scholarly journals A way to compute a greatest common divisor in the Galois field (GF (2^n ))

2019 ◽  
Vol 16 ◽  
pp. 8317-8321
Author(s):  
Waleed Eltayeb Ahmed
Keyword(s):  
Galois Field ◽  
Euclidean Algorithm ◽  
Greatest Common Divisor ◽  
Multiplicative Inverse

This paper presents how the steps that used to determine a multiplicative inverse by method based on the Euclidean algorithm, can be used to find a greatest common divisor for polynomials in the Galois field (2^n ).

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 New S-box calculation approach for Rijndael-AES based on an artificial neural network

2017 ◽  
Vol 6 (2) ◽  
pp. E1-1-E1-21
Author(s):  
Jaime David Rios Arrañaga ◽  
◽  
Janneth Alejandra Salamanca Chavarin ◽  
Juan José Raygoza Panduro ◽  
Edwin Christian Becerra Alvarez ◽  
...  
Keyword(s):  
Neural Network ◽  
Artificial Neural Network ◽  
Galois Field ◽  
Block Ciphers ◽  
Multiplicative Inverse ◽  
Symmetric Key ◽  
Artificial Neural ◽  
Hidden Layer ◽  
The Relationship

The S-box is a basic important component in symmetric key encryption, used in block ciphers to confuse or hide the relationship between the plaintext and the ciphertext. In this paper a way to develop the transformation of an input of the S-box specified in AES encryption system through an artificial neural network and the multiplicative inverse in Galois Field is presented. With this implementation more security is achieved since the values of the S-box remain hidden and the inverse table serves as a distractor since it would appear to be the complete S-box. This is implemented on MATLAB and HSPICE using a network of perceptron neurons with a hidden layer and null error.

scholarly journals New Constructions of Balanced Boolean Functions with Maximum Algebraic Immunity, High Nonlinearity and Optimal Algebraic Degree

2020 ◽  
Vol 17 (7) ◽  
pp. 639-654
Author(s):  
Dheeraj Kumar SHARMA ◽  
Rajoo PANDEY
Keyword(s):  
Lower Bound ◽  
Boolean Function ◽  
Boolean Functions ◽  
Primitive Element ◽  
Galois Field ◽  
Algebraic Degree ◽  
Algebraic Immunity ◽  
Greatest Common Divisor ◽  
Primitive Elements ◽  
High Nonlinearity

This paper consists of proposal of two new constructions of balanced Boolean function achieving a new lower bound of nonlinearity along with high algebraic degree and optimal or highest algebraic immunity. This construction has been made by using representation of Boolean function with primitive elements. Galois Field,  used in this representation has been constructed by using powers of primitive element such that greatest common divisor of power and  is 1. The constructed balanced  variable Boolean functions achieve higher nonlinearity, algebraic degree of , and algebraic immunity of   for odd ,  for even . The nonlinearity of Boolean function obtained in the proposed constructions is better as compared to existing Boolean functions available in the literature without adversely affecting other properties such as balancedness, algebraic degree and algebraic immunity.

The Euclidean Algorithm as a Matrix Transformation

1972 ◽  
Vol 65 (3) ◽  
pp. 228-229
Author(s):  
Aziz Ibrahim ◽  
Edward Gucker
Keyword(s):  
Euclidean Algorithm ◽  
Matrix Transformation ◽  
Greatest Common Divisor ◽  
Repeated Application ◽  
Division Algorithm ◽  
Positive Integers

The algorithm of Euclid for finding the greatest common divisor of two positive integers is based on repeated application of the division algorithm.

Mathematical Method to Search for Monic Irreducible Polynomials with Decimal Equivalents of Polynomials over Galois Field GF(pq)

2017 ◽  
Vol 2 (11) ◽  
pp. 17-22
Author(s):  
Sankhanil Dey ◽  
Ranjan Ghosh
Keyword(s):  
Mathematical Method ◽  
Execution Time ◽  
Galois Field ◽  
Polynomial Multiplication ◽  
Irreducible Polynomials ◽  
Multiplicative Inverse ◽  
Advance Encryption Standard ◽  
Cipher Text ◽  
Substitution Boxes ◽  
Encryption And Decryption

Substitution boxes or S-boxes play a significant role in encryption and decryption of bit level plaintext and cipher-text respectively. Irreducible Polynomials (IPs) have been used to construct 4-bit or 8-bit substitution boxes in many cryptographic block ciphers. In Advance Encryption Standard the 8-bit the elements S-box have been obtained from the Multiplicative Inverse (MI) of elemental polynomials (EPs) of the 1st IP over Galois field GF(28) by adding an additive element. In this paper a mathematical method and the algorithm of the said method with the discussion of the execution time of the algorithm, to obtain monic IPs over Galois field GF(pq) have been illustrated with example. The method is very similar to polynomial multiplication of two polynomials over Galois field GF(pq) but has a difference in execution. The decimal equivalents of polynomials have been used to identify Basic Polynomials (BPs), EPs, IPs and Reducible polynomials (RPs). The monic RPs have been determined by this method and have been cancelled out to produce monic IPs. The non-monic IPs have been obtained with multiplication of α where α GF(pq) and assume values from 2 to (p-1) to monic IPs.

Microcomputer-assisted Discoveries: Euclidean Algorithm and Continued Fractions

1983 ◽  
Vol 76 (7) ◽  
pp. 510-548
Author(s):  
Clark Kimberling
Keyword(s):  
Continued Fractions ◽  
Euclidean Algorithm ◽  
Greatest Common Divisor ◽  
Positive Integers

Students can use microcomputers to cut through algorithms and computations to gain mathematical insights. This approach is especially true for the Euclidean algorithm, so often used to find the greatest common divisor (GCD) of two positive integers. The Euclidean algorithm also yields continued fractions, at least far enough for students to find patterns and discover truths about numbers.

A Greatest Common Divisor Algorithm

1998 ◽  
Vol 08 (05) ◽  
pp. 617-623 ◽  
Author(s):  
Ari Belenkiy ◽  
Raimundas Vidunas
Keyword(s):  
Simple Proof ◽  
Correct Answer ◽  
Computer Programming ◽  
Euclidean Algorithm ◽  
Greatest Common Divisor ◽  
Principal Role ◽  
Large Numbers ◽  
Rational Arithmetic ◽  
Computational Systems

Algorithms of computation of the Greatest Common Divisor (GCD) of two integers play a principal role in all computational systems dealing with rational arithmetic. The simplest one (Euclidean) is not the best for large numbers (see D. E. Knuth's book "The Art of Computer Programming" for details). One improvement was suggested by D. H. Lehmer in 1938 who noticed that it is possible to run the Euclidean algorithm with a few leading digits of large numbers and, with some care, still obtain the correct answer. In the 70's G. E. Collins pointed out that Lehmer's algorithm simultaneously analyzed two similar sequences of numbers and hence did twice as much work as necessary. Collins found a way to work with only one sequence of numbers together with a verification of a certain inequality. The proof of the validity of this inequality is, perhaps, too complicated. We present a similar but softer inequality and give a short and simple proof thereof.

1. Numbers and algebra

2015 ◽  
pp. 1-10
Author(s):  
Peter M. Higgins
Keyword(s):  
Fundamental Theorem ◽  
Common Factor ◽  
Number System ◽  
Rational Numbers ◽  
Euclidean Algorithm ◽  
Greatest Common Divisor ◽  
Natural Numbers ◽  
Division Algorithm ◽  
Fundamental Theorem Of Arithmetic ◽  
And Algebra

‘Numbers and algebra’ introduces the number system and explains several terms used in algebra, including natural numbers, positive and negative integers, rational numbers, number factorization, the Fundamental Theorem of Arithmetic, Euclid’s Lemma, the Division Algorithm, and the Euclidean Algorithm. It proves that any common factor c of a and b is also a factor of any number of the form ax + by, and since the greatest common divisor (gcd) of a and b has this form, which may be found by reversing the steps of the Euclidean Algorithm, it follows that any common factor c of a and b divides their gcd d.

scholarly journals Frequency Analysis of 32-bit Modular Divider Based on Extended GCD Algorithm for Different FPGA chips

2018 ◽  
Vol 17 (1) ◽  
pp. 7133-7139 ◽  
Author(s):  
Qasem Abu Al-Hiaja ◽  
Abdullah AlShuaibi ◽  
Ahmad Al Badawi
Keyword(s):  
Frequency Analysis ◽  
Delay Period ◽  
Euclidean Algorithm ◽  
Fundamental Operation ◽  
Multiplicative Inverse ◽  
Public Key Cryptosystems ◽  
Field Programmable ◽  
Minimum Delay ◽  
Performance Results ◽  
Modular Inversion

Modular inversion with large integers and modulus is a fundamental operation in many public-key cryptosystems. Extended Euclidean algorithm (XGCD) is an extension of Euclidean algorithm (GCD) used to compute the modular multiplicative inverse of two coprime numbers. In this paper, we propose a Frequency Analysis study of 32-bit modular divider based on extended-GCD algorithm targeting different chips of field-programmable gate array (FPGA). The experimental results showed that the design recorded the best performance results when implemented using Kintex7 (xc7k70t-2-fbg676) FPGA kit with a minimum delay period of 50.63 ns and maximum operating frequency of 19.5 MHz. Therefore, the proposed work can be embedded with many FPGA based cryptographic applications.

Sign in / Sign up

Export Citation Format

Share Document