Number Theory Commons^™

A Cryptographic Attack: Finding The Discrete Logarithm On Elliptic Curves Of Trace One, Tatiana Bradley

Scripps Senior Theses

The crux of elliptic curve cryptography, a popular mechanism for securing data, is an asymmetric problem. The elliptic curve discrete logarithm problem, as it is called, is hoped to be generally hard in one direction but not the other, and it is this asymmetry that makes it secure.

This paper describes the mathematics (and some of the computer science) necessary to understand and compute an attack on the elliptic curve discrete logarithm problem that works in a special case. The algorithm, proposed by Nigel Smart, renders the elliptic curve discrete logarithm problem easy in both directions for elliptic curves of …

Polynomials With Prescribed Bad Primes, David P. Roberts

Mathematics Publications

We tabulate polynomials in ℚ[t] with a given factorization partition, bad reduction entirely within a given set of primes, and satisfying auxiliary conditions associated to 0, 1, and ∞. We explain how these polynomials are of particular interest because of their role in the construction of nonsolvable number fields of arbitrarily large degree and bounded ramification.

Deconstructing The Welch Equation Using P-Adic Methods, Abigail Mann, Adelyn Yeoh

Mathematical Sciences Technical Reports (MSTR)

The Welch map x -> g^x-1+c is similar to the discrete exponential map x -> g^x, which is used in many cryptographic applications including the ElGamal signature scheme. This paper analyzes the number of solutions to the Welch equation: g^x-1+c = x (mod p^e) where p is a prime, and looks at other patterns of the equation that could possibly exploited in a similar cryptographic system. Since the equation is modulo p^e, where p is a prime number, p-adic methods of analysis are used in counting the number of solutions modulo p …

Deconstructing The Welch Equation Using P-Adic Methods, Abigail Mann, Adelyn Yeoh

Rose-Hulman Undergraduate Research Publications

The Welch map x -> g^x-1+c is similar to the discrete exponential map x -> g^x, which is used in many cryptographic applications including the ElGamal signature scheme. This paper analyzes the number of solutions to the Welch equation: g^x-1+c = x (mod p^e) where p is a prime, and looks at other patterns of the equation that could possibly exploited in a similar cryptographic system. Since the equation is modulo p^e, where p is a prime number, p-adic methods of analysis are used in counting the number of solutions modulo p …

The Tame-Wild Principle For Discriminant Relations For Number Fields, John W. Jones, David P. Roberts

Mathematics Publications

Consider tuples ( K₁ , … , K_r ) of separable algebras over a common local or global number field F₁, with the K_i related to each other by specified resolvent constructions. Under the assumption that all ramification is tame, simple group-theoretic calculations give best possible divisibility relations among the discriminants of K_i ∕ F . We show that for many resolvent constructions, these divisibility relations continue to hold even in the presence of wild ramification.

Polynomial Factoring Algorithms And Their Computational Complexity, Nicholas Cavanna

Honors Scholar Theses

Finite fields, and the polynomial rings over them, have many neat algebraic properties and identities that are very convenient to work with. In this paper we will start by exploring said properties with the goal in mind of being able to use said properties to efficiently irreducibly factorize polynomials over these fields, an important action in the fields of discrete mathematics and computer science. Necessarily, we must also introduce the concept of an algorithm’s speed as well as particularly speeds of basic modular and integral arithmetic opera- tions. Outlining these concepts will have laid the groundwork for us to introduce …

Computing Local Constants For Cm Elliptic Curves, Sunil Chetty, Lung Li

Mathematics Faculty Publications

Let E/k be an elliptic curve with CM by O. We determine a formula for (a generalization of) the arithmetic local constant of Mazur-Rubin at almost all primes of good reduction. We apply this formula to the CM curves defined over Q and are able to describe extensions F/Q over which the O-rank of E grows.

A Frobenius Question Related To Actions On Curves In Characteristic P, Darren B. Glass

Math Faculty Publications

We consider which integers g can occur as the genus and of a curve defined over a field of characteristic p which admits an automorphism of degree pq, where p and q are distinct primes. This investigation leads us to consider a certain family of three-dimensional Frobenius problems and prove explicit formulas giving their solution in many cases.

New Research On Neutrosophic Algebraic Structures, Florentin Smarandache, Mumtaz Ali, Muhammad Shabir

Branch Mathematics and Statistics Faculty and Staff Publications

In this book, we define several new neutrosophic algebraic structures and their related properties. The main focus of this book is to study the important class of neutrosophic rings such as neutrosophic LA-semigroup ring, neutrosophic loop ring, neutrosophic groupoid ring and so on. We also construct their generalization in each case to study these neutrosophic algebraic structures in a broader sense. The indeterminacy element “ I “ gives rise to a more bigger algebraic structure than the classical algebraic structures. It mainly classifies the algebraic structures in three categories: such as neutrosophic algebraic structures, strong neutrosophic algebraic structures, and classical …

Special Pseudo Linear Algebras Using [0,N), Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book we introduce some special type of linear algebras called pseudo special linear algebras using the interval [0, n). These new types of special pseudo interval linear algebras has several interesting properties. Special pseudo interval linear algebras are built over the subfields in Zn where Zn is a S-ring. We study the substructures of them. The notion of Smarandache special interval pseudo linear algebras and Smarandache strong special pseudo interval linear algebras are introduced. The former Sspecial interval pseudo linear algebras are built over the Sring itself. Study in this direction has yielded several interesting results. S-strong special …

Soft Neutrosophic Algebraic Structures And Their Generalization - Vol. 1, Florentin Smarandache, Mumtaz Ali, Muhammad Shabir

Branch Mathematics and Statistics Faculty and Staff Publications

In this book the authors introduced the notions of soft neutrosophic algebraic structures. These soft neutrosophic algebraic structures are basically defined over the neutrosophic algebraic structures which means a parameterized collection of subsets of the neutrosophic algebraic structure. For instance, the existence of a soft neutrosophic group over a neutrosophic group or a soft neutrosophic semigroup over a neutrosophic semigroup, or a soft neutrosophic field over a neutrosophic field, or a soft neutrosophic LA-semigroup over a neutrosophic LAsemigroup, or a soft neutosophic loop over a neutrosophic loop. It is interesting to note that these notions are defined over finite and …

Interval Neutrosophic Logic, Haibin Wang, Florentin Smarandache, Yan-Qing Zhang, Rajshekhar Sunderraman

Branch Mathematics and Statistics Faculty and Staff Publications

Interval Neutrosophic Logic

Intuitionistic Neutrosophic Soft Set Over Rings, Florentin Smarandache, Said Broumi, Pabitra Kumar Maji

Branch Mathematics and Statistics Faculty and Staff Publications

S.Broumi and F.Smarandache introduced the concept of intuitionistic neutrosophic soft set as an extension of the soft set theory. In this paper we have applied the concept of intuitionistic neutrosophic soft set to rings theory .The notion of intuitionistic neutrosophic soft set over ring (INSSOR for short ) is introduced and their basic properties have been investigated.The definitions of intersection, union, AND, and OR operations over ring (INSSOR) have also been defined. Finally, we have defined the product of two intuitionistic neutrosophic soft set over ring.

On Crittenden And Vanden Eynden's Conjecture, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

It is possible to cover all (positive) integers with n geometrical progressions of integers? Find a necessary and sufficient condition for a general class of positive integer sequences such that, for a fixed n , there are n (distinct) sequences of this class which cover all integers.

Neutrosophic Crisp Sets & Neutrosophic Crisp Topological Spaces, A. A. Salama, Florentin Smarandache, Valeri Kroumov

Branch Mathematics and Statistics Faculty and Staff Publications

In this paper, we generalize the crisp topological space to the notion of neutrosophic crisp topological space, and we construct the basic concepts of the neutrosophic crisp topology. In addition to these, we introduce the definitions of neutrosophic crisp continuous function and neutrosophic crisp compact spaces. Finally, some characterizations concerning neutrosophic crisp compact spaces are presented and one obtains several properties. Possible application to GIS topology rules are touched upon.

Multi-Criteria Decision Making Based On Dsmt-Ahp, Jean Dezert, Jean Marc Tacnet, Mireille Batton-Hubert, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

In this paper, we present an extension of the multicriteria decision making based on the Analytic Hierarchy Process (AHP) which incorporates uncertain knowledge matrices for generating basic belief assignments (bba’s). The combination of priority vectors corresponding to bba’s related to each (sub)- criterion is performed using the Proportional Conflict Redistribution rule no. 5 proposed in Dezert-Smarandache Theory (DSmT) of plausible and paradoxical reasoning. The method presented here, called DSmT-AHP, is illustrated on very simple examples.

Algebraic Structures On The Fuzzy Interval [0, 1), Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book we introduce several algebraic structures on the special fuzzy interval [0, 1). This study is different from that of the algebraic structures using the interval [0, n) n ≠ 1, as these structures on [0, 1) has no idempotents or zero divisors under ×. Further [0, 1) under product × is only a semigroup. However by defining min(or max) operation in [0, 1); [0, 1) is made into a semigroup. The semigroup under × has no finite subsemigroups but under min or max we have subsemigroups of order one, two and so on. [0, 1) under + …

Solving Diophantine Equations, Florentin Smarandache, Octavian Cira

Branch Mathematics and Statistics Faculty and Staff Publications

In recent times, we witnessed an explosion of Number Theory problems that are solved using mathematical software and powerful computers. The observation that the number of transistors packed on integrated circuits doubles every two years made by Gordon E. Moore in 1965 is still accurate to this day. With ever increasing computing power more and more mathematical problems can be tacked using brute force. At the same time the advances in mathematical software made tools like Maple, Mathematica, Matlab or Mathcad widely available and easy to use for the vast majority of the mathematical research community. This tools don’t only …

Lower And Upper Soft Interval Valued Neutrosophic Rough Approximations Of An Ivnss-Relation, Said Broumi, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

In this paper, we extend the lower and upper soft interval valued intuitionistic fuzzy rough approximations of IVIFS –relations proposed by Anjan et al. to the case of interval valued neutrosophic soft set relation(IVNSS-relation for short)

Algebraic Structures On Finite Complex Modulo Integer Interval C([0, N)), Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors introduce the notion of finite complex modulo integer intervals. Finite complex modulo integers was introduced by the authors in 2011. Now using this finite complex modulo integer intervals several algebraic structures are built. Further the concept of finite complex modulo integers itself happens to be new and innovative for in case of finite complex modulo integers the square value of the finite complex number varies with varying n of Zn. In case of finite complex modulo integer intervals also we can have only pseudo ring as the distributive law is not true, in general in C([0, …