How the Number of Points of An Elliptic Curve Over a Fixed Prime Field Varies

1968 ◽  
Vol s1-43 (1) ◽  
pp. 57-60 ◽  
Author(s):  
B. J. Birch
Keyword(s):  
Elliptic Curve ◽  
Prime Field

MicroBlaze-Based Multiprocessor Embedded Cryptosystem on FPGA for Elliptic Curve Scalar Multiplication Over Fp

2019 ◽  
Vol 28 (03) ◽  
pp. 1950037 ◽  
Author(s):  
A. Bellemou ◽  
N. Benblidia ◽  
M. Anane ◽  
M. Issad
Keyword(s):  
Elliptic Curve ◽  
Scalar Multiplication ◽  
Security Level ◽  
Prime Field ◽  
Montgomery Modular Multiplication ◽  
Parallel Implementations ◽  
High Radix ◽  
System On Programmable Chip ◽  
Xilinx Fpga ◽  
Elliptic Curve Scalar Multiplication

In this paper, we present Microblaze-based parallel architectures of Elliptic Curve Scalar Multiplication (ECSM) computation for embedded Elliptic Curve Cryptosystem (ECC) on Xilinx FPGA. The proposed implementations support arbitrary Elliptic Curve (EC) forms defined over large prime field ([Formula: see text]) with different security-level sizes. ECSM is performed using Montgomery Power Ladder (MPL) algorithm in Chudnovsky projective coordinates system. At the low abstraction level, Montgomery Modular Multiplication (MMM) is considered as the critical operation. It is implemented within a hardware Accelerator MMM (AccMMM) core based on the modified high radix, [Formula: see text] MMM algorithm. The efficiency of our parallel implementations is achieved by the combination of the mixed SW/HW approach with Multi Processor System on Programmable Chip (MPSoPC) design. The integration of multi MicroBlaze processor in single architecture allows not only the flexibility of the overall system but also the exploitation of the parallelism in ECSM computation with several degrees. The Virtex-5 parallel implementations of 256-bit and 521-bis ECSM computations run at 100[Formula: see text]MHZ frequency and consume between 2,739 and 6,533 slices, 22 and 72 RAMs and between 16 and 48 DSP48E cores. For the considered security-level sizes, the delays to perform single ECSM are between 115[Formula: see text]ms and 14.72[Formula: see text]ms.

scholarly journals A High-Speed Elliptic Curve Cryptography Processor for Teleoperated Systems Security

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Yong Xiao ◽  
Weibin Lin ◽  
Yun Zhao ◽  
Chao Cui ◽  
Ziwen Cai
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curve Cryptography ◽  
High Speed ◽  
High Performance ◽  
Prime Field ◽  
Practical Applications ◽  
Cell Library ◽  
Cryptographic Algorithm ◽  
Human Operators ◽  
Key Length

Teleoperated robotic systems are those in which human operators control remote robots through a communication network. The deployment and integration of teleoperated robot’s systems in the medical operation have been hampered by many issues, such as safety concerns. Elliptic curve cryptography (ECC), an asymmetric cryptographic algorithm, is widely applied to practical applications because its far significantly reduced key length has the same level of security as RSA. The efficiency of ECC on GF (p) is dictated by two critical factors, namely, modular multiplication (MM) and point multiplication (PM) scheduling. In this paper, the high-performance ECC architecture of SM2 is presented. MM is composed of multiplication and modular reduction (MR) in the prime field. A two-stage modular reduction (TSMR) algorithm in the SCA-256 prime field is introduced to achieve low latency, which avoids more iterative subtraction operations than traditional algorithms. To cut down the run time, a schedule is put forward when exploiting the parallelism of multiplication and MR inside PM. Synthesized with a 0.13 um CMOS standard cell library, the proposed processor consumes 341.98k gate areas, and each PM takes 0.092 ms.

scholarly journals An Efficient Elliptic Curve based Key Management Scheme for Distributed Sensor Networks

2019 ◽  
Vol 4 (6) ◽  
pp. 111-116
Author(s):  
Porkodi Chinniah ◽  
Sangavai Krishnamoorthi
Keyword(s):  
Sensor Networks ◽  
Elliptic Curve ◽  
Key Management ◽  
Key Distribution ◽  
Secret Key ◽  
Distributed Sensor Networks ◽  
Prime Field ◽  
Distributed Sensor ◽  
Management Scheme ◽  
Key Management Scheme

Distributed Sensor Networks are broadly used in many applications and key distribution is a challenging task. In this work, a key management scheme is developed for distributed sensor networks based on elliptic curve cryptography over prime field. Key distribution among the nodes and interactive as well as non interactive protocols for agreement of common secret key for message transmission between two nodes are discussed. The probability for connectivity of the network generated according to the proposed key distribution scheme is discussed in detail. The implementation of the proposed scheme is done using NetSim interfaced with MATLAB. Connectivity of the network is also checked through eigenvalues of the Laplacian matrix of the network.   

scholarly journals The Arithmetic Of elliptic Curve for Prime Curve Secp-384r1 Using One Variable Polynomial Division for Security of Transport Layer Protocol

2019 ◽  
Vol 8 (2) ◽  
pp. 4770-4774
Keyword(s):  
Elliptic Curve ◽  
Transport Layer ◽  
Mathematical Function ◽  
Secure Socket Layer ◽  
Prime Field ◽  
Security Services ◽  
Transport Layer Security ◽  
Digital Signature Algorithm ◽  
Ssl Protocol ◽  
Transport Layer Protocol

In this paper, we present a new method for solving multivariate polynomial elliptic curve equations over a finite field. The arithmetic of elliptic curve is implemented using the mathematical function trace of finite fields. We explain the approach which is based on one variable polynomial division. This is achieved by identifying the plane p with the extension of and transforming elliptic curve equations as well as line equations arising in point addition or point doubling into one variable polynomial. Hence the intersection of the line with the curve is analogous to the roots of the division between these polynomials. Hence this is the different way of computing arithmetic of elliptic curve.Transport layer security provides endto-end security services for applications that use a reliable transport layer protocol such as TCP. Two Protocols are dominant today for providing security at the transport layer, the secure socket layer (SSL) protocol and transport layer security (TLS) protocol. One of the goals of these protocols is to provide server and client authentication, data confidentiality and data integrity. The above goals are achieved by establishing the keys between server and client, the algorithm is called elliptic curve digital signature algorithm (ECDSA) and elliptic curve DiffieHellman (ECDH). These algorithms are implemented using standard for efficient cryptography(SEC) prime field elliptic curve secp-384r1 currently specified in NSA Suite B Cryptography. The algorithm is verified on elliptic curve secp384r1and is shown to be adaptable to perform computation

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