scholarly journals Twisted Edwards Elliptic Curves for Zero-Knowledge Circuits

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3022
Author(s):  
Marta Bellés-Muñoz ◽  
Barry Whitehat ◽  
Jordi Baylina ◽  
Vanesa Daza ◽  
Jose Luis Muñoz-Tapia
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Elliptic Curve Cryptography ◽  
Public Key Cryptography ◽  
Deterministic Algorithm ◽  
Public Key ◽  
Security Attacks ◽  
Zero Knowledge ◽  
Prime Field ◽  
Encryption Schemes

Circuit-based zero-knowledge proofs have arose as a solution to the implementation of privacy in blockchain applications, and to current scalability problems that blockchains suffer from. The most efficient circuit-based zero-knowledge proofs use a pairing-friendly elliptic curve to generate and validate proofs. In particular, the circuits are built connecting wires that carry elements from a large prime field, whose order is determined by the number of elements of the pairing-friendly elliptic curve. In this context, it is important to generate an inner curve using this field, because it allows to create circuits that can verify public-key cryptography primitives, such as digital signatures and encryption schemes. To this purpose, in this article, we present a deterministic algorithm for generating twisted Edwards elliptic curves defined over a given prime field. We also provide an algorithm for checking the resilience of this type of curve against most common security attacks. Additionally, we use our algorithms to generate Baby Jubjub, a curve that can be used to implement elliptic-curve cryptography in circuits that can be validated in the Ethereum blockchain.

Public-Key Encryption

2017 ◽  
Author(s):  
Keith M. Martin
Keyword(s):  
Elliptic Curve ◽  
Public Key Cryptography ◽  
Public Key ◽  
Public Key Encryption ◽  
Hybrid Encryption ◽  
Public Key Cryptosystems ◽  
Encryption And Decryption ◽  
Encryption Schemes ◽  
Hard Problems ◽  
Set Up

In this chapter, we introduce public-key encryption. We first consider the motivation behind the concept of public-key cryptography and introduce the hard problems on which popular public-key encryption schemes are based. We then discuss two of the best-known public-key cryptosystems, RSA and ElGamal. For each of these public-key cryptosystems, we discuss how to set up key pairs and perform basic encryption and decryption. We also identify the basis for security for each of these cryptosystems. We then compare RSA, ElGamal, and elliptic-curve variants of ElGamal from the perspectives of performance and security. Finally, we look at how public-key encryption is used in practice, focusing on the popular use of hybrid encryption.

Study on the Safety Application of Public Key Cryptography in Electronic Commerce

2014 ◽  
Vol 1079-1080 ◽  
pp. 856-859
Author(s):  
Yu Zhong Zhang
Keyword(s):  
Electronic Commerce ◽  
Elliptic Curve ◽  
Communication Technology ◽  
Elliptic Curve Cryptography ◽  
Public Key Cryptography ◽  
Public Key ◽  
Electronic Transactions ◽  
Safety Application

With the progress of computer and communication technology, electronic commerce flourished. Security is a key problem in the development of electronic commerce. This paper discusses the principle of elliptic curve cryptography and its safety application in electronic transactions.

scholarly journals A COMPARATIVE ANALYSIS OF ECDSA V/S RSA ALGORITHM

2014 ◽  
pp. 7-9
Author(s):  
AMANPREET KAUR ◽  
VIKAS GOYAL
Keyword(s):  
Elliptic Curve ◽  
Finite Fields ◽  
Elliptic Curve Cryptography ◽  
Public Key Cryptography ◽  
Cryptographic Primitives ◽  
Prime Field ◽  
Rsa Algorithm ◽  
Encryption And Decryption ◽  
Security Problem ◽  
Accurate Observation

Elliptic curve Cryptography with its various protocols implemented in terms of accuracy and fast observation of results for better security solution. ECC applied on two finite fields: prime field and binary field. Because it is public key cryptography so, it also focus on generation of elliptic curve and shows why finite fields are introduced. But for accurate observation we do analysis on category of cryptographic primitives used to solve given security problem. RSA & ECDSA both have basic criteria of production of keys and method of encryption and decryption in basic application as per security and other properties which are authentication, non-repudiation, privacy, integrity.

scholarly journals Scalable Normal Basis Arithmetic Unit for Elliptic Curve Cryptography

10.14311/688 ◽  
2005 ◽  
Vol 45 (2) ◽  
Author(s):  
J. Schmidt ◽  
M. Novotný
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curve Cryptography ◽  
Additional Data ◽  
Public Key Cryptography ◽  
Normal Basis ◽  
Public Key ◽  
Arithmetic Unit ◽  
Inversion Algorithm ◽  
Implementation Data ◽  
And Inversion

The design of a scalable arithmetic unit for operations over elements of GF(2m) represented in normal basis is presented. The unit is applicable in public-key cryptography. It comprises a pipelined Massey-Omura multiplier and a shifter. We equipped the multiplier with additional data paths to enable easy implementation of both multiplication and inversion in a single arithmetic unit. We discuss optimum design of the shifter with respect to the inversion algorithm and multiplier performance. The functionality of the multiplier/inverter has been tested by simulation and implemented in Xilinx Virtex FPGA.We present implementation data for various digit widths which exhibit a time minimum for digit width D = 15.

scholarly journals Implementation Methodology of ECC to Overcome Side Channel Attacks

2018 ◽  
Vol 7 (3.27) ◽  
pp. 421
Author(s):  
M Maheswari ◽  
R A. Karthika ◽  
Anuska Chatterjee
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curve Cryptography ◽  
Public Key Cryptography ◽  
Public Key ◽  
Side Channel ◽  
Side Channel Attacks ◽  
Private Key ◽  
Wide Range ◽  
Cryptographic Algorithms ◽  
Implementation Methodology

Elliptic Curve Cryptography (ECC) is a form of public-key cryptography. This implies that there is the involvement of a private key and a public key for the purpose of cryptography. ECC can be used for a wide range of applications. The keys used are much smaller than the non-ECC cryptographic algorithms. 256 bit and 384 bit ECC are used by NSA for storage of classified intel as ECC is considered to be a part of suit B cryptography by the NSA. When it comes to normal usage, other versions of ECC are used. So, many of the applications protected by ECC are vulnerable to side channel attacks. So, the objective is to modify the existing method of implementation of ECC is some regular domains like media, smart grid, etc., such that the side-channel attacks [7], [3] vulnerabilities are fixed.  

scholarly journals Cryptography in Terms of Triangular Neutrosophic Numbers with Real Life Applications

2020 ◽  
pp. 39-52
Author(s):  
admin admin ◽  
◽  
◽  
◽  
◽  
...  
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curve Cryptography ◽  
Information Transfer ◽  
Real Life ◽  
Public Key Cryptography ◽  
Public Key ◽  
Encryption And Decryption ◽  
Basic Concepts ◽  
Triangular Neutrosophic Numbers

In this article, our main focus is to put forward the concept of Cryptography in terms of triangular neutrosophic numbers. This kind of cryptography is really reliable, manual, secure, and based on few simple steps. All the encryption and decryption are easy to proceed (mention below). As we know, Public-key cryptography as an indefatigable defender for human privacy and use as information transfer from the ages. various concepts are available with regard to cryptography e.g. Elliptic curve cryptography. TNNC (Triangular neutrosophic numbers cryptography) is familiar with basic concepts of math as well as applicable in different situations e.g. code cryptography, detailed view cryptography, and Graph cryptography encryption facilitate.

scholarly journals Model Proyeksi (X/Z2, Y/Z2) pada Kurva Hesian Secara Paralel Menggunakan Mekanisme Kriptografi Kurva Eliptik

2012 ◽  
Vol 12 (1) ◽  
pp. 65
Author(s):  
Winsy Weku
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Elliptic Curve Cryptography ◽  
Galois Field ◽  
Public Key ◽  
Public Key Cryptosystem ◽  
Projection Model ◽  
Galois Fields ◽  
Time Operation ◽  
Modular Inversion

MODEL PROYEKSI (X/Z2, Y/Z2) PADA KURVA HESIAN SECARA PARALEL MENGGUNAKAN MEKANISME KRIPTOGRAFI KURVA ELIPTIKABSTRAK Suatu kunci publik, Elliptic Curve Cryptography (ECC) dikenal sebagai algoritma yang paling aman yang digunakan untuk memproteksi informasi sepanjang melakukan transmisi.  ECC dalam komputasi aritemetika didapatkan berdasarkan operasi inversi modular. Inversi modular adalah operasi aritmetika dan operasi yang sangat panjang yang didapatkan berdasar ECC crypto-processor. Penggunaan koordinat proyeksi untuk menentukan Kurva Eliptik/ Elliptic Curves pada kenyataannya untuk memastikan koordinat proyeksi yang sebelumnya telah ditentukan oleh kurva eliptik E: y2 = x3 + ax + b yang didefinisikan melalui Galois field GF(p)untuk melakukan operasi aritemtika dimana dapat diketemukan bahwa terdapat beberapa multiplikasi yang dapat diimplementasikan secara paralel untuk mendapatkan performa yang tinggi. Pada penelitian ini, akan dibahas tentang sistem koordinat proyeksi Hessian (X/Z2, Y,Z2) untuk meningkatkan operasi penggandaan ECC dengan menggunakan pengali paralel untuk mendapatkan paralel yang maksimum untuk mendapatkan hasil maksimal. Kata kunci: Elliptic Curve Cryptography, Public-Key Cryptosystem, Galois Fields of Primes GF(p PROJECTION MODEL (X/Z2, Y/Z2) ON PARALLEL HESIAN CURVE USING CRYPTOGRAPHY ELIPTIC CURVE MECHANISM ABSTRACT As a public key cryptography, Elliptic Curve Cryptography (ECC) is well known to be the most secure algorithms that can be used to protect information during the transmission. ECC in its arithmetic computations suffers from modular inversion operation. Modular Inversion is a main arithmetic and very long-time operation that performed by the ECC crypto-processor. The use of projective coordinates to define the Elliptic Curves (EC) instead of affine coordinates replaced the inversion operations by several multiplication operations. Many types of projective coordinates have been proposed for the elliptic curve E: y2 = x3 + ax + b which is defined over a Galois field GF(p) to do EC arithmetic operations where it was found that these several multiplications can be implemented in some parallel fashion to obtain higher performance. In this work, we will study Hessian projective coordinates systems (X/Z2, Y,Z2) over GF (p) to perform ECC doubling operation by using parallel multipliers to obtain maximum parallelism to achieve maximum gain. Keywords: Elliptic Curve Cryptography , Public-Key Cryptosystem , Galois Fields of  Primes GF(p)

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