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Optimizing the selection of the secret parameters for public key cryptosystems by using interval linear programming and fully fuzzy linear programming

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ENGINEERING APPLICATIONS OF ARTIFICIAL INTELLIGENCE 2025年 144卷

作者： Bas, Esra Istanbul Tech Univ Dept Ind Engn TR-34367 Istanbul Turkiye

One of the basic tasks in public key cryptosystems is to determine the secret parameters. This paper proposes multi-objective interval linear programming and multi-objective fully fuzzy linear programming (FFLP) models to determine the optimum interval numbers and triangular fuzzy numbers (TFNs) of the secret parameters in the public key cryptosystems. The public and private (secret) objective function(s) and constraints can be considered such that the optimum interval numbers and optimum TFNs for the secret parameters can be obtained from which the random selection of the secret parameters can be made for each algorithm's execution. Adapting the general model to the Rivest, Shamir, Adleman (RSA) public key cryptosystem is also illustrated. In the interval programming model for the RSA, the linear combination method combines multiple objective functions into a single objective function, and nonlinear constraints are linearized using the Taylor series expansion method. No multiple objective functions are obtained in the FFLP model of the RSA, but the nonlinear constraints are again linearized using the Taylor series expansion method. The numerical application for the RSA public key cryptosystem is made for both models by considering five cases and twenty-four subcases, and the results are analyzed. To the best of our knowledge, there is no study in the literature that considers interval linear programming and/or FFLP to obtain the optimum interval numbers and TFNs for the secret parameters of the public key cryptosystems. The proposed methodology is systematic and generic and can also be adapted for other public key cryptosystems easily.

关键词： Public key cryptosystem Rivest Shamir Adleman public key cryptosystem Interval linear programming fully fuzzy linear programming Multi-objective optimization

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Solving fully fuzzy linear programming problems with flexible constraints based on a new order relation

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JOURNAL OF INTELLIGENT & fuzzy SYSTEMS 2015年第4期29卷 1539-1550页

作者： Yang, Xiao-Peng Cao, Bing-Yuan Zhou, Xue-Gang Guangzhou Univ Sch Math & Informat Sci Key Lab Math & Interdisciplinary Sci Guangdong Hi Guangzhou 510006 Guangdong Peoples R China Hanshan Normal Univ Dept Math & Stat Chaozhou Peoples R China Guangzhou Vocat Coll Sci & Technol Guangzhou Guangdong Peoples R China Guangdong Univ Finance Dept Appl Math Guangzhou Guangdong Peoples R China

In this paper, a new order relation for ranking fuzzy numbers and the fully fuzzy linear programming (FFLP) problems with flexible constraints are proposed. Based on the new order relation, three algorithms are developed to deal with FFLP problems by converting them into some equivalent crisp linear programming problems. Algorithm 1 is employed for solving FFLP problem without flexible constraint, while Algorithms 2 and 3 are developed to deal with FFLP problem with flexible constraints. A numerical example is given to illustrate the feasibility and efficiency of our proposed algorithms.

关键词： fully fuzzy linear programming flexible constraint order relation expected value

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A method to find the unique optimal fuzzy value of fully fuzzy linear programming problems with inequality constraints having unrestricted L-R fuzzy parameters and decision variables

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EXPERT SYSTEMS WITH APPLICATIONS 2019年 123卷 256-269页

作者： Perez-Canedo, Boris Rene Concepcion-Morales, Eduardo Univ Cienfuegos Fac Econ & Business Sci Dept Math Cienfuegos 59430 Cuba Univ Cienfuegos Fac Engn Dept Informat Cienfuegos 59430 Cuba Metropolitan Univ UMET Sch Informat Syst Quito 170523 Pichincha Ecuador

In the literature, a wide variety of methods exist for solving fully fuzzy linear programming (FFLP) problems. However, there is still no method to find the unique optimal fuzzy value of FFLP problems with inequality constraints having unrestricted L-R fuzzy parameters and decision variables. Alternatively, some researchers introduce non-negative fuzzy slack and surplus variables to transform the FFLP problems with inequality constraints into FFLP problems having only equality constraints. Others, replace each fuzzy inequality constraint with a set of crisp linear inequalities. However, these two approaches cannot guarantee solutions with unique optimal fuzzy values and may lead to unfeasible problems. In this paper, based on the total order properties of a lexicographic criterion for ranking L-R fuzzy numbers, a method to find the unique optimal fuzzy value of FFLP problems with equality and inequality constraints having unrestricted L-R fuzzy parameters and decision variables is proposed. By following the steps of the proposed method, the FFLP problem is transformed into an equivalent mixed 0-1 lexicographic non-linear programming (MLNLP) problem. A numerical example is provided to illustrate the proposed method, and the results are compared with those obtained by other alternative methods, showing that the proposed method overcomes the shortcomings and limitations of the existing ones. (C) 2019 Elsevier Ltd. All rights reserved.

关键词： fully fuzzy linear programming L-R fuzzy number Lexicographic ranking criterion

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An analysis of fully fuzzy linear programming with fuzzy decision variables through logistics network design problem

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KNOWLEDGE-BASED SYSTEMS 2015年第Dec.期90卷 165-184页

作者： Baykasoglu, Adil Subulan, Kemal Dokuz Eylul Univ Fac Engn Dept Ind Engn Izmir Turkey

Recently, there is a growing attention by the researchers to solve and interpret the analysis of fully fuzzy linear programming problems in which all of the parameters as well as the decision variables are considered as fuzzy numbers. Under a fully uncertain environment where all of the data are stated as fuzzy, presenting the reasonable range of values for the decision variables may be comparatively better than the currently available crisp solutions so as to provide ranges of flexibility to decision makers. However, there is still a scarcity of solution methodologies on fuzzy mathematical programs with fuzzy decision variables. Based on this motivation, a new parametric method which is mainly based on alpha-cut representation of fuzzy intervals is proposed in this paper by incorporating the decision maker's attitude toward risk. In order to illustrate validity and practicality of the proposed method, it is applied to a generic reverse logistics network design model including fuzzy decision variables. To the best of our knowledge, this is the first study in the literature which presents fuzzy efficient solutions and analysis for a fully fuzzy reverse logistics network design problem with fuzzy decision variables. The provided solutions by the proposed method are also compared to the available solution methodologies from the literature in terms of computational efficiency, solution quality and ease of use. By using the proposed method, the decision makers can be supported by yielding fuzzy efficient solutions under different uncertainty levels and risk attitudes. The computational results have also shown that more reliable and necessarily precise solutions can be generated by the proposed method for a risk-averse decision maker. (C) 2015 Elsevier B.V. All rights reserved.

关键词： fully fuzzy linear programming Reverse logistics network design fuzzy decision variables fuzzy mathematical programming Risk attitude

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Exact fuzzy optimal solution of fully fuzzy linear programming problems with unrestricted fuzzy variables

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APPLIED INTELLIGENCE 2012年第1期37卷 145-154页

作者： Kaur, Jagdeep Kumar, Amit Thapar Univ Sch Math & Comp Applicat Patiala 147004 Punjab India

Kumar et al. (Appl. Math. Model. 35:817-823, 2011) pointed out that there is no method in literature to find the exact fuzzy optimal solution of fully fuzzy linear programming (FFLP) problems and proposed a new method to find the fuzzy optimal solution of FFLP problems with equality constraints having non-negative fuzzy variables and unrestricted fuzzy coefficients. There may exist several FFLP problems with equality constraints in which no restriction can be applied on all or some of the fuzzy variables but due to the limitation of the existing method these types of problems can not be solved by using the existing method. In this paper a new method is proposed to find the exact fuzzy optimal solution of FFLP problems with equality constraints having non-negative fuzzy coefficients and unrestricted fuzzy variables. The proposed method can also be used to solve the FFLP problems with equality constraints having non-negative fuzzy variables and unrestricted fuzzy coefficients. To show the advantage of the proposed method over existing method the results of some FFLP problems with equality constraints, obtained by using the existing and proposed method, are compared. Also, to show the application of proposed method a real life problem is solved by using the proposed method.

关键词： fully fuzzy linear programming Exact fuzzy optimal solution Trapezoidal fuzzy number

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A nonlinear model for fully fuzzy linear programming with fully unrestricted variables and parameters

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ALEXANDRIA ENGINEERING JOURNAL 2016年第3期55卷 2589-2595页

作者： Najafi, H. Saberi Edalatpanah, S. A. Dutta, H. Univ Guilan Dept Math Fac Sci Univ Campus 2 Rasht Iran Gauhati Univ Dept Math Gauhati 781014 India

Several methods currently exist for solving fuzzy linear programming problems under nonnegative fuzzy variables and restricted fuzzy coefficients. However, due to the limitation of these methods, they cannot be applied for solving fully fuzzy linear programming (FFLP) with unrestricted fuzzy coefficients and fuzzy variables. In this paper a new efficient method for FFLP has been proposed, in order to obtain the fuzzy optimal solution with unrestricted variables and parameters. This proposed method is based on crisp nonlinear programming and has a simple structure. To show the efficiency of our proposed method some numerical examples have been illustrated. (C) 2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://***/licenses/by-nc-nd/4.0/).

关键词： fully fuzzy linear programming Crisp nonlinear programming fuzzy arithmetic fuzzy optimal solution Triangular fuzzy numbers

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A mathematical model for solving fully fuzzy linear programming problem with trapezoidal fuzzy numbers

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APPLIED INTELLIGENCE 2017年第3期46卷 509-519页

作者： Das, Sapan Kumar Mandal, T. Edalatpanah, S. A. Natl Inst Technol Jamshedpur Dept Math Jamshedpur 831014 Jharkhand India Islamic Azad Univ Lahijan Branch Dept Appl Math Lahijan Iran Islamic Azad Univ Lahijan Branch Young Researchers Club Lahijan Iran

In this paper, an efficient method is introduced to solve fully fuzzy linear programming problems. The proposed method is derived from the multi-objective linear programming problem and lexicographic ordering method. Theoretical analysis for the proposed method has been provided. Moreover, some numerical experiments are given to show the preference of the proposed methods and are compared with some available methods.

关键词： fully fuzzy linear programming Trapezoidal fuzzy numbers Multi-objective linear programming

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A note on "A new method for solving fully fuzzy linear programming problems"

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APPLIED MATHEMATICAL MODELLING 2013年第14-15期37卷 7865-7867页

作者： Najafi, H. Saberi Edalatpanah, S. A. Islamic Azad Univ Lahijan Branch Dept Appl Math Lahijan Iran Islamic Azad Univ Lahijan Branch Young Researchers Club Lahijan Iran

This note shows that solving fully fuzzy linear programming (FFLP) model presented by Kumar et al. [A. Kumar, J. Kaur, P. Singh, A new method for solving fully fuzzy linear programming problems, Appl. Math. Model. 35 (2011) 817-823] needs some corrections to make the model well in general. A new version is provided in this note. A simple example is also presented to demonstrate the new form. (C) 2013 Elsevier Inc. All rights reserved.

关键词： fully fuzzy linear programming Triangular fuzzy numbers Nonnegative fuzzy numbers

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An effective computational attempt for solving fully fuzzy linear programming using MOLP problem

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JOURNAL OF INDUSTRIAL AND PRODUCTION ENGINEERING 2019年第2期36卷 59-69页

作者： Ebrahimnejad, Ali Islamic Azad Univ Dept Math Qaemshahr Branch Qaemshahr Iran

This paper is concerned with the solution procedure of a fully fuzzy linear programming (FFLP) problem in which all parameters are represented in terms of triangular fuzzy numbers. According to the existing solution approach and based a new lexicographic ordering on triangular fuzzy numbers, the FFLP problem under consideration is converted into a multi-objective linear programming (MOLP) problem with three-objective functions. This paper provides a new technique for reducing the computational complexity of some current FFLP problems for situations that all imprecise data are represented as nonnegative triangular fuzzy numbers. Finally, using a simple example and one case study the results obtained using the proposed algorithm and those obtained using with the existing methods are compared, demonstrating the reliability and applicability of our algorithm.

关键词： fully fuzzy linear programming multi-objective linear programming problem lexicographic ordering on triangular fuzzy numbers computational complexity

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Multi-Objective fully fuzzy linear programming Problems With Triangular fuzzy Numbers 11

Multi-Objective Fully Fuzzy Linear Programming Problems With...

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11th International Conference on fuzzy Systems and Knowledge Discovery (FSKD)

作者： Yang, Xiao-Peng Cao, Bing-Yuan Lin, Hai-Tao Guangzhou Univ Sch Math & Informat Sci Guangzhou Guangdong Peoples R China Hanshan Normal Univ Dept Math & Stat Chaozhou Peoples R China Guangzhou Univ Sch Math & Informat Sci Key Lab Math & Interdisciplinary Ciences Guangdon Guangzhou Guangdong Peoples R China Hanshan Normal Univ Coll Ceram Chaozhou Peoples R China

ISBN: (纸本)9781479951482

For checking the optimality of the objective function, we introduce a lexicographic order relation to compare two arbitrary triangular fuzzy numbers. Based on the order relation, the multi-objective fully fuzzy linear programming problem is converted into a crisp multi-level multi-objective linear programming problem, which may be solved by using the Min operator step by step. A numerical example is given to illustrate the validity of the proposed method.

关键词： fuzzy number fully fuzzy linear programming multi-level multi-objective linear programming lexicographic order relation

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