Elliptic Curve Digital Signature Algorithm Signature Research Articles

Implementation of Elliptic Curve Cryptosystem with Bitcoin Curves on SECP256k1, NIST256p, NIST521p, and LLL

Very recent attacks like ladder leaks demonstrated the feasibility of recovering private keys with side-channel attacks using just one bit of secret nonce. ECDSA nonce bias can be exploited in many ways. Some attacks on ECDSA involve complicated Fourier analysis and lattice mathematics. This paper will enable cryptographers to identify efficient ways in which ECDSA can be cracked on curves NIST256p, SECP256k1, NIST521p, and weak nonce, kind of attacks that can crack ECDSA and how to protect yourself. Initially, we begin with an ECDSA signature to sign a message using the private key and validate the generated signature using the shared public key. Then we use a nonce or a random value to randomize the generated signature. Every time we sign, a new verifiable random nonce value is created, and a way in which the intruder can discover the private key if the signer leaks any one of the nonce values. Then we use Lenstra–Lenstra–Lovasz (LLL) method as a black box, we will try to attack signatures generated from bad nonce or bad random number generator (RAG) on NIST256p, SECP256k1 curves. The combination of nonce generation, post-message signing, and validation in ECDSA helps achieve Uniqueness, Authentication, Integrity, and Non-Repudiation. The analysis is performed by considering all three curves for the implementation of the Elliptic Curve Digital Signature Algorithm (ECDSA). The comparative analysis for each of the selected curves in terms of computational time is done with the leak of nonce and with the Lenstra–Lenstra–Lovasz method to crack ECDSA. The average computational costs to break ECDSA with curves NIST256p, NIST521p, and SECP256k1 are 0.016, 0.34,0.46 respectively which is almost zero depicting the strength of the algorithm. The average computational costs to break ECDSA with curves SECP256K1 and NIST256p using LLL are 2.9 and 3.4 respectively

A Multi-Party Functional Signatures Scheme for Private Blockchain

Digital signature technology is essential for ensuring the authenticity and unforgeability of transactions in a private blockchain framework. In some scenarios, transactions require verification from multiple parties, each of whom needs to authenticate different parts of the transaction. To address this issue, researchers have developed multi-party ECDSA (Elliptic Curve Digital Signature Algorithm) signature schemes. However, these schemes either need to consider the authentication of different parts of the transaction or generate an aggregated signature. This paper proposes a novel solution that combines functional signatures and multi-party ECDSA signatures to create a multi-party functional signature for private blockchains. Compared to previous constructions, the proposed scheme ensures that each part of the transaction is verified. Furthermore, when the aggregate signature of the entire transaction cannot be verified, this scheme identifies the specific part of the transaction for which the signature authentication fails instead of rejecting the entire transaction. This paper uses a smart contract to securely deploy the proposed scheme and authenticate the f in functional signatures. The constructed scheme also provides security under the existential unforgeability of the ECDSA signature, even if n−1 parties are corrupted, assuming a total of n parties. The scheme of this paper successfully conducted experiments on a personal computer, with three users taking approximately 343 ms, six users taking 552 ms, and nine users taking 791 ms.

Research on Elliptic Curve Crypto System with Bitcoin Curves – SECP256k1, NIST256p, NIST521p and LLL

Very recent attacks like ladder leak demonstrated feasibility to recover private key with side channel attacks using just one bit of secret nonce. ECDSA nonce bias can be exploited in many ways. Some attacks on ECDSA involve complicated Fourier analysis and lattice mathematics. In this paper will enable cryptographers to identify efficient ways in which ECDSA can be cracked on curves NIST256p, SECP256k1, NIST521p and weak nonce, kind of attacks that can crack ECDSA and how to protect yourself. Initially we begin with ECDSA signature to sign a message using private key and validate the generated signature using the shared public key. Then we use a nonce or a random value to randomize the generated signature. Every time we sign, a new verifiable random nonce value is created and way in which the intruder can discover the private key if the signer leaks any one of the nonce value. Then we use Lenstra–Lenstra–Lovasz (LLL) method as a black box, we will try to attack signatures generated from bad nonce or bad random number generator (RAG) on NIST256p, SECP256k1 curves. The analysis is performed by considering all the three curves for implementation of Elliptic Curve Digital Signature Algorithm (ECDSA).The comparative analysis for each of the selected curves in terms of computational time is done with leak of nonce and with Lenstra–Lenstra–Lovasz method to crack ECDSA. The average computational costs to break ECDSA with curves NIST256p, NIST521p and SECP256k1 are 0.016, 0.34, 0.46 respectively which is almost to zero depicts the strength of algorithm. The average computational costs to break ECDSA with curves SECP256K1 and NIST256p using LLL are 2.9 and 3.4 respectively.

Hybrid approach of parallel implementation on CPU–GPU for high-speed ECDSA verification

Since the advent of deep belief network deep learning technology in 2006, artificial intelligence technology has been utilized in various convergence areas, such as autonomous driving and medical care. Some services requiring fast decision making and action typically work seamlessly with edge computing service model. In autonomous driving of a connected vehicle with vehicle-to-everything (V2X) communication, roadside unit (RSU) acts as an edge computing device and it will make safer service by processing V2X messages fast, sent by vehicles or other devices. IEEE 1609.2 standard provides application message security technology to ensure the security and reliability of vehicle-to-vehicle communication messages. It uses elliptic curve digital signature algorithm (ECDSA) signatures based on the NIST p256 curve for message authenticity. In this paper, we investigate that RSU should be able to verify 3500 ECDSA signatures per second considering the expected maximum number of vehicles on nearby roads (e.g., during rush hour), message transmission rate, and IEEE 802.11p wireless channel capacity. RSU should satisfy this requirement without assistance of hardware-based cryptographic accelerator. For the requirement, we propose a hybrid approach of parallel ECDSA signature verification at high speed by using CPU and GPU, simultaneously. Moreover, we implemented the proposed method in various modern computing environments for RSU and edge computing devices. Through the experiments, we reach the conclusion that GPU can contribute to the required performance of ECDSA signature verification in RSU platform, which could not satisfy the above throughput only with CPU unit. The target platform with Intel Pentium E6500 CPU and GeForce GTX650 GPU can verify 5668 signatures per second with 30% utilization, while CPU in the platform can process only 2640 signatures. Even in a higher-performance edge computing device, we examine experimentally that the performance can be further improved by using the proposed hybrid approach.

Low-Latency ECDSA Signature Verification—A Road Toward Safer Traffic

Car-to-car and car-to-infrastructure messages exchanged in intelligent transportation systems can reach reception rates over 1000 messages per second. As these messages contain elliptic curve digital signature algorithm (ECDSA) signatures, this puts a very heavy load onto the verification hardware. In fact, the load is so high that, currently, it can only be achieved by implementations running on high-end CPUs and field-programmable gate arrays. These implementations are far from cost-effective or energy efficient. In this paper, we present an application-specified integrated circuit implementation of a dedicated ECDSA verification engine that can reach verification rates of up to 27 000 verifications per second, which is by far the fastest implementation on a single core reported in the literature.

Compact ECDSA engine for IoT applications

Security problems introduced with rapid increase in deployment of Internet-of-Things devices can be overcome only with lightweight cryptographic schemes and modules. A compact prime field (GF(p)) elliptic curve digital signature algorithm (ECDSA) engine suitable for use in such applications is presented. Generic architecture of the engine makes it suitable for other elliptic curve (EC) based schemes (EC Diffie–Hellman key exchange, EC integrated encryption, EC factoring etc.) with slight modifications. The presented engine is composed of a simple microcoded controller and application-specific processing units. It can work with ECs of up to 256 bits, while 160-bit ECDSA signature generation takes 490 K cycles. The engine is implemented as an intellectual property (IP) in a 180 nm process. However, its architecture allows it to be implemented on any application specific integrated circuit (ASIC) or FPGA platform with dual-port memory support. In view of its gate count of 11,366 gate equivalents, the presented work is the most compact ECDSA engine with capability for a wide range of curves and different applications.

Optical identity authentication scheme based on elliptic curve digital signature algorithm and phase retrieval algorithm

An optical identity authentication scheme based on the elliptic curve digital signature algorithm (ECDSA) and phase retrieval algorithm (PRA) is proposed. In this scheme, a user's certification image and the quick response code of the user identity's keyed-hash message authentication code (HMAC) with added noise, serving as the amplitude and phase restriction, respectively, are digitally encoded into two phase keys using a PRA in the Fresnel domain. During the authentication process, when the two phase keys are presented to the system and illuminated by a plane wave of correct wavelength, an output image is generated in the output plane. By identifying whether there is a match between the amplitude of the output image and all the certification images pre-stored in the database, the system can thus accomplish a first-level verification. After the confirmation of first-level verification, the ECDSA signature is decoded from the phase part of the output image and verified to allege whether the user's identity is legal or not. Moreover, the introduction of HMAC makes it almost impossible to forge the signature and hence the phase keys thanks to the HMAC's irreversible property. Theoretical analysis and numerical simulations both validate the feasibility of our proposed scheme.

Communication-Computation Trade-off in Executing ECDSA in a Contactless Smartcard

Emerging standards specify a communication rate between a contactless smartcard and a terminal that is of the same order of magnitude as the internal clock rate in the card. This gives a natural ground for the known card-terminal communication-computation trade-off, where non-secure operations should rather be performed by the terminal and not in the card. In this paper we treat an implementation of Elliptic Curve Digital Signature Algorithm (ECDSA), the most cost effective digital signature algorithm, which has a potential of being executed under the heavy constraints imposed by a contactless smartcard environment. This algorithm heavily relies on numerous calculations of modular multiplicative inverses. It is shown in this paper that, based on communicating with the terminal, each modular inverse operation needed to be executed in the card during ECDSA signature generation requires only two modular multiplications in the card. Each modular inverse operation performed during signature verification requires only one modular multiplication in the card. A complete ECDSA implementation over integers or over GF(2n) is then treated in detail

High-speed implementation of an ECC-based wireless authentication protocol on an ARM microprocessor

The results of the implementation of elliptic curve cryptography (ECC) over the field GF(p) on an 80 MHz, 32-bit ARM microprocessor are presented. A practical software library has been produced which supports variable length implementation of the elliptic curve digital signature algorithm (ECDSA). The ECDSA and a recently proposed ECC-based wireless authentication protocol are implemented using the library. Timing results show that the 160-bit ECDSA signature generation and verification operations take around 46 ms and 94 ms, respectively. With these timings, the execution of the ECC-based wireless authentication protocol takes around 140 ms on the ARM7TDMI processor, which is a widely used, low-power core processor for wireless applications.