scholarly journals Multiplication over Extended Galois Field: A New Approach to Find Monic Irreducible Polynomials over Galois Field GF(p^q).

2017 ◽  
Author(s):  
Sankhanil Dey ◽  
Ranjan Ghosh
Keyword(s):  
Time Complexity ◽  
Complexity Analysis ◽  
Galois Field ◽  
Irreducible Polynomials ◽  
Galois Fields ◽  
New Approach ◽  
Multiplication Algorithm

Searching for Monic Irreducible Polynomials (IPs) over extended Galois Field GF(p^q) for large value of prime moduli p and extension to Galois Field q is a well needed solution in the field of Cryptography. In this paper a new algorithm to obtain Monic IPs over extended Galois Fields GF(p^q) for large value of p and q has been introduced. The algorithm has been based on Multiplication algorithm over Galois Field GF(p^q).Time complexity analysis of the said algorithm has also been executed that ensures the algorithm to be less time consuming.

scholarly journals Multiplication over Extended Galois Field: A New Approach to Find Monic Irreducible Polynomials over Galois Field GF(p^q).

2017 ◽  
Author(s):  
Sankhanil Dey ◽  
Ranjan Ghosh
Keyword(s):  
Time Complexity ◽  
Complexity Analysis ◽  
Galois Field ◽  
Irreducible Polynomials ◽  
Galois Fields ◽  
New Approach ◽  
Multiplication Algorithm

Searching for Monic Irreducible Polynomials (IPs) over extended Galois Field GF(p^q) for large value of prime moduli p and extension to Galois Field q is a well needed solution in the field of Cryptography. In this paper a new algorithm to obtain Monic IPs over extended Galois Fields GF(p^q) for large value of p and q has been introduced. The algorithm has been based on Multiplication algorithm over Galois Field GF(p^q).Time complexity analysis of the said algorithm has also been executed that ensures the algorithm to be less time consuming.

scholarly journals Multiplication and Division over Extended Galois Field GF(p^q): A new Approach to find Monic Irreducible Polynomials over any Galois Field GF(p^q).

2017 ◽  
Author(s):  
Sankhanil Dey ◽  
Ranjan Ghosh
Keyword(s):  
Efficient Algorithm ◽  
Time Complexity ◽  
Complexity Analysis ◽  
Galois Field ◽  
Division Algorithm ◽  
Irreducible Polynomials ◽  
New Approach ◽  
Multiplication Algorithm ◽  
Research Article ◽  
Substitution Boxes

Irreducible Polynomials (IPs) have been of utmost importance in generation of substitution boxes in modern cryptographic ciphers. In this paper an algorithm entitled Composite Algorithm using both multiplication and division over Galois fields have been demonstrated to generate all monic IPs over extended Galois Field GF(p^q) for large value of both p and q. A little more efficient Algorithm entitled Multiplication Algorithm and more too Division Algorithm have been illustrated in this Paper with Algorithms to find all Monic IPs over extended Galois Field GF(p^q) for large value of both p and q. Time Complexity Analysis of three algorithms with comparison to Rabin’s Algorithms has also been exonerated in this Research Article.

scholarly journals Multiplication and Division over Extended Galois Field GF(p^q): A new Approach to find Monic Irreducible Polynomials over any Galois Field GF(p^q).

2017 ◽  
Author(s):  
Sankhanil Dey ◽  
Ranjan Ghosh
Keyword(s):  
Efficient Algorithm ◽  
Time Complexity ◽  
Complexity Analysis ◽  
Galois Field ◽  
Division Algorithm ◽  
Irreducible Polynomials ◽  
New Approach ◽  
Multiplication Algorithm ◽  
Research Article ◽  
Substitution Boxes

Irreducible Polynomials (IPs) have been of utmost importance in generation of substitution boxes in modern cryptographic ciphers. In this paper an algorithm entitled Composite Algorithm using both multiplication and division over Galois fields have been demonstrated to generate all monic IPs over extended Galois Field GF(p^q) for large value of both p and q. A little more efficient Algorithm entitled Multiplication Algorithm and more too Division Algorithm have been illustrated in this Paper with Algorithms to find all Monic IPs over extended Galois Field GF(p^q) for large value of both p and q. Time Complexity Analysis of three algorithms with comparison to Rabin’s Algorithms has also been exonerated in this Research Article.

scholarly journals Construction of Irreducible Polynomials in Galois elds, GF(2m) Using Normal Bases

2019 ◽  
pp. 1-15
Author(s):  
Abraham Aidoo ◽  
Kwasi Baah Gyam
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
General Rule ◽  
Galois Field ◽  
Irreducible Polynomials ◽  
Galois Fields ◽  
Normal Bases ◽  
Primitive Polynomials

This thesis is about Construction of Polynomials in Galois fields Using Normal Bases in finite fields. In this piece of work, we discussed the following in the text; irreducible polynomials, primitive polynomials, field, Galois field or finite fields, and the order of a finite field. We found the actual construction of polynomials in GF(2m) with degree less than or equal to m − 1 and also illustrated how this construction can be done using normal bases. Finally, we found the general rule for construction of GF(pm) using normal bases and even the rule for producing reducible polynomials.

scholarly journals Galois Field Instructions in the Sandblaster 2.0 Architectrue

2009 ◽  
Vol 2009 ◽  
pp. 1-5 ◽  
Author(s):  
Mayan Moudgill ◽  
Andrei Iancu ◽  
Daniel Iancu
Keyword(s):  
Irreducible Polynomial ◽  
Galois Field ◽  
Galois Fields ◽  
New Approach ◽  
Novel Approach ◽  
Simd Architecture

This paper presents a novel approach to implementing multiplication of Galois Fields with . Elements of GF() can be represented as polynomials of degree less than N over GF(2). Operations are performed modulo an irreducible polynomial of degree n over GF(2). Our approach splits a Galois Field multiply into two operations, polynomial-multiply and polynomial-remainder over GF(2). We show how these two operations can be implemented using the same hardware. Further, we show that in many cases several polynomial-multiply operations can be combined before needing to a polynomial-remainder. The Sandblaster 2.0 is a SIMD architecture. It has SIMD variants of the poly-multiply and poly-remainder instructions. We use a Reed-Solomon encoder and decoder to demonstrate the performance of our approach. Our new approach achieves speedup of 11.5x compared to the standard SIMD processor of 8x.

scholarly journals Traps to the BGJT-algorithm for discrete logarithms

2014 ◽  
Vol 17 (A) ◽  
pp. 218-229 ◽  
Author(s):  
Qi Cheng ◽  
Daqing Wan ◽  
Jincheng Zhuang
Keyword(s):  
Finite Fields ◽  
Polynomial Time ◽  
Time Complexity ◽  
Complexity Analysis ◽  
Discrete Logarithm ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
New Approach ◽  
Discrete Logarithms ◽  
Polynomial Time Complexity

AbstractIn the recent breakthrough paper by Barbulescu, Gaudry, Joux and Thomé, a quasi-polynomial time algorithm is proposed for the discrete logarithm problem over finite fields of small characteristic. The time complexity analysis of the algorithm is based on several heuristics presented in their paper. We show that some of the heuristics are problematic in their original forms, in particular when the field is not a Kummer extension. We propose a fix to the algorithm in non-Kummer cases, without altering the heuristic quasi-polynomial time complexity. Further study is required in order to fully understand the effectiveness of the new approach.

scholarly journals A sieve algorithm based on overlattices

2014 ◽  
Vol 17 (A) ◽  
pp. 49-70 ◽  
Author(s):  
Anja Becker ◽  
Nicolas Gama ◽  
Antoine Joux
Keyword(s):  
Heuristic Algorithm ◽  
Orthonormal Basis ◽  
Complexity Analysis ◽  
Computer Architectures ◽  
High Dimensions ◽  
Average Case ◽  
Solution Vector ◽  
New Approach ◽  
Bounded Norm

AbstractIn this paper, we present a heuristic algorithm for solving exact, as well as approximate, shortest vector and closest vector problems on lattices. The algorithm can be seen as a modified sieving algorithm for which the vectors of the intermediate sets lie in overlattices or translated cosets of overlattices. The key idea is hence no longer to work with a single lattice but to move the problems around in a tower of related lattices. We initiate the algorithm by sampling very short vectors in an overlattice of the original lattice that admits a quasi-orthonormal basis and hence an efficient enumeration of vectors of bounded norm. Taking sums of vectors in the sample, we construct short vectors in the next lattice. Finally, we obtain solution vector(s) in the initial lattice as a sum of vectors of an overlattice. The complexity analysis relies on the Gaussian heuristic. This heuristic is backed by experiments in low and high dimensions that closely reflect these estimates when solving hard lattice problems in the average case.This new approach allows us to solve not only shortest vector problems, but also closest vector problems, in lattices of dimension$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}n$in time$2^{0.3774\, n}$using memory$2^{0.2925\, n}$. Moreover, the algorithm is straightforward to parallelize on most computer architectures.

scholarly journals DIFFERENTIATION OF POLYNOMIALS IN SEVERAL VARIABLES OVER GALOIS FIELDS OF FUZZY CARDINALITY AND APPLICATIONS TO REED-MULLER CODES

2018 ◽  
Vol 18 (3) ◽  
pp. 339-348
Author(s):  
V. M. Deundyak ◽  
N. S. Mogilevskaya
Keyword(s):  
Coding Theory ◽  
Communication Systems ◽  
Communication Channels ◽  
Galois Field ◽  
Second Order ◽  
Galois Fields ◽  
Practical Applications ◽  
Several Variables ◽  
Confidential Data ◽  
Partial Distortion

Introduction. Polynomials in several variables over Galois fields provide the basis for the Reed-Muller coding theory, and are also used  in a number of cryptographic problems. The properties of such polynomials specified over the derived Galois fields of fuzzy cardinality are studied. For the results obtained,  two  real-world  applications  are  proposed: partitioning scheme and Reed-Muller code decoder.Materials and Methods. Using linear algebra, theory of Galois fields, and general theory of polynomials in several variables, we have obtained results related to the differentiation and integration  of polynomials  in  several  variables  over  Galois fields of fuzzy cardinality. An analog of the differentiation operator is constructed and studied for vectors.Research Results. On the basis of the obtained results on the differentiation and integration of polynomials, a new decoder for Reed-Muller codes of the second order is given, and a scheme for organizing the partitioned transfer of confidential data is proposed. This is a communication system in which the source data on the sender is divided into several parts and, independently of one  another,  transmitted  through  different communication channels, and then, on the receiver, the initial data is restored of the parts retrieved. The proposed scheme feature is that it enables to protect data, both from the nonlegitimate access, and from unintentional errors; herewith, one  and  the  same  mathematical  apparatus  is  used  in  both cases. The developed decoder for the second-order Reed-Muller codes prescribed over the derived odd Galois field may have a constraint to the recoverable error level; however, its use is advisable for a number of the communication channels.Discussion    and    Conclusions.    The    proposed    practical applications   of   the   results   obtained   are   useful   for   the organization of reliable communication systems. In future, it is planned  to  study  the  restoration  process  of  the  original polynomial by its derivatives, in case of their partial distortion, and the development of appropriate applications.

Real-time engine model development based on time complexity analysis

2021 ◽  
pp. 146808742110397
Author(s):  
Haotian Chen ◽  
Kun Zhang ◽  
Kangyao Deng ◽  
Yi Cui
Keyword(s):  
Real Time ◽  
Time Complexity ◽  
Complexity Analysis ◽  
High Accuracy ◽  
The Real ◽  
Cycle Simulation ◽  
Crank Angle ◽  
Time Requirements ◽  
Improved Model ◽  
Simulation Time

Real-time simulation models play an important role in the development of engine control systems. The mean value model (MVM) meets real-time requirements but has limited accuracy. By contrast, a crank-angle resolved model, such as the filling -and-empty model, can be used to simulate engine performance with high accuracy but cannot meet real-time requirements. Time complexity analysis is used to develop a real-time crank-angle resolved model with high accuracy in this study. A method used in computer science, program static analysis, is used to theoretically determine the computational time for a multicylinder engine filling-and-empty (crank-angle resolved) model. Then, a prediction formula for the engine cycle simulation time is obtained and verified by a program run test. The influence of the time step, program structure, algorithm and hardware on the cycle simulation time are analyzed systematically. The multicylinder phase shift method and a fast calculation method for the turbocharger characteristics are used to improve the crank-angle resolved filling-and-empty model to meet real-time requirements. The improved model meets the real-time requirement, and the real-time factor is improved by 3.04 times. A performance simulation for a high-power medium-speed diesel engine shows that the improved model has a max error of 5.76% and a real-time factor of 3.93, which meets the requirement for a hardware-in-the-loop (HIL) simulation during control system development.

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