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Power Digraphs in Number Theory

Thesis Info

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Author

Uzma Ahmad

Program

PhD

Institute

National University of Computer and Emerging Sciences

City

Lahore

Province

Punjab

Country

Pakistan

Thesis Completing Year

2013

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/bitstream/123456789/2622/1/2656S.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676726982494

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The modular exponentiation is considered to be one of the renowned problems in number theory and is of paramount importance in the field of cryptography. Now a days many security systems are based on powerful cryptographic algorithms. Most of them are designed by using the exponentiation x k ≡ y (mod n) as in RSA, Diffie- Hellman key exchange, Pseudo-random number generators etc. For the last two decades, this problem is being studied by associating the power digraphs with modular exponentiation. For the fixed values of n and k, a power digraph G(n, k) is formed by taking Z n as the set of vertices and the directed edges (x, y) from x to y if x k ≡ y (mod n) for the vertices x and y. These digraphs make a novel connection between three disciplines of discrete mathematics namely number theory, graph theory and cryptography. The objective of this dissertation is to generalize the results on symmetry, heights, isolated fixed points, the number of components of a power digraph and the primality of Fermat numbers. To obtain the desired goal, a power digraph is decomposed into the direct product of smaller power digraphs by using the Chinese Remainder Theorem. The method of elimination is adopted to discard those values of n and k which do not provide desired results. During the entire course of research, the Carmichael lambda-function λ(n) is used for developing the relations between the properties of a power digraph and the parameters n, k. For any prime divisor p of n, the concept of equivalence classes has been used to discuss the symmetry of order p of G(n, k). The general rules to determine the heights are formulated by comparing the prime factorizations of k, λ(n) and the orders of vertices. Some necessary and sufficient conditions for the existence of symmetric power digraphs G(n, k), where n = p α q 1 q 2 · · · q m such that p, q i are distinct primes and α > 1, of order p are established. Explicit formulae for the determination of the heights of the vertices and components of a power digraph in terms of n, k, λ(n) and the orders of vertices are formulated. An expression for the number of vertices at a specific height is established. The power digraphs in which each vertex of indegree 0 of a certain subdigraph is at height q ≥ 1 are characterized. The necessary and sufficient conditions on n and k for a digraph to have at least one isolated fixed point are obtained. The work ends with the complete classification of the power digraphs with exactly two components.
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نیاز ؔفتح پوری

نیاز فتح پوری
گزشتہ مہینہ نیاز فتح پوری کی وفات اکیاسی (۸۱) سال کی عمر میں کراچی میں ہوگئی، معارف کو ان کی پچھلی زندگی میں ان کے بعض دلآزار مذہبی مضامین سے بڑا اختلاف رہا، لیکن انھوں نے اظہار انابت کرکے آخر میں مذہبی دلآزادی چھوڑ دی تھی، ان کے علمی ذوق میں بڑی رنگارنگی تھی، رسالہ نگار کے اڈیٹر ہونے کے ساتھ، مذہب تاریخ، سوانح، ناول نگاری، افسانہ نویسی اور شعر و ادب پر بھی طبع آزمائی کرتے رہے اپنے بعض مذہبی مضامین کی وجہ سے تو مطعون ہوئے، اچھے مورخ اور اچھے سوانح نگار بھی نہ ہوسکے، لیکن ان کا نام اچھے ناول نگار عمدہ افسانہ نویس اور شعرو ادب کے بلندپایہ نقاد کی حیثیت سے اردو زبان کی تاریخ میں خصوصیت کے ساتھ برابر لیا جائے گا، وہ اپنے رسالہ نگار کے ذریعہ جو علمی و ادبی خدمت انجام دیتے رہے وہ بھی ان کے اہم کارناموں میں شمار ہوگا، دعا ہے کہ اﷲ تبارک و تعالیٰ ان کی کمزوریوں کو اپنے دامن عفو میں جگہ دیں اور ان کو اپنی رحمت و مغفرت سے سرفراز فرمائیں، آمین۔
(صباح الدین عبدالرحمن، جون ۱۹۶۶ء)

 

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