Edwards Curves Research Articles

Mobile Payment Transfer Using Edwards Curve Digital Signature Algorithm Based on the Proof of Work in Blockchain

Mobile Payment Transfer Using Edwards Curve Digital Signature Algorithm Based on the Proof of Work in Blockchain

Solving Elliptic Curve Discrete Logarithm Problem on Twisted Edwards Curves Using Quantum Annealing and Index Calculus Method

Abstract This paper presents an approach to solving the elliptic curve discrete logarithm problem on alternative curve models over prime fields using a quantum annealing and index calculus method. Part of the algorithm, relation searching, is transformed into the Quadratic Unconstrained Boolean Optimization (QUBO) problem and then is efficiently solved using the D-Wave computer by quantum annealing. As Faugère et al. showed, twisted Edwards curves, because of their symmetric shape, allow us to obtain solutions of relations searching step using Groebner basis faster than in the case of Weierstrass curves. Because of symmetries, a system of equations of relations searching step for twisted Edwards curves has many symmetric solutions. Using the Groebner basis and having many system solutions makes it easier to find any of them. The same is true using quantum annealing - it is easier to find any solution to the QUBO problem if many are correct. In this paper, we used this observation to find out that a properly constructed QUBO problem for the relations searching step for twisted Edwards curves allows us to find a solution faster for the same size of the base field than in the case of Weierstrass curves. Using the presented approach, we solved the discrete logarithm problem using quantum annealing and index calculus method for elliptic curve discrete logarithm problem defined on twisted Edwards curve over a field 𝔽1021 with order equal to 4 · 241. It is now the biggest field and size of the group, where the elliptic curve discrete logarithm problem was solved using quantum methods.

Efficient Commutative PQC Algorithms on Isogenies of Edwards Curves

The article presents the author’s works in the field of modifications and modeling of the Post-Quantum Cryptography (PQC) Commutative Supersingular Isogeny Diffie-Hellman (CSIDH) algorithm on non-cyclic supersingular Edwards curves and its predecessor Couveignes-Rostovtsev-Stolbunov (CRS) scheme on ordinary non-cyclic Edwards curves are reviewed. Lower estimates of the computational speed gains of the modified algorithms over the original ones are obtained. The most significant results were obtained by choosing classes of non-cyclic Edwards curves connected as quadratic twist pairs instead of cyclic complete Edwards curves, as well as the method of algorithm randomization as an alternative to “constant time CSIDH”. It is shown that in the CSIDH and Commutative Supersingular Isogeny Key Encapsulation (CSIKE) algorithms, there are two independent cryptosystems with the possibility of parallel computation, eliminating the threat of side-channel attacks. There are four such cryptosystems for the CRS scheme. Integral lower bound estimates of the performance gain of the modified CSIDH algorithm are obtained at 1.5 × 29, and for the CRS scheme are 3 × 29.

Base of exponent representation matters -- more efficient reduction of discrete logarithm problem and elliptic curve discrete logarithm problem to the QUBO problem

This paper presents further improvements in the transformation of the Discrete Logarithm Problem (DLP) and Elliptic Curve Discrete Logarithm Problem (ECDLP) over prime fields to the Quadratic Unconstrained Binary Optimization (QUBO) problem. This is significant from a cryptanalysis standpoint, as QUBO problems may be solved using quantum annealers, and the fewer variables the resulting QUBO problem has, the less time is expected to obtain a solution.The main idea presented in the paper is allowing the representation of the exponent in different bases than the typically used base 2 (binary representation). It is shown that in such cases, the reduction of the discrete logarithm problem over the prime field \(\mathbb{F}_p\) to the QUBO problem may be obtained using approximately \(1.89n^2\) logical variables for \(n\) being the bitlength of prime \(p\), instead of the \(2n^2\) which was previously the best-known reduction method. The paper provides a practical example using the given method to solve the discrete logarithm problem over the prime field \(\mathbb{F}_{47}\). Similarly, for the elliptic curve discrete logarithm problem over the prime field \(\mathbb{F}_p\), allowing the representation of the exponent in different bases than typically used base two results in a lower number of required logical variables for \(n\) being the bitlength of prime \(p\), from \(3n^3\) to \(\frac{6n^3}{\log_{2}\left(\frac{3}{4}n\right)}\) logical variables, in the case of Edwards curves.

Edwards curve digital signature algorithm for video integrity verification on blockchain framework

Within the field of forensic science, video integrity is an essential component that ensures the validity and dependability of visual evidence that is essential for court cases. Maintaining the integrity of justice in a time when digital modification tools are easily accessible requires making sure that video recordings are unaltered. Video footage is carefully inspected by forensic analysts who use advanced tools to look for indications of manipulation or change. This paper presents a novel, lightweight approach for verifying the integrity of video data. The proposed method utilizes the blockchain and the Edwards Curve Digital Signature Algorithm, coupled with the BLAKE2b hash function. Signatures are generated for video clips with a predetermined size called video segments and stored in blocks chronologically. The signature of the previous block is stored in the current block to add another layer of security. At the time of validation, these signatures are verified. Experimental results show that the proposed method outperforms state-of-the-art methods in both speed and security. Our method is capable of identifying any type of forgery on any video file, by anyone, at any given moment, with insignificant additional storage requirements. Furthermore, our security analysis shows that our method is resilient against various types of attacks, including collision attacks, key substitution attacks, side-channel attacks, and chosen message attacks. The proposed lightweight video integrity verification method is better suited for use in resource constrained devices.

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.

Transformation of Elliptic Curve Discrete Logarithm Problem to QUBO Using Direct Method in Quantum Annealing Applications

This paper investigates how to reduce the elliptic curve discrete logarithm problem over prime fields to the quadratic unconstrained binary optimization (QUBO) problem in order to obtain as few logical qubits as possible. In the best case scenario, if n is the bitlength of a characteristic of prime field Fp, approximately 3n³ logical qubits are required for such a reduction in the Edwards curve case. We present a practical attack on an elliptic curve discrete logarithm problem over the 3-bit prime field F7 for an elliptic curve with the subgroup of order 8. We solved this problem using the D-Wave Advantage QPU. To the best of the authors' knowledge, no one has made, so far, a practical attack on the elliptic curve discrete logarithm over a prime field using the direct quantum method.

Armed with Faster Crypto: Optimizing Elliptic Curve Cryptography for ARM Processors

Elliptic curve cryptography is a widely deployed technology for securing digital communication. It is the basis of many cryptographic primitives such as key agreement protocols, digital signatures, and zero-knowledge proofs. Fast elliptic curve cryptography relies on heavily optimised modular arithmetic operations, which are often tailored to specific micro-architectures. In this article, we study and evaluate optimisations of the popular elliptic curve Curve25519 for ARM processors. We specifically target the ARM NEON single instruction, multiple data (SIMD) architecture, which is a popular architecture for modern smartphones. We introduce a novel representation for 128-bit NEON SIMD vectors, optimised for SIMD parallelisation, to accelerate elliptic curve operations significantly. Leveraging this representation, we implement an extended twisted Edwards curve Curve25519 back-end within the popular Rust library “curve25519-dalek”. We extensively evaluate our implementation across multiple ARM devices using both cryptographic benchmarks and the benchmark suite available for the Signal protocol. Our findings demonstrate a substantial back-end speed-up of at least 20% for ARM NEON, along with a noteworthy speed improvement of at least 15% for benchmarked Signal functions.

Exp function for Edwards curves over local fields

We extend the map Exp for elliptic curves in short Weierstrass form over $ \mathbb{C} $ to Edwards curves over local fields. Subsequently, we compute the map Exp for Edwards curves over the local field $ \mathbb{Q}_{p} $ of $ p $-adic numbers.

Efficient ECC Processor Designs for IoT Using Edwards Curves and Exploiting FPGA Embedded Components

Efficient ECC Processor Designs for IoT Using Edwards Curves and Exploiting FPGA Embedded Components

Software for processing of long-term strength data of materials by use of edwards curves

A description of the developed software for saving and analyzing data on high-temperature deformation and long-term strength of metallic materials is provided. The open source Laravel web framework was chosen to implement in the application. The Voyager solution is used together with it as a graphical interface for interacting with the database. A JavaScript programming language solution using Canvas technology is used to build plots. Basic classes and components are described. With the help of diagrams of use cases, the possibility of using the web application in the engineering practice of data processing of experimental high-temperature creep and fracture studies is shown. An approach and an algorithm for approximating long-term strength curves of general appearance, including different types of failure, using Edwards curves are proposed. With their use, an analytical nonlinear expression for the functional dependence between the time to failure and the failure stress was obtained. Approximations of various types of the long-term strength curve corresponding to the processes of accumulation of damage in the material by various types of physical mechanisms are discussed. The undesirable possibility of obtaining an overestimated value of the time to failure in the case of small and medium stress values without using information on the areas of intergranular failure, failure due to oxidation or corrosion, and overaging on the long-term strength curve is demonstrated. The use of the developed software, approach and method of analytical representation of the functional dependence of the values of failure stress on time in the evolution equation for the damage parameter allows you to carry out refined calculations of structural elements operating under high-temperature load conditions

An Optimized Hardware Implementation of a Non-Adjacent Form Algorithm Using Radix-4 Multiplier for Binary Edwards Curves

Binary Edwards Curves (BEC) play a pivotal role in modern cryptographic processes and applications, offering a combination of robust security as well as computational efficiency. For robust security, this article harnesses the inherent strengths of BEC for the cryptographic point multiplication process by utilizing the Non-Adjacent Form (NAF) algorithm. For computational efficiency, a hardware architecture for the NAF algorithm is proposed. Central to this architecture is an Arithmetic Logic Unit (ALU) designed for streamlined execution of essential operations, including addition, squaring, and multiplication. One notable innovation in our ALU design is the integration of multiplexers, which maximize ALU efficiency with minimal additional hardware requirements. Complementing the optimized ALU, the proposed architecture incorporates a radix-4 multiplier, renowned for its efficiency in both multiplication and reduction. It eliminates resource-intensive divisions, resulting in a substantial boost to overall computational speed. The architecture is implemented on Xilinx Virtex series Field-Programmable Gate Arrays (FPGAs). It achieves throughput-to-area ratios of 14.819 (Virtex-4), 25.5 (Virtex-5), 34.58 (Virtex-6), and 37.07 (Virtex-7). These outcomes underscore the efficacy of our optimizations, emphasizing an equilibrium between computational performance and area utilization.

Fuzzy Perspective of Online Games by Using Cryptography and Cooperative Game Theory

In this paper, we mathematically associate crypto-cloud computing with cooperative game theory in the presence of fuzzy uncertainty. For this purpose, we retrieve information from the database of Amazon Web Service. Then we construct a secure crypto-cloud game theoretical model and apply this model to online games under fuzzy uncertainty. Further, we suggest some fuzzy game theoretical solutions by proposing a novel twisted Edwards curve pairing-based scheme over finite fields having the property of indistinguishable under chosen ciphertext attacks. Finally, it is seen that costs of the numerical fuzzy solutions are reduced and their efficiency is increased compared with the numerical crisp ones.

PQC CSIKE Algorithm on Non-Cyclic Edwards Curves

PQC CSIKE Algorithm on Non-Cyclic Edwards Curves

How to construct CSIDH on Edwards curves

CSIDH is an isogeny-based key exchange protocol proposed by Castryck et al. in 2018. It is based on the ideal class group action on Fp-isomorphism classes of Montgomery curves. The original CSIDH algorithm requires a calculation over Fp by representing points as x-coordinate over Montgomery curves. There is a special coordinate on Edwards curves (the w-coordinate) to calculate group operations and isogenies. If we try to calculate the class group action on Edwards curves by using the w-coordinate in a similar way on Montgomery curves, we have to consider points defined over Fp4. Therefore, calculating the class group action on Edwards curves with w-coordinates over only Fp is not a trivial task.In this paper, we prove some theorems about the properties of Edwards curves. We construct the new CSIDH algorithm using these theorems on Edwards curves with w-coordinates over Fp. This algorithm is as fast as (or a little bit faster than) the algorithm proposed by Meyer and Reith.This paper is an extended version of [29]. We added the construction of a technique similar to Elligator on Edwards curves. This technique contributes to the efficiency of the constant-time CSIDH algorithm. We also added the construction of new formulas to compute isogenies in O˜(ℓ) time on Edwards curves. It is based on formulas on Montgomery curves proposed by Bernstein et al. (élu's formulas). In our analysis, these formulas on Edwards curves are a little bit faster than those on Montgomery curves.We finally implemented CSIDH, élu's formulas, and CTIDH [3] (faster constant-time CSIDH) on Edwards curves. Each result shows the efficiency of algorithms on Edwards curves.

Efficient Construction of CGL Hash Function Using Legendre Curves

The CGL hash function is a provably secure hash function using walks on isogeny graphs of supersingular elliptic curves. A dominant cost of its computation comes from iterative computations of power roots over quadratic extension fields. In this paper, we reduce the necessary number of power root computations by almost half, by applying and also extending an existing method of efficient isogeny sequence computation on Legendre curves (Hashimoto and Nuida, CASC 2021). We also point out some relationship between 2-isogenies for Legendre curves and those for Edwards curves, which is of independent interests, and develop a method of efficient computation for 2e-th roots in quadratic extension fields.

A Crypto Accelerator of Binary Edward Curves for Securing Low-Resource Embedded Devices

This research presents a novel binary Edwards curve (BEC) accelerator designed specifically for resource-constrained embedded systems. The proposed accelerator incorporates the fixed window algorithm, a two-stage pipelined architecture, and the Montgomery radix-4 multiplier. As a result, it achieves remarkable performance improvements in throughput and resource utilization. Experimental results, conducted on various Xilinx Field Programmable Gate Arrays (FPGAs), demonstrate impressive throughput/area ratios observed for GF(2233). The achieved ratios for Virtex-4, Virtex-5, Virtex-6, and Virtex-7 are 12.2, 19.07, 36.01, and 38.39, respectively. Furthermore, the processing time for one-point multiplication on a Virtex-7 platform is 15.87 µs. These findings highlight the effectiveness of the proposed accelerator for improved throughput and optimal resource utilization.

A hybrid elliptic curve cryptography (HECC) technique for fast encryption of data for public cloud security

Cloud computing provides the users a centralized place for data storage and other commercial applications. To guarantee everything is secure, cloud security has to handle all the devices, apps, and data connected to the cloud. Because of the cloud's robust security, the data and applications may be accessed by the relevant people. The usage of public cloud security ensures that customers always have a trustworthy method of accessing apps and data, which in turn enables service providers to resolve any potential security problems as soon as they arise. This paper presents a public cloud security technique using Hybrid Elliptic Curve Cryptography (HECC). The proposed approach makes keys using a lightweight Edwards curve. The user's Identity Based Encryption is used to change the generated private keys. The proposed key reduction method is used to make the keys even shorter, which speeds up the Advanced Encryption standard (AES) encryption process. The public keys are exchanged using Diffie Hellman key exchange. Throughput and key generation, encryption, and decryption times are used to evaluate how well the suggested model performs. The suggested model fared better than the current models in every way. The suggested method's key creation process takes 0.000025 s, while the resulting encryption process takes 0.00349 s. The achieved throughput is 693.10 kB/s.

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.

Faster Montgomery multiplication and Multi-Scalar-Multiplication for SNARKs

The bottleneck in the proving algorithm of most of elliptic-curve-based SNARK proof systems is the Multi-Scalar-Multiplication (MSM) algorithm. In this paper we give an overview of a variant of the Pippenger MSM algorithm together with a set of optimizations tailored for curves that admit a twisted Edwards form. We prove that this is the case for SNARK-friendly chains and cycles of elliptic curves, which are useful for recursive constructions. Our contribution is twofold: first, we optimize the arithmetic of finite fields by improving on the well-known Coarsely Integrated Operand Scanning (CIOS) modular multiplication. This is a contribution of independent interest that applies to many different contexts. Second, we propose a new coordinate system for twisted Edwards curves tailored for the Pippenger MSM algorithm.Accelerating the MSM over these curves is critical for deployment of recursive proof< systems applications such as proof-carrying-data, blockchain rollups and blockchain light clients. We implement our work in Go and benchmark it on two different CPU architectures (x86 and arm64). We show that our implementation achieves a 40-47% speedup over the state-of-the-art implementation (which was implemented in Rust). This MSM implementation won the first place in the ZPrize competition in the open division “Accelerating MSM on Mobile” and will be deployed in two real-world applications: Linea zkEVM by ConsenSys and probably Celo network.