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 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.

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.

scholarly journals The Effect of the Primitive Irreducible Polynomial on the Quality of Cryptographic Properties of Block Ciphers

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
Sajjad Shaukat Jamal ◽  
Dawood Shah ◽  
Abdulaziz Deajim ◽  
Tariq Shah
Keyword(s):  
Secure Communication ◽  
Irreducible Polynomial ◽  
Galois Field ◽  
Block Ciphers ◽  
Affine Transformations ◽  
Differential Approximation ◽  
Irreducible Polynomials ◽  
Substitution Boxes ◽  
Nonlinearity Test ◽  
Galois Field Extensions

Substitution boxes are the only nonlinear component of the symmetric key cryptography and play a key role in the cryptosystem. In block ciphers, the S-boxes create confusion and add valuable strength. The majority of the substitution boxes algorithms focus on bijective Boolean functions and primitive irreducible polynomial that generates the Galois field. For binary field F2, there are exactly 16 primitive irreducible polynomials of degree 8 and it prompts us to construct 16 Galois field extensions of order 256. Conventionally, construction of affine power affine S-box is based on Galois field of order 256, depending on a single degree 8 primitive irreducible polynomial over ℤ2. In this manuscript, we study affine power affine S-boxes for all the 16 distinct degree 8 primitive irreducible polynomials over ℤ2 to propose 16 different 8×8 substitution boxes. To perform this idea, we introduce 16 affine power affine transformations and, for fixed parameters, we obtained 16 distinct S-boxes. Here, we thoroughly study S-boxes with all possible primitive irreducible polynomials and their algebraic properties. All of these boxes are evaluated with the help of nonlinearity test, strict avalanche criterion, bit independent criterion, and linear and differential approximation probability analyses to measure the algebraic and statistical strength of the proposed substitution boxes. Majority logic criterion results indicate that the proposed substitution boxes are well suited for the techniques of secure communication.

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 Efficient Algorithm For L(3,2,1)-Labeling of Cartesian Product Between Some Graphs

10.29007/x3qf ◽  
2019 ◽  
Author(s):  
Sumonta Ghosh ◽  
Prakhar Pogde ◽  
Narayan C. Debnath ◽  
Anita Pal
Keyword(s):  
Communication Network ◽  
Assignment Problem ◽  
Efficient Algorithm ◽  
Time Complexity ◽  
Cartesian Product ◽  
Frequency Assignment ◽  
Frequency Assignment Problem ◽  
The Difference

L(h,k) Labeling in graph came into existence as a solution to frequency assignment problem. To reduce interference a frequency in the form of non negative integers is assigned to each radio or TV transmitters located at various places. After L(h,k) labeling, L(h,k, j) labeling is introduced to reduce noise in the communication network. We investigated the graph obtained by Cartesian Product betweenCompleteBipartiteGraphwithPathandCycle,i. e.,Km,n×Pr andKm,n×Cr byapplying L(3,2,1)Labeling. The L(3,2,1) Labeling of a graph G is the difference between the highest and the lowest labels used in L(3,2,1) and is denoted by λ3,2,1(G) In this paper we have designed three suitable algorithms to label the graphs Km,n × Pr and Km,n × Cr . We have also analyzed the time complexity of each algorithm with illustration.

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.

Sign in / Sign up

Export Citation Format

Share Document