Solving discrete logarithms on a 170-bit MNT curve by pairing reduction

Pairing based cryptography is in a dangerous position following the breakthroughs on discrete logarithms computations in finite fields of small characteristic. Remaining instances are built over finite fields of large characteristic and their security relies on the fact that the embedding field of the underlying curve is relatively large. How large is debatable. The aim of our work is to sustain the claim that the combination of degree 3 embedding and too small finite fields obviously does not provide enough security. As a computational example, we solve the DLP on a 170-bit MNT curve, by exploiting the pairing embedding to a 508-bit, degree-3 extension of the base field.Comment: to appear in the Lecture Notes in Computer Science (LNCS

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

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

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

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

Cryptographic Key Distribution In Wireless Sensor Networks Using Bilinear Pairings

It is envisaged that the use of cheap and tiny wireless sensors will soon bring a third wave of evolution in computing systems. Billions of wireless senor nodes will provide a bridge between information systems and the physical world. Wireless nodes deployed around the globe will monitor the surrounding environment as well as gather information about the people therein. It is clear that this revolution will put security solutions to a great test. Wireless Sensor Networks (WSNs) are a challenging environment for applying security services. They differ in many aspects from traditional fixed networks, and standard cryptographic solutions cannot be used in this application space. Despite many research efforts, key distribution in WSNs still remains an open problem. Many of the proposed schemes suffer from high communication overhead and storage costs, low scalability and poor resilience against different types of attacks. The exclusive usage of simple and energy efficient symmetric cryptography primitives does not solve the security problem. On the other hand a full public key infrastructure which uses asymmetric techniques, digital signatures and certificate authorities seems to be far too complex for a constrained WSN environment. This thesis investigates a new approach to WSN security which addresses many of the shortcomings of existing mechanisms. It presents a detailed description on how to provide practical Public Key Cryptography solutions for wireless sensor networks. The contributions to the state-of-the-art are added on all levels of development beginning with the basic arithmetic operations and finishing with complete security protocols. This work includes a survey of different key distribution protocols that have been developed for WSNs, with an evaluation of their limitations. It also proposes Identity- Based Cryptography (IBC) as an ideal technique for key distribution in sensor networks. It presents the first in-depth study of the application and implementation of Pairing- Based Cryptography (PBC) to WSNs. This is followed by a presentation of the state of the art on the software implementation of Elliptic Curve Cryptography (ECC) on typical WSNplatforms. New optimized algorithms for performing multiprecision multiplication on a broad range of low-end CPUs are introduced as well. Three novel protocols for key distribution are proposed in this thesis. Two of these are intended for non-interactive key exchange in flat and clustered networks respectively. A third key distribution protocol uses Identity-Based Encryption (IBE) to secure communication within a heterogeneous sensor network. This thesis includes also a comprehensive security evaluation that shows that proposed schemes are resistant to various attacks that are specific to WSNs. This work shows that by using the newest achievements in cryptography like pairings and IBC it is possible to deliver affordable public-key cryptographic solutions and to apply a sufficient level of security for the most demanding WSN applications