RSA-Public Key Cryptosystems Based on Quadratic Equations in Finite Field

2014 ◽  
pp. 238-258
Author(s):  
Sattar B. Sadkhan Al Maliky ◽  
Luay H. Al-Siwidi
Keyword(s):  
Finite Field ◽  
Chinese Remainder Theorem ◽  
Quadratic Equation ◽  
Public Key ◽  
Factorization Problem ◽  
Quadratic Equations ◽  
Public Key Cryptosystems ◽  
Encryption And Decryption ◽  
Mathematical Problems ◽  
One Way Function

The importance of Public Key Cryptosystems (PKCs) in the cryptography field is well known. They represent a great revolution in this field. The PKCs depend mainly on mathematical problems, like factorization problem, and a trapdoor one-way function problem. Rivest, Shamir, and Adleman (RSA) PKC systems are based on factorization mathematical problems. There are many types of RSA cryptosystems. Rabin's Cryptosystem is considered one example of this type, which is based on using the square order (quadratic equation) in encryption function. Many cryptosystems (since 1978) were implemented under such a mathematical approach. This chapter provides an illustration of the variants of RSA-Public Key Cryptosystems based on quadratic equations in Finite Field, describing their key generation, encryption, and decryption processes. In addition, the chapter illustrates a proposed general formula for the equation describing these different types and a proposed generalization for the Chinese Remainder Theorem.

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.

scholarly journals Preimage Selective Trapdoor Function: How to Repair an Easy Problem

2014 ◽  
Vol 2014 ◽  
pp. 1-18
Author(s):  
Baocang Wang
Keyword(s):  
Public Key Cryptography ◽  
Public Key ◽  
Constructive Method ◽  
Key Recovery ◽  
Public Key Cryptosystems ◽  
Cryptographic Primitive ◽  
The One ◽  
Trapdoor Function ◽  
One Way Function ◽  
Encryption Function

Public key cryptosystems are constructed by embedding a trapdoor into a one-way function. So, the one-wayness and the trapdoorness are vital to public key cryptography. In this paper, we propose a novel public key cryptographic primitive called preimage selective trapdoor function. This scenario allows to use exponentially many preimage to hide a plaintext even if the underlying function is not one-way. The compact knapsack problem is used to construct a probabilistic public key cryptosystem, the underlying encryption function of which is proven to be preimage selective trapdoor one-way functions under some linearization attack models. The constructive method can guarantee the noninjectivity of the underlying encryption function and the unique decipherability for ciphertexts simultaneously. It is heuristically argued that the security of the proposal cannot be compromised by a polynomial-time adversary even if the compact knapsack is easy to solve. We failed to provide any provable security results about the proposal; however, heuristic illustrations show that the proposal is secure against some known attacks including brute force attacks, linearization attacks, and key-recovery attacks. The proposal turns out to have acceptable key sizes and performs efficiently and hence is practical.

scholarly journals A Public Key Cryptosystem Using Cyclotomic Matrices

2021 ◽  
Author(s):  
Md. Helal Ahmed ◽  
Jagmohan Tanti ◽  
Sumant Pushp
Keyword(s):  
Computational Complexity ◽  
Communication Technology ◽  
Public Key ◽  
Public Key Cryptosystem ◽  
Privileged Information ◽  
Diophantine System ◽  
Encryption And Decryption ◽  
Information And Communication ◽  
Asymptotic Complexity ◽  
One Way Function

Confidentiality and Integrity are two paramount objectives in the evaluation of information and communication technology. In this chapter, we propose an arithmetic approach for designing asymmetric key cryptography. Our method is based on the formulation of cyclotomic matrices correspond to a diophantine system. The strategy uses in cyclotomic matrices to design a one-way function. The result of a one-way function that is efficient to compute, however, is hard to process its inverse except if privileged information about the hidden entry is known. Also, we demonstrate that encryption and decryption can be efficiently performed with the asymptotic complexity of Oe2.373. Finally, we study the computational complexity of the cryptosystem.

Key Authentication Scheme-based on Discrete Logarithms and Chinese Remainder Theorem

2016 ◽  
Vol 66 (6) ◽  
pp. 590
Author(s):  
P. Kumaraswamy ◽  
C.V. Guru Rao ◽  
V. Janaki ◽  
K.V.T.K.N. Prashanth
Keyword(s):  
Chinese Remainder Theorem ◽  
Authentication Scheme ◽  
Third Party ◽  
Public Key ◽  
Online Services ◽  
Trusted Third Party ◽  
Discrete Logarithms ◽  
The Public ◽  
Public Key Cryptosystems

<p>Public key cryptosystems are secure only when the authenticity of the public key is assured. Shao proposed<br />a new scheme to overcome the problems of the existing schemes, which suffers from two major drawbacks. The<br />first drawback is the availability of users’ passwords in plaintext format in key server which are prone to attacks<br />by ill-minded users. The second one is depending on the key server blindly for certificate generation, without<br />further verification by the user. To overcome these severe drawbacks, we proposed an improved key authentication<br />scheme based on Chinese remainder theorem and discrete logarithms. Our scheme allows the user to generate his/<br />her certificate without the help of any trusted third party. This scheme is intended for online services, military and<br />defense applications to exchange keys securely.<br /> </p>

scholarly journals Analysis Aryabhata Remainder Theorem On Rivest Shamir Adleman (RSA) Cryptography

2020 ◽  
Vol 5 (2) ◽  
pp. 88-93
Author(s):  
Eko Hariyanto Hariyanto ◽  
Muhammad Zarlis ◽  
Darmeli Nasution Nasution ◽  
Sri Wahyuni
Keyword(s):  
Time Complexity ◽  
Prime Factor ◽  
Chinese Remainder Theorem ◽  
Public Key Cryptography ◽  
Public Key ◽  
Security Level ◽  
Encryption And Decryption ◽  
Rsa Cryptography

 Security of RSA cryptography lied in the difficulty of factoring the number p and q bacame the prime factor. The greater the value of p and q used, the better the security level was. However, this could result in a very slow decryption process. The most commonly used and discussed method of speeding up the encryption and decryption process in RSA was the Chinese Remainder Theorem (CRT). Beside that method, there was another method with the same concept with CRT namely Aryabhata Remainder Theorem which was also relevant used in public key cryptography such as RSA. The purpose of this study was to obtain an effective method to RSA especially in the decryption process based on the calculation of the time complexity of computing.

A Novel RFID Anti-Counterfeiting Based on Bisectional Multivariate Quadratic Equations

2018 ◽  
Vol 6 (2) ◽  
pp. 1-9
Author(s):  
Xiaoyi Zhou ◽  
Jixin Ma ◽  
Xiaoming Yao ◽  
Honglei Li
Keyword(s):  
Finite Field ◽  
Data Encryption ◽  
Public Key ◽  
Generation Process ◽  
Key Generation ◽  
Quadratic Equations ◽  
Private Key ◽  
Message Digest ◽  
The One ◽  
N Vector

This article proposes a novel scheme for RFID anti-counterfeiting by applying bisectional multivariate quadratic equations (BMQE) system into an RF tag data encryption. In the key generation process, arbitrarily choose two matrix sets (denoted as A and B) and a base RAB such that [(AB) ⃗ ]=λ〖R_AB〗^T, and generate 2n BMQ polynomials (denoted as ρ) over finite field F_q. Therefore, (F_q, ρ) is taken as a public key and (A,B,λ) as a private key. In the encryption process, the EPC code is hashed into a message digest d_m. Then d_m is padded to d_m^' which is a non-zero 2n×2n matrix over F_q. With (A,B,λ)and d_m^', s_m is formed as an n-vector over F_2. Unlike the existing anti-counterfeit scheme, the one the authors proposed is based on quantum cryptography, thus it is robust enough to resist the existing attacks and has high security.

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