∗ TRIPLE SYSTEM AND FANO PLANE STRUCTURE IN Zn Dennis Kinoti Gikunda A research project submitted in partial fulfilment of the requirements for the award of the degree of Masters of Science (Pure Mathematics) in the School of Pure and Applied Sciences, Kenyatta University December 2020 ii DECLARATION Declaration by the Candidate This project is my original work and has not been published or presented for a degree award in any other university or any other award. Signature............................. Date..................................... Dennis Kinoti Gikunda I56/37638/2017 Declaration by the supervisor This project has been submitted for examination with my approval as the University supervisor. Signature................................ Date........................................ Dr.Benard Kivunge Mathematics Department, Kenyatta University, Kenya. iii DEDICATION This work is dedicated to the community of researchers, in the field of modern number theory, combinatorics and computer science. It's my hope that this work will be put into use, in building secure algorithms for cryptography. iv ACKNOWLEDGEMENT I sincerely acknowledge with thanks, my family members, for their unwavering support in the entire period of this study. No words can express my gratitude to you for your precious support. You're my rock { I am blessed to have you. Thanks a lot. Church family, your prayers and encouragement have born fruits. The Gospel is indeed bearing much fruit of its own inherent power. To my friends, I'm grateful for cheering me on, through this course of life. I sincerely thank you for always giving me an extra push { through words of encouragement and applauding my efforts after every successful step completed. To my colleagues Waweru, Maina, Tanui, Kioko, Ogeto and Kinyanjui, I am grateful that you made the struggles bearable. I now understand why they say, \on the road to success, there is always we, not me." We were a rare unit of support. In my next level, I hope to do it again with you, or better still, colleagues like you. In a world where academic supervisors are largely blamed for delayed research, I wish to sincerely appreciate my supervisor, Dr. Bernard Kivunge for valued support through out my study. You are a true leader. I proudly applause you as my academic mentor. When I start to practise, I definitely want to be like you. This appreciation extends to the other lecturers from Mathematics Department, Kenyatta University, for rightly equipping us. Finally and most importantly, all glory to the Almighty God for the opportunity, strength and provision of all that I needed for this journey. Indeed, You are a Faithful God. v ABSTRACT A triple system is an absolutely fascinating concept in projective geometry. This project is an extension of previously done work on triple systems, specifically the triples that fit into a Fano plane and the (i, j, k) triples of the quaternion group. Here, we have explored and determined ∗ m the existence of triple systems in Zn for n = p, n = pq, n = 2 p and n = pqr with m 2 N, ∗ p; q; r 2 P, and p > q > r. A triple system in Zn has been denoted by (k1; k2; k3) where there 2 exists ki > 1, i = 1; 2; 3, such that ki ≡ 1(mod n) with k1k2 ≡ k3(mod n), k1k3 ≡ k2 (mod ∗ n) and k2k3 ≡ k1 (mod n). We have successfully proved that there exists no triples in Zn, for ∗ n = p and n = 2p, p 2 P. Further, we have established the existence of triples in Zn, for m n = pq, n = 2 p and n = pqr, where m 2 N, p; q; r are odd primes and p > q > r. Finally, ∗ m we have fitted the triples of Zn, n = 2 p and n = pqr into Fano Planes. TABLE OF CONTENTS DECLARATION ..................................... ii DEDICATION ...................................... iii ACKNOWLEDGEMENT ................................ iv ABSTRACT ........................................ v TABLE OF CONTENTS ................................ vi LIST OF TABLES . viii LIST OF FIGURES ................................... ix NOTATIONS ........................................ x Chapter 1. INTRODUCTION 1 1.1 Background Information . 1 1.2 Definition of Terms . 3 1.3 Statement of the Problem . 7 1.4 Objectives of the Study . 7 1.4.1 Main Objective . 7 1.4.2 Specific Objectives . 7 1.5 Significance of the Study . 8 Chapter 2. LITERATURE REVIEW 9 2.1 Introduction . 9 2.2 Triple Systems . 9 2.3 Steiner Triple Systems and Fano Planes . 11 2.4 Euler's Totient Function . 12 Chapter 3. RESEARCH METHODOLOGY 14 3.1 Introduction . 14 3.2 Research Design . 14 vi vii 3.3 Methodology . 14 Chapter 4. MAIN RESULTS 16 ∗ 4.1 Triple System and Fano Plane Structure in Zn, for n = p, p 2 P: . 16 ∗ m 4.2 Triple System and Fano Plane Structure in Zn, for n = 2 p, p 2 P, m 2 N . 18 ∗ 4.3 Triple System and Fano Plane Structure in Zn, for n = pq, p; q 2 P, with p > q > 2 30 ∗ 4.4 Triple System and Fano Plane Structure in Zn, for n = pqr, p; q; r 2 P, with p > q > r ........................................ 45 Chapter 5. CONCLUSIONS AND RECOMMENDATIONS 58 5.1 Conclusion . 58 5.2 Recommendations . 59 REFERENCES ...................................... 60 List of Tables 4.1 Multiplication table for k2(mod p) .......................... 17 4.2 Multiplication table for k2(mod 2p).......................... 19 4.3 Triple systems in n = 22p ............................... 21 4.4 Triple systems in n = 2mp ............................... 23 4.5 Triple systems in n = 3p ............................... 33 4.6 Triple systems in n = 5p ............................... 35 4.7 Triple systems in n = 7p ............................... 37 4.8 Triple systems in n = 11p ............................... 39 4.9 Triple systems in n = 13p ............................... 42 4.10 Triple systems in n = pqr ............................... 47 viii List of Figures ∗ 4.1 Fano Plane Structure for Z24 ............................. 25 ∗ 4.2 Fano Plane Structure for Z80 ............................. 26 ∗ 4.3 Fano Plane Structure for Z224 ............................. 28 ∗ 4.4 Fano Plane Structure for Z704 ............................. 29 ∗ 4.5 Fano Plane Structure for Z105 ............................. 49 ∗ 4.6 Fano Plane Structure for Z165 ............................. 51 ∗ 4.7 Fano Plane Structure for Z195 ............................. 53 ∗ 4.8 Fano Plane Structure for Z455 ............................. 55 ∗ 4.9 Fano Plane Structure for Z1001 ............................ 57 ix x NOTATIONS P - Set of prime numbers N - Set of natural numbers Z - Set of integers R - Set of real numbers C - Set of complex numbers R - A ring R∗ - The set of units in a ring R H - The quaternion group Zn - The set of integers modulo n ∗ Zn - The set of units in Zn also known as the multiplicative group of Zn ' - Eulers phi function. Chapter 1. INTRODUCTION 1.1 Background Information Any natural number n can be expressed uniquely as a product of primes, for n 6= 0; 1. This implies that prime factors are the `building blocks' for any n 2 N. Mathematicians find delight in understanding the compositional properties of the primes and try to figure out the structure of their sequence (if any exists). The nature of their existence makes them useful in puzzles, cryptography and generation of security codes. Most of the common private and public key cryptographic systems rely on prime numbers and their role in decomposing other numbers into prime components by factorisation. Considering a case study of RSA (Rivest-Shamir-Adleman), which is an algorithm used to encrypt and decrypt messages by modern computers, a pair of same size prime numbers is needed for every key pair. RSA system is popularly used, an example is SSL (Secure Socket Layer) which is used in https (secure http). While using \secure sites" on the internet, the computer generates both private and public keys for the user. The private key ensures that all personal details like passwords and credit card details remain a secret while on the sites. Examples of such sites include emails and online-pay services. Thanks to Euclid's proof that there exist infinitely many prime numbers, in cryptography there are many choice numbers to use as keys or use in generating of keys. This makes it hard for a third party to accurately guess by generating all keys especially those equal to or above 512 − bit, hence guaranteeing secure algorithms. The prime number theorem has an interesting corollary: Let x be a random \large" number. The possibility of x being a prime number is 1=ln x. Taking a random string of size n − bit and setting 1 as the least significant bit so that it's always an odd number, the probability of having n as a prime is 1=n (ln 2). From this formula, if you use a 1024 − bit prime, the odds of getting the right code is 1 in 700. With larger primes, the odds get lower. This makes it 1 2 suitable and more secure to use prime numbers as keys or components for key generation, provided there is an effective way of testing whether a number is prime. Galbreath (2003) discusses Miller-Rabin test and the APR test as the common probabilistic tests for primality. In recent years, mathematicians have made a considerably great progress in a sub-branch of mathematics that concern finite geometries. Herein, we find the concept of triple systems referring to a vector space V over a field K together with K- trilinear mapping V ⊗ V ⊗ V ! V . Girard Desargues (1591 − 1661) discovered the projective geometry derived from Euclidean geometry, which involved 3- dimension finite geometry. An Italian mathematician, Gino Fano (1871 − 1952) later discussed the 3-dimension finite geometry with 3 points in each line and 7 points on each plane. The total number of points was 15, with 35 lines and 15 planes. The 3 points on each line form unique triple systems. Another interesting triple is the (i; j; k) triples of the quaternion group, denoted by H = {±1; ±i; ±j; ±kg.