Implementation Of Elliptic Curve Cryptography Research Articles

Hardware Implementations of Elliptic Curve Cryptography Using Shift-Sub Based Modular Multiplication Algorithms

Elliptic curve cryptography (ECC) over prime fields relies on scalar point multiplication realized by point addition and point doubling. Point addition and point doubling operations consist of many modular multiplications of large operands (256 bits for example), especially in projective and Jacobian coordinates which eliminate the modular inversion required in affine coordinates for every point addition or point doubling operation. Accelerating modular multiplication is therefore important for high-performance ECC. This paper presents the hardware implementations of modular multiplication algorithms, including (1) interleaved modular multiplication (IMM), (2) Montgomery modular multiplication (MMM), (3) shift-sub modular multiplication (SSMM), (4) SSMM with advance preparation (SSMMPRE), and (5) SSMM with CSAs and sign detection (SSMMCSA) algorithms, and evaluates their execution time (the number of clock cycles and clock frequency) and required hardware resources (ALMs and registers). Experimental results show that SSMM is 1.80 times faster than IMM, and SSMMCSA is 3.27 times faster than IMM. We also present the ECC hardware implementations based on the Secp256k1 protocol in affine, projective, and Jacobian coordinates using the IMM, SSMM, SSMMPRE, and SSMMCSA algorithms, and investigate their cost and performance. Our ECC implementations can be applied to the design of hardware security module systems.

Power/Area-Efficient ECC Processor Implementation for Resource-Constrained Devices

The use of resource-constrained devices is rising nowadays, and these devices mostly operate with sensitive data. Consequently, security is a key issue for these devices. In this paper, we propose a compact ECC (elliptic curve cryptography) architecture for resource-constrained devices based on López–Dahab (LD) projective point arithmetic operations on GF(2m). To achieve an efficient area-power hardware ECC implementation, an efficient digit-serial multiplier is developed. The proposed multiplier is built on a Bivariate Polynomial Basis representation and a modified Radix-n Interleaved Multiplication (mRnIM) method (for area and power complexities reduction). Furthermore, the LD-Montgomery point multiplication algorithm is adjusted for accurate scheduling in the compact ECC architecture to eliminate data reliance and improve signal management. Meanwhile, the area complexity is reduced by reuse of resources, and clock gating and asynchronous counter are exploited to reduce the power consumption. Finally, the proposed compact ECC architecture is implemented over GF(2m) (m = 163, 233, 283, 409, and 571) on Xilinx FPGAs’ (Field-Programmable Gate Array) Virtex 5, Virtex 6, and Virtex 7, showing that the efficiency of this design outperforms to date when compared to reported works individually. It utilizes less area and consumes low power. The FPGA results clearly demonstrate that the proposed ECC architecture is appropriate for constraint-resources devices.

Topgun: An ECC Accelerator for Private Set Intersection

Elliptic Curve Cryptography (ECC), one of the most widely used asymmetric cryptographic algorithms, has been deployed in Transport Layer Security (TLS) protocol, blockchain, secure multiparty computation, and so on. As one of the most secure ECC curves, Curve25519 is employed by some secure protocols, such as TLS 1.3 and Diffie-Hellman Private Set Intersection (DH-PSI) protocol. High-performance implementation of ECC is required, especially for the DH-PSI protocol used in privacy-preserving platform. Point multiplication, the chief cryptographic primitive in ECC, is computationally expensive. To improve the performance of DH-PSI protocol, we propose Topgun, a novel and high-performance hardware architecture for point multiplication over Curve25519. The proposed architecture features a pipelined Finite-field Arithmetic Unit and a simple and highly efficient instruction set architecture. Compared to the best existing work on Xilinx Zynq 7000 series FPGA, our implementation with one Processing Element can achieve 3.14× speedup on the same device. To the best of our knowledge, our implementation appears to be the fastest among the state-of-the-art works. We also have implemented our architecture consisting of 4 Compute Groups, each with 16 PEs, on an Intel Agilex AGF027 FPGA. The measured performance of 4.48 Mops/s is achieved at the cost of 86 Watts power, which is the record-setting performance for point multiplication over Curve25519 on FPGAs.

Review on E-Health Privacy and Security through ECC, SHA-256, and Multi-Authority Approaches

The rapid digitization of healthcare services has led to the proliferation of electronic health (E-Health) systems, providing convenience and improved patient care. However, the increasing use of E-Health platforms also raises concerns about the privacy and security of sensitive medical information. This review paper explores the challenges surrounding E-Health privacy and security and presents a comprehensive approach to addressing these concerns through the implementation of Elliptic Curve Cryptography (ECC), Secure Hash Algorithm 256 (SHA-256), and multi-authority schemes. The synergistic application of these cryptographic techniques not only fortifies the protection of patient data but also establishes a robust framework for secure data sharing and management in E-Health systems.

Improved authenticated elliptic curve cryptography scheme for resource starve applications

Elliptic curve cryptography (ECC) remains the best approach to asymmetric cryptography when it comes to securing communication among communication partners in low-computing devices such as wireless sensor networks (WSN) and the Internet of Things (IoT) due to its effectiveness in generating small keys with a strong encryption mechanism. The ECC cuts down on power use and improves device performance, so it can be used in a wide range of devices that don't have a lot of resources. However, most of the existing ECC implementations suffer from implementation flaws that make them vulnerable to cryptanalysis attacks. In this study, flaws in the existing implementation of ECC are identified. A new scheme where the identified flaws are remedied was developed. The results of the security analysis show that the new scheme is an indistinguishable authenticated adaptive chosen ciphertext attack (IND-CCA3), resistant to malleability and man-in-the-middle attacks (MIMA). The results of comparative security analysis show that the mapping scheme employed in the new scheme maps any blocks of plaintext to distinct points on an elliptic curve, which makes it resistant to all attacks that the existing schemes are vulnerable to without having a negative effect on its encryption and decryption time, throughput, or power consumption.

Improved authenticated elliptic curve cryptography scheme for resource starve applications

Elliptic curve cryptography (ECC) remains the best approach to asymmetric cryptography when it comes to securing communication among communication partners in low-computing devices such as wireless sensor networks (WSN) and the Internet of Things (IoT) due to its effectiveness in generating small keys with a strong encryption mechanism. The ECC cuts down on power use and improves device performance, so it can be used in a wide range of devices that don't have a lot of resources. However, most of the existing ECC implementations suffer from implementation flaws that make them vulnerable to cryptanalysis attacks. In this study, flaws in the existing implementation of ECC are identified. A new scheme where the identified flaws are remedied was developed. The results of the security analysis show that the new scheme is an indistinguishable authenticated adaptive chosen ciphertext attack (IND-CCA3), resistant to malleability and man-in-the-middle attacks (MIMA). The results of comparative security analysis show that the mapping scheme employed in the new scheme maps any blocks of plaintext to distinct points on an elliptic curve, which makes it resistant to all attacks that the existing schemes are vulnerable to without having a negative effect on its encryption and decryption time, throughput, or power consumption.

Implementation and Performance Analysis of Elliptic Curve Cryptography using an Efficient Multiplier

In Elliptic Curve Cryptography (ECC), modular multiplication in a finite field is a resource-consuming operation. The design of an efficient multiplier for ECC implementation in FPGA is an ever-evolving research problem. In this paper, we provide an efficient implementation of ECC for GF (2<SUP>233</SUP>) in FPGA. We propose a hybrid Karatsuba multiplier which is found to be occupying lesser slices compared to other optimized Karatsuba multiplier in [1]. This, in turn, amends the efficiency of the ECC processor in the FPGA platform. The proposed processor is implemented in ZedBoard containing Xilinx XC7Z020-1CLG484C Zynq-7000 AP SoC. In comparison with other similar works, the implementation results of ECC show consequential competitiveness in hardware efficiency and we obtain an efficient multiplier and ECC processor realized in FPGA. From the application point of view, ZedBoard has an ARM Processor and hence this can be utilized in vehicle On-Board Units (OBU) because of its lightweight properties.

Survey on performance and security problems of countermeasures for passive side-channel attacks on ECC

This paper presents an analysis on the state of the art of several proposals for algorithmic countermeasures to prevent passive side-channel attacks (SCA) on elliptic curve cryptography (ECC) defined over prime fields. One of the main applications of ECC is in the field of Internet of Things, where the interconnection of devices requires public-key technology with small key sizes and high security levels. Since the secure implementation of ECC on embedded devices involves many challenges for security and efficiency, this work evaluates the trade-offs between security and performance of side-channel attack countermeasures for scalar multiplication algorithms without precomputation. The main contribution is to present a panorama of explicit solutions that may be used for the implementation of ECC mechanisms suitable for embedded devices. In addition to the security problems, some countermeasures are also analyzed.

A Novel Authentication and Key Agreement Scheme for Countering MITM and Impersonation Attack in Medical Facilities

Authentication isused to enfold the privacy of the patient to implement security onto the communication between patients and service providers. Several types of research have proposed support for anonymity for contextual privacy in medical systems that are still vulnerable to impersonation attack and Man-in-the-middle attack. By using powerful technology that is used in medical facilities, it can help in building an advanced system. However, the same powerful tools can also be used by the attackers to gain personal profits and to cause chaos. The proposed countermeasure that is to be taken to prevent this kind of attacks is by implementing mutual authentication between users, their devices/mobile devices, and the system’s cloud server, and also a key agreement scheme together with the help of Elliptic Curve Cryptography (ECC). A novel authentication scheme which consists of two phases, a signature generation, and authentication process. The ECC implementation is to ensure that the keys are thoroughly secured and is not copy- able, together with a Key generation scheme that shields the system against impersonation attacks. The usage of Elliptic Curve Digital Signature Algorithm (ECDSA), in a signature generation, on the other hand, provides users more secure way to hide the user private key and bring additional security layer before proceeding to authentication phase due to the existence of extra elements of domain parameters. Authentication is still considered as a crucial component in maintaining the security of any critical facilities that require the CIA tried and non-repudiation as a need to maintain their data. It does not only apply to medical centers, but any organizations that possess valuable data that is needed to be protected also requires strong authentication protocols. Thus, the trend for the need of novel authentication protocols will keep on rising as technology gets fancier and fancier.

DPF-ECC: A Framework for Efficient ECC With Double Precision Floating-Point Computing Power

Used ubiquitously in a huge amount of security protocols or applications, elliptic curve cryptography (ECC) is one of the most important cryptographic primitives, featuring efficiency and short key size compared with other public-key cryptosystems such as DSA and RSA. However, as a computation-intensive public-key cryptographic primitive, ECC arithmetic is still the bottleneck that restrains the overall performance of the end applications. In this paper, instead of the conventional and straightforward integer-based methods, we present a general framework to accelerate ECC schemes over prime field, called DPF-ECC, that deeply exploits double precision floating-point (DPF) computing power. The DPF-ECC framework finely manages each bit of the DPF numbers and minimizes the overhead brought by additional data format conversion, by making use of the DPF representation, the rounding operations, and fused multiply-add instruction supported by the IEEE 754 floating point standard. We also conduct two comprehensive case studies on Crandall primes and Solinas primes to demonstrate how the DPF-ECC framework is applied to the prevailing ECC schemes. To evaluate the proposed DPF-ECC framework in the real world, leveraging the floating-point computing power of GPUs, we implement Curve25519/448 and Edwards25519/448, the popular ECC schemes widely used in TLS 1.3, SSH, etc. The experimental result in Tesla P100 achieves a record-setting performance that outperforms the existing fastest integer work with 2x to 3x throughput. With dependency only on the very commonly supported IEEE 754 floating point standard, DPF-ECC framework can be a very competent and promising candidate for ECC implementation in most of general-purpose platforms.

Fast Implementation of NIST P-256 Elliptic Curve Cryptography on 8-Bit AVR Processor

In this paper, we present a highly optimized implementation of elliptic curve cryptography (ECC) over NIST P-256 curve for an 8-bit AVR microcontroller. For improving the performance of ECC implementation, we focus on optimizing field arithmetics. In particular, we optimize the modular multiplication and squaring method exploiting the state-of-the-art optimization technique, namely range shifted representation (RSR). With optimized field arithmetics, we significantly improve the performance of scalar multiplication and set the speed record for execution time of variable base scalar multiplication over NIST P-256 curve. When compared with previous works, we achieve a performance gain of 17.3% over the best previous result on the same platform. Moreover, the execution time of our result is even faster than that over the NIST P-192 curve of the well-known TinyECC library. Our result shows that RSR can be applied to all field arithmetics and evaluate the impact of the adoption of RSR over the performance of scalar multiplication. Additionally, our implementation provides a high degree of regularity to withstand side-channel attacks.

ITrust-A Trustworthy and Efficient Mapping Scheme in Elliptic Curve Cryptography.

Recently, many platforms have outsourced tasks to numerous smartphone devices known as Mobile Crowd-sourcing System (MCS). The data is collected and transferred to the platform for further analysis and processing. These data needs to maintain confidentiality while moving from smartphones to the platform. Moreover, the limitations of computation resources in smartphones need to be addressed to balance the confidentiality of the data and the capabilities of the devices. For this reason, elliptic curve cryptography (ECC) is accepted, widespread, and suitable for use in limited resources environments such as smartphone devices. ECC reduces energy consumption and maximizes devices’ efficiency by using small crypto keys with the same strength of the required cryptography of other cryptosystems. Thus, ECC is the preferred approach for many environments, including the MCS, Internet of Things (IoT) and wireless sensor networks (WSNs). Many implementations of ECC increase the process of encryption and/or increase the space overhead by, for instance, incorrectly mapping points to EC with extra padding bits. Moreover, the wrong mapping method used in ECC results in increasing the computation efforts. This study provides comprehensive details about the mapping techniques used in the ECC mapping phase, and presents performance results about widely used elliptic curves. In addition, it suggests an optimal enhanced mapping method and size of padding bit to secure communications that guarantee the successful mapping of points to EC and reduce the size of padding bits.

Performance Evaluation of Scalar Multiplication in Elliptic Curve Cryptography Implementation using Different Multipliers Over Binary Field GF (2233)

This paper presents a point multiplication processor over the binary field GF (2233) with internal registers integrated within the point-addition architecture to enhance the Performance Index (PI) of scalar multiplication. The proposed design uses one of two types of finite field multipliers, either the Montgomery multiplier or the interleaved multiplier supported by the additional layer of internal registers. Lopez Dahab coordinates are used for the computation of point multiplication on Koblitz Curve (K-233bit). In contrast, the metric used for comparison of the implementations of the design on different types of FPGA platforms is the Performance Index.&#x0D; The first approach attains a performance index of approximately 0.217610202 when its realization is over Virtex-6 (6vlx130tff1156-3). It uses an interleaved multiplier with 3077 register slices, 4064 lookup tables (LUTs), 2837 flip-flops (FFs) at a maximum frequency of 221.6Mhz. This makes it more suitable for high-frequency applications. The second approach, which uses the Montgomery multiplier, produces a PI of approximately 0.2228157 when its implementation is on Virtex-4 (6vlx130tff1156-3). This approach utilizes 3543 slices, 2985 LUTs, 3691 FFs at a maximum frequency of 190.47MHz. Thus, it is found that the implementation of the second approach on Virtex-4 is more suitable for applications with a low frequency of about 86.4Mhz and a total number of slices of about 12305.

Demystifying elliptic curve cryptography: Curve selection, implementation and countermeasures to attacks

Elliptic curve cryptography (ECC) has been the main choice for the security experts from some time now. As the IoT has enveloped the world around us; ECC has became more popular as security solution for the resource-constrained devices which covers the physical layer of IoT. Different libraries are available for ECC. In this paper we have discussed various important aspects in the implementation of ECC, such as finite fields, selection of curve model and curves along with the attacks on ECC. Implementation of ECC has been carried out on NIST curve P-256 with ElGamal encryption.

High-performance Implementation of Elliptic Curve Cryptography Using Vector Instructions

Elliptic curve cryptosystems are considered an efficient alternative to conventional systems such as DSA and RSA. Recently, Montgomery and Edwards elliptic curves have been used to implement cryptosystems. In particular, the elliptic curves Curve25519 and Curve448 were used for instantiating Diffie-Hellman protocols named X25519 and X448. Mapping these curves to twisted Edwards curves allowed deriving two new signature instances, called Ed25519 and Ed448, of the Edwards Digital Signature Algorithm. In this work, we focus on the secure and efficient software implementation of these algorithms using SIMD parallel processing. We present software techniques that target the Intel AVX2 vector instruction set for accelerating prime field arithmetic and elliptic curve operations. Our contributions result in a high-performance software library for AVX2-ready processors. For example, our library computes digital signatures 19% (for Ed25519) and 29% (for Ed448) faster than previous optimized implementations. Also, our library improves by 10% and 20% the execution time of X25519 and X448, respectively.

Lightweight Implementations of NIST P-256 and SM2 ECC on 8-bit Resource-Constraint Embedded Device

Elliptic Curve Cryptography (ECC) now is one of the most important approach to instantiate asymmetric encryption and signature schemes, which has been extensively exploited to protect the security of cyber-physical systems. With the advent of the Internet of Things (IoT), a great deal of constrained devices may require software implementations of ECC operations. Under this circumstances, the SM2, a set of public key cryptographic algorithms based on elliptic curves published by Chinese Commercial Cryptography Administration Office, was standardized at ISO in 2017 to enhance the cyber-security. However, few research works on the implementation of SM2 for constrained devices have been conducted. In this work, we fill this gap and propose our efficient, secure, and compact implementation of scalar multiplication on a 256-bit elliptic curve recommended by the SM2, as well as a comparison implementation of scalar multiplication on the same bit-length elliptic curve recommended by NIST. We re-design some existent techniques to fit the low-end IoT platform, namely 8-bit AVR processors, and our implementations evaluated on the desired platform show that the SM2 algorithms have competitive efficiency and security with NIST, which would work well to secure the IoT world.

High-performance ECC processor architecture design for IoT security applications

In recent years, the usage of elliptic curve cryptography (ECC) in IoT applications is steadily increasing. The end nodes in IoT applications demand optimized device performance in terms of reduced power consumption and improved computing speed while not compromising on the security of the connected devices. ECC provides better security standards compared with many conventional cryptographic algorithms providing further scope to optimize the performance parameters. This work focuses on improving the key parameters like computing speed, area required for hardware implementation of ECC and demonstrates an efficient way of using the hardware resource sharing and scheduling mechanisms in elliptic curve group operations in affine coordinates which is crucial for implementation of scalar multiplication over prime field $$\mathbb {F}_{p}$$ . With the proposed scalar multiplication hardware architecture, we have achieved a good area-delay product and a significant reduction in cycle count when compared with other reported designs using the same affine coordinates. The proposed architecture has been implemented with 256 bits in both Xilinx Kintex-7 and Virtex-7 FPGA devices. The FPGA synthesis results show that a throughput of 68.52 kbps at a clock frequency of 124.2 MHz is achieved for $$\mathbb {F}_{256}$$ and the computation time is reduced around 1 ms without using any DSP slices.

ECC based Authentication Scheme for Securing Data Contents over Open Wireless Network Systems

Multimedia contents have been increasingly available over the Internet as wireless networks systems are continuously growing popular. Unlimited access from various users has led to unauthorized access of third parties or adversaries. This paper deals with the implementation of elliptic curve cryptography (ECC) based user authentication for securing multimedia contents over the Internet. The ECC technique has been incorporated with the advanced encryption standard (AES) algorithm to ensure the complexity of the proposed authentication scheme and to guarantee authenticity of multimedia services.

Dynamic Scalable Elliptic Curve Cryptographic Scheme and Its Application to In-Vehicle Security

The design of unified, efficient, and lightweight cryptographic platform for resource-constrained on-board devices such as sensors, microcontrollers, and actuators in the context of Internet of Vehicles remains an open and challenging problem, for both academic and industry. Elliptic curve cryptography (ECC) is considered as a promising encryption algorithm for the next generation communications, as it could provide the same strong security level using relatively smaller key size when compared to the currently used Rivest–Shamir–Adleman algorithm. However, traditional ECCs have the disadvantage of using a fixed curve, making it very easy to be intensively analyzed while being hard to construct a united platform for on-board devices with processors of different instruction lengths. To mitigate the above problem, this paper suggests a dynamic scalable elliptic curve cryptosystem. To synchronize the curve in use, a curve list of different security levels is generated and preserved on both parties. Since both parties randomly choose the curve and the prime number, a extra security level could be provided, so that the security level can still remain the same even using smaller key sizes, while the computation efficiency will be enhanced and the power consumption will be reduced, which is especially suitable for the application in on-board embedded devices. Detailed experimental results illustrate that the presented scheme improves the efficiency by 30% in average when compared with traditional ECC implementations on a similar security level. Therefore, the proposed scalable ECC scheme as a unified cryptographic platform is more economic for these on-board devices in vehicles.

Feed-Forward Back-Propagation Neural Networks in Side-Channel Information Characterization

The safety of cryptosystems, mainly based on algorithmic improvement, is still vulnerable to side-channel attacks (SCA) based on machine learning. Multi-class classification based on neural networks and principal components analysis (PCA) can be powerful tools for pattern recognition and classification of side-channel information. In this paper, an experimental investigation was conducted to explore the efficiency of various architectures of feed-forward back-propagation (FFBP) neural networks and PCA against side-channel attacks. The experiment is performed on the data leakage of an FPGA implementation of elliptic curve cryptography (ECC). Our results show that the proposed method is a promising method for SCA with an overall accuracy of 88% correct classification.