Ordinary Pairing Friendly Curve of Embedding Degree 3 Whose Order Has Two Large Prime Factors

This paper proposes a method for generating a certain composite order ordinary pairing–friendly elliptic curve of embedding degree 3. In detail, the order has two large prime factors such as the modulus of RSA cryptography. The method is based on the property that the order of the target pairing–friendly curve is given by a polynomial as r(X) of degree 2 with respect to the integer variable X. When the bit size of the prime factors is about 500 bits, the proposed method averagely takes about 15 minutes on Core 2 Quad (2.66Hz) for generating one

Developing an Automatic Generation Tool for Cryptographic Pairing Functions

Pairing-Based Cryptography is receiving steadily more attention from industry, mainly because of the increasing interest in Identity-Based protocols. Although there are plenty of applications, efficiently implementing the pairing functions is often difficult as it requires more knowledge than previous cryptographic primitives. The author presents a tool for automatically generating optimized code for the pairing functions which can be used in the construction of such cryptographic protocols. In the following pages I present my work done on the construction of pairing function code, its optimizations and how their construction can be automated to ease the work of the protocol implementer. Based on the user requirements and the security level, the created cryptographic compiler chooses and constructs the appropriate elliptic curve. It identifies the supported pairing function: the Tate, ate, R-ate or pairing lattice/optimal pairing, and its optimized parameters. Using artificial intelligence algorithms, it generates optimized code for the final exponentiation and for hashing a point to the required group using the parametrisation of the chosen family of curves. Support for several multi-precision libraries has been incorporated: Magma, MIRACL and RELIC are already included, but more are possible

Performance Evaluation of Optimal Ate Pairing on Low-Cost Single Microprocessor Platform

The framework of low-cost interconnected devices forms a new kind of cryptographic environment with diverse requirements. Due to the minimal resource capacity of the devices, light-weight cryptographic algorithms are favored. Many applications of IoT work autonomously and process sensible data, which emphasizes security needs, and might also cause a need for specific security measures. A bilinear pairing is a mapping based on groups formed by elliptic curves over extension fields. The pairings are the key-enabler for versatile cryptosystems, such as certificateless signatures and searchable encryption. However, they have a major computational overhead, which coincides with the requirements of the low-cost devices. Nonetheless, the bilinear pairings are the only known approach for many cryptographic protocols so their feasibility should certainly be studied, as they might turn out to be necessary for some future IoT solutions. Promising results already exist for high-frequency CPU:s and platforms with hardware extensions. In this work, we study the feasibility of computing the optimal ate pairing over the BN254 curve, on a 64 MHz Cortex-M33 based platform by utilizing an optimized open-source library. The project is carried out for the company Nordic Semiconductor. As a result, the pairing was effectively computed in under 26* 10^6 cycles, or in 410 ms. The resulting pairing enables a limited usage of pairing-based cryptography, with a capacity of at most few cryptographic operations, such as ID-based key verifications per second. Referring to other relevant works, a competent pairing application would require either a high-frequency - and thus high consuming - microprocessor, or a customized FPGA. Moreover, it is noted that the research in efficient pairing-based cryptography is constantly taking steps forward in every front-line: efficient algorithms, protocols, and hardware-solutions

Pairing computation on elliptic curves with efficiently computable endomorphism and small embedding degree

Scott uses an efficiently computable isomorphism in order to optimize pairing computation on a particular class of curves with embedding degree 2. He points out that pairing implementation becomes thus faster on these curves than on their supersingular equivalent, originally recommended by Boneh and Franklin for Identity Based Encryption. We extend Scott\u27s method to other classes of curves with small embedding degree and efficiently computable endomorphism

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

Machine-Level Software Optimization of Cryptographic Protocols

This work explores two methods for practical cryptography on mobile devices. The first method is a quantum-resistant key-exchange protocol proposed by Jao et al.. As the use of mobile devices increases, the deployment of practical cryptographic protocols designed for use on these devices is of increasing importance. Furthermore, we are faced with the possible development of a large-scale quantum computer in the near future and must take steps to prepare for this possibility. We describe the key-exchange protocol of Jao et al. and discuss their original implementation. We then describe our modifications to their scheme that make it suitable for use in mobile devices. Our code is between 18-26% faster (depending on the security level). The second is an highly optimized implementation of Miller's algorithm that efficiently computes the Optimal Ate pairing over Barreto-Naehrig curves proposed by Grewal et al.. We give an introduction to cryptographic pairings and describe the Tate pairing and its variants. We then proceed to describe Grewal et al.'s implementation of Miller's algorithm, along with their optimizations. We describe our use of hand-optimized assembly code to increase the performance of their implementation. For the Optimal Ate pairing over the BN-446 curve, our code is between 7-8% faster depending on whether the pairing uses affine or projective coordinates

Efficient Pairings on Various Platforms

Pairings have found a range of applications in many areas of cryptography. As such, to utilize the enormous potential of pairing-based protocols one needs to efficiently compute pairings across various computing platforms. In this thesis, we give an introduction to pairing-based cryptography and describe the Tate pairing and its variants. We then describe some recent work to realize efficient computation of pairings. We further extend these optimizations and implement the O-Ate pairing on BN-curves on ARM and x86-64 platforms. Specifically, we extend the idea of lazy reduction to field inversion, optimize curve arithmetic, and construct efficient tower extensions to optimize field arithmetic. We also analyze the use of affine coordinates for pairing computation leading us to the conclusion that they are a competitive choice for fast pairing computation on ARM processors, especially at high security level. Our resulting implementation is more than three times faster than any previously reported implementation on ARM processors

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

Theory and Practice of Cryptography and Network Security Protocols and Technologies

In an age of explosive worldwide growth of electronic data storage and communications, effective protection of information has become a critical requirement. When used in coordination with other tools for ensuring information security, cryptography in all of its applications, including data confidentiality, data integrity, and user authentication, is a most powerful tool for protecting information. This book presents a collection of research work in the field of cryptography. It discusses some of the critical challenges that are being faced by the current computing world and also describes some mechanisms to defend against these challenges. It is a valuable source of knowledge for researchers, engineers, graduate and doctoral students working in the field of cryptography. It will also be useful for faculty members of graduate schools and universities