Reconfigurable elliptic curve cryptography

Elliptic Curve Cryptosystems (ECC) have been proposed as an alternative to other established public key cryptosystems such as RSA (Rivest Shamir Adleman). ECC provide more security per bit than other known public key schemes based on the discrete logarithm problem. Smaller key sizes result in faster computations, lower power consumption and memory and bandwidth savings, thus making ECC a fast, flexible and cost-effective solution for providing security in constrained environments. Implementing ECC on reconfigurable platform combines the speed, security and concurrency of hardware along with the flexibility of the software approach. This work proposes a generic architecture for elliptic curve cryptosystem on a Field Programmable Gate Array (FPGA) that performs an elliptic curve scalar multiplication in 1.16milliseconds for GF (2163), which is considerably faster than most other documented implementations. One of the benefits of the proposed processor architecture is that it is easily reprogrammable to use different algorithms and is adaptable to any field order. Also through reconfiguration the arithmetic unit can be optimized for different area/speed requirements. The mathematics involved uses binary extension field of the form GF (2n) as the underlying field and polynomial basis for the representation of the elements in the field. A significant gain in performance is obtained by using projective coordinates for the points on the curve during the computation process

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 Side-Channel Aware Elliptic Curve Cryptosystems over Prime Fields

Elliptic Curve Cryptosystems (ECCs) are utilized as an alternative to traditional public-key cryptosystems, and are more suitable for resource limited environments due to smaller parameter size. In this dissertation we carry out a thorough investigation of side-channel attack aware ECC implementations over finite fields of prime characteristic including the recently introduced Edwards formulation of elliptic curves, which have built-in resiliency against simple side-channel attacks. We implement Joye\u27s highly regular add-always scalar multiplication algorithm both with the Weierstrass and Edwards formulation of elliptic curves. We also propose a technique to apply non-adjacent form (NAF) scalar multiplication algorithm with side-channel security using the Edwards formulation. Our results show that the Edwards formulation allows increased area-time performance with projective coordinates. However, the Weierstrass formulation with affine coordinates results in the simplest architecture, and therefore has the best area-time performance as long as an efficient modular divider is available

Efficient Arithmetic for the Implementation of Elliptic Curve Cryptography

The technology of elliptic curve cryptography is now an important branch in public-key based crypto-system. Cryptographic mechanisms based on elliptic curves depend on the arithmetic of points on the curve. The most important arithmetic is multiplying a point on the curve by an integer. This operation is known as elliptic curve scalar (or point) multiplication operation. A cryptographic device is supposed to perform this operation efficiently and securely. The elliptic curve scalar multiplication operation is performed by combining the elliptic curve point routines that are defined in terms of the underlying finite field arithmetic operations. This thesis focuses on hardware architecture designs of elliptic curve operations. In the first part, we aim at finding new architectures to implement the finite field arithmetic multiplication operation more efficiently. In this regard, we propose novel schemes for the serial-out bit-level (SOBL) arithmetic multiplication operation in the polynomial basis over F_2^m. We show that the smallest SOBL scheme presented here can provide about 26-30\% reduction in area-complexity cost and about 22-24\% reduction in power consumptions for F_2^{163} compared to the current state-of-the-art bit-level multiplier schemes. Then, we employ the proposed SOBL schemes to present new hybrid-double multiplication architectures that perform two multiplications with latency comparable to the latency of a single multiplication. Then, in the second part of this thesis, we investigate the different algorithms for the implementation of elliptic curve scalar multiplication operation. We focus our interest in three aspects, namely, the finite field arithmetic cost, the critical path delay, and the protection strength from side-channel attacks (SCAs) based on simple power analysis. In this regard, we propose a novel scheme for the scalar multiplication operation that is based on processing three bits of the scalar in the exact same sequence of five point arithmetic operations. We analyse the security of our scheme and show that its security holds against both SCAs and safe-error fault attacks. In addition, we show how the properties of the proposed elliptic curve scalar multiplication scheme yields an efficient hardware design for the implementation of a single scalar multiplication on a prime extended twisted Edwards curve incorporating 8 parallel multiplication operations. Our comparison results show that the proposed hardware architecture for the twisted Edwards curve model implemented using the proposed scalar multiplication scheme is the fastest secure SCA protected scalar multiplication scheme over prime field reported in the literature

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

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

Side-Channel Analysis: Countermeasures and Application to Embedded Systems Debugging

Side-Channel Analysis plays an important role in cryptology, as it represents an important class of attacks against cryptographic implementations, especially in the context of embedded systems such as hand-held mobile devices, smart cards, RFID tags, etc. These types of attacks bypass any intrinsic mathematical security of the cryptographic algorithm or protocol by exploiting observable side-effects of the execution of the cryptographic operation that may exhibit some relationship with the internal (secret) parameters in the device. Two of the main types of side-channel attacks are timing attacks or timing analysis, where the relationship between the execution time and secret parameters is exploited; and power analysis, which exploits the relationship between power consumption and the operations being executed by a processor as well as the data that these operations work with. For power analysis, two main types have been proposed: simple power analysis (SPA) which relies on direct observation on a single measurement, and differential power analysis (DPA), which uses multiple measurements combined with statistical processing to extract information from the small variations in power consumption correlated to the data. In this thesis, we propose several countermeasures to these types of attacks, with the main themes being timing analysis and SPA. In addition to these themes, one of our contributions expands upon the ideas behind SPA to present a constructive use of these techniques in the context of embedded systems debugging. In our first contribution, we present a countermeasure against timing attacks where an optimized form of idle-wait is proposed with the goal of making the observable decryption time constant for most operations while maintaining the overhead to a minimum. We show that not only we reduce the overhead in terms of execution speed, but also the computational cost of the countermeasure, which represents a considerable advantage in the context of devices relying on battery power, where reduced computations translates into lower power consumption and thus increased battery life. This is indeed one of the important themes for all of the contributions related to countermeasures to side- channel attacks. Our second and third contributions focus on power analysis; specifically, SPA. We address the issue of straightforward implementations of binary exponentiation algorithms (or scalar multiplication, in the context of elliptic curve cryptography) making a cryptographic system vulnerable to SPA. Solutions previously proposed introduce a considerable performance penalty. We propose a new method, namely Square-and-Buffered- Multiplications (SABM), that implements an SPA-resistant binary exponentiation exhibiting optimal execution time at the cost of a small amount of storage --- O(\sqrt(\ell)), where \ell is the bit length of the exponent. The technique is optimal in the sense that it adds SPA-resistance to an underlying binary exponentiation algorithm while introducing zero computational overhead. We then present several new SPA-resistant algorithms that result from a novel way of combining the SABM method with an alternative binary exponentiation algorithm where the exponent is split in two halves for simultaneous processing, showing that by combining the two techniques, we can make use of signed-digit representations of the exponent to further improve performance while maintaining SPA-resistance. We also discuss the possibility of our method being implemented in a way that a certain level of resistance against DPA may be obtained. In a related contribution, we extend these ideas used in SPA and propose a technique to non-intrusively monitor a device and trace program execution, with the intended application of assisting in the difficult task of debugging embedded systems at deployment or production stage, when standard debugging tools or auxiliary components to facilitate debugging are no longer enabled in the device. One of the important highlights of this contribution is the fact that the system works on a standard PC, capturing the power traces through the recording input of the sound card

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 from many researchers in different institutions because these algorithms provide security and high performance when being used in many areas such as electronic-healthcare, electronic-banking, electronic-commerce, electronic-vehicular, and electronic-governance. These algorithms heighten security against various attacks and the same time improve performance to obtain efficiencies (time, memory, reduced computation complexity, and energy saving) in an environment of constrained source and large systems. This paper presents detailed and a comprehensive survey of an update of the ECDSA algorithm in terms of performance, security, and applications