Point Addition Operations Research Articles

Improved Arithmetic on Koblitz Curves over Binary Field

The longest process in ECC is the elliptic curve scalar multiplication. The structure of this operation involves three mathematical levels; this work aims to study issues that arise in the efficient implementation of this operation, specifically targeting the point arithmetic level for the Koblitz curve over a binary field. Theorems have been made for a speedy point doubling and point addition operation, in these theorems Jacobian coordinate modification has been considered, where these coordinates represent each point: refers to a point on a curve over . This occurs when a coordinate system represents any point on a Koblitz curve over a binary field. By choosing the right coordinate system, it is possible to speed up the elliptic curve scalar multiplication using this method.

An Optimized Point Multiplication Strategy in Elliptic Curve Cryptography for Resource-Constrained Devices

Elliptic curve cryptography (ECC) is widely acknowledged as a method for implementing public key cryptography on devices with limited resources thanks to its use of small keys. A crucial and complex operation in ECC calculations is scalar point multiplication. To improve its execution time and computational complexity in low-power devices, such as embedded systems, several algorithms have been suggested for scalar point multiplication, with each featuring different techniques and mathematical formulas. In this research, we focused on combining some techniques to produce a scalar point multiplication algorithm for elliptic curves over finite fields. The employed methodology involved mathematical analysis to investigate commonly used point multiplication methods. The aim was to propose an efficient algorithm that combined the best computational techniques, resulting in lower computational requirements. The findings show that the proposed method can overcome certain implementation issues found in other multiplication algorithms. In certain scenarios, the proposed method offers a more efficient approach by reducing the number of point doubling and point addition operations on elliptic curves using the inverse of the targeted point.

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.

On k-Connected Γ-Extensions of Binary Matroids

Slater introduced the point-addition operation on graphs to classify 4-connected graphs. The Γ-extension operation on binary matroids is a generalization of the point-addition operation. In this paper, we obtain necessary and sufficient conditions to preserve k-connectedness of a binary matroid under the Γ-extension operation. We also obtain a necessary and sufficient condition to get a connected matroid from a disconnected binary matroid using the Γ-extension operation.

Accelerating Digital Forensics through Parallel Computing

Digital crimes in the era of big data and cloud computing imposes significant challenges in digital forensics. Cloud environment provides low cost, easy management and reasonable solutions. Moreover, it supports big data structures and solutions (i.e., security, privacy and digital forensics). In order to achieve a secure digital forensics analysis in cloud environment, researchers have proposed solutions with expensive communication cost and computation overheads. Among these solutions Nasereldin et al. proposed a protocol which solves the problem of authenticity and integrity of evidence using signcryption technique. This leads to low communication and implementation overheads. Furthermore, identity-based cryptography is used to solve Public Key Infrastructure (PKI) problems. In addition, it is characterized by the ability to divide the message into small messages which is suitable for pipelining techniques. Nasreldin et al.'s signcryption protocol is based on Elliptic Curve Cryptography (ECC) which is implemented by using different mathematical operations. In this protocol, ECC mathematical operations take huge time during the execution of the algorithm. ECC consists of point doubling and point addition operations. These operations require the execution of many Montgomery modular multiplications that consume time. In this study, we introduce a technique to speed up ECC operations in order to enhance the efficiency of Nasreldin et al. protocol. In particular, we propose a multi-stage parallel design which consists of three stages. First, we speed up the point doubling and point addition operation. Secondly, we enhance the execution time of Montgomery multiplications. Finally, pipelining is used to obtain a better performance. The results show that the proposed design enhances Nasreldin et al. protocol’s execution time by 47.1, 64.7, 73.5 and 79.4%, assuming that the number of nodes is 2, 4, 6 and 12, respectively.

{0, 1, 3}-NAF Representation and Algorithms for Lightweight Elliptic Curve Cryptosystem in Lopez Dahab Model

Elliptic curve scalar multiplications is the most time-consuming and costly operation in elliptic curve cryptosystem. The scalar multiplication involves computation of Q = kP where k is a scalar multiplier, and P and Q are points on an elliptic curve. This computation can be improved by reducing the Hamming weight of the scalar multiplier k. The Hamming weight of k represents the number of nonzero digits in the scalar multiplier. This paper proposes a new scalar representation in non-adjacent form (NAF) using the digits 0, 1 and 3. This paper also proposes an algorithm for converting from a binary to {0,1,3}-NAF representation. Comparative analysis between the proposed NAF and the traditional NAF with digit {-1,0,1} is carried out. At average case, the proposes {0,1,3}-NAF representation has a lower Hamming weight than the traditional NAF. In our analysis, we use the {0,1,3}-NAF representation in the scalar multiplication operation. The average number of point addition operations in the scalar multiplication is considerably reduced compared to the addition-subtraction scalar multiplication algorithm.

Improved Arithmetic on Elliptic Curves over Prime Field

A fast point doubling and point addition operations on an elliptic curve over prime field are proposed. This occur when we use a special coordinates system (to represent any point on elliptic curve over prime field. Using this system improved the elliptic curve point arithmetic by reducing the computation cost for point doubling and point addition operation.

Γ-Extension of Binary Matroids

We extend the notion of a point-addition operation from graphs to binary matroids. This operation can be expressed in terms of element-addition operation and splitting operation. We consider a special case of this construction and study its properties. We call the resulting matroid of this special case a Γ-extension of the given matroid. We characterize circuits and bases of the resulting matroids and explore the effect of this operation on the connectivity of the matroids.

High speed and Low space complexity FPGA based ECC processor

Elliptic Curve Cryptography is one of the most interested research topic in VLSI. Network security is becoming more and more crucial as the volume of data being exchanged on the Internet increases. ECC offers high security for networking and communication. FPGA based architecture for elliptic curve cryptography coprocessor ,which has promising performance in terms of both Space Complexity and Time Complexity is proposed in this paper. The coprocessor consists of three levels of operations, which are elements operations over binary finite field, point adding and doubling operations on elliptic curve and scalar multiplication. In this work on coprocessor, Point addition operation is performed with mixed coordinates to improve the performance of scalar multiplication . VHDL codes were developed for all these operations. The modules are simulated using Modelsim SE software and synthesized using Xilinx ISE 9.2i software. Experimental results show that ECC coprocessor realized in this architecture can speed up an elliptic curve scalar multiplication suitable for low area constraint applications and very high speed applications.