Point Multiplication Operation Research Articles

GMC-crypto: Low latency implementation of ECC point multiplication for generic Montgomery curves over GF(p)

Elliptic Curve Cryptography (ECC) is the front-runner among available public key cryptography (PKC) schemes due to its potential to offer higher security per key bit. All ECC-based cryptosystems heavily rely on point multiplication operation where its efficient realization has attained notable focus in the research community. Low latency implementation of the point multiplication operation is frequently required in high-speed applications such as online authentication and web server certification. This paper presents a low latency ECC point multiplication architecture for Montgomery curves over generic prime filed GF(p). The proposed architecture is able to operate for a general prime modulus without any constraints on its structure. It is based on a new novel pipelined modular multiplier developed using the Montgomery multiplication and the Karatsuba-Offman technique with a four-part splitting methodology. The Montgomery ladder approach is adopted on a system level, where a high-speed scheduling strategy to efficiently execute GF(p) operations is also presented. Due to these circuit and system-level optimizations, the proposed design delivers low-latency results without a significant increase in resource consumption. The proposed architecture is described in Verilog-HDL for 256-bit key lengths and implemented on Virtex-7 and Virtex-6 FPGA platforms using Xilinx ISE Design Suite. On the Virtex-7 FPGA platform, it performs a 256-bit point multiplication operation in just 110.9 us with a throughput of almost 9017 operations per second. The implementation results demonstrate that despite its generic nature, it produces low latency as compared to the state-of-the-art. Therefore, it has prominent prospects to be used in high-speed authentication and certification applications.

Cost-Effective Signcryption for Securing IoT: A Novel Signcryption Algorithm Based on Hyperelliptic Curves

Security and efficiency remain a serious concern for Internet of Things (IoT) environments due to the resource-constrained nature and wireless communication. Traditional schemes are based on the main mathematical operations, including pairing, pairing-based scalar multiplication, bilinear pairing, exponential operations, elliptic curve scalar multiplication, and point multiplication operations. These traditional operands are cost-intensive and require high computing power and bandwidth overload, thus affecting efficiency. Due to the cost-intensive nature and high resource requirements, traditional approaches are not feasible and are unsuitable for resource-limited IoT devices. Furthermore, the lack of essential security attributes in traditional schemes, such as unforgeability, public verifiability, non-repudiation, forward secrecy, and resistance to denial-of-service attacks, puts data security at high risk. To overcome these challenges, we have introduced a novel signcryption algorithm based on hyperelliptic curve divisor multiplication, which is much faster than other traditional mathematical operations. Hence, the proposed methodology is based on a hyperelliptic curve, due to which it has enhanced security with smaller key sizes that reduce computational complexity by 38.16% and communication complexity by 62.5%, providing a well-balanced solution by utilizing few resources while meeting the security and efficiency requirements of resource-constrained devices. The proposed strategy also involves formal security validation, which provides confidence for the proposed methodology in practical implementations.

A Study of Elliptic Curve Cryptography and Its Applications

This paper aims to provide a comprehensive review on Elliptic Curve Cryptography (ECC), a public key cryptographic system and its applications. The paper discusses important mathematical properties and operations of elliptic curves, like point addition and multiplication operations and its implementation in cryptographic methods such as encryption and decryption. This paper provides a detailed workout on important mathematical problems on elliptic curves and ECC which provides insight into working of essential cryptographic techniques in ECC. And the paper also provides a literature review of research works based on ECC in various fields such as Internet of Things (IoT), Cloud computing, Blockchain and Image Security. And the paper further provides insight into the recent applications of ECC in fields like IoT and Blockchain by comprehensively discussing the proposed mechanism for each of the recent applications and also briefly discussing the security of the proposed mechanism.

Efficient Hardware Implementation of Elliptic-Curve Diffie–Hellman Ephemeral on Curve25519

Hardware architecture optimized for implementing the elliptic-curve Diffie–Hellman ephemeral (ECDHE) on 256-bit Montgomery elliptic curves presents unique challenges, particularly for resource-constrained IoT and mobile devices. This work aims to provide an efficient hardware implementation of ECDHE on Curve25519, including a dedicated finite state machine (FSM) designed to handle point multiplication and ECDHE operations, utilizing constant-time algorithms and a unified memory block for resource management. Additionally, we introduce an optimized modular computation unit that covers modular addition, subtraction, multiplication, and inversion. Our proposed hardware architecture enhances the efficiency of ECDHE operations while maintaining low resource utilization, considerably reduced latency, and low power consumption. Synthesized on the Xilinx Artix-7 platform, our design boasts 64,000 Slices and a clock speed of 102 MHz, and it computes an ECDHE scalar multiplication operation in 1.1 ms, consuming 117 mW. The proposed hardware design can be applied to various platforms, including mobile devices and IoT systems.

Area‐time efficient point multiplication architecture on twisted Edwards curve over general prime field GF(p)

AbstractElliptic curve point multiplication is the main primitive required in almost all security schemes using elliptic curve cryptography (ECC). It is the leading computationally intensive operation that sets the overall performance of the associated cryptosystem. This work presents a highly novel area–time efficient elliptic curve point multiplier over a general prime field . It is based on an efficient radix‐23 parallel multiplier, which performs a ‐bit multiplication in clock cycles. On the system level, the twisted Edwards curves with unified point addition using projective coordinates are adopted, where an efficient scheduling technique is presented to schedule several operations on deployed modular arithmetic units. Due to the introduced optimization at different stages of the design, latency, hardware resource requirement, and total clock cycle count are reduced significantly. Synthesis, and implementation of the proposed design over different Xilinx FPGA platforms are completed using the Xilinx ISE Design Suite tool for key sizes of 192, 224, and 256 bits. The 256‐bit Xilinx Virtex‐7 FPGA implementation reveals that it completes a single point multiplication operation in 0.8 ms and occupies 6.7K FPGA slices in a clock cycle count of 132.2K. It produces significantly better area–time product and throughput per slice than the contemporary designs. The proposed design also has the potential to counter simple power analysis and timing attacks. Thus, it is an elegant solution to develop ECC‐based cryptosystems for applications, where both speed and hardware resource consumption are important.

FPGA Implementation for Elliptic Curve Cryptography Algorithm and Circuit with High Efficiency and Low Delay for IoT Applications.

The Internet of Things requires greater attention to the security and privacy of the network. Compared to other public-key cryptosystems, elliptic curve cryptography can provide better security and lower latency with shorter keys, rendering it more suitable for IoT security. This paper presents a high-efficiency and low-delay elliptic curve cryptographic architecture based on the NIST-p256 prime field for IoT security applications. A modular square unit utilizes a fast partial Montgomery reduction algorithm, demanding just a mere four clock cycles to complete a modular square operation. The modular square unit can be computed simultaneously with the modular multiplication unit, consequently improving the speed of point multiplication operations. Synthesized on the Xilinx Virtex-7 FPGA platform, the proposed architecture completes one PM operation in 0.08 ms using 23.1 k LUTs at 105.3 MHz. These results show significantly better performance compared to that in previous works.

A Hybrid Approach for Efficient and Secure Point Multiplication on Binary Edwards Curves

The focus of this article is to present a novel crypto-accelerator architecture for a resource-constrained embedded system that utilizes elliptic curve cryptography (ECC). The architecture is built around Binary Edwards curves (BEC) to provide resistance against simple power analysis (SPA) attacks. Furthermore, the proposed architecture incorporates several optimizations to achieve efficient hardware resource utilization for the point multiplication process over GF(2m). This includes the use of a Montgomery radix-2 multiplier and the projective coordinate hybrid algorithm (combination of Montgomery ladder and double and add algorithm) for scalar multiplication. A two-stage pipelined architecture is employed to enhance throughput. The design is modeled in Verilog HDL and verified using Vivado and ISE design suites from Xilinx. The obtained results demonstrate that the proposed BEC accelerator offers significant performance improvements compared to existing solutions. The obtained throughput over area ratio for GF(2233) on Virtex-4, Virtex-5, Virtex-6, and Virtex-7 Xilinx FPGAs are 9.43, 14.39, 26.14, and 28.79, respectively. The computation time required for a single point multiplication operation on the Virtex-7 device is 19.61 µs. These findings indicate that the proposed architecture has the potential to address the challenges posed by resource-constrained embedded systems that require high throughput and efficient use of available resources.

High performance HITA based Binary Edward Curve Crypto processor for FPGA platforms

In an embedded and resource-constrained environment, Elliptic Curve Cryptography (ECC) has been noted as an efficient and suitable methodology for achieving information security via public-key cryptography. However, the drawback of ECC is its lack of unifiedness in point operation that makes it prone to side-channel attack. Also, ECC does not satisfy the completeness property due to which the addition formula is not defined for all the pairs of input points. Edward curve, with its unified addition law and completeness property, proved to be the answer to aforementioned flaws. High throughput while maintaining low resource is a key issue for elliptic curve cryptography (ECC) hardware implementations in many applications. This paper presents the implementation of a Binary Edward curve Crypto processor over GF(2233) for FPGA platforms. The architecture is modified to perform scalar multiplication in a parallel manner using two hybrid Karatsuba field multipliers. Field inversion being one of the most tedious operations while reconversion, is also performed in a parallel manner using an efficient Hex Itoh-Tsujii inversion algorithm. The hardware resources are shared for performing point operations and inversion. Exploiting parallelism in point and inversion operations has resulted in reduction of the clock cycles consumed and the resultant architecture is more efficient in terms of throughput over area. The design takes 0.038 ms on Xilinx Virtex-4 and 0.031 ms on Virtex-7 FPGA platforms to perform a 233-bit point multiplication operation. It takes 73.57%, 13.71%, 14.76% and 48.76% more efficient than existing scalar multiplication with BEC. This proposed scalable, side-channel attack resilient design outperforms the existing techniques with respect to throughput over area.

A Flexible Architecture for Cryptographic Applications: ECC and PRESENT

This work presents a flexible/unified hardware architecture of Elliptic-curve Cryptography (ECC) and PRESENT for cryptographic applications. The features of the proposed work are (i) computation of only the point multiplication operation of ECC over for a 163-bit key generation, (ii) execution of only the variant of an 80-bit PRESENT block cipher for data encryption & decryption and (iii) execution of point multiplication operation (ECC algorithm) along with the data encryption and decryption (PRESENT algorithm). To establish an area overhead for the flexible design, dedicated hardware architectures of ECC and PRESENT are implemented in the first step, and a sum of their hardware area is computed. Then, the implementation of the proposed flexible architecture for ECC and PRESENT algorithms is presented. Implementation results regarding the area, clock cycles, latency, clock frequency, and power after the place-and-route level on Xilinx Virtex-5, Virtex-6, and Virtex-7 FPGA devices are presented. Hence, the implementation results and comparisons show that the proposed architecture suits applications demanding flexible implementation of cryptographic applications.

Lightweight anonymous authentication protocol for resource-constrained smart home devices based on elliptic curve cryptography

The communication channel between the smart home devices and the remote users is susceptible to numerous privacy and security compromise attacks. To address these issues, many authentication protocols have been developed. However, majority of these security schemes have vulnerabilities that may still be exploited to wreck havoc in smart homes. For instance, protocols based on low entropy passwords can be broken by polynomial time adversaries. Apart from security and privacy challenges, efficiency of the entire authentication process is another challenge that needs to be solved. To this end, most of the conventional smart home authentication protocols incur extensive storage, computation as well as communication overheads which are unsuitable for resource limited smart home devices. In this paper, an anonymous lightweight protocol is developed, based on one-way hashing and elliptic curve point multiplication operations. Formal verification of this protocol is executed using ProVerif while its informal security analysis demonstrates its robustness against majority of the smart home privacy and security attacks. In terms of operational efficiency, comparative analysis is carried out which shows that it incurs relatively low computation, storage and communication overheads.

Classification for Fruit Fly Images Using Convolutional Neural Networks

The discriminative information of fruit fly images often exists in a very fine region, resulting in hard extracting the desired features from the entire image level. Hence, it is a tough task to obtain the discriminative features in the fine-grained image for the classification of fruit fly images. To address this, here proposed a convolutional neural network of bilinear pooling based on feature fusion for the classification of fruit fly images. Firstly, Gaussian blur is performed on fruit fly images to reduce the detailed level of fruit fly images, which conveniently extracts high-level features. In the fruit fly images processed by Gaussian blur, the convolutional layers in the proposed model (i.e., consisting of CNN A and CNN B) are used for the features extraction from the processed images. Then, these extracted features are fused by using the matrix dot multiplication in the Bilinear layers. According to these fused features, the softmax layer performs the classification of fruit fly images. Experimental results on the 4000 image set containing different fruit fly morphology show that the propose method not only outperforms the state-of-the-art methods in the classification accuracy of fruit fly images, but also the precision, recall and score are above 94.87% on testing set and validation set. We find that image feature fusion has positive effects on promoting the accuracy of image classification. We also demonstrate that using feature connection operation after performing the matrix point multiplication operation is beneficial to feature fusion, instead of using feature connection operation directly. In addition, our findings indicates that convolutional neural networks easily obtain the desired features from the images performed by Gaussian blur.

FPGA Implementation of High-Efficiency ECC Point Multiplication Circuit

The authentication of Internet of Things (IoT) devices based on the Physical Unclonable Function (PUF) is widely adopted in the information security domain. However, the leakage of PUF responses in an authentication system reduces its privacy and security. To improve its security, we can utilize the Elliptic Curve Cryptography (ECC) algorithm with different key lengths to encrypt the PUF response arbitrarily. Point multiplication is the most time-consuming operation in ECC because of its complex calculation process, which seriously affects the efficiency of the PUF response encryption. In order to solve this problem, a point multiplier based on binary field with reconfigurable key lengths of 233, 283, 409 and 571 is designed in this paper. In our method, by reusing the underlying computing units, the resources needed for point multiplication are effectively reduced. What it is more innovative is that double point multiplication operations with a key length of less than 283 bits can be performed simultaneously in the elaborate designed point multiplication circuit, which can effectively speed up the encryption process of ECC. The circuit is implemented on Xilinx Virtex-6 FPGA. The experiment results show the single point multiplication times of 233, 283, 409 and 571 key lengths are 19.33, 22.36, 41.36 and 56.5 μs, respectively, under the clock frequency of 135 MHz. In addition, it only needs 19.33 μs to perform two-point multiplication operations when the key length is 233 bits at the same time. When the key length is 283 bits, the point multiplication operation can be performed twice in 22.36 μs.

A Low-Complexity Edward-Curve Point Multiplication Architecture

The Binary Edwards Curves (BEC) are becoming more and more important, as compared to other forms of elliptic curves, thanks to their faster operations and resistance against side channel attacks. This work provides a low-complexity architecture for point multiplication computations using BEC over GF(2233). There are three major contributions in this article. The first contribution is the reduction of instruction-level complexity for unified point addition and point doubling laws by eliminating multiple operations in a single instruction format. The second contribution is the optimization of hardware resources by minimizing the number of required storage elements. Finally, the third contribution is to reduce the number of required clock cycles by incorporating a 32-bit finite field digit-parallel multiplier in the datapath. As a result, the achieved throughput over area ratio over GF(2233) on Virtex-4, Virtex-5, Virtex-6 and Virtex-7 Xilinx FPGA (Field Programmable Gate Array) devices are 2.29, 19.49, 21.5 and 20.82, respectively. Furthermore, on the Virtex-7 device, the required computation time for one point multiplication operation is 18 µs, while the power consumption is 266 mW. This reveals that the proposed architecture is best suited for those applications where the optimization of both area and throughput parameters are required at the same time.

Image Encryption and Authentication With Elliptic Curve Cryptography and Multidimensional Chaotic Maps

Many researchers have used the properties of the popular Elliptic Curve Cryptography(ECC) to devise a stronger and faster image encryption algorithm to assure the secrecy of images during online transmission. In this paper, a robust Elliptic curve based image encryption and authentication model for both grayscale and color images has been proposed. The model uses the secure Elliptic Curve Diffie-Hellman(ECDH) key exchange to compute a shared session key along with the improved ElGamal encoding scheme. 3D and 4D Arnold Cat maps are used to effectively scramble and transform the values of plain image pixels. A well-structured digital signature is used to verify the authenticity of the encrypted image prior to decryption. The model produces good-quality cipher images with an average entropy of 7.9993 for grayscale and 7.99925 for the individual components of color images. The model has high average NPCR of 99.6%, average UACI of 33.3% and low correlation for both grayscale and color images. The model has low computational costs with minimized point multiplication operations. The proposed model is robust with high resilience against statistical, differential, chosen-plaintext(CPA), known-plaintext(KPA) and occlusion attacks.

ECS‐CP‐ABE: A lightweight elliptic curve signcryption scheme based on ciphertext‐policy attribute‐based encryption to secure downlink multicast communication in edge envisioned advanced metering infrastructure networks

AbstractAs an integral part of the smart grid, advanced metering infrastructure (AMI) implements two‐way communication to enable the smart grid achieving the desired performance improvements over the legacy grid. Integrating communication in addition to the inherited weakness of the power grid paves the way to countless number of security threats to target its communication networks, therefore ensuring secure data exchange is a key concern for AMI networks. In this regard, many schemes are proposed in the literature to secure this two‐way communication, however, very little effort has been focused on multicast transmissions carried out between the control center and a group of smart meters. In order to empower efficient, authentic, and confidential data exchange, this article proposes a new signcryption scheme for multicast downlink communication in edge envisioned AMI networks. The proposed scheme is constructed based on ciphertext‐policy attribute‐based encryption to attain secure fine‐grained access control for the multirecipient communication between the utility control center and a group of smart meters. The novelty of our scheme lies behind the fact that it is an elliptic curve cryptosystem that is pairing free, in opposite to most ABE schemes. Instead of the complex bilinear pairing operations, we exploited faster elliptic curve point multiplication operations to construct the proposed signcryption model. we demonstrate the strength of our protocol in defending against passive, collusion, reply, data modification, and signature forgery attacks. Simulation results prove the efficiency of our protocol in terms of messages size and computation complexity when compared with other schemes.

A privacy protection scheme of smart meter for decentralized smart home environment based on consortium blockchain

In order to ensure the reasonable dispatch and utilization of power in smart homes, decentralized smart homes need to send power information to the power control center, and the power control center need to send control instructions to smart homes. However, the bi-directional interaction between smart homes and the power control center may lead to a series of security problems, such as smart meters of smart homes collect fine-grained real-time power consumption data and upload it to the control center, which will lead to the leakage of users’ privacy. The existing schemes protect users’ privacy by signcryption and data aggregation usually, and elliptic curve bilinear pairing or exponential operation are used in most of them, whose computation costs is high. In addition, the existing schemes for centralized storage of power consumption data may result in some security problems, such as single point failure, malicious tampering of data, etc. Aiming at the problems above, a privacy protection scheme of smart meter for decentralized smart home environment based on consortium blockchain is proposed, elliptic curve point multiplication operation is used in the certificateless aggregated ring signcryption algorithm, which reduces the computing and communication costs; distributed storage of users’ data with consortium blockchain is used to solve the problem of single point failure and data tampering. Theoretical analysis proves that the scheme not only guarantees users’ privacy, but also has characteristics of confidentiality and unforgeability, and experiments show that the scheme has lower computing and communication costs.

Conditional Privacy-Preserving Authentication Scheme Without Using Point Multiplication Operations Based on Elliptic Curve Cryptography (ECC)

Existing conditional privacy-preserving authentication schemes utilized in Vehicular Ad-hoc Networks (VANETs) to satisfy security and privacy requirements essentially depend on point multiplication operations. Achieving repaid verification method of the message is commonly suffer performance efficiency from resulting overheads. We propose a conditional privacy-preserving authentication scheme to secure communication and perform better performance efficiency in this article. The proposed scheme only depends on an elliptic curve cryptography (ECC) based on a point addition operation instead of a point multiplication operation during signing and verifying messages. In the joining phase of the proposed scheme, the vehicle requires the joining process for the broadcasting traffic-related message to others or nearby RSU within its communication range. After obtaining the pseudonym and secret key from RSU, the vehicle is considered as a registered node in VANET. This article utilizes a Burrows-Abadi-Needham (BAN) logic to evidence that the proposed scheme fulfill successfully mutual authentication. The formal security phase shows that security and privacy requirements are satisfied by the proposed scheme. The performance efficiency shows that our proposed scheme has lower overhead in terms of computation cost compared with other recent schemes since a point multiplication operations based o ECC are not used. Therefore, the computation costs of the message signing, individual-authentication and batch-authentication in our proposed scheme are decreased by 99.3%, 99.7% and 98.1%, respectively.

Secure outsourcing of modular inverses and scalar multiplications on elliptic curves

Currently, pricey data analysis urges outsourcing computations, such as modular inverse and point multiplication in elliptic curve cryptosystem. However, it raises security concerns on the untrusted cloud. Thus, we propose two outsourcing protocols for these two operations, respectively. For efficiency, in the first protocol, only three modular multiplications required by the client to outsource a modular inverse operation. In the second protocol, a point multiplication operation can be carried out just by doing two addition operations on elliptic curves. For security, in both protocols, the input is randomly split into two parts to meet both input and output privacy. The security is theoretically proved under a single server and two non-colluding servers' models, respectively. Furthermore, the cheating behaviours of the servers can be detected by probabilities of 100% and 75%, respectively. Besides the achieved security goal, experiments confirm the significant improvement on efficiency in comparison with the traditional ones.

Secure outsourcing of modular inverses and scalar multiplications on elliptic curves

Currently, pricey data analysis urges outsourcing computations, such as modular inverse and point multiplication in elliptic curve cryptosystem. However, it raises security concerns on the untrusted cloud. Thus, we propose two outsourcing protocols for these two operations, respectively. For efficiency, in the first protocol, only three modular multiplications required by the client to outsource a modular inverse operation. In the second protocol, a point multiplication operation can be carried out just by doing two addition operations on elliptic curves. For security, in both protocols, the input is randomly split into two parts to meet both input and output privacy. The security is theoretically proved under a single server and two non-colluding servers' models, respectively. Furthermore, the cheating behaviours of the servers can be detected by probabilities of 100% and 75%, respectively. Besides the achieved security goal, experiments confirm the significant improvement on efficiency in comparison with the traditional ones.

Redundant-Signed-Digit-Based High Speed Elliptic Curve Cryptographic Processor

In this paper, a high speed elliptic curve cryptographic (ECC) processor for National Institute of Standards and Technology (NIST) recommended prime [Formula: see text] is proposed. The modular arithmetic components in the proposed ECC processor are highly optimized at both architectural level and circuit level. Redundant-signed-digit (RSD) arithmetic is adopted in the modular arithmetic components to avoid lengthy carry propagation delay. A high speed modular multiplier is designed based on an efficient segmentation and pipelining strategy. The clock cycle count is reduced as result of the segmentation, whereas operating frequency and throughput are significantly increased due to the pipelining. An optimized pipelined architecture for modular division is also presented which is suitable for the design of ECC processor using projective coordinates. The Joye’s double and add (DAA) algorithm based on [Formula: see text]-only common [Formula: see text] (co-[Formula: see text]) coordinate is adopted at the system level for its regular and efficient behavior. The proposed ECC processor is flexible and can be implemented using any field programmable gate array (FPGA) family or standard cell libraries. The proposed ECC processor executes a single elliptic curve (EC) point multiplication (PM) operation in 0.47[Formula: see text]ms at a maximum frequency of 327[Formula: see text]MHz on Virtex-6 FPGA. The implementation results demonstrate that the proposed ECC processor outperforms the other contemporary designs reported in the literature in terms of speed and [Formula: see text] metrics.