Computing multiplicative inverses in finite fields by long division

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.

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

INTERNATIONAL JOURNAL OF MATHEMATICS AND COMPUTER RESEARCH ◽

10.47191/ijmcr/v9i9.02 ◽

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.

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

Information Sciences ◽

10.1016/j.ins.2019.10.007 ◽

2020 ◽

Vol 512 ◽

pp. 641-660 ◽

Cited By ~ 6

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

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Public-key cryptosystem based on invariants of diagonalizable groups

Groups – Complexity – Cryptology ◽

10.1515/gcc-2017-0003 ◽

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).

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On a parallel extended Euclidean algorithm

Proceedings ACS/IEEE International Conference on Computer Systems and Applications ◽

10.1109/aiccsa.2001.933982 ◽

2002 ◽

Cited By ~ 1

Author(s):

S.M. Sedjelmaci

Keyword(s):

Euclidean Algorithm ◽

Extended Euclidean Algorithm

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Atomic Latin Squares based on Cyclotomic Orthomorphisms

The Electronic Journal of Combinatorics ◽

10.37236/1919 ◽

2005 ◽

Vol 12 (1) ◽

Cited By ~ 16

Author(s):

Ian M. Wanless

Keyword(s):

Finite Fields ◽

Complete Graph ◽

Prime Order ◽

Automorphism Groups ◽

Latin Square ◽

Latin Squares ◽

Established Method ◽

Cyclic Groups ◽

Complete Bipartite Graphs ◽

First Time

Atomic latin squares have indivisible structure which mimics that of the cyclic groups of prime order. They are related to perfect $1$-factorisations of complete bipartite graphs. Only one example of an atomic latin square of a composite order (namely 27) was previously known. We show that this one example can be generated by an established method of constructing latin squares using cyclotomic orthomorphisms in finite fields. The same method is used in this paper to construct atomic latin squares of composite orders 25, 49, 121, 125, 289, 361, 625, 841, 1369, 1849, 2809, 4489, 24649 and 39601. It is also used to construct many new atomic latin squares of prime order and perfect $1$-factorisations of the complete graph $K_{q+1}$ for many prime powers $q$. As a result, existence of such a factorisation is shown for the first time for $q$ in $\big\{$529, 2809, 4489, 6889, 11449, 11881, 15625, 22201, 24389, 24649, 26569, 29929, 32041, 38809, 44521, 50653, 51529, 52441, 63001, 72361, 76729, 78125, 79507, 103823, 148877, 161051, 205379, 226981, 300763, 357911, 371293, 493039, 571787$\big\}$. We show that latin squares built by the 'orthomorphism method' have large automorphism groups and we discuss conditions under which different orthomorphisms produce isomorphic latin squares. We also introduce an invariant called the train of a latin square, which proves to be useful for distinguishing non-isomorphic examples.

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