Parallelizing Explicit Formula for Arithmetic in the Jacobian of Hyperelliptic Curves

A subexponential algorithm for discrete logarithms over hyperelliptic curves of large genus over GF( q)

The Diffie-Hellman problem as a cryptographic primitive plays an important role in modern cryptology. The Bit Security or Hard-Core Bits of Diffie-Hellman problem in arbitrary finite cyclic group is a long-standing open problem in cryptography. Until now, only few groups have been studied. Hyperelliptic curve cryptography is an alternative to elliptic curve cryptography. Due to the recent cryptanalytic results that the best known algorithms to attack hyperelliptic curve cryptosystems of genus \(g<3\) are the generic methods and the recent implementation results that hyperelliptic curve cryptography in genus 2 has the potential to be competitive with its elliptic curve cryptography counterpart. In this paper, we generalize Boneh and Shparlinksi’s method and result about elliptic curve to the case of Jacobians of hyperelliptic curves. We prove that the least significant bit of each coordinate of hyperelliptic curves Diffie-Hellman secret value in genus 2 is hard as the entire Diffie-Hellman value, and then we also show that any bit is hard as the entire Diffie-Hellman value. Finally, we extend our techniques and results to hyperelliptic curves of any genus.

Due to short operand size, Hyperelliptic Curve Cryptosystem (HECC) of genus 3 is well suited for all kinds of embedded processor architectures, where resources such as storage, time or power are constrained. In the implementation of HECC, a significant step is the selection of secure hyperelliptic curves on which the Jacobian is constructed and speed up the scalar multiplications in the Jacobians of hyperelliptic curves. In this paper, we have explored various possible attacks to the discrete logarithm in the Jacobian of a Hyperelliptic Curve (HEC) that are to be considered to establish a secure HEC, analysed addition and doubling of divisor which are the prime steps of scalar multiplication and then proposed certain improvements in the existing explicit formula that will result in a performance gain for HECC of genus 3.

This paper presents the design and implementation of a hyperelliptic curve cryptography (HECC) coprocessor over affine and projective coordinates, along with measurements of its performance, hardware complexity, and power consumption. We applied several design techniques, including parallelism, pipelining, and loop unrolling, in designing field arithmetic units, group operation units, and scalar multiplication units to improve the performance and power consumption. Our affine and projective coordinate-based HECC processors execute in 0.436 ms and 0.531 ms, respectively, based on the underlying field GF(2 89 ). These results are about five times faster than those for previous hardware implementations and at least 13 times better in terms of area-time products. Further results suggest that neither case is superior to the other when considering the hardware complexity and performance. The characteristics of our proposed HECC coprocessor show that it is applicable to high-speed network applications as well as resource-constrained environments, such as PDAs, smart cards, and so on.

Our main contribution is an efficient implementation of the Gaudry - Schost and Galbraith - Ruprai point-counting algorithms on Jacobians of hyperelliptic curves. Both of them are low memory variants of Matsuo - Chao - Tsujii (MCT) Baby-Step Giant-Step-like algorithm. We present an optimal memory restriction (a time-memory tradeoff) that minimizes the runtime of the algorithms. This tradeoff allows us to get closer in practical computations to theoretical bounds of expected runtime at 2.45√N and 2.38a√N for the Gaudry - Schost and Galbraith - Ruprai algorithms, respectively. Here N is the size of the 2-dimensional searching space, which is as large as the Jacobian group order, divided by small modulus m, precomputed by using other techniques. Our implementation profits from the multithreaded regime and we provide some performance statistics of operation on different size inputs. This is the first open-source parallel implementation of 2-dimensional Galbraith - Ruprai algorithm.

Due to the unprecedented growth of data, edge computing is transforming our lives in a better manner by increasing the performance and reducing the latency. It is the key framework deployed close to the data for processing effectively by connecting millions of devices together. Though it handles accelerated growth of data efficiently in real-time, data security and edge device authentication are still a nightmare. On the contrary, it is not possible to use complex protocols for authentication of edge devices due to its resource-constrained in nature. To address this issue, a state-of-art Hyperelliptic Curve Diffie–Hellman-based two-server Password-Only Authenticated Key Exchange Protocol is proposed in this paper by utilizing the properties of a geometrical shape, namely, tetrahedron. As a primary mandate, edge devices are authenticated to centralize cloud servers that store password information as tetrahedron properties using Hyperelliptic Curve Cryptography and Diffie–Hellman key exchange mechanisms. The proposed work is compared with similar existing works and proves its strength by considering communication cost, computational cost, security defense rate, and attack resistance rate as its core evaluation parameters.

Vehicular ad hoc networks (VANETs) have become an essential part of the intelligent transportation system because they provide secure communication among vehicles, enhance vehicle safety, and improve the driving experience. However, due to the openness and vulnerability of wireless networks, the participating vehicles in a VANET system are prone to a variety of cyberattacks. To secure the privacy of vehicles and assure the authenticity, integrity, and nonrepudiation of messages, numerous signature schemes have been employed in the literature on VANETs. The majority of these solutions, however, are either not fully secured or entail high computational costs. To address the above issues and to enable secure communication between the vehicle and the roadside unit (RSU), we propose a certificateless aggregate signature (CLAS) scheme based on hyperelliptic curve cryptography (HECC). This scheme enables participating vehicles to share their identities with trusted authorities via an open wireless channel without revealing their identities to unauthorized participants. Another advantage of this approach is its capacity to release the partial private key to participating devices via an open wireless channel while keeping its identity secret from any other third parties. A provable security analysis through the random oracle model (ROM), which relies on the hyperelliptic curve discrete logarithm problem, is performed, and we have proven that the proposed scheme is unforgeable against Type 1 (FGR1) and Type 2 (FGR2) forgers. The proposed scheme is compared with relevant schemes in terms of computational cost and communication overhead, and the results demonstrate that the proposed scheme is more efficient than the existing schemes in maintaining high-security levels.

In this book chapter, the authors discuss some important communication issues to obtain a highly scalable computing system. They consider the CGM (Coarse-Grained Multicomputer) model, a realistic computing model to obtain scalable parallel algorithms. The communication cost is modeled by the number of communication rounds and the objective is to design algorithms that require the minimum number of communication rounds. They discuss some important issues and make considerations of practical importance, based on our previous experience in the design and implementation of parallel algorithms. The first issue is the amount of data transmitted in a communication round. For a practical implementation to be successful they should attempt to minimize this amount, even when it is already within the limit allowed by the CGM model. The second issue concerns the trade-off between the number of communication rounds which the CGM attempts to minimize and the overall communication time taken in the communication rounds. Sometimes a larger number of communication rounds may actually reduce the total amount of data transmitted in the communications rounds. These two issues have guided us to present efficient parallel algorithms for the string similarity problem, used as an illustration.

Biomedical Engineering, usually known as Bioengineering, is among the fastest developing and one of most important interdisciplinary fields today. It connects natural and technical sciences, for all of which biological and medical phenomena, computation, and data management play important roles in science and industry. Distributed computing and parallel algorithms have proved to be effective in solving the problem with high computational complexity in a wide range of domains, including areas of computational bioengineering. This special edition collects high-quality research papers in the field of application of distributed computing systems in bioengineering applications. This issue is the achievement of the developments of modern computer and biomedical technology. Most bioengineering or bioinformatics methods need to access and analyses large amounts of data. For that it is necessary to achieve an effective way that from Big data creates knowledge and useful applications. The nature of such scientific data/information/knowledge demands the use of a powerful computing and intelligent database management systems. Areas of current and future research include advanced topics in this special issue. The paper from Juhasz et al. present a novel, GPU-based streaming architecture that has the potential for drastically reducing execution times and at the same time providing simultaneous 2D and 3D visualization facilities. The system uses a highly-optimised and re-configurable pipeline of CPU and GPU cores that attempts to exploit the tremendous computing power whenever possible. The system can process live data arriving from an EEG device or data stored in EEG data files. The computer drives a large display wall system of four 46 monitors that provides a 4K-resolution drawing surface for visualizing raw EEG data, potential maps and various 3D views of the patient's head. Authors from the group of M. Gusev presents the scalable balancer in Cloud computing extended to Dew Computing level. Given a successful implementation of scalable low level load balancer, implemented on the network layer. The scalability is proved with series of experiments. The experiments showed that it adds small latency of several milliseconds and thus it slightly reduces the performance when the distributed system is underutilized. Rybicki et al. demonstrate how to realize a Software-as-a-Service solution for a variety of science software using container technologies. The presented solution utilized DARIAH-DE research infrastructure based on OpenStack and UNICORE grid to deliver an extensible solution for the digital humanities domain. The paper from authors Forer et al. describe the extension of Cloudflow to support ApacheSpark without any adaptions to already implemented pipelines. The described performance evaluation demonstrates that Spark can bring an additional boost for analyzing next generation sequencing(NGS) data to the eld of genetics. The Cloudflow framework is open source and freely available at https://github.com/genepi/cloudflow . The paper by Memon et al. describes an approach for executing computational jobs across HPC and HTC resources while operating on geographically dispersed data via a global federated file system. The solution is realized as a new framework that integrates UNICORE and GFFS to provide a standards-based environment to support large scale data intensive computations frequent in today's biomedical analyses. The paper Parallel algorithm for the state space exploration, authored by L. Allal et al., proposes a new automatic verification technique based on model checking that determines whether a given system satisfies its specification. Such a technique suffers from the state explosion problem when traversing all possible states of systems. A new synchronized parallel algorithm (SPA) of exploration is proposed based on a fixed number of threads. Exhaustive comparative studies between the standard parallel exploration algorithm in SPIN and the new SPA show that the SPA performs slightly better regarding the execution time and memory complexity. Smart systems in telemedicine frequently use intelligent sensor devices at large scale. Practitioners can monitor non-stop the vital parameters of hundreds of patients in real-time. The most important pillars of remote patient monitoring services are communication and data processing. Large scale data processing is done mainly using workflows. In the paper Eszter et al. give a brief overview of the different check pointing techniques and propose two new provenance based check pointing algorithms which uses the information stored in the workflow structure to dynamically change the frequency of check pointing and can be efficiently used for dynamic health care smart systems.

Cryptographic algorithms are used in a large variety of different applications to ensure security services. It is, thus, very interesting to investigate various implementation platforms. Hyperelliptic curve schemes are cryptographic primitives to which a lot of attention was recently given due to the short operand size compared to other algorithms. They are specifically interesting for special-purpose hardware. This paper provides a comprehensive investigation of high-efficient HEC architectures.We propose a genus-2 hyperelliptic curve cryptographic coprocessor using affine coordinates. We implemented a special class of hyperelliptic curves, namely using the parameter h(x)=x and f=x 5+f 1 x+f 0 and the base field GF(289). In addition, we only consider the most frequent case in our implementation and assume that the other cases are handled, e.g. by the protocol.We provide three different implementations ranging from high speed to moderate area. Hence, we provide a solution for a variety of applications. Our high performance HECC coprocessor is 78.5% faster than the best previous implementation and our low area implementation utilizes only 22.7% of the area that the smallest published design uses. Taking into account both area and latency, our coprocessor is an order of magnitude more efficient than previous implementations. We hope that the work at hand provides a step towards introducing HEC systems in practical applications.

Unlocking massively parallel spectral proper orthogonal decompositions in the PySPOD package

An explicit formula concerning curve intersections equivalent to the time evolution of the periodic discrete Toda lattice (pdTL) is presented. First, the time evolution is realized as a point addition on a hyperelliptic curve, which is the spectral curve of the pdTL, then the point addition is translated into curve intersections. Next, it is shown that the curves which appear in the curve intersections are explicitly given by using the conserved quantities of the pdTL. Finally, the formulation is lifted to the framework of tropical geometry and a tropical geometric realization of the periodic box–ball system is constructed via tropical curve intersections.

We propose a reduced equation for hyperelliptic curves of genus 2 over finite fields ? q of q elements with characteristic different from 2 and 5. We determine the number of isomorphism classes of genus-2 hyperelliptic curves having an ? q -rational Weierstrass point. These results have applications to hyperelliptic curve cryptography.

Elliptic and Hyper Elliptic Curve Cryptography Over Finite Field Fp

Side-channel attacks are a prominent threat to the security of cryptographic implementations. Differently from the traditional black-box attacks, which exploit the inputs and outputs of cryptographic schemes, side-channel attacks partially access the inner working of the scheme as well, by observing the physical leakage emitted by the device executing cryptographic algorithms. A notable example is the class of power-analysis attacks, which exploits the power consumption of the underlying device to recover the secret keys of the implemented cryptosystem. Since their first presentation in the late 1990s, the problem of securing cryptographic systems in the presence of side-channel leakages received significant attention by the cryptographic community. In particular, the theory community has concentrated its research efforts on formally modeling the side-channel leakages and on designing cryptographic schemes \textit{provably} secure in such leakage models. However, conceiving a formal model that realistically captures possible physical leakages and constructing primitives sufficiently efficient to be deployed in practice are still challenging research objectives. This research area is usually referred to as leakage-resilient cryptography. In this dissertation, we examine three main research directions. The first one focuses on the technique of masking, which is the most popular approach to counteract power-analysis attacks. In this context, our work analyzes possible methods to improve the efficiency of masked implementations, with the goal of overcoming the decrease in performance, which is a common drawback of masking. In particular, we provide new and more efficient algorithms for the computation of basic operations in two kinds of masking schemes: the Boolean masking and the Inner Product masking. The second research direction of this work addresses the problem of formalizing security requirements for hardware implementations. While provable security of software-oriented masking has been studied since the origin of leakage resilient cryptography, the more complex task of formally proving the security of hardware-oriented masking is receiving attention by the community only recently. We discuss the different challenges of this field, and we introduce a new theoretical model, the robust probing model, capturing the conditions required for securely implementing complex algorithms in practice, providing a guide to practical implementers. Finally, the last part of this work is concerned with the protection of algorithms employed in lattice-based constructions, which are studied in the field of post-quantum cryptography. While several works investigate the application of masking schemes to standard cryptosystems, for most post-quantum schemes this is still an ongoing research. We contribute to this research area, by applying the masking countermeasure to a binomial sampler which is at the base of many lattice-based cryptosystems.