre curvas elípticas permitiu ainda sua flexibilização e a construção de sistemas criptográficos com propriedades inovadoras, como sistemas baseados em identidades e suas variantes. Porém, o custo computacional de criptossistemas baseados em emparelhamentos ainda permanece significativamente maior do que os assimétricos tradicionais, representando um obstáculo para sua adoção, especialmente em dispositivos com recursos limitados. As contribuições deste trabalho objetivam aprimorar o desempenho de criptossistemas baseados em curvas elípticas e emparelhamentos bilineares e consistem em: (i) implementação eficiente de corpos binários em arquiteturas embutidas de 8 bits (microcontroladores presentes em sensores sem fio); (ii) formulação eficiente de aritmética em corpos binários para conjuntos vetoriais de arquiteturas de 64 bits e famílias mais recentes de processadores desktop dotadas de suporte nativo à multiplicação em corpos binários; (iii) técnicas para implementação serial e paralela de curvas elípticas binárias e emparelhamentos bilineares simétricos e assimétricos definidos sobre corpos primos ou binários. Estas contribuições permitiram obter significativos ganhos de desempenho e, conseqüentemente, uma série de recordes de velocidade para o cálculo de diversos algoritmos criptográficos relevantes em arquiteturas modernas que vão de sistemas embarcados de 8 bits a processadores com 8 coresAbstract: The development of asymmetric or public key cryptography made possible new applications of cryptography such as digital signatures and electronic commerce. Cryptography is now a vital component for providing confidentiality and authentication in communication infra-structures. Elliptic Curve Cryptography is among the most efficient public-key methods because of its low storage and computational requirements. The relatively recent advent of Pairing-Based Cryptography allowed the further construction of flexible and innovative cryptographic solutions like Identity-Based Cryptography and variants. However, the computational cost of pairing-based cryptosystems remains significantly higher than traditional public key cryptosystems and thus an important obstacle for adoption, specially in resource-constrained devices. The main contributions of this work aim to improve the performance of curve-based cryptosystems, consisting of: (i) efficient implementation of binary fields in 8-bit microcontrollers embedded in sensor network nodes; (ii) efficient formulation of binary field arithmetic in terms of vector instructions present in 64-bit architectures, and on the recently-introduced native support for binary field multiplication in the latest Intel microarchitecture families; (iii) techniques for serial and parallel implementation of binary elliptic curves and symmetric and asymmetric pairings defined over prime and binary fields. These contributions produced important performance improvements and, consequently, several speed records for computing relevant cryptographic algorithms in modern computer architectures ranging from embedded 8-bit microcontrollers to 8-core processorsDoutoradoCiência da ComputaçãoDoutor em Ciência da Computaçã

    Counting Points on Genus 2 Curves with Real Multiplication

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    We present an accelerated Schoof-type point-counting algorithm for curves of genus 2 equipped with an efficiently computable real multiplication endomorphism. Our new algorithm reduces the complexity of genus 2 point counting over a finite field (\F_{q}) of large characteristic from (\widetilde{O}(\log^8 q)) to (\widetilde{O}(\log^5 q)). Using our algorithm we compute a 256-bit prime-order Jacobian, suitable for cryptographic applications, and also the order of a 1024-bit Jacobian

    Efficient Implementations of Pairing-Based Cryptography on Embedded Systems

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    Many cryptographic applications use bilinear pairing such as identity based signature, instance identity-based key agreement, searchable public-key encryption, short signature scheme, certificate less encryption and blind signature. Elliptic curves over finite field are the most secure and efficient way to implement bilinear pairings for the these applications. Pairing based cryptosystems are being implemented on different platforms such as low-power and mobile devices. Recently, hardware capabilities of embedded devices have been emerging which can support efficient and faster implementations of pairings on hand-held devices. In this thesis, the main focus is optimization of Optimal Ate-pairing using special class of ordinary curves, Barreto-Naehring (BN), for different security levels on low-resource devices with ARM processors. Latest ARM architectures are using SIMD instructions based NEON engine and are helpful to optimize basic algorithms. Pairing implementations are being done using tower field which use field multiplication as the most important computation. This work presents NEON implementation of two multipliers (Karatsuba and Schoolbook) and compare the performance of these multipliers with different multipliers present in the literature for different field sizes. This work reports the fastest implementation timing of pairing for BN254, BN446 and BN638 curves for ARMv7 architecture which have security levels as 128-, 164-, and 192-bit, respectively. This work also presents comparison of code performance for ARMv8 architectures

    Pairing computation on hyperelliptic curves of genus 2

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    Bilinear pairings have been recently used to construct cryptographic schemes with new and novel properties, the most celebrated example being the Identity Based Encryption scheme of Boneh and Franklin. As pairing computation is generally the most computationally intensive part of any painng-based cryptosystem, it is essential to investigate new ways in which to compute pairings efficiently. The vast majority of the literature on pairing computation focuscs solely on using elliptic curves. In this thesis we investigate pairing computation on supersingular hyperelliptic curves of genus 2 Our aim is to provide a practical alternative to using elliptic curves for pairing based cryptography. Specifically, we illustrate how to implement pairings efficiently using genus 2 curves, and how to attain performance comparable to using elliptic curves. We show that pairing computation on genus 2 curves over F2m can outperform elliptic curves by using a new variant of the Tate pairing, called the r¡j pairing, to compute the fastest pairing implementation in the literature to date We also show for the first time how the final exponentiation required to compute the Tate pairing can be avoided for certain hyperelliptic curves. We investigate pairing computation using genus 2 curves over large prime fields, and detail various techniques that lead to an efficient implementation, thus showing that these curves are a viable candidate for practical use

    The projective translation equation and unramified 2-dimensional flows with rational vector fields

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    Let X=(x,y). Previously we have found all rational solutions of the 2-dimensional projective translation equation, or PrTE, (1-z)f(X)=f(f(Xz)(1-z)/z); here f(X)=(u(x,y),v(x,y)) is a pair of two (real or complex) functions. Solutions of this functional equation are called projective flows. A vector field of a rational flow is a pair of 2-homogenic rational functions. On the other hand, only special pairs of 2-homogenic rational functions give rise to rational flows. In this paper we are interested in all non-singular (satisfying the boundary condition) and unramified (without branching points, i.e. single-valued functions in C^2\{union of curves}) projective flows whose vector field is still rational. We prove that, up to conjugation with 1-homogenic birational plane transformation, these are of 6 types: 1) the identity flow; 2) one flow for each non-negative integer N - these flows are rational of level N; 3) the level 1 exponential flow, which is also conjugate to the level 1 tangent flow; 4) the level 3 flow expressable in terms of Dixonian (equianharmonic) elliptic functions; 5) the level 4 flow expressable in terms of lemniscatic elliptic functions; 6) the level 6 flow expressable in terms of Dixonian elliptic functions again. This reveals another aspect of the PrTE: in the latter four cases this equation is equivalent and provides a uniform framework to addition formulas for exponential, tangent, or special elliptic functions (also addition formulas for polynomials and the logarithm, though the latter appears only in branched flows). Moreover, the PrTE turns out to have a connection with Polya-Eggenberger urn models. Another purpose of this study is expository, and we provide the list of open problems and directions in the theory of PrTE; for example, we define the notion of quasi-rational projective flows which includes curves of arbitrary genus.Comment: 34 pages, 2 figure
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