scholarly journals Generic Efficient Arithmetic Algorithms for PAFFs (Processor Adequate Finite Fields) and Related Algebraic Structures

2004 ◽  
pp. 320-334 ◽  
Author(s):  
Roberto Maria Avanzi ◽  
Preda Mihăilescu
Keyword(s):  
Finite Fields ◽  
Algebraic Structures

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 SRIM and SCRIM factors of xn + 1 over finite fields and their applications

2020 ◽  
pp. 2050098 ◽  
Author(s):  
Arunwan Boripan ◽  
Somphong Jitman
Keyword(s):  
Finite Fields ◽  
Algebraic Structures ◽  
Negacyclic Codes

Self-Reciprocal Irreducible Monic (SRIM) and Self-Conjugate-Reciprocal Irreducible Monic (SCRIM) factors of [Formula: see text] over finite fields have become of interest due to their rich algebraic structures and wide applications. In this paper, these notions are extended to factors of [Formula: see text] over finite fields. Characterization and enumeration of SRIM and SCRIM factors of [Formula: see text] over finite fields are established. Simplification and recessive formulas for the number of such factors are given. Finally, applications in the study of complementary negacyclic codes are discussed.

Algebraic Structures II (Vector Spaces and Finite Fields)

2019 ◽  
pp. 191-224
Author(s):  
R. Balakrishnan ◽  
Sriraman Sridharan
Keyword(s):  
Finite Fields ◽  
Vector Spaces ◽  
Algebraic Structures

scholarly journals Determinants of some special matrices over commutative finite chain rings

2020 ◽  
Vol 8 (1) ◽  
pp. 242-256
Author(s):  
Somphong Jitman
Keyword(s):  
Finite Fields ◽  
Principal Ideal ◽  
Residue Field ◽  
Circulant Matrices ◽  
Finite Chain ◽  
Algebraic Structures ◽  
Open Problems ◽  
Finite Chain Rings ◽  
Positive Integers ◽  
Finite Principal Ideal Rings

AbstractCirculant matrices over finite fields and over commutative finite chain rings have been of interest due to their nice algebraic structures and wide applications. In many cases, such matrices over rings have a closed connection with diagonal matrices over their extension rings. In this paper, the determinants of diagonal and circulant matrices over commutative finite chain rings R with residue field 𝔽q are studied. The number of n × n diagonal matrices over R of determinant a is determined for all elements a in R and for all positive integers n. Subsequently, the enumeration of nonsingular n × n circulant matrices over R of determinant a is given for all units a in R and all positive integers n such that gcd(n, q) = 1. In some cases, the number of singular n × n circulant matrices over R with a fixed determinant is determined through the link between the rings of circulant matrices and diagonal matrices. As applications, a brief discussion on the determinants of diagonal and circulant matrices over commutative finite principal ideal rings is given. Finally, some open problems and conjectures are posted

Finite Fields

1996 ◽  
Author(s):  
Rudolf Lidl ◽  
Harald Niederreiter
Keyword(s):  
Finite Fields

Maximum Distance Separable Hankel Rhotrices over Finite Fields

2018 ◽  
Vol 43 (1-4) ◽  
pp. 13-45
Author(s):  
Prof. P. L. Sharma ◽  
◽  
Mr. Arun Kumar ◽  
Mrs. Shalini Gupta ◽  
◽  
...  
Keyword(s):  
Finite Fields ◽  
Maximum Distance ◽  
Maximum Distance Separable

Notes on three-pass protocol

2020 ◽  
Vol 25 (4) ◽  
pp. 4-9
Author(s):  
Yerzhan R. Baissalov ◽  
Ulan Dauyl
Keyword(s):  
Finite Fields ◽  
Matrix Algebras ◽  
Associative Structures

The article discusses primitive, linear three-pass protocols, as well as three-pass protocols on associative structures. The linear three-pass protocols over finite fields and the three-pass protocols based on matrix algebras are shown to be cryptographically weak.

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