Elliptic Curve Scalar Multiplication Research Articles

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.

Secure ECDSA SRAM-PUF Based on Universal Single/Double Scalar Multiplication Architecture.

Physically unclonable functions (PUFs) are crucial for enhancing cybersecurity by providing unique, intrinsic identifiers for electronic devices, thus ensuring their authenticity and preventing unauthorized cloning. The SRAM-PUF, characterized by its simple structure and ease of implementation in various scenarios, has gained widespread usage. The soft-decision Reed-Muller (RM) code, an error correction code, is commonly employed in these designs. This paper introduces the design of an RM code soft-decision attack algorithm to reveal its potential security risks. To address this problem, we propose a soft-decision SRAM-PUF structure based on the elliptic curve digital signature algorithm (ECDSA). To improve the processing speed of the proposed secure SRAM-PUF, we propose a custom ECDSA scheme. Further, we also propose a universal architecture for the critical operations in ECDSA, elliptic curve scalar multiplication (ECSM), and elliptic curve double scalar multiplication (ECDSM) based on the differential addition chain (DAC). For ECSMs, iterations can be performed directly; for ECDSMs, a two-dimensional DAC is constructed through precomputation, followed by iterations. Moreover, due to the high similarity of ECSM and ECDSM data paths, this universal architecture saves hardware resources. Our design is implemented on a field-programmable gate array (FPGA) and an application-specific integrated circuit (ASIC) using a Xilinx Virtex-7 and an TSMC 40 nm process. Compared to existing research, our design exhibits a lower bit error rate (2.7×10-10) and better area-time performance (3902 slices, 6.615 μs ECDSM latency).

Enhanced Scalar Multiplication Algorithm over Prime Field Using Elliptic Net

Scalar multiplication in elliptic curve cryptography is the most expensive and time-consuming operation. The elliptic curve cryptography attracted interest due to the development of modern technology since it could offer the equivalent high level of security with a reduced length of key. Therefore, improving elliptic curve scalar multiplication performance has always been the primary goal of cryptography. In this paper, a novel scalar multiplication algorithm based on the modified double and double add via elliptic net with Karatsuba method was proposed in order to enhance the efficiency of scalar multiplication. In the experimental results, the elliptic net equivalence sequence was applied to the Twisted Edwards curve together with safe curves of numsp384t1 and numsp512t1. At the point operational level, the proposed method reduced the cost of multiplication by 46.15% and 42.30% for double and double add, respectively, when compared to elliptic net using eight blocks method. The proposed double lowered the multiplication cost by 12.5% and the squaring cost by 20% when compared to elliptic net using ten temporary variables method. Following this, proposed double add cost reductions of 6.25% and 20% were obtained to multiplication and squaring. At the field operational level, in comparison to the binary method, the eight-block elliptic net method, and the elliptic net method with ten temporary variables for the 384 bits scenario, the developed scalar multiplication algorithm obtained cost reductions of 57.6%, 31.3%, and 13.2%, respectively. On 512 bits with similar comparison, the designed algorithm exhibited better performance by averages 59.2%, 31.0% and 13.2%. The results signified that the designed algorithm over prime field performed better at the point and field operational levels with larger scalar bit size.

Non-Profiled Unsupervised Horizontal Iterative Attack against Hardware Elliptic Curve Scalar Multiplication Using Machine Learning

While IoT technology makes industries, cities, and homes smarter, it also opens the door to security risks. With the right equipment and physical access to the devices, the attacker can leverage side-channel information, like timing, power consumption, or electromagnetic emanation, to compromise cryptographic operations and extract the secret key. This work presents a side channel analysis of a cryptographic hardware accelerator for the Elliptic Curve Scalar Multiplication operation, implemented in a Field-Programmable Gate Array and as an Application-Specific Integrated Circuit. The presented framework consists of initial key extraction using a state-of-the-art statistical horizontal attack and is followed by regularized Artificial Neural Networks, which take, as input, the partially incorrect key guesses from the horizontal attack and correct them iteratively. The initial correctness of the horizontal attack, measured as the fraction of correctly extracted bits of the secret key, was improved from 75% to 98% by applying the iterative learning.

A High Speed and Area Efficient Processor for Elliptic Curve Scalar Point Multiplication for GF($2^m$)

A High Speed and Area Efficient Processor for Elliptic Curve Scalar Point Multiplication for GF($2^m$)

An area-efficient and low-latency elliptic curve scalar multiplication accelerator over prime field

For the elliptic curve cryptography (ECC) over prime field, this paper presents an area-efficient and low-latency elliptic curve scalar multiplication (ECSM) accelerator, which supports to process any prime number p ≤ 256 bits. The proposed accelerator is based on a parallel hardware implementation of iterative digit-digit Montgomery multiplication and an optimized data flow architecture for elliptic curve point operations. The proposed design is implemented in Xilinx Virtex-7 FPGA. The experimental results show that the proposed architecture occupies 23.7k look-up tables (LUTs) and performs single 256-bit scalar multiplication in 0.56 ms. The area-time product of the proposed architecture is only 13.36. Compared with other state-of-the-art ECSM implementations, the proposed architecture has obvious advantage in terms of the area-time product.

Elliptic Curve Scalar Multiplication Operation: a Survey Study

Scalar multiplication is the fundamental operation in the elliptic curve cryptosystem. It involves calculating the integer multiple of a specific elliptic curve point. It involves three levels: field, point, and scalar arithmetic. Scalar multiplication will be significantly more efficient overall if the final level is improved. By reducing the hamming weight or the number of operations in the scalar representation, one can raise the level of scalar arithmetic. This paper reviews some of the algorithms and techniques that improve the elliptic curve scalar multiplication in terms of the third level.

E$${^2}$$CSM: efficient FPGA implementation of elliptic curve scalar multiplication over generic prime field GF(p)

E$${^2}$$CSM: efficient FPGA implementation of elliptic curve scalar multiplication over generic prime field GF(p)

High-Performance Implementation of the Identity-Based Signature Scheme in IEEE P1363 on GPU

Identity-based cryptography is proposed to solve the complicated certificate management of traditional public-key cryptography. The pairing computation and high-level tower extension field arithmetic turn out to be the performance bottleneck of pairing-based signature schemes. Graphics processing units have been increasingly popular for general-purpose computing in recent years. They have shown a lot of promise in speeding up cryptographic schemes such as AES, RSA, and ECDSA. However, to our knowledge, the research on parallel implementation of pairings and identity-based cryptographic schemes on graphics processing units is somewhat outdated. Therefore, in this article, we implement the identity-based signature scheme in the IEEE P1363 Standard on a modern NVIDIA RTX 3060 card. We convert the pairing computation in signature verification into a product of pairings with fixed arguments and therefore avoid the scalar multiplication in 𝔾 2 . Then we employ the precomputation technique to improve the elliptic curve scalar multiplication, exponentiation in \(\mathbb {F}_{p^{12}}\) and the pairing computation. We also apply PTX ISA to multiple-precision arithmetic. Experiments demonstrate that our implementation can perform 43,856/46,753/39,798 pairings/sec for the Optimal Ate pairing, the pairing with a fixed argument, and two pairings with fixed arguments, respectively. Peak throughputs of signature generation and verification can achieve 322.6 and 40.6 kops/sec over the BN254 curve.

Stealing Keys From Hardware Wallets: A Single Trace Side-Channel Attack on Elliptic Curve Scalar Multiplication Without Profiling

Stealing Keys From Hardware Wallets: A Single Trace Side-Channel Attack on Elliptic Curve Scalar Multiplication Without Profiling

Secure and Compact Elliptic Curve Scalar Multiplication with Optimized Inversion

Abstract Elliptic curve cryptography (ECC) is a typical public key cryptography technique that can ensure equivalent security with considerably smaller key sizes than Rivest-Shamir-Adleman (RSA). Hence, various implementations based on ECC are recommended for block chain and Internet of Things (IoT) devices. Because elliptic curve scalar multiplication (ECSM) is a fundamental computation in ECC, enhancing the security and efficiency of ECSM is important. ECSM specifies a scalar multiplication algorithm and elliptic curve addition formulae. Elliptic curve addition formulae on affine coordinates are compact from a memory cost perspective but weak against side channel attacks. Elliptic curve complete addition (CA) formulae can achieve secure ECSMs but are inefficient. A newly proposed secure ECSM, which uses a right-to-left (RL) scalar multiplication algorithm and (extended) affine coordinates, takes advantage of elliptic curve addition formulae on affine coordinates. However, it can only scan the input scalars from right to left. We propose new ECSMs, which can scan the input scalars from left to right (LR) based on (extended) affine coordinates. We also prove that our LR ECSMs satisfy secure generality without requiring exceptional computations. We enhance the efficiency of both LR and RL ECSMs with optimized inversion. Our LR ECSM with a memory of 12 field elements reduces that of the Montgomery ladder and Joye’s LR with CA formulae by 36.84% and that of 2-ary RL with (extended) affine coordinates by 14.29%, respectively. Our compact ECSMs are fit for applications on IoT devices and block chain, with a critical memory requirement.

An improvement of both security and reliability for elliptic curve scalar multiplication Montgomery algorithm

An improvement of both security and reliability for elliptic curve scalar multiplication Montgomery algorithm

Lightweight Architecture for Elliptic Curve Scalar Multiplication over Prime Field

In this paper, we present a novel lightweight elliptic curve scalar multiplication architecture for random Weierstrass curves over prime field Fp. The elliptic curve scalar multiplication is executed in Jacobian coordinates based on the Montgomery ladder algorithm with (X,Y)-only common Z coordinate arithmetic. At the finite field operation level, the adder-based modular multiplier and modular divider are optimized by the pre-calculation method to reduce the critical path while maintaining low resource consumption. At the group operation level, the point addition and point doubling methods in (X,Y)-only common Z coordinate arithmetic are modified to improve computation parallelism. A compact scheduling method is presented to improve the architecture’s performance, which includes appropriate scheduling of finite field operations and specific register connections. Compared with existing works, our design is implemented on the FPGA platform without using DSPs or BRAMs for higher portability. It utilizes 6.4~6.5k slices in Kintex-7, Virtex-7, and ZYNQ FPGA and executes an elliptic curve scalar multiplication for a field size of 256-bit in 1.73 ms, 1.70 ms, and 1.80 ms, respectively. Additionally, our design is resistant to timing attacks, simple power analysis attacks, and safe-error attacks. This architecture outperforms most state-of-the-art lightweight designs in terms of area-time products.

Design of an Integrated Cryptographic SoC Architecture for Resource-Constrained Devices

One of the active research areas in recent years that has seen researchers from numerous related fields converging and sharing ideas and developing feasible solutions is the area of hardware security. The hardware security discipline deals with the protection from vulnerabilities by way of physical devices such as hardware firewalls or hardware security modules rather than installed software programs. These hardware security modules use physical security measures, logical security controls, and strong encryption to protect sensitive data that is in transit, in use, or stored from unauthorized interferences. Without mechanisms to circumvent the ever-evolving attacking strategies on hardware devices and the data that they process or store, billions of dollars will always be lost to attackers who ply their trade by targeting such vulnerable devices. This paper, therefore, proposes an integrated cryptographic SoC architecture solution to this menace. The proposed architecture provides security by way of key exchange, management, and encryption. The proposed architecture is based on a True Random Number generator core that generates secret keys that are used in Elliptic Curve Diffie-Hellman Key Exchange to perform elliptic curve scalar multiplication to obtain public and shared keys after the exchange of the public keys. The proposed architecture further relies on a Key Derivation Function based on the CubeHash algorithm to obtain Derived Keys that provide the needed security using the ChaCha20_Poly1305 Authenticated Encryption with Associated (AEAD) Data Core. The proposed Integrated SoC architecture is interconnected by AMBA AHB-APB on-chip bus and the system is scheduled and controlled using the PicoRV32 opensource RISC-V processor. The proposed architecture is tested and verified on the Virtex-4 FPGA board using a custom-designed GUI desktop application.

Adversarial Attack Protection Scalar Multiplication for WSNs Resistance Machine-Learning Side-channel Attack

Wireless sensor networks (WSNs) have limited storage and low processing capabilities. However, these devices may be captured by opponents, so the security requirements are particularly strict. With the development of side-channel attacks based on the machine-learning, designing a safe elliptic curve scalar multiplication for computing-limited devices like WSNs has become a major challenge. Based on the adversarial attack technology, a secure scalar multiplication is proposed in this article. The main contributions are: (1) We propose an efficient non-zero form (NZF) encoding algorithm that can be applied to various types of elliptic curves; (2) we have designed a secure scalar multiplication algorithm that can resist against conventional side-channel attacks such as SPA, DA, DPA, RPA, and ZPA; and (3) we propose an adversarial protection mechanism based on blind point technology and NZF coding, which can prevent side-channel attacks based on machine learning. The algorithm has no precomputation and is suitable for low communication frequency, low calculation amount, and high security requirements. Especially, it can be applied to lightweight equipment such as WSN and IoT.

Optimization of elliptic curve scalar multiplication using constraint based scheduling

Elliptic Curve Cryptography is public key cryptography that features smaller keys, ciphertexts, and signatures and is faster than RSA at the same security level. Scalar multiplication is the main and the most compute-intensive operation in the generation of keys. Point Addition, Doubling and Inversion are the basic operations for scalar multiplication. Inversion is a very expensive operation as compared to multiplication, addition and squaring in the finite fields with an affine coordinate system. López-Dahab coordinates are the best alternative to reduce the inversion overhead in scalar computation. Area, Delay and Power trade-offs are the main constraints in hardware implementations of scalar multiplication. In this paper, optimization of elliptic curve scalar multiplication using constraint-based scheduling for the López-Dahab coordinate system is proposed. Data dependency graphs of point addition and doubling are modified for optimization of area and delay. The proposed architecture is implemented on Altera Stratix-II FPGA. The constraint is applied on the field multiplication operation and the considerable area is reduced. The proposed architecture computes scalar multiplication in 11.43 μs and takes 9856 ALMs. The performance comparison with state of the art shows that area is reduced by 41.21%, delay is reduced by 2.4% and Area-Delay-Product is improved.

An Efficient and Secure Revocation-Enabled Attribute-Based Access Control for eHealth in Smart Society.

The ever-growing ecosystem of the Internet of Things (IoT) integrating with the ever-evolving wireless communication technology paves the way for adopting new applications in a smart society. The core concept of smart society emphasizes utilizing information and communication technology (ICT) infrastructure to improve every aspect of life. Among the variety of smart services, eHealth is at the forefront of these promises. eHealth is rapidly gaining popularity to overcome the insufficient healthcare services and provide patient-centric treatment for the rising aging population with chronic diseases. Keeping in view the sensitivity of medical data, this interfacing between healthcare and technology has raised many security concerns. Among the many contemporary solutions, attribute-based encryption (ABE) is the dominant technology because of its inherent support for one-to-many transfer and fine-grained access control mechanisms to confidential medical data. ABE uses costly bilinear pairing operations, which are too heavy for eHealth’s tiny wireless body area network (WBAN) devices despite its proper functionality. We present an efficient and secure ABE architecture with outsourcing intense encryption and decryption operations in this work. For practical realization, our scheme uses elliptic curve scalar point multiplication as the underlying technology of ABE instead of costly pairing operations. In addition, it provides support for attribute/users revocation and verifiability of outsourced medical data. Using the selective-set security model, the proposed scheme is secure under the elliptic curve decisional Diffie–Hellman (ECDDH) assumption. The performance assessment and top-ranked value via the help of fuzzy logic’s evaluation based on distance from average solution (EDAS) method show that the proposed scheme is efficient and suitable for access control in eHealth smart societies.

New Regular Sliding Window Algorithms for Elliptic Curve Scalar Point Multiplication

New Regular Sliding Window Algorithms for Elliptic Curve Scalar Point Multiplication

Regular SPA Resistant Fixed-base Comb Method for Elliptic Curve Scalar Multiplication

Regular SPA Resistant Fixed-base Comb Method for Elliptic Curve Scalar Multiplication

Online Template Attacks: Revisited

An online template attack (OTA) is a powerful technique previously used to attack elliptic curve scalar multiplication algorithms. This attack has only been analyzed in the realm of power consumption and EM side channels, where the signals leak related to the value being processed. However, microarchitecture signals have no such feature, invalidating some assumptions from previous OTA works.In this paper, we revisit previous OTA descriptions, proposing a generic framework and evaluation metrics for any side-channel signal. Our analysis reveals OTA features not previously considered, increasing its application scenarios and requiring a fresh countermeasure analysis to prevent it.In this regard, we demonstrate that OTAs can work in the backward direction, allowing to mount an augmented projective coordinates attack with respect to the proposal by Naccache, Smart and Stern (Eurocrypt 2004). This demonstrates that randomizing the initial targeted algorithm state does not prevent the attack as believed in previous works.We analyze three libraries libgcrypt, mbedTLS, and wolfSSL using two microarchitecture side channels. For the libgcrypt case, we target its EdDSA implementation using Curve25519 twist curve. We obtain similar results for mbedTLS and wolfSSL with curve secp256r1. For each library, we execute extensive attack instances that are able to recover the complete scalar in all cases using a single trace.This work demonstrates that microarchitecture online template attacks are also very powerful in this scenario, recovering secret information without knowing a leakage model. This highlights the importance of developing secure-by-default implementations, instead of fix-on-demand ones.