Refinements of Miller's Algorithm over Weierstrass Curves Revisited

In 1986 Victor Miller described an algorithm for computing the Weil pairing in his unpublished manuscript. This algorithm has then become the core of all pairing-based cryptosystems. Many improvements of the algorithm have been presented. Most of them involve a choice of elliptic curves of a \emph{special} forms to exploit a possible twist during Tate pairing computation. Other improvements involve a reduction of the number of iterations in the Miller's algorithm. For the generic case, Blake, Murty and Xu proposed three refinements to Miller's algorithm over Weierstrass curves. Though their refinements which only reduce the total number of vertical lines in Miller's algorithm, did not give an efficient computation as other optimizations, but they can be applied for computing \emph{both} of Weil and Tate pairings on \emph{all} pairing-friendly elliptic curves. In this paper we extend the Blake-Murty-Xu's method and show how to perform an elimination of all vertical lines in Miller's algorithm during Weil/Tate pairings computation on \emph{general} elliptic curves. Experimental results show that our algorithm is faster about 25% in comparison with the original Miller's algorithm.Comment: 17 page

Efficient software implementation of elliptic curves and bilinear pairings

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

Faster computation of the Tate pairing

This paper proposes new explicit formulas for the doubling and addition step in Miller's algorithm to compute the Tate pairing. For Edwards curves the formulas come from a new way of seeing the arithmetic. We state the first geometric interpretation of the group law on Edwards curves by presenting the functions which arise in the addition and doubling. Computing the coefficients of the functions and the sum or double of the points is faster than with all previously proposed formulas for pairings on Edwards curves. They are even competitive with all published formulas for pairing computation on Weierstrass curves. We also speed up pairing computation on Weierstrass curves in Jacobian coordinates. Finally, we present several examples of pairing-friendly Edwards curves.Comment: 15 pages, 2 figures. Final version accepted for publication in Journal of Number Theor

Novel Area-Efficient and Flexible Architectures for Optimal Ate Pairing on FPGA

While FPGA is a suitable platform for implementing cryptographic algorithms, there are several challenges associated with implementing Optimal Ate pairing on FPGA, such as security, limited computing resources, and high power consumption. To overcome these issues, this study introduces three approaches that can execute the optimal Ate pairing on Barreto-Naehrig curves using Jacobean coordinates with the goal of reaching 128-bit security on the Genesys board. The first approach is a pure software implementation utilizing the MicroBlaze processor. The second involves a combination of software and hardware, with key operations in $F_{p}$ and $F_{p^{2}}$ being transformed into IP cores for the MicroBlaze. The third approach builds on the second by incorporating parallelism to improve the pairing process. The utilization of multiple MicroBlaze processors within a single system offers both versatility and parallelism to speed up pairing calculations. A variety of methods and parameters are used to optimize the pairing computation, including Montgomery modular multiplication, the Karatsuba method, Jacobean coordinates, the Complex squaring method, sparse multiplication, squaring in $G_{\phi 6}F_{p^{12}}$, and the addition chain method. The proposed systems are designed to efficiently utilize limited resources in restricted environments, while still completing tasks in a timely manner.Comment: 13 pages, 8 figures, and 5 table

Squaring Algorithm Efficient for Cubic Extension Field Derived with Pseudo Gauss Period Normal Basis

Recently, pairing–based cryptographies have attracted much attention. For fast pairing calculation, not only pairing algorithms but also arithmetic operations in extension field should be efficient. Especially for final exponentiation included in pairing calculation, squaring is more important than multiplication. This paper considers squaring algorithms efficient for cubic extension field which is often used for pairing implementaions

Efficient Implementations of Pairing-Based Cryptography on Embedded Systems

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

Elliptic Curve Cryptography on Modern Processor Architectures

Abstract Elliptic Curve Cryptography (ECC) has been adopted by the US National Security Agency (NSA) in Suite "B" as part of its "Cryptographic Modernisation Program ". Additionally, it has been favoured by an entire host of mobile devices due to its superior performance characteristics. ECC is also the building block on which the exciting field of pairing/identity based cryptography is based. This widespread use means that there is potentially a lot to be gained by researching efficient implementations on modern processors such as IBM's Cell Broadband Engine and Philip's next generation smart card cores. ECC operations can be thought of as a pyramid of building blocks, from instructions on a core, modular operations on a finite field, point addition & doubling, elliptic curve scalar multiplication to application level protocols. In this thesis we examine an implementation of these components for ECC focusing on a range of optimising techniques for the Cell's SPU and the MIPS smart card. We show significant performance improvements that can be achieved through of adoption of EC