COURSE STRUCTURE Course Code Course Category Professional Core Course Title Mathematical Foundation for Security Teaching Scheme and Credits Lecture Tutorial Laboratory Credits Weekly load hrs 3 1 3+0+1=4 Pre-requisites: Discrete Mathematics, Fundamentals of Statistics and Probability Course Objectives: 1. To identify and reproduce the theoretical framework underlying basic number theory. 2. To understand the basic concepts of algebraic structures in abstract algebra. 3. To learn discrete and continuous probability distributions models 4. To learn the concepts of Tests of Significance and Stochastic process Course Outcomes: On completion of the course, students should be able to 1. Apply the number theory concepts and results in various engineering problems. 2. Use the modular arithmetic operations mostly used in cryptography. 3. Compute orthonormal basis for inner product spaces. 4. Use the techniques studied in statistics & probability to solve the problems in domains such as data mining, machine learning and network analysis. 5. Use the concept of Tests of Significance Course Contents: Introduction to Number Theory Introduction, Well ordering Principle, Divisibility in the set of integers, Division Algorithm, Greatest common divisor, Euclidean algorithm, Modular Arithmetic, Prime numbers ,Cardinality of Primes, Fundamental theorem of arithmetic, Euclid’s Lemma, Congruence , Fermat’s and Euler’s Theorem, Chinese Reminder Theorem, Quadratic Congruence. Applications of number theory in security Algebraic Structures and Finite Fields Groups, Subgroups, Cyclic groups, Isomorphism of groups, Cosets , Modulo groups , Introduction of Rings and Fields, Finite Fields , Polynomial Arithmetic, Finite Fields of the Form GF(2n), Application of finite fields in cryptography Vector Space Introduction to Vector space & subspace, Inner products, Cauchy-Schwartz inequality, Orthogonal Projections, Gram-Schmidt process of Orthogonalization, Matrix representation of inner product, Least square solutions. Statistics and Probability Statistics: Moments, Skewness and Kurtosis, Correlation, Partial correlation, regression. Probability: Basic probability theory, Bayes’ theorem, Discrete and continuous random variables, probability mass function, probability density function, probability distributions: Binomial, Poisson and Normal distributions. Tests of Significance and Stochastic process Test of Significance: Parameter and Statistic, Hypothesis testing, Parametric and non-parametric tests, Chi-square Test for goodness of fit, student’s t-distribution test, F- Test. Stochastic Process: Introduction and classification of stochastic processes, Markov process, Transition probability, Transition probability matrix, Markov Chain. Learning Resources: Reference Books: 1. Ivan Nivam & H. S. Zuckerman, “An introduction to Number Theory”, Wiley Eastern Limited. 2. T. M. Apostol, “An Introduction to Analytical Number Theory”, Springer International Student’s edition. 3. Joseph Gallion, “Contemporary Abstract Algebra”, Narosa Publishing house. 4. I. N. Herstein,”Topics in Algebra”, Wiley Eastern Limited. 5. J. B. Fraleigh,”A First course in Abstract Algebra”, Narosa Publishing House. 6. Trivedi Kishor S., “Probability & Statistics with Reliability, Queuing and Computer Science Applications”, Second Edition, Wiley Student Edition, 2012. 7. Ross Sheldon M., “Introduction to Probability and Statistics for Engineers and Scientists”, Fifth Edition, 2014. 8. Gupta S. C. and Kapoor V. K., “Fundamentals of Applied Statistics”, Third Edition, S. Chand and Sons, New Delhi, 1987. 9. DeGroot Morris H. and Schervish Mark J., “Probability and Statistics”, Fourth Edition, Pearson New International Edition, 2010 10. William Stallings, Computer Security : Principles and Practices, Pearson 6 th Ed, ISBN: 978-013-335469-0 11. Bruice Schneier, Applied Cryptography- Protocols, Algorithms and Source code in C, Algorithms, Wiely India Pvt Ltd, 2nd Edition, ISBN 978-81-265-1368-0. Supplementary Reading: 1. Montgometry Douglas C.and Runger George C., “Applied Statistics and Probability for Engineers”, Third Edition, Wiley Student Edition, 2008. 2. Victor Shoup , “A Computational Introduction to Number Theory and Algebra “(Version 1) Web Resources: Web links: https://www.tutorialspoint.com/statistics/probability.htm MOOCs: http://nptel.ac.in/courses/111105090/16 http://nptel.ac.in/courses/111105090/46 http://nptel.ac.in/courses/111105090/76 Pedagogy: ● ● ● Chalk and Board PPT Two Teacher Method Assessment Scheme: Class Continuous Assessment (CCA)- 60 Marks (100%) Mid Term Quiz Assignments Exam 20% 40% 40% 10 marks 30 marks 20 marks Total 100% 60 marks Laboratory Continuous Assessment (LCA): 50 Marks Regularity and punctuality 5 Understanding of objective 5 Understanding of procedure 20 Experimental skills 20 Ethics Term End Examination: 40 marks (100%) Syllabus: Theory: Workload in Hrs Module Contents No. Theory Lab Assess Introduction to Number Theory Introduction, Well ordering Principle, Divisibility in the set of integers, Division Algorithm, Greatest common divisor, Euclidean algorithm, Modular Arithmetic, 1 08 --Prime numbers ,Cardinality of Primes, Fundamental theorem of arithmetic, Euclid’s Lemma, Congruence , Fermat’s and Euler’s Theorem, Chinese Reminder Theorem, Quadratic Congruence. Applications of number theory in security 2 Algebraic Structures and Finite Fields Groups, Subgroups, Cyclic groups, Isomorphism of groups, Cosets , Modulo groups , Introduction of Rings and Fields, Finite Fields , Polynomial Arithmetic, Finite Fields of the Form GF(2n), Application of finite fields in cryptography Vector Space 3 Introduction to Vector space & subspace, Inner products, Cauchy-Schwartz inequality, Orthogonal Projections, GramSchmidt process of Orthogonalization, Matrix representation of inner product, Least square solutions. 4 Statistics and Probability 08 - - 07 - - Statistics: Moments, Skewness and Kurtosis, Correlation, Partial correlation, regression. Probability: Basic probability theory, Bayes’ theorem, Discrete and continuous random variables, probability mass function, probability density function, probability distributions: Binomial, Poisson and Normal distributions. Tests of Significance and Stochastic process Test of Significance: Parameter and Statistic, Hypothesis testing, Parametric and non-parametric tests, Chi-square Test for 5 goodness of fit, student’s t-distribution test, F- Test. Stochastic Process: Introduction and classification of stochastic processes, Markov process, Transition probability, Transition probability matrix, Markov Chain. Laboratory: Mathematical Foundation for Security 07 - - The course faculty should frame the suitable assignments/problem statements based on the concern theory subject. Concerned faculty member may add/modify the assignment list as per the need of the course. There will be continuous evaluation of these assignments during the Trimester. Student has to submit a Journal/report consisting of suitable write up in the prescribed format. Softcopy of journal/report and code is to be maintained at department/institute in digital repository. Faculty advisor/ laboratory instructor suggested language/platform/framework is to be used for completing assignments/mini-project. Guidelines for Term Work Assessment Continuous assessment of laboratory work is done based on performance of student. Each assignment/ mini project assessment to be done based on parameters with appropriate weightage. Faculty should do the overall assessment as well as mini project assessment be based on the suggested parameters. Assessment Scheme: Laboratory Continuous Assessment (LCA) 50 marks (100%) Design and Implementation Performance of Experiment Result analysis and Reporting 30% 30% 20% Class Participation/GD/QUIZ/ Total Discipline/ Initiative/ Behaviour 20% 100% Laboratory Assignments: Module No. Contents Workload in Hrs Theory Lab Assess Mathematical Foundation for Security 1 2 3 4 5 6 7 8 9 Introduction and installation of software tools : R , SCILAB 4 To find the greatest common divisor of two integers 𝑎 & 𝑏 using Euclidean algorithm. Also find 𝑥, 𝑦 ∈ 𝑍, such that 𝑎𝑥 + 𝑏𝑦 = gcd (𝑎, 𝑏) 4 To find a unique solution to simultaneous linear congruences with co-prime moduli using Chinese remainder theorem. Gram-Schmidt process of orthogonalisation Solution of Linear system using least square method Analyze the given data set statistically using moments , skewness and kurtosis Fit a regression line for the given data sets such as Air Quality , Life Cycle saving, World phones etc. Solve problems on Tests of significance using suitable programing techniques Students t- distribution, F-Test 4 4 3 4 2 2 3 Prepared By Checked By Approved By Prof. Vaishali M. Joshi Assistant Professor School of Mathematics & Statistics, MITWPU, Pune Prof. Ramaa Sandu Associate Professor School of Mathematics & Statistics, MITWPU, Pune Prof. Dr Prashant Malavadkar HOS, Mathematics and Statistics, MITPU, Pune.
