How to securely outsource the extended euclidean algorithm for large-scale polynomials over finite fields

2020 ◽  
Vol 512 ◽  
pp. 641-660 ◽  
Author(s):  
Qiang Zhou ◽  
Chengliang Tian ◽  
Hanlin Zhang ◽  
Jia Yu ◽  
Fengjun Li
Keyword(s):  
Finite Fields ◽  
Large Scale ◽  
Euclidean Algorithm ◽  
Extended Euclidean Algorithm ◽  
Polynomials Over Finite Fields

scholarly journals GCD of Aunu Binary Polynomials of Cardinality Seven Using Extended Euclidean Algorithm

2021 ◽  
Vol 09 (09) ◽  
Author(s):  
Ibrahim A. A. ◽  
Keyword(s):  
Finite Fields ◽  
Coding Theory ◽  
Error Correcting Codes ◽  
Euclidean Algorithm ◽  
Greatest Common Divisor ◽  
Algebraic Structures ◽  
Extended Euclidean Algorithm ◽  
Encryption Schemes ◽  
Theory Error

Finite fields is considered to be the most widely used algebraic structures today due to its applications in cryptography, coding theory, error correcting codes among others. This paper reports the use of extended Euclidean algorithm in computing the greatest common divisor (gcd) of Aunu binary polynomials of cardinality seven. Each class of the polynomial is permuted into pairs until all the succeeding classes are exhausted. The findings of this research reveals that the gcd of most of the pairs of the permuted classes are relatively prime. This results can be used further in constructing some cryptographic architectures that could be used in design of strong encryption schemes.

scholarly journals Computing multiplicative inverses in finite fields by long division

2018 ◽  
Vol 69 (5) ◽  
pp. 400-402
Author(s):  
Otokar Grošek ◽  
Tomáš Fabšič
Keyword(s):  
Finite Fields ◽  
Prime Order ◽  
Euclidean Algorithm ◽  
Fixed Integer ◽  
Extended Euclidean Algorithm ◽  
Asymptotic Complexity ◽  
Long Division ◽  
Field Element

Abstract We study a method of computing multiplicative inverses in finite fields using long division. In the case of fields of a prime order p, we construct one fixed integer d(p) with the property that for any nonzero field element a, we can compute its inverse by dividing d(p) by a and by reducing the result modulo p. We show how to construct the smallest d(p) with this property. We demonstrate that a similar approach works in finite fields of a non-prime order, as well. However, we demonstrate that the studied method (in both cases) has worse asymptotic complexity than the extended Euclidean algorithm.

Some primitive polynomials over finite fields

2001 ◽  
Vol 21 (3) ◽  
pp. 412-416 ◽  
Author(s):  
Seunghwan Chang ◽  
June Bok Lee
Keyword(s):  
Finite Fields ◽  
Primitive Polynomials ◽  
Polynomials Over Finite Fields

scholarly journals Public-key cryptosystem based on invariants of diagonalizable groups

2017 ◽  
Vol 9 (1) ◽  
Author(s):  
František Marko ◽  
Alexandr N. Zubkov ◽  
Martin Juráš
Keyword(s):  
Linear Algebra ◽  
Finite Fields ◽  
Number Fields ◽  
Euclidean Algorithm ◽  
Public Key ◽  
Public Key Cryptosystem ◽  
Finite Rings

AbstractWe develop a public-key cryptosystem based on invariants of diagonalizable groups and investigate properties of such a cryptosystem first over finite fields, then over number fields and finally over finite rings. We consider the security of these cryptosystem and show that it is necessary to restrict the set of parameters of the system to prevent various attacks (including linear algebra attacks and attacks based on the Euclidean algorithm).

scholarly journals Factorization of a class of polynomials over finite fields

2012 ◽  
Vol 18 (1) ◽  
pp. 108-122 ◽  
Author(s):  
Henning Stichtenoth ◽  
Alev Topuzoğlu
Keyword(s):  
Finite Fields ◽  
Polynomials Over Finite Fields

scholarly journals Value sets of polynomials over finite fields

1993 ◽  
Vol 119 (3) ◽  
pp. 711-711 ◽  
Author(s):  
Da Qing Wan ◽  
Peter Jau-Shyong Shiue ◽  
Ching Shyang Chen
Keyword(s):  
Finite Fields ◽  
Polynomials Over Finite Fields ◽  
Value Sets
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