CONSTRUCTION OF BOOLEAN FUNCTIONS AND SOLUTIONS OF A SYSTEM OF LINEAR INEQUALITIES BY NSGA-II

Abstract

The research work presented in the thesis is study of methods developed to construct desired Boolean functions and solutions of system of linear inequalities. The thesis consists of seven chapters. The chapter-wise summary of the thesis is as follows: Chapter 1 is introductory in nature. In this chapter, we have de ned the relevant supporting theory of Boolean functions. In particular, we have provided numerous def- initions and theorems for various aspects of the theory. The necessary cryptographic properties which are used to analyze the strength of Boolean functions have been also de ned and discussed, and inter-relations between pairs of selected properties are also discussed. Finally, we have presented a brief summary of major cryptanalytic attacks against Boolean functions and cipher systems. In Chapter 2, we have developed a new evolutionary method to optimize the Boolean functions' properties by two objective optimization method. In this Chapter , we have taken balancedness, nonlinearity and resiliency, and developed an evolutionary method to construct Boolean functions having these properties at optimal level. We have got the desired functions and compared our results with previous results. Our results are as good as previous results. In Chapter 3 also, we have developed a new evolutionary method to optimize the Boolean functions' properties by two objective optimization method but here we have taken balancedness, nonlinearity and autocorrelation, and developed an evolutionary method to construct desired Boolean functions. We have got the desired functions and compared our results with previous results. Our results are at least as better as previous results. i In Chapter 4, we have developed a new method to optimize the Boolean func- tions' properties by three objective optimization method. In this chapter, we have taken balancedness, nonlinearity, resiliency and autocorrelation simultaneously to op- timize these properties. We have got the desired functions and compared our results with previous results and found that our results are at least as better as available in the literature. In Chapter 5, we have introduced the concept of biasedness in the proposed method and developed a new method based on biasedness to construct Boolean func- tions and got the desired results. In this chapter, we got the Boolean functions of 7 and 8 variables that could not be possible by the methods developed in Chapters 2, 3 and 4. We also compared our results with previous results and found that our results are at least as better as available in the literature. In Chapter 6, we have developed a new method based on NSGA-II to solve a system of linear inequalities. This method is applicable for all types of inequalities. We have generated three examples of di erent types and solved them by the developed method. The developed method gives better spread of solutions. Consequently, our method is better than previous methods to solve the system of linear inequalities. In Chapter 7, based on the study carried out in the thesis, conclusions are drawn and future scope of the research work is suggested