Fermat's Last Theorem for Regular Primes
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An Analysis of Primality Testing and Its Use in Cryptographic Applications
An Analysis of Primality Testing and Its Use in Cryptographic Applications Jake Massimo Thesis submitted to the University of London for the degree of Doctor of Philosophy Information Security Group Department of Information Security Royal Holloway, University of London 2020 Declaration These doctoral studies were conducted under the supervision of Prof. Kenneth G. Paterson. The work presented in this thesis is the result of original research carried out by myself, in collaboration with others, whilst enrolled in the Department of Mathe- matics as a candidate for the degree of Doctor of Philosophy. This work has not been submitted for any other degree or award in any other university or educational establishment. Jake Massimo April, 2020 2 Abstract Due to their fundamental utility within cryptography, prime numbers must be easy to both recognise and generate. For this, we depend upon primality testing. Both used as a tool to validate prime parameters, or as part of the algorithm used to generate random prime numbers, primality tests are found near universally within a cryptographer's tool-kit. In this thesis, we study in depth primality tests and their use in cryptographic applications. We first provide a systematic analysis of the implementation landscape of primality testing within cryptographic libraries and mathematical software. We then demon- strate how these tests perform under adversarial conditions, where the numbers being tested are not generated randomly, but instead by a possibly malicious party. We show that many of the libraries studied provide primality tests that are not pre- pared for testing on adversarial input, and therefore can declare composite numbers as being prime with a high probability. -
Methodology and Metaphysics in the Development of Dedekind's Theory
Methodology and metaphysics in the development of Dedekind’s theory of ideals Jeremy Avigad 1 Introduction Philosophical concerns rarely force their way into the average mathematician’s workday. But, in extreme circumstances, fundamental questions can arise as to the legitimacy of a certain manner of proceeding, say, as to whether a particular object should be granted ontological status, or whether a certain conclusion is epistemologically warranted. There are then two distinct views as to the role that philosophy should play in such a situation. On the first view, the mathematician is called upon to turn to the counsel of philosophers, in much the same way as a nation considering an action of dubious international legality is called upon to turn to the United Nations for guidance. After due consideration of appropriate regulations and guidelines (and, possibly, debate between representatives of different philosophical fac- tions), the philosophers render a decision, by which the dutiful mathematician abides. Quine was famously critical of such dreams of a ‘first philosophy.’ At the oppos- ite extreme, our hypothetical mathematician answers only to the subject’s internal concerns, blithely or brashly indifferent to philosophical approval. What is at stake to our mathematician friend is whether the questionable practice provides a proper mathematical solution to the problem at hand, or an appropriate mathematical understanding; or, in pragmatic terms, whether it will make it past a journal referee. In short, mathematics is as mathematics does, and the philosopher’s task is simply to make sense of the subject as it evolves and certify practices that are already in place. -
Arxiv:Math/0412262V2 [Math.NT] 8 Aug 2012 Etrgae Tgte Ihm)O Atnscnetr and Conjecture fie ‘Artin’S Number on of Cojoc Me) Domains’
ARTIN’S PRIMITIVE ROOT CONJECTURE - a survey - PIETER MOREE (with contributions by A.C. Cojocaru, W. Gajda and H. Graves) To the memory of John L. Selfridge (1927-2010) Abstract. One of the first concepts one meets in elementary number theory is that of the multiplicative order. We give a survey of the lit- erature on this topic emphasizing the Artin primitive root conjecture (1927). The first part of the survey is intended for a rather general audience and rather colloquial, whereas the second part is intended for number theorists and ends with several open problems. The contribu- tions in the survey on ‘elliptic Artin’ are due to Alina Cojocaru. Woj- ciec Gajda wrote a section on ‘Artin for K-theory of number fields’, and Hester Graves (together with me) on ‘Artin’s conjecture and Euclidean domains’. Contents 1. Introduction 2 2. Naive heuristic approach 5 3. Algebraic number theory 5 3.1. Analytic algebraic number theory 6 4. Artin’s heuristic approach 8 5. Modified heuristic approach (`ala Artin) 9 6. Hooley’s work 10 6.1. Unconditional results 12 7. Probabilistic model 13 8. The indicator function 17 arXiv:math/0412262v2 [math.NT] 8 Aug 2012 8.1. The indicator function and probabilistic models 17 8.2. The indicator function in the function field setting 18 9. Some variations of Artin’s problem 20 9.1. Elliptic Artin (by A.C. Cojocaru) 20 9.2. Even order 22 9.3. Order in a prescribed arithmetic progression 24 9.4. Divisors of second order recurrences 25 9.5. Lenstra’s work 29 9.6. -
MATH 404: ARITHMETIC 1. Divisibility and Factorization After Learning To
MATH 404: ARITHMETIC S. PAUL SMITH 1. Divisibility and Factorization After learning to count, add, and subtract, children learn to multiply and divide. The notion of division makes sense in any ring and much of the initial impetus for the development of abstract algebra arose from problems of division√ and factoriza- tion, especially in rings closely related to the integers such as Z[ d]. In the next several chapters we will follow this historical development by studying arithmetic in these and similar rings. √ To see that there are some serious issues to be addressed notice that in Z[ −5] the number 6 factorizes as a product of irreducibles in two different ways, namely √ √ 6 = 2.3 = (1 + −5)(1 − −5). 1 It isn’t hard to show these factors are irreducible but we postpone√ that for now . The key point for the moment is to note that this implies that Z[ −5] is not a unique factorization domain. This is in stark contrast to the integers. Since our teenage years we have taken for granted the fact that every integer can be written as a product of primes, which in Z are the same things as irreducibles, in a unique way. To emphasize the fact that there are questions to be understood, let us mention that Z[i] is a UFD. Even so, there is the question which primes in Z remain prime in Z[i]? This is a question with a long history: Fermat proved in ??? that a prime p in Z remains prime in Z[i] if and only if it is ≡ 3(mod 4).√ It is then natural to ask for each integer d, which primes in Z remain prime in Z[ d]? These and closely related questions lie behind much of the material we cover in the next 30 or so pages. -
A Brief History of Ring Theory
A Brief History of Ring Theory by Kristen Pollock Abstract Algebra II, Math 442 Loyola College, Spring 2005 A Brief History of Ring Theory Kristen Pollock 2 1. Introduction In order to fully define and examine an abstract ring, this essay will follow a procedure that is unlike a typical algebra textbook. That is, rather than initially offering just definitions, relevant examples will first be supplied so that the origins of a ring and its components can be better understood. Of course, this is the path that history has taken so what better way to proceed? First, it is important to understand that the abstract ring concept emerged from not one, but two theories: commutative ring theory and noncommutative ring the- ory. These two theories originated in different problems, were developed by different people and flourished in different directions. Still, these theories have much in com- mon and together form the foundation of today's ring theory. Specifically, modern commutative ring theory has its roots in problems of algebraic number theory and algebraic geometry. On the other hand, noncommutative ring theory originated from an attempt to expand the complex numbers to a variety of hypercomplex number systems. 2. Noncommutative Rings We will begin with noncommutative ring theory and its main originating ex- ample: the quaternions. According to Israel Kleiner's article \The Genesis of the Abstract Ring Concept," [2]. these numbers, created by Hamilton in 1843, are of the form a + bi + cj + dk (a; b; c; d 2 R) where addition is through its components 2 2 2 and multiplication is subject to the relations i =pj = k = ijk = −1. -
GRACEFUL and PRIME LABELINGS -Algorithms, Embeddings and Conjectures
GRACEFUL AND PRIME LABELINGS-Algorithms, Embeddings and Conjectures Group Discussion (KREC, Surathkal: Aug.16-25, 1999). Improved Version. GRACEFUL AND PRIME LABELINGS -Algorithms, Embeddings and Conjectures Suryaprakash Nagoji Rao* Exploration Business Group MRBC, ONGC, Mumbai-400 022 Abstract. Four algorithms giving rise to graceful graphs from a known (non-)graceful graph are described. Some necessary conditions for a graph to be highly graceful and critical are given. Finally some conjectures are made on graceful, critical and highly graceful graphs. The Ringel-Rosa-Kotzig Conjecture is generalized to highly graceful graphs. Mayeda-Seshu Tree Generation Algorithm is modified to generate all possible graceful labelings of trees of order p. An alternative algorithm in terms of integers modulo p is described which includes all possible graceful labelings of trees of order p and some interesting properties are observed. Optimal and graceful graph embeddings (not necessarily connected) are given. Alternative proofs for embedding a graph into a graceful graph as a subgraph and as an induced subgraph are included. An algorithm to obtain an optimal graceful embedding is described. A necessary condition for a graph to be supergraceful is given. As a consequence some classes of non-supergraceful graphs are obtained. Embedding problems of a graph into a supergraceful graph are studied. A catalogue of super graceful graphs with at most five nodes is appended. Optimal graceful and supergraceful embeddings of a graph are given. Graph theoretical properties of prime and superprime graphs are listed. Good upper bound for minimum number of edges in a nonprime graph is given and some conjectures are proposed which in particular includes Entringer’s prime tree conjecture. -
Appendix a Solutions and Hints to Exercises
Appendix A Solutions and Hints to Exercises Chapter 1 1.1 1. Suppose that there are only finitely many prime p ≡ 3 (mod 4), and label them p1,...,pn. Then consider N = 4p1 ...pn − 1. This number is clearly odd, and has remainder 3 after dividing by 4. So its prime divisors are all odd, and they can’t all be 1 (mod 4), as the product of numbers which are 1 (mod 4) is again 1 (mod 4). So N has a prime divisor p ≡ 3(mod4).But p must be different from all the pi , since p|N but pi N for all i = 1,...,n. So there must be infinitely many primes p ≡ 3(mod4). 2. Suppose that there are only finitely many prime p ≡ 5 (mod 6), and label them p1,...,pn. Then consider N = 6p1 ...pn − 1. This number is clearly odd, and has remainder 5 after dividing by 6. So (6, N) = 1, and the prime divisors of N can’t all be 1 (mod 6), as the product of numbers which are 1 (mod 6) is again 1 (mod 6). So N has a prime divisor p ≡ 5(mod6).But p must be different from all the pi , since p|N but pi N for all i = 1,...,n. So there must be infinitely many primes p ≡ 5(mod6). 1.2 1. Suppose that p|x2 +1. If x is even, clearly p is odd. Then x2 ≡−1(modp) (and as p = 2, we have x2 ≡ 1(modp)), and so x4 ≡ 1(modp). -
13 Lectures on Fermat's Last Theorem
I Paulo Ribenboim 13 Lectures on Fermat's Last Theorem Pierre de Fermat 1608-1665 Springer-Verlag New York Heidelberg Berlin Paulo Ribenboim Department of Mathematics and Statistics Jeffery Hall Queen's University Kingston Canada K7L 3N6 Hommage a AndrC Weil pour sa Leqon: goat, rigueur et pCnCtration. AMS Subiect Classifications (1980): 10-03, 12-03, 12Axx Library of Congress Cataloguing in Publication Data Ribenboim, Paulo. 13 lectures on Fermat's last theorem. Includes bibliographies and indexes. 1. Fermat's theorem. I. Title. QA244.R5 512'.74 79-14874 All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag. @ 1979 by Springer-Verlag New York Inc. Printed in the United States of America. 987654321 ISBN 0-387-90432-8 Springer-Verlag New York ISBN 3-540-90432-8 Springer-Verlag Berlin Heidelberg Preface Fermat's problem, also called Fermat's last theorem, has attracted the attention of mathematicians for more than three centuries. Many clever methods have been devised to attack the problem, and many beautiful theories have been created with the aim of proving the theorem. Yet, despite all the attempts, the question remains unanswered. The topic is presented in the form of lectures, where I survey the main lines of work on the problem. In the first two lectures, there is a very brief description of the early history, as well as a selection of a few of the more representative recent results. In the lectures which follow, I examine in suc- cession the main theories connected with the problem. -
Math 331-2 Lecture Notes
Math 331-2: Abstract Algebra Northwestern University, Lecture Notes Written by Santiago Ca~nez These are notes which provide a basic summary of each lecture for Math 331-2, the second quarter of \MENU: Abstract Algebra", taught by the author at Northwestern University. The book used as a reference is the 3rd edition of Abstract Algebra by Dummit and Foote. Watch out for typos! Comments and suggestions are welcome. Contents Lecture 1: Introduction to Rings 2 Lecture 2: Matrix and Polynomial Rings 6 Lecture 3: More Ring Constructions 8 Lecture 4: Quotient Rings 14 Lecture 5: More on Quotients and Ideals 18 Lecture 6: Maximal Ideals 21 Lecture 7: Prime Ideals 25 Lecture 8: Geometry via Algebra 28 Lecture 9: Rings of Fractions 32 Lecture 10: More on Fractions 37 Lecture 11: Chinese Remainder Theorem 43 Lecture 12: Euclidean Domains 47 Lecture 13: More on Euclidean 51 Lecture 14: Principal Ideal Domains 54 Lecture 15: Unique Factorization 57 Lecture 16: More on UFDs 60 Lecture 17: Polynomials Revisited 63 Lecture 18: Gauss's Lemma 66 Lecture 19: Eisenstein's Criterion 69 Lecture 20: Modules over Rings 72 Lecture 21: Module Homomorphisms 77 Lecture 22: Sums and Free Modules 80 Lecture 23: Modules over PIDs 85 Lecture 24: Structure Theorems 89 Lecture 25: More on Structure Theorems 92 Lecture 26: Jordan Normal Form 97 Lecture 27: More on Normal Forms 103 Lecture 1: Introduction to Rings We continue our study of abstract algebra, this quarter focusing on rings and (to a lesser extent) modules. Rings provide a general setting in which \arithmetic" and the notion of \number" makes sense, and indeed the development of ring theory was borne out of attempts to understand general properties analogous to those of integers. -
Dedekind's 1871 Version of the Theory of Ideals∗
Dedekind's 1871 version of the theory of ideals¤ Translated by Jeremy Avigad March 19, 2004 Translator's introduction By the middle of the nineteenth century, it had become clear to mathe- maticians that the study of ¯nite ¯eld extensions of the rational numbers is indispensable to number theory, even if one's ultimate goal is to understand properties of diophantine expressions and equations in the ordinary integers. It can happen, however, that the \integers" in such extensions fail to satisfy unique factorization, a property that is central to reasoning about the or- dinary integers. In 1844, Ernst Kummer observed that unique factorization fails for the cyclotomic integers with exponent 23, i.e. the ring Z[³] of inte- gers of the ¯eld Q(³), where ³ is a primitive twenty-third root of unity. In 1847, he published his theory of \ideal divisors" for cyclotomic integers with prime exponent. This was to remedy the situation by introducing, for each such ring of integers, an enlarged domain of divisors, and showing that each integer factors uniquely as a product of these. He did not actually construct these integers, but, rather, showed how one could characterize their behav- ior qua divisibility in terms of ordinary operations on the associated ring of integers. Richard Dedekind and Leopold Kronecker later took up the task of ex- tending the theory to the integers in arbitrary ¯nite extensions of the ratio- nals. Despite their common influences and goals, however, the theories they ¤Work on this translation has been supported by a New Directions fellowship from the Andrew W. -
A Performant, Misuse-Resistant API for Primality Testing
A Performant, Misuse-Resistant API for Primality Testing Jake Massimo1 and Kenneth G. Paterson2 1 Information Security Group, Royal Holloway, University of London [email protected] 2 Department of Computer Science, ETH Zurich [email protected] Abstract. Primality testing is a basic cryptographic task. But developers today are faced with complex APIs for primality testing, along with documentation that fails to clearly state the reliability of the tests being performed. This leads to the APIs being incorrectly used in practice, with potentially disastrous consequences. In an effort to overcome this, we present a primality test having a simplest-possible API: the test accepts a number to be tested and returns a Boolean indicating whether the input was composite or probably prime. For all inputs, the output is guaranteed to be correct with probability at least 1 − 2−128. The test is performant: on random, odd, 1024-bit inputs, it is faster than the default test used in OpenSSL by 17%. We investigate the impact of our new test on the cost of random prime generation, a key use case for primality testing. The OpenSSL developers have adopted our suggestions in full; our new API and primality test are scheduled for release in OpenSSL 3.0. 1 Introduction Primality testing, and closely related tasks like random prime generation and testing of Diffie-Hellman parameters, are core cryptographic tasks. Primality testing is by now very well understood mathematically; there is a clear distinction between accuracy and running time of different tests in settings that are malicious (i.e. -
Fermat's Last Theorem for Regular Primes: Dinesh S. Thakur
Fermat's last theorem for regular primes Dinesh S. Thakur The so-called Fermat's last theorem (FLT for short) that `there are no integral solutions to xn + yn = zn, with n > 2 and xyz 6= 0 (a non-triviality condition)' is now proved by Wiles. The proof is outside the scope of this summer school, though we will sketch some ideas in the later lectures. Now we will prove the so-called `regular prime case' of FLT, following Kummer in essence, but connecting with later ideas and simplifications by other mathematicians. It may be advisable to first learn the proof for n = 3 from Hardy and Wright or from Ireland-Rosen pp. 285-286, which prove even stronger result, using only the basics of quadratic fields that we have seen. Since FLT for the exponent n implies FLT for multiples of n and since we proved it for n = 4, it is enough to prove FLT for n = p an odd prime. The basic strategy is the same as before: try to get an infinite descent and thus a contradiction starting with an assumed non-trivial solution. We have p Q i a factorization z = (x + ζpy) and we try to conclude that each factor (apart from GCD) is a p-th power. This conclusion is justified in the case of unique factorization domains. But since we only have unique factorization in ideals in general, we can conclude that each factor is a p-th power of an ideal, which can be further assumed to be principal, if p does not divide h, the class number of Q(ζp).