On the Optimal Pre-Computation of Window τ\tauNAF for Koblitz Curves

Koblitz curves have been a nice subject of consideration for both theoretical and practical interests. The window $\tau$-adic algorithm of Solinas (window $\tau$NAF) is the most powerful method for computing point multiplication for Koblitz curves. Pre-computation plays an important role in improving the performance of point multiplication. In this paper, the concept of optimal pre-computation for window $\tau$NAF is formulated. In this setting, an optimal pre-computation has some mathematically natural and clean forms, and requires $2^{w-2}-1$ point additions and two evaluations of the Frobenius map $\tau$, where $w$ is the window width. One of the main results of this paper is to construct an optimal pre-computation scheme for each window width $w$ from $4$ to $15$ (more than practical needs). These pre-computations can be easily incorporated into implementations of window $\tau$NAF. The ideas in the paper can also be used to construct other suitable pre-computations. This paper also includes a discussion of coefficient sets for window $\tau$NAF and the divisibility by powers of $\tau$ through different approaches

Multi-Base Chains for Faster Elliptic Curve Cryptography

This research addresses a multi-base number system (MBNS) for faster elliptic curve cryptography (ECC). The emphasis is on speeding up the main operation of ECC: scalar multiplication (tP). Mainly, it addresses the two issues of using the MBNS with ECC: deriving optimized formulas and choosing fast methods. To address the first issue, this research studies the optimized formulas (e.g., 3P, 5P) in different elliptic curve coordinate systems over prime and binary fields. For elliptic curves over prime fields, affine Weierstrass, Jacobian Weierstrass, and standard twisted Edwards coordinate systems are reviewed. For binary elliptic curves, affine, Lambda-projective, and twisted mu4-normal coordinate systems are reviewed. Additionally, whenever possible, this research derives several optimized formulas for these coordinate systems. To address the second issue, this research theoretically and experimentally studies the MBNS methods with respect to the average chain length, the average chain cost, and the average conversion cost. The reviewed MBNS methods are greedy, ternary/binary, multi-base NAF, tree-based, and rDAG-based. The emphasis is on these methods\u27 techniques to convert integer t to multi-base chains. Additionally, this research develops bucket methods that advance the MBNS methods. The experimental results show that the MBNS methods with the optimized formulas, in general, have good improvements on the performance of scalar multiplication, compared to the single-base number system methods

Pre-computation in Width-w τ-adic NAF Implementations on Koblitz Curves

This paper examines scalar multiplication on Koblitz curves employing the Frobenius endomorphism. We examine simple binary scalar multiplication, binary Non Adjacent Formats or NAF\u27s, followed by τ-NAF methods. We pay particular attention to width-τ-NAF where we focus on pre-computation. We present alternative pre-computation arrangements for αu for width sizes of 5 and 6 which are better than any previously published results since they: involve a single power of τ are based on least norms; and have a maximum of 2w - 2 - 1 elliptic curve operations. We then study widths of 7 and 8 producing efficient arrangements. Arrangements for width sizes of 7 and 8 have never before appeared in the literature. Furthermore, we introduce a simplified rounding technique for reduction modulo (τm - 1)/(τ - 1) relaxing the requirement of least norms. Lastly, we discuss an O(n) technique for finding arbitrary powers of &tau in software

Fast Scalar Multiplication for Elliptic Curves over Binary Fields by Efficiently Computable Formulas

This paper considers efficient scalar multiplication of elliptic curves over binary fields with a twofold purpose. Firstly, we derive the most efficient $3P$ formula in $\lambda$-projective coordinates and $5P$ formula in both affine and $\lambda$-projective coordinates. Secondly, extensive experiments have been conducted to test various multi-base scalar multiplication methods (e.g., greedy, ternary/binary, multi-base NAF, and tree-based) by integrating our fast formulas. The experiments show that our $3P$ and $5P$ formulas had an important role in speeding up the greedy, the ternary/binary, the multi-base NAF, and the tree-based methods over the NAF method. We also establish an efficient $3P$ formula for Koblitz curves and use it to construct an improved set for the optimal pre-computation of window TNAF

Efficient and Secure ECDSA Algorithm and its Applications: A Survey

Public-key cryptography algorithms, especially elliptic curve cryptography (ECC)and elliptic curve digital signature algorithm (ECDSA) have been attracting attention frommany researchers in different institutions because these algorithms provide security andhigh performance when being used in many areas such as electronic-healthcare, electronicbanking,electronic-commerce, electronic-vehicular, and electronic-governance. These algorithmsheighten security against various attacks and the same time improve performanceto obtain efficiencies (time, memory, reduced computation complexity, and energy saving)in an environment of constrained source and large systems. This paper presents detailedand a comprehensive survey of an update of the ECDSA algorithm in terms of performance,security, and applications

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

Efficient implementation of elliptic curve cryptography.

Elliptic Curve Cryptosystems (ECC) were introduced in 1985 by Neal Koblitz and Victor Miller. Small key size made elliptic curve attractive for public key cryptosystem implementation. This thesis introduces solutions of efficient implementation of ECC in algorithmic level and in computation level. In algorithmic level, a fast parallel elliptic curve scalar multiplication algorithm based on a dual-processor hardware system is developed. The method has an average computation time of n3 Elliptic Curve Point Addition on an n-bit scalar. The improvement is n Elliptic Curve Point Doubling compared to conventional methods. When a proper coordinate system and binary representation for the scalar k is used the average execution time will be as low as n Elliptic Curve Point Doubling, which makes this method about two times faster than conventional single processor multipliers using the same coordinate system. In computation level, a high performance elliptic curve processor (ECP) architecture is presented. The processor uses parallelism in finite field calculation to achieve high speed execution of scalar multiplication algorithm. The architecture relies on compile-time detection rather than of run-time detection of parallelism which results in less hardware. Implemented on FPGA, the proposed processor operates at 66MHz in GF(2 167) and performs scalar multiplication in 100muSec, which is considerably faster than recent implementations.Dept. of Electrical and Computer Engineering. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis2004 .A57. Source: Masters Abstracts International, Volume: 44-03, page: 1446. Thesis (M.A.Sc.)--University of Windsor (Canada), 2005

Existence and Optimality of ww-Non-adjacent Forms with an Algebraic Integer Base

We consider digital expansions in lattices with endomorphisms acting as base. We focus on the $w$-non-adjacent form ($w$-NAF), where each block of $w$ consecutive digits contains at most one non-zero digit. We prove that for sufficiently large $w$ and an expanding endomorphism, there is a suitable digit set such that each lattice element has an expansion as a $w$-NAF. If the eigenvalues of the endomorphism are large enough and $w$ is sufficiently large, then the $w$-NAF is shown to minimise the weight among all possible expansions of the same lattice element using the same digit system

Diseño de criptoprocesadores de curva elíptica sobre gf(2^163) usando bases normales gaussianas

This paper presents the efficient hardware implementation of cryptoprocessors that carry out the scalar multiplication kP over finite field GF(2163) using two digit-level multipliers. The finite field arithmetic operations were implemented using Gaussian normal basis (GNB) representation, and the scalar multiplication kP was implemented using Lopez-Dahab algorithm, 2-NAF halve-and-add algorithm and w-tNAF method for Koblitz curves. The processors were designed using VHDL description, synthesized on the Stratix-IV FPGA using Quartus II 12.0 and verified using SignalTAP II and Matlab. The simulation results show that the cryptoprocessors present a very good performance to carry out the scalar multiplication kP. In this case, the computation times of the multiplication kP using Lopez-Dahab, 2-NAF halve-and-add and 16-tNAF for Koblitz curves were 13.37 µs, 16.90 µs and 5.05 µs, respectively.En este trabajo se presenta la implementación eficiente en hardware de criptoprocesadores que permiten llevar a cabo la multiplicación escalar kP sobre el campo finito GF(2163) usando dos multiplicadores a nivel de digito. Las operaciones aritméticas de campo finito fueron implementadas usando la representación de bases normales Gaussianas (GNB), y la multiplicación escalar kP fue implementada usando el algoritmo de López-Dahab, el algoritmo de bisección de punto 2-NAF y el método w-tNAF para curvas de Koblitz. Los criptoprocesadores fueron diseñados usando descripción VHDL, sintetizados en el FPGA Stratix-IV usando Quartus II 12.0 y verificados usando SignalTAP II y Matlab. Los resultados de simulación muestran que los criptoprocesadores presentan un muy buen desempeño para llevar a cabo la multiplicación escalar kP. En este caso, los tiempos de computo de la multiplicación kP usando Lopez-Dahab, bisección de punto 2-NAF y 16-tNAF para curvas de Koblitz fueron 13.37 µs, 16.90 µs and 5.05 µs, respectivamente