Koblitz curves over quadratic fields

In this work, we retake an old idea that Koblitz presented in his landmark paper, where he suggested the possibility of defining anomalous elliptic curves over the base field F4. We present a careful implementation of the base and quadratic field arithmetic required for computing the scalar multiplication operation in such curves. We also introduce two ordinary Koblitz-like elliptic curves defined over F4 that are equipped with efficient endomorphisms. To the best of our knowledge these endomorphisms have not been reported before. In order to achieve a fast reduction procedure, we adopted a redundant trinomial strategy that embeds elements of the field F4^m, with m a prime number, into a ring of higher order defined by an almost irreducible trinomial. We also present a number of techniques that allow us to take full advantage of the native vector instructions of high-end microprocessors. Our software library achieves the fastest timings reported for the computation of the timing-protected scalar multiplication on Koblitz curves, and competitive timings with respect to the speed records established recently in the computation of the scalar multiplication over binary and prime fields

High Speed and Low-Complexity Hardware Architectures for Elliptic Curve-Based Crypto-Processors

The elliptic curve cryptography (ECC) has been identified as an efficient scheme for public-key cryptography. This thesis studies efficient implementation of ECC crypto-processors on hardware platforms in a bottom-up approach. We first study efficient and low-complexity architectures for finite field multiplications over Gaussian normal basis (GNB). We propose three new low-complexity digit-level architectures for finite field multiplication. Architectures are modified in order to make them more suitable for hardware implementations specially focusing on reducing the area usage. Then, for the first time, we propose a hybrid digit-level multiplier architecture which performs two multiplications together (double-multiplication) with the same number of clock cycles required as the one for one multiplication. We propose a new hardware architecture for point multiplication on newly introduced binary Edwards and generalized Hessian curves. We investigate higher level parallelization and lower level scheduling for point multiplication on these curves. Also, we propose a highly parallel architecture for point multiplication on Koblitz curves by modifying the addition formulation. Several FPGA implementations exploiting these modifications are presented in this thesis. We employed the proposed hybrid multiplier architecture to reduce the latency of point multiplication in ECC crypto-processors as well as the double-exponentiation. This scheme is the first known method to increase the speed of point multiplication whenever parallelization fails due to the data dependencies amongst lower level arithmetic computations. Our comparison results show that our proposed multiplier architectures outperform the counterparts available in the literature. Furthermore, fast computation of point multiplication on different binary elliptic curves is achieved

Generation, Verification, and Attacks on Elliptic Curves and their Applications in Signal Protocol

Elliptic curves (EC) are widely studied due to their mathematical and cryptographic properties. Cryptographers have used the properties of EC to construct elliptic curve cryptosystems (ECC). ECC are based on the assumption of hardness of special instances of the discrete logarithm problem in EC. One of the strong merits of ECC is providing the same cryptographic strength with smaller key size compared to other public key cryptosystems. A 256 bit ECC can provide similar cryptographic strength as a 3072 bit RSA cryptosystem. Due to smaller key sizes, elliptic curves are an attractive option in devices with limited storage capacity. It is therefore essential to understand how to generate these curves, verify their correctness and assure that they are resistant against attacks. The security of an EC cryptosystem is determined by the choice of the curve that is used in that cryptosystem. Over the years, a number of elliptic curves were introduced for cryptographic use. Elliptic curves such as FRP256V1, NIST P-256, Secp256k1 or SM2 curve are widely used in many applications like cryptocurrencies, transport layer protocol and Internet messaging applications. Another type of popular curves are Curve25519 introduced by Dan Bernstein and Curve448 introduced by Mike Hamburg, which are used in an end to end encryption protocol called Signal. This protocol is used in popular messaging applications like WhatsApp, Signal Messenger and Facebook Messenger. Recently, there has been a growing distrust among security researchers against the previously standardized curves. We have seen backdoors in the elliptic curve cryptosystems like the DUAL_EC_DRBG function that was standardized by NIST, and suspicious random seeds that were used in NIST P-curves. We can say that many of the previously standardized curves lack transparency in their generation and verification. We focus on transparent generation and verification of elliptic curves. We generate curves based on NIST standards and propose new standards to generate special types of elliptic curves. We test their resistance against the known attacks that target the ECC. Finally, we demonstrate ECDLP attacks on small curves with weak structure

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

Elliptic curve cryptography: Generation and validation of domain parameters in binary Galois Fields

Elliptic curve cryptography (ECC) is an increasingly popular method for securing many forms of data and communication via public key encryption. The algorithm utilizes key parameters, referred to as the domain parameters. These parameters must adhere to specific characteristics in order to be valid for use in the algorithm. The American National Standards Institute (ANSI), in ANSI X9.62, provides the process for generating and validating these parameters. The National Institute of Standards and Technology (NIST) has identified fifteen sets of parameters; five for prime fields, five for binary fields, and five for Koblitz curves. The parameter generation and validation processes have several key issues. The first is the fast reduction within the proper modulus. The modulus chosen is an irreducible polynomial having degree greater than 160. Choosing irreducible polynomials of a particular order is less critical since they have isomorphic properties, mathematically. However, since there are differences in performance, there are standards that determine the specific polynomials chosen. The NIST standards are also based on word lengths of 32 bits. Processor architecture, primality, and validation of irreducibility are other important characteristics. The area of ECC that is researched is the generation and validation processes, as they are specified for binary Galois Fields F (2m). The rationale for the parameters, as computed for 32 bit and 64 bit computer architectures, and the algorithms used for implementation, as specified by ANSI, NIST and others, are examined. The methods for fast reduction are also examined as a baseline for understanding these parameters. Another aspect of the research is to determine a set of parameters beyond the 571-bit length that meet the necessary criteria as determined by the standards

Group law computations on Jacobians of hyperelliptic curves

We derive an explicit method of computing the composition step in Cantor’s algorithm for group operations on Jacobians of hyperelliptic curves. Our technique is inspired by the geometric description of the group law and applies to hyperelliptic curves of arbitrary genus. While Cantor’s general composition involves arithmetic in the polynomial ring F_q[x], the algorithm we propose solves a linear system over the base field which can be written down directly from the Mumford coordinates of the group elements. We apply this method to give more efficient formulas for group operations in both affine and projective coordinates for cryptographic systems based on Jacobians of genus 2 hyperelliptic curves in general form

A Survey Report On Elliptic Curve Cryptography

The paper presents an extensive and careful study of elliptic curve cryptography (ECC) and its applications. This paper also discuss the arithmetic involved in elliptic curve  and how these curve operations is crucial in determining the performance of cryptographic systems. It also presents  different forms of elliptic curve in various coordinate system , specifying which is most widely used and why. It also explains how isogenenies between elliptic curve  provides the secure ECC. Exentended form of elliptic curve i.e hyperelliptic curve has been presented here with its pros and cons. Performance of ECC and HEC is also discussed based on scalar multiplication and DLP. Keywords: Elliptic curve cryptography (ECC), isogenies, hyperelliptic curve (HEC) , Discrete Logarithm Problem (DLP), Integer  Factorization , Binary Field, Prime FieldDOI:http://dx.doi.org/10.11591/ijece.v1i2.8