Elliptic Curve Scalar Multiplication Combining Yao’s Algorithm and Double Bases

Abstract. In this paper we propose to take one step back in the use of double base number systems for elliptic curve point scalar multiplication. Using a mod-ified version of Yao’s algorithm, we go back from the popular double base chain representation to a more general double base system. Instead of representing an integer k as Pn i=1 2 bi3ti where (bi) and (ti) are two decreasing sequences, we only set a maximum value for both of them. Then, we analyze the efficiency of our new method using different bases and optimal parameters. In particular, we pro-pose for the first time a binary/Zeckendorf representation for integers, providing interesting results. Finally, we provide a comprehensive comparison to state-of-the-art methods, including a large variety of curve shapes and latest point addition formulae speed-ups

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

Fast Elliptic Curve Cryptography Using Optimal Double-Base Chains

In this work, we propose an algorithm to produce the double-base chains that optimize the time used for computing an elliptic curve cryptosystem. The double-base chains is the representation that combining the binary and ternary representation. By this method, we can reduce the Hamming weight of the expansion, and reduce the time for computing the scalar point multiplication (Q = rS), that is the bottleneck operation of the elliptic curve cryptosystem. This representation is very redundant, i.e. we can present a number by many expansions. Then, we can select the way that makes the operation fastest. However, the previous works on double-bases chain have used a greedy algorithm, and their solutions are not optimized. We propose the algorithm based on the dynamic programming scheme that outputs the optimized the double-bases chain. The experiments show that we have reduced the time for computing the scalar multiplication by 3.88-3.95%, the multi-scalar multiplication by 2.55-4.37%, and the multi-scalar multiplication on the larger digit set by 3.5-12%

A Survey of Fast Scalar Multiplication on Elliptic Curve Cryptography for Lightweight Embedded Devices

Elliptic curve cryptography (ECC) is one of the most famous asymmetric cryptographic schemes which offers the same level of security with much shorter keys than the other widely used asymmetric cryptographic algorithm, Rivest, Shamir, and Adleman (RSA). In ECC, the main and most heavily used operation is the scalar multiplication kP, where the scalar value k is a private integer and must be secured. Various methods for fast scalar multiplication are based on the binary/ternary representation of the scalar. In this chapter, we present various methods to make fast scalar multiplication on ECC over prime field for lightweight embedded devices like wireless sensor network (WSN) and Internet of Things (IoT)

Optimizing scalar multiplication for koblitz curves using hybrid FPGAs

Elliptic curve cryptography (ECC) is a type of public-key cryptosystem which uses the additive group of points on a nonsingular elliptic curve as a cryptographic medium. Koblitz curves are special elliptic curves that have unique properties which allow scalar multiplication, the bottleneck operation in most ECC cryptosystems, to be performed very efficiently. Optimizing the scalar multiplication operation on Koblitz curves is an active area of research with many proposed algorithms for FPGA and software implementations. As of yet little to no research has been reported on using the capabilities of hybrid FPGAs, such as the Xilinx Virtex-4 FX series, which would allow for the design of a more flexible single-chip system that performs scalar multiplication and is not constrained by high communication costs between hardware and software. While the results obtained in this thesis were competitive with many other FPGA implementations, the most recent research efforts have produced significantly faster FPGA based systems. These systems were created by utilizing new and interesting approaches to improve the runtime of performing scalar multiplication on Koblitz curves and thus significantly outperformed the results obtained in this thesis. However, this thesis also functioned as a comparative study of the usage of different basis representations and proved that strict polynomial basis approaches can compete with strict normal basis implementations when performing scalar multiplication on Koblitz curves

High-Speed Elliptic Curve and Pairing-Based Cryptography

Elliptic Curve Cryptography (ECC), independently proposed by Miller [Mil86] and Koblitz [Kob87] in mid 80’s, is finding momentum to consolidate its status as the public-key system of choice in a wide range of applications and to further expand this position to settings traditionally occupied by RSA and DL-based systems. The non-existence of known subexponential attacks on this cryptosystem directly translates to shorter keylengths for a given security level and, consequently, has led to implementations with better bandwidth usage, reduced power and memory requirements, and higher speeds. Moreover, the dramatic entry of pairing-based cryptosystems defined on elliptic curves at the beginning of the new millennium has opened the possibility of a plethora of innovative applications, solving in some cases longstanding problems in cryptography. Nevertheless, public-key cryptography (PKC) is still relatively expensive in comparison with its symmetric-key counterpart and it remains an open challenge to reduce further the computing cost of the most time-consuming PKC primitives to guarantee their adoption for secure communication in commercial and Internet-based applications. The latter is especially true for pairing computations. Thus, it is of paramount importance to research methods which permit the efficient realization of Elliptic Curve and Pairing-based Cryptography on the several new platforms and applications. This thesis deals with efficient methods and explicit formulas for computing elliptic curve scalar multiplication and pairings over fields of large prime characteristic with the objective of enabling the realization of software implementations at very high speeds. To achieve this main goal in the case of elliptic curves, we accomplish the following tasks: identify the elliptic curve settings with the fastest arithmetic; accelerate the precomputation stage in the scalar multiplication; study number representations and scalar multiplication algorithms for speeding up the evaluation stage; identify most efficient field arithmetic algorithms and optimize them; analyze the architecture of the targeted platforms for maximizing the performance of ECC operations; identify most efficient coordinate systems and optimize explicit formulas; and realize implementations on x86-64 processors with an optimal algorithmic selection among all studied cases. In the case of pairings, the following tasks are accomplished: accelerate tower and curve arithmetic; identify most efficient tower and field arithmetic algorithms and optimize them; identify the curve setting with the fastest arithmetic and optimize it; identify state-of-the-art techniques for the Miller loop and final exponentiation; and realize an implementation on x86-64 processors with optimal algorithmic selection. The most outstanding contributions that have been achieved with the methodologies above in this thesis can be summarized as follows: • Two novel precomputation schemes are introduced and shown to achieve the lowest costs in the literature for different curve forms and scalar multiplication primitives. The detailed cost formulas of the schemes are derived for most relevant scenarios. • A new methodology based on the operation cost per bit to devise highly optimized and compact multibase algorithms is proposed. Derived multibase chains using bases {2,3} and {2,3,5} are shown to achieve the lowest theoretical costs for scalar multiplication on certain curve forms and for scenarios with and without precomputations. In addition, the zero and nonzero density formulas of the original (width-w) multibase NAF method are derived by using Markov chains. The application of “fractional” windows to the multibase method is described together with the derivation of the corresponding density formulas. • Incomplete reduction and branchless arithmetic techniques are optimally combined for devising high-performance field arithmetic. Efficient algorithms for “small” modular operations using suitably chosen pseudo-Mersenne primes are carefully analyzed and optimized for incomplete reduction. • Data dependencies between contiguous field operations are discovered to be a source of performance degradation on x86-64 processors. Three techniques for reducing the number of potential pipeline stalls due to these dependencies are proposed: field arithmetic scheduling, merging of point operations and merging of field operations. • Explicit formulas for two relevant cases, namely Weierstrass and Twisted Edwards curves over and , are carefully optimized employing incomplete reduction, minimal number of operations and reduced number of data dependencies between contiguous field operations. • Best algorithms for the field, point and scalar arithmetic, studied or proposed in this thesis, are brought together to realize four high-speed implementations on x86-64 processors at the 128-bit security level. Presented results set new speed records for elliptic curve scalar multiplication and introduce up to 34% of cost reduction in comparison with the best previous results in the literature. • A generalized lazy reduction technique that enables the elimination of up to 32% of modular reductions in the pairing computation is proposed. Further, a methodology that keeps intermediate results under Montgomery reduction boundaries maximizing operations without carry checks is introduced. Optimized formulas for the popular tower are explicitly stated and a detailed operation count that permits to determine the theoretical cost improvement attainable with the proposed method is carried out for the case of an optimal ate pairing on a Barreto-Naehrig (BN) curve at the 128-bit security level. • Best algorithms for the different stages of the pairing computation, including the proposed techniques and optimizations, are brought together to realize a high-speed implementation at the 128-bit security level. Presented results on x86-64 processors set new speed records for pairings, introducing up to 34% of cost reduction in comparison with the best published result. From a general viewpoint, the proposed methods and optimized formulas have a practical impact in the performance of cryptographic protocols based on elliptic curves and pairings in a wide range of applications. In particular, the introduced implementations represent a direct and significant improvement that may be exploited in performance-dominated applications such as high-demand Web servers in which millions of secure transactions need to be generated

Revisiting ECM on GPUs

Modern public-key cryptography is a crucial part of our contemporary life where a secure communication channel with another party is needed. With the advance of more powerful computing architectures – especially Graphics Processing Units (GPUs) – traditional approaches like RSA and Diffie-Hellman schemes are more and more in danger of being broken. We present a highly optimized implementation of Lenstra’s ECM algorithm customized for GPUs. Our implementation uses state-of-the-art elliptic curve arithmetic and optimized integer arithmetic while providing the possibility of arbitrarily scaling ECM’s parameters allowing an application even for larger discrete logarithm problems. Furthermore, the proposed software is not limited to any specific GPU generation and is to the best of our knowledge the first implementation supporting multiple device computation. To this end, for a bound of B1=8,192 and a modulus size of 192 bit, we achieve a throughput of 214 thousand ECM trials per second on a modern RTX 2080 Ti GPU considering only the first stage of ECM. To solve the Discrete Logarithm Problem for larger bit sizes, our software can easily support larger parameter sets such that a throughput of 2,781 ECM trials per second is achieved using B1=50,000, B2=5,000,000, and a modulus size of 448 bit

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