On the Computation of the Optimal Ate Pairing at the 192-bit Security Level

Barreto, Lynn and Scott elliptic curves of embedding degree 12 denoted BLS12 have been proven to present fastest results on the implementation of pairings at the 192-bit security level [1]. The computation of pairings in general involves the execution of the Miller algorithm and the final exponentiation. In this paper, we improve the complexity of these two steps up to 8% by searching an appropriate parameter. We compute the optimal ate pairing on BLS curves of embedding degree 12 and we also extend the same analysis to BLS curves with embedding degree 24. Furthermore, as many pairing based protocols are implemented on memory constrained devices such as SIM or smart cards, we describe an efficient algorithm for the computation of the final exponentiation less memory intensive with an improvement up to 25% with respect to the previous work

Adequate Elliptic Curve for Computing the Product of n Pairings

Many pairing-based protocols require the computation of the product and/or of a quotient of n pairings where n > 1 is a natural integer. Zhang et al.[1] recently showed that the Kachisa-Schafer and Scott family of elliptic curves with embedding degree 16 denoted KSS16 at the 192-bit security level is suitable for such protocols comparatively to the Baretto- Lynn and Scott family of elliptic curves of embedding degree 12 (BLS12). In this work, we provide important corrections and improvements to their work based on the computation of the optimal Ate pairing. We focus on the computation of the nal exponentiation which represent an important part of the overall computation of this pairing. Our results improve by 864 multiplications in Fp the computations of Zhang et al.[1]. We prove that for computing the product or the quotient of 2 pairings, BLS12 curves are the best solution. In other cases, specially when n > 2 as mentioned in [1], KSS16 curves are recommended for computing product of n pairings. Furthermore, we prove that the curve presented by Zhang et al.[1] is not resistant against small subgroup attacks. We provide an example of KSS16 curve protected against such attacks

Optimal TNFS-secure pairings on elliptic curves with composite embedding degree

In this paper we present a comprehensive comparison between pairing-friendly elliptic curves, considering di erent curve forms and twists where possible. We de ne an additional measure of the e- ciency of a parametrized pairing-friendly family that takes into account the number eld sieve (NFS) attacks (unlike the -value). This measure includes an approximation of the security of the discrete logarithm problem in F pk , computed via the method of Barbulescu and Duquesne [4]. We compute the security of the families presented by Fotiadis and Konstantinou in [14], compute some new families, and compare the eciency of both of these with the (adjusted) BLS, KSS, and BN families, and with the new families of [20]. Finally, we recommend pairing-friendly elliptic curves for security levels 128 and 192

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

Still Wrong Use of Pairings in Cryptography

Several pairing-based cryptographic protocols are recently proposed with a wide variety of new novel applications including the ones in emerging technologies like cloud computing, internet of things (IoT), e-health systems and wearable technologies. There have been however a wide range of incorrect use of these primitives. The paper of Galbraith, Paterson, and Smart (2006) pointed out most of the issues related to the incorrect use of pairing-based cryptography. However, we noticed that some recently proposed applications still do not use these primitives correctly. This leads to unrealizable, insecure or too inefficient designs of pairing-based protocols. We observed that one reason is not being aware of the recent advancements on solving the discrete logarithm problems in some groups. The main purpose of this article is to give an understandable, informative, and the most up-to-date criteria for the correct use of pairing-based cryptography. We thereby deliberately avoid most of the technical details and rather give special emphasis on the importance of the correct use of bilinear maps by realizing secure cryptographic protocols. We list a collection of some recent papers having wrong security assumptions or realizability/efficiency issues. Finally, we give a compact and an up-to-date recipe of the correct use of pairings.Comment: 25 page

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

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

Secure and Efficient Delegation of Elliptic-Curve Pairing

Many public-key cryptosystems and, more generally, cryp- tographic protocols, use pairings as important primitive operations. To expand the applicability of these solutions to computationally weaker devices, it has been advocated that a computationally weaker client del- egates such primitive operations to a computationally stronger server. Important requirements for such delegation protocols include privacy of the client's pairing inputs and security of the client's output, in the sense of detecting, except for very small probability, any malicious server's at- tempt to convince the client of an incorrect pairing result. In this paper we show that the computation of bilinear pairings in all known pairing-based cryptographic protocols can be eciently, privately and securely delegated to a single, possibly malicious, server. Our tech- niques provides eciency improvements over past work in all input sce- narios, regardless on whether inputs are available to the parties in an oine phase or only in the online phase, and on whether they are public or have privacy requirements. The client's online runtime improvement is, for some of our protocols almost 1 order of magnitude, no matter which practical elliptic curve, among recently recommended ones, is used for the pairing realization