FACTORING. n = fall in the arithmetic sequence


 Gary Wilkerson
 2 years ago
 Views:
Transcription
1 FACTORING The claim that factorization is harder than primality testing (or primality certification) is not currently substantiated rigorously. As some sort of backward evidence that factoring is hard, this writeup briefly introduces some factoring methods. 1. Futility of Trial Division Trial division attempts to divide a given integer n by integers 2 through n. Either we find a proper factor of n, or n is prime. A crude estimate shows the hopelessness of this method for factoring integers large enough to be relevant to publickey cryptology. Suppose that n , and either n is prime or n = pq where p, q are primes. Determining whether n is prime or n = pq will take about trial divisions. If internet hosts each carried out trials per second, then since there are fewer than 10 8 seconds per year, even this massively parallel attack would require some years. The skeptics may generate ever larger probable primes and then test trial division on them. The process will become hopeless long before 200 digits. 2. The Euler Fermat Trick Consider a positive integer of the special form n = b e 1 For such n, if p is a prime divisor of n then either p b d 1 for some proper divisor d of e (possibly d = 1), or p = 1 mod e. This follows from the fact that p n then b is an eth root of unity in Z/pZ. In formulas, the argument that p b d 1 for some divisor d of e (possibly d = e) is: p b e 1 = b e = 1 mod p = b d = 1 mod p for some minimal positive d, d e = p b d 1 for the same d Now, either d is a proper divisor of e, or d = e. But if d = e then e is the order of b in (Z/pZ), a cyclic group of order p 1, and so e p 1, i.e., p = 1 mod e. Thus the statement in italics is justified. It reduces by a significant constant factor the amount of required trial division to check the primality of n. Similarly, if p is a prime divisor of the Fermat number n = 2 2q + 1 then 2 would be a 2 q th root of 1 in Z/pZ. Thus 2 would be a primitive 2 q+1 st root of unity in Z/pZ, forcing 2 q+1 p 1, i.e., p = 1 mod 2 q+1. Euler used this idea to show that possible prime factors of the fifth Fermat number fall in the arithmetic sequence n = {64k + 1} = {65, 129,..., 641,... }, 1
2 2 FACTORING and in fact 641 is a factor. The difference between five trial divisions (641 is the fifth prime in the arithmetic sequence) and 116 trial divisions (641 is the 116th prime) is significant. Similarly, this method finds a sixdigit prime factor of n = in 150 trial divisions. The modern Lucas Lehmer test is much faster, but it does not actually find a prime factor. 3. Pollard s Rho Method This method quickly finds relatively small factors p of composite numbers in perhaps p steps, and it uses very little memory. [Pollard s rho method for discrete logarithms would make a good project topic. See his 1978 paper in Mathematics of Computation.] It is simple to implement, and it demonstrates the suprising power of algorithms that are irremedially probabilistic. Although an obvious heuristic suggests the reasonableness of Pollard s rho, it is difficult to prove that it works as well as it does. [But see the 1991 paper by E. Bach in Information and Computation.] Here is the method: Given n, define a function f : {0,..., n 1} {0,..., n 1}, f(x) = x % n. Then Initialize x = 2, y = f(x). While gcd(x y, n) = 1, replace (x, y) by (f(x), f(f(y))). Now 1 < gcd(x y, n) n. If 1 < gcd(x y, n) < n then the gcd is a proper factor of n, but if gcd(x y, n) then the test has failed. When the test fails it can be tried again with a different starting value of x, or with a different function f. Not only does the test work surprisingly well, but it is hardly matters what starting value x or what function f is used. The most compelling explanation of Pollard s rho factorization method is heuristic, and attempts to be more rigorous do not succeed easily. The two critical components of the heuristic explanation are the birthday paradox and Floyd s cycledetection method The Birthday Paradox and the First Idea of Pollard s Rho. The number d of draws (with replacement) from n objects required to make the probability of at least two pair matching exceed 1/2 is essentially 1.2 n. This is perhaps a suprisingly small number. Indeed, the probability of no match is d 1 P d = (1 i/n). So the logarithm of the probability is d 1 ( log P d = log 1 i ) n d 1 i n = (d 1)d 2n d2 2n. So if d 2 > 2 log 2 n then log(p d ) log 2 = log(1/2), i.e., P d 1/2. Since P d is complementary to the probability of a match, we are done. And the condition is essentially d > n. (If we want to be more finicky, we can note that a precise estimate of the nomatch probability is log P d (d 1) 2 /(2n), and so a precise condition to make the match probability exceed 1/2 is d > 2 log 2 n + 1.)
3 FACTORING 3 Returning to Pollard s rho method, suppose that n has a proper divisor p that is much smaller than n. If we have more than p integers x 1,..., x t, then with probability greater than 1/2, two of the x i should agree modulo p. Since p is much smaller than n, if the sequence {x i } behaves somewhat randomly then two x i values should agree modulo p long before any two agree modulo n. If, say, x i = x j mod p then gcd(x i x j, n) will be divisible by p. The gcd will be a divisor of n greater than 1 and probably smaller than n. But computing gcd(x i x j, n) for every pair {x i, x j } as we go along until we find a match could take roughly p p n comparisons, and this is as bad as trial division. To compute many fewer gcds with confidence of nonetheless finding a match quickly, Pollard s rho method is using a second idea beyond the birthday paradox Floyd s Cycle Detection Method and the Second Idea of Pollard s Rho Method. Pretend that the function f : {0,..., n 1} {0,..., n 1}, is a random map of Z/nZ to itself, and define a sequence x 1 = 2, x i = f(x i 1 ) for i > 1. f(x) = x % n Note that if ever x j = x i (where j > i) then also x j+k = x i+k for all k 0. That is, as soon as the sequence reaches its first value that will repeat, it settles into a cycle of length j i, and so the sequence takes the form of the greek letter ρ. Consequently, x t = x t k(j i) whenever k 1 and t k(j i) i. Pollard s method compares only pairs of terms (x s, y s ) = (x s, x 2s ). That is, it searches for an equality as in the previous display but where t = 2s for some s and where the right subscript is s itself, x 2s = x 2s k(j i) whenever 2s k(j i) = s i. Such an equality will hold under the conditions s = k(j i) for some k 1, s i. Since Pollard s rho method compares x s and x 2s for s = 1, 2,..., the analysis here proves that it will find a match. 4. Pollard s p 1 Method Call a positive integer smooth if it divisible only by small primes. Pollard s p 1 method finds prime factors p of a given n with the property that p 1 is smooth. (Such primes p are called weak.) More specifically, let B some bound. Then positive integers of the form m = q eq q<b are called Bsmooth. Pollard s p 1 method finds a prime factor p of n such that p 1 is Bsmooth using O(B ln n/ ln b) multiplications modulo n. But the algorithm can fail. In practice, because the value of B must be kept too small to find all possible factors, the algorithm is used just a little hoping for luck or for
4 4 FACTORING negligence on the part of an adversary. The question of how large we should expect the prime factors of a randomly chosen number to be is not trivial. Here is the method: Given n, known to be composite but not a prime power, and given a smoothness bound B, proceed as follows. Choose a random integer b, with 2 b n 1. Let g = gcd(b, n). If g 2 then g is a proper factor of n, so stop. Otherwise let p 1,..., p t be the primes that are at most B. For i = 1,..., t, do the following (1) Let q = p i. (2) Let l = ln n/ ln q. (3) Replace b by b ql % n. (4) Compute g = gcd(b 1, n). (5) If 1 < g < n then g is a proper factor of n, so stop. If g = n then the algorithm has failed. If g = 1 then continue to the next value of i. Why does this work? Suppose that n has a prime factor p such that p 1 is Bsmooth, t p = 1 + p ei i. For any b such that gcd(b, n) = 1, Fermat s Little Theorem says that b p 1 = 1 mod p, i.e., b pe 1 1 p e t t = 1 mod p. The calculation ln n ln p > ln j p ej j = j e j ln p j e i ln p i shows that the quantity l i = ln n/ ln p i is at least e i. Thus p 1 = i pei i diviides the quantity T = i pli i, and so bt = 1 mod p for any integer b coprime to p. That is, p gcd(b T 1, n). The format of the algorithm shows that the bound B serves only to ensure that the algorithm terminates after a while. Instead, one could try successively larger primes until deciding to quit by some criterion such as total time invested. Example. Factor Initialize b = 3. The exponent for 2 is 46, the exponent for 3 is 29, and the exponent for 5 is 20. Replace b by b 246 % = Since gcd(b 1, n) = 1, continue. Replace b by b 329 % = Since gcd(b 1, n) = 1, continue. Replace b by b 520 % = This time the gcd is 54001, giving a proper factor. And indeed, is {2, 3, 5} weak because = is {2, 3, 5}smooth.
5 FACTORING 5 There is an analogous p + 1 method due to Williams in Math Comp 39 (1982). More generally, for any cyclotomic polynomial ϕ n (x) = x n 1/ ϕ d (x) d n, d<n there is a ϕ n (p) method. The case n = 1 is Pollard s p 1, and the case n = 2 is Williams s p + 1.
Factoring & Primality
Factoring & Primality Lecturer: Dimitris Papadopoulos In this lecture we will discuss the problem of integer factorization and primality testing, two problems that have been the focus of a great amount
More informationPrimality Testing and Factorization Methods
Primality Testing and Factorization Methods Eli Howey May 27, 2014 Abstract Since the days of Euclid and Eratosthenes, mathematicians have taken a keen interest in finding the nontrivial factors of integers,
More informationFactoring Algorithms
Factoring Algorithms The p 1 Method and Quadratic Sieve November 17, 2008 () Factoring Algorithms November 17, 2008 1 / 12 Fermat s factoring method Fermat made the observation that if n has two factors
More informationPrimality  Factorization
Primality  Factorization Christophe Ritzenthaler November 9, 2009 1 Prime and factorization Definition 1.1. An integer p > 1 is called a prime number (nombre premier) if it has only 1 and p as divisors.
More informationFactoring Algorithms
Institutionen för Informationsteknologi Lunds Tekniska Högskola Department of Information Technology Lund University Cryptology  Project 1 Factoring Algorithms The purpose of this project is to understand
More informationU.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra
U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory
More informationFACTORING LARGE NUMBERS, A GREAT WAY TO SPEND A BIRTHDAY
FACTORING LARGE NUMBERS, A GREAT WAY TO SPEND A BIRTHDAY LINDSEY R. BOSKO I would like to acknowledge the assistance of Dr. Michael Singer. His guidance and feedback were instrumental in completing this
More informationI. Introduction. MPRI Cours 2122. Lecture IV: Integer factorization. What is the factorization of a random number? II. Smoothness testing. F.
F. Morain École polytechnique MPRI cours 2122 20132014 3/22 F. Morain École polytechnique MPRI cours 2122 20132014 4/22 MPRI Cours 2122 I. Introduction Input: an integer N; logox F. Morain logocnrs
More informationElementary factoring algorithms
Math 5330 Spring 013 Elementary factoring algorithms The RSA cryptosystem is founded on the idea that, in general, factoring is hard. Where as with Fermat s Little Theorem and some related ideas, one can
More informationAn Overview of Integer Factoring Algorithms. The Problem
An Overview of Integer Factoring Algorithms Manindra Agrawal IITK / NUS The Problem Given an integer n, find all its prime divisors as efficiently as possible. 1 A Difficult Problem No efficient algorithm
More informationCryptography and Network Security Chapter 8
Cryptography and Network Security Chapter 8 Fifth Edition by William Stallings Lecture slides by Lawrie Brown (with edits by RHB) Chapter 8 Introduction to Number Theory The Devil said to Daniel Webster:
More informationInteger Factorization using the Quadratic Sieve
Integer Factorization using the Quadratic Sieve Chad Seibert* Division of Science and Mathematics University of Minnesota, Morris Morris, MN 56567 seib0060@morris.umn.edu March 16, 2011 Abstract We give
More informationELEMENTARY THOUGHTS ON DISCRETE LOGARITHMS. Carl Pomerance
ELEMENTARY THOUGHTS ON DISCRETE LOGARITHMS Carl Pomerance Given a cyclic group G with generator g, and given an element t in G, the discrete logarithm problem is that of computing an integer l with g l
More informationOn Generalized Fermat Numbers 3 2n +1
Applied Mathematics & Information Sciences 4(3) (010), 307 313 An International Journal c 010 Dixie W Publishing Corporation, U. S. A. On Generalized Fermat Numbers 3 n +1 Amin Witno Department of Basic
More information2 Primality and Compositeness Tests
Int. J. Contemp. Math. Sciences, Vol. 3, 2008, no. 33, 16351642 On Factoring R. A. Mollin Department of Mathematics and Statistics University of Calgary, Calgary, Alberta, Canada, T2N 1N4 http://www.math.ucalgary.ca/
More informationComputer and Network Security
MIT 6.857 Computer and Networ Security Class Notes 1 File: http://theory.lcs.mit.edu/ rivest/notes/notes.pdf Revision: December 2, 2002 Computer and Networ Security MIT 6.857 Class Notes by Ronald L. Rivest
More informationMA2C03 Mathematics School of Mathematics, Trinity College Hilary Term 2016 Lecture 59 (April 1, 2016) David R. Wilkins
MA2C03 Mathematics School of Mathematics, Trinity College Hilary Term 2016 Lecture 59 (April 1, 2016) David R. Wilkins The RSA encryption scheme works as follows. In order to establish the necessary public
More informationFaster deterministic integer factorisation
David Harvey (joint work with Edgar Costa, NYU) University of New South Wales 25th October 2011 The obvious mathematical breakthrough would be the development of an easy way to factor large prime numbers
More informationFactorization Methods: Very Quick Overview
Factorization Methods: Very Quick Overview Yuval Filmus October 17, 2012 1 Introduction In this lecture we introduce modern factorization methods. We will assume several facts from analytic number theory.
More informationModern Factoring Algorithms
Modern Factoring Algorithms Kostas Bimpikis and Ragesh Jaiswal University of California, San Diego... both Gauss and lesser mathematicians may be justified in rejoicing that there is one science [number
More informationFactoring. Factoring 1
Factoring Factoring 1 Factoring Security of RSA algorithm depends on (presumed) difficulty of factoring o Given N = pq, find p or q and RSA is broken o Rabin cipher also based on factoring Factoring like
More informationStudy of algorithms for factoring integers and computing discrete logarithms
Study of algorithms for factoring integers and computing discrete logarithms First IndoFrench Workshop on Cryptography and Related Topics (IFW 2007) June 11 13, 2007 Paris, France Dr. Abhijit Das Department
More informationLecture 13  Basic Number Theory.
Lecture 13  Basic Number Theory. Boaz Barak March 22, 2010 Divisibility and primes Unless mentioned otherwise throughout this lecture all numbers are nonnegative integers. We say that A divides B, denoted
More informationCHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY
January 10, 2010 CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY The set of polynomials over a field F is a ring, whose structure shares with the ring of integers many characteristics.
More informationArithmetic algorithms for cryptology 5 October 2015, Paris. Sieves. Razvan Barbulescu CNRS and IMJPRG. R. Barbulescu Sieves 0 / 28
Arithmetic algorithms for cryptology 5 October 2015, Paris Sieves Razvan Barbulescu CNRS and IMJPRG R. Barbulescu Sieves 0 / 28 Starting point Notations q prime g a generator of (F q ) X a (secret) integer
More information8 Divisibility and prime numbers
8 Divisibility and prime numbers 8.1 Divisibility In this short section we extend the concept of a multiple from the natural numbers to the integers. We also summarize several other terms that express
More informationHomework 5 Solutions
Homework 5 Solutions 4.2: 2: a. 321 = 256 + 64 + 1 = (01000001) 2 b. 1023 = 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = (1111111111) 2. Note that this is 1 less than the next power of 2, 1024, which
More informationElements of Applied Cryptography Public key encryption
Network Security Elements of Applied Cryptography Public key encryption Public key cryptosystem RSA and the factorization problem RSA in practice Other asymmetric ciphers Asymmetric Encryption Scheme Let
More informationALGEBRAIC APPROACH TO COMPOSITE INTEGER FACTORIZATION
ALGEBRAIC APPROACH TO COMPOSITE INTEGER FACTORIZATION Aldrin W. Wanambisi 1* School of Pure and Applied Science, Mount Kenya University, P.O box 55350100, Kakamega, Kenya. Shem Aywa 2 Department of Mathematics,
More information(x + a) n = x n + a Z n [x]. Proof. If n is prime then the map
22. A quick primality test Prime numbers are one of the most basic objects in mathematics and one of the most basic questions is to decide which numbers are prime (a clearly related problem is to find
More informationMATH 168: FINAL PROJECT Troels Eriksen. 1 Introduction
MATH 168: FINAL PROJECT Troels Eriksen 1 Introduction In the later years cryptosystems using elliptic curves have shown up and are claimed to be just as secure as a system like RSA with much smaller key
More informationCS 103X: Discrete Structures Homework Assignment 3 Solutions
CS 103X: Discrete Structures Homework Assignment 3 s Exercise 1 (20 points). On wellordering and induction: (a) Prove the induction principle from the wellordering principle. (b) Prove the wellordering
More informationLUC: A New Public Key System
LUC: A New Public Key System Peter J. Smith a and Michael J. J. Lennon b a LUC Partners, Auckland UniServices Ltd, The University of Auckland, Private Bag 92019, Auckland, New Zealand. b Department of
More informationToday s Topics. Primes & Greatest Common Divisors
Today s Topics Primes & Greatest Common Divisors Prime representations Important theorems about primality Greatest Common Divisors Least Common Multiples Euclid s algorithm Once and for all, what are prime
More informationRecent Breakthrough in Primality Testing
Nonlinear Analysis: Modelling and Control, 2004, Vol. 9, No. 2, 171 184 Recent Breakthrough in Primality Testing R. Šleževičienė, J. Steuding, S. Turskienė Department of Computer Science, Faculty of Physics
More informationAdvanced Cryptography
Family Name:... First Name:... Section:... Advanced Cryptography Final Exam July 18 th, 2006 Start at 9:15, End at 12:00 This document consists of 12 pages. Instructions Electronic devices are not allowed.
More informationSmooth numbers and the quadratic sieve
Algorithmic Number Theory MSRI Publications Volume 44, 2008 Smooth numbers and the quadratic sieve CARL POMERANCE ABSTRACT. This article gives a gentle introduction to factoring large integers via the
More information1 Digital Signatures. 1.1 The RSA Function: The eth Power Map on Z n. Crypto: Primitives and Protocols Lecture 6.
1 Digital Signatures A digital signature is a fundamental cryptographic primitive, technologically equivalent to a handwritten signature. In many applications, digital signatures are used as building blocks
More informationThe application of prime numbers to RSA encryption
The application of prime numbers to RSA encryption Prime number definition: Let us begin with the definition of a prime number p The number p, which is a member of the set of natural numbers N, is considered
More information8 Primes and Modular Arithmetic
8 Primes and Modular Arithmetic 8.1 Primes and Factors Over two millennia ago already, people all over the world were considering the properties of numbers. One of the simplest concepts is prime numbers.
More informationTHE MATHEMATICS OF PUBLIC KEY CRYPTOGRAPHY.
THE MATHEMATICS OF PUBLIC KEY CRYPTOGRAPHY. IAN KIMING 1. Forbemærkning. Det kan forekomme idiotisk, at jeg som dansktalende og skrivende i et danskbaseret tidsskrift med en (formentlig) primært dansktalende
More informationCryptography: RSA and the discrete logarithm problem
Cryptography: and the discrete logarithm problem R. Hayden Advanced Maths Lectures Department of Computing Imperial College London February 2010 Public key cryptography Assymmetric cryptography two keys:
More informationOutline. Computer Science 418. Digital Signatures: Observations. Digital Signatures: Definition. Definition 1 (Digital signature) Digital Signatures
Outline Computer Science 418 Digital Signatures Mike Jacobson Department of Computer Science University of Calgary Week 12 1 Digital Signatures 2 Signatures via Public Key Cryptosystems 3 Provable 4 Mike
More informationELLIPTIC CURVES AND LENSTRA S FACTORIZATION ALGORITHM
ELLIPTIC CURVES AND LENSTRA S FACTORIZATION ALGORITHM DANIEL PARKER Abstract. This paper provides a foundation for understanding Lenstra s Elliptic Curve Algorithm for factoring large numbers. We give
More informationCryptography and Network Security Number Theory
Cryptography and Network Security Number Theory XiangYang Li Introduction to Number Theory Divisors b a if a=mb for an integer m b a and c b then c a b g and b h then b (mg+nh) for any int. m,n Prime
More informationMath 319 Problem Set #3 Solution 21 February 2002
Math 319 Problem Set #3 Solution 21 February 2002 1. ( 2.1, problem 15) Find integers a 1, a 2, a 3, a 4, a 5 such that every integer x satisfies at least one of the congruences x a 1 (mod 2), x a 2 (mod
More informationRevised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m)
Chapter 23 Squares Modulo p Revised Version of Chapter 23 We learned long ago how to solve linear congruences ax c (mod m) (see Chapter 8). It s now time to take the plunge and move on to quadratic equations.
More informationRuntime and Implementation of Factoring Algorithms: A Comparison
Runtime and Implementation of Factoring Algorithms: A Comparison Justin Moore CSC290 Cryptology December 20, 2003 Abstract Factoring composite numbers is not an easy task. It is classified as a hard algorithm,
More informationDiscrete Mathematics Lecture 3 Elementary Number Theory and Methods of Proof. Harper Langston New York University
Discrete Mathematics Lecture 3 Elementary Number Theory and Methods of Proof Harper Langston New York University Proof and Counterexample Discovery and proof Even and odd numbers number n from Z is called
More informationIntroduction to Finite Fields (cont.)
Chapter 6 Introduction to Finite Fields (cont.) 6.1 Recall Theorem. Z m is a field m is a prime number. Theorem (Subfield Isomorphic to Z p ). Every finite field has the order of a power of a prime number
More informationElementary Number Theory We begin with a bit of elementary number theory, which is concerned
CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,
More informationCIS 5371 Cryptography. 8. Encryption 
CIS 5371 Cryptography p y 8. Encryption  Asymmetric Techniques Textbook encryption algorithms In this chapter, security (confidentiality) is considered in the following sense: Allornothing secrecy.
More informationFactoring pq 2 with Quadratic Forms: Nice Cryptanalyses
Factoring pq 2 with Quadratic Forms: Nice Cryptanalyses Phong Nguyễn http://www.di.ens.fr/~pnguyen & ASIACRYPT 2009 Joint work with G. Castagnos, A. Joux and F. Laguillaumie Summary Factoring A New Factoring
More informationInternational Journal of Information Technology, Modeling and Computing (IJITMC) Vol.1, No.3,August 2013
FACTORING CRYPTOSYSTEM MODULI WHEN THE COFACTORS DIFFERENCE IS BOUNDED Omar Akchiche 1 and Omar Khadir 2 1,2 Laboratory of Mathematics, Cryptography and Mechanics, Fstm, University of Hassan II MohammediaCasablanca,
More informationThe RSA Algorithm. Evgeny Milanov. 3 June 2009
The RSA Algorithm Evgeny Milanov 3 June 2009 In 1978, Ron Rivest, Adi Shamir, and Leonard Adleman introduced a cryptographic algorithm, which was essentially to replace the less secure National Bureau
More informationDiscrete Mathematics, Chapter 4: Number Theory and Cryptography
Discrete Mathematics, Chapter 4: Number Theory and Cryptography Richard Mayr University of Edinburgh, UK Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 4 1 / 35 Outline 1 Divisibility
More informationPrimes in Sequences. Lee 1. By: Jae Young Lee. Project for MA 341 (Number Theory) Boston University Summer Term I 2009 Instructor: Kalin Kostadinov
Lee 1 Primes in Sequences By: Jae Young Lee Project for MA 341 (Number Theory) Boston University Summer Term I 2009 Instructor: Kalin Kostadinov Lee 2 Jae Young Lee MA341 Number Theory PRIMES IN SEQUENCES
More informationFaster Cryptographic Key Exchange on Hyperelliptic Curves
Faster Cryptographic Key Exchange on Hyperelliptic Curves No Author Given No Institute Given Abstract. We present a key exchange procedure based on divisor arithmetic for the real model of a hyperelliptic
More informationPrimes, Factoring, and RSA A Return to Cryptography. Table of contents
Primes, Factoring, and RSA A Return to Cryptography Foundations of Cryptography Computer Science Department Wellesley College Table of contents Introduction Generating Primes RSA Assumption A classic hard
More informationDiscrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 5
CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 5 Modular Arithmetic One way to think of modular arithmetic is that it limits numbers to a predefined range {0,1,...,N
More informationPublicKey Cryptanalysis 1: Introduction and Factoring
PublicKey Cryptanalysis 1: Introduction and Factoring Nadia Heninger University of Pennsylvania July 21, 2013 Adventures in Cryptanalysis Part 1: Introduction and Factoring. What is publickey crypto
More informationThe Quadratic Sieve Factoring Algorithm
The Quadratic Sieve Factoring Algorithm Eric Landquist MATH 488: Cryptographic Algorithms December 14, 2001 1 Introduction Mathematicians have been attempting to find better and faster ways to factor composite
More informationNEW DIGITAL SIGNATURE PROTOCOL BASED ON ELLIPTIC CURVES
NEW DIGITAL SIGNATURE PROTOCOL BASED ON ELLIPTIC CURVES Ounasser Abid 1, Jaouad Ettanfouhi 2 and Omar Khadir 3 1,2,3 Laboratory of Mathematics, Cryptography and Mechanics, Department of Mathematics, Fstm,
More informationLibrary (versus Language) Based Parallelism in Factoring: Experiments in MPI. Dr. Michael Alexander Dr. Sonja Sewera.
Library (versus Language) Based Parallelism in Factoring: Experiments in MPI Dr. Michael Alexander Dr. Sonja Sewera Talk 20071019 Slide 1 of 20 Primes Definitions Prime: A whole number n is a prime number
More informationLecture 13: Factoring Integers
CS 880: Quantum Information Processing 0/4/0 Lecture 3: Factoring Integers Instructor: Dieter van Melkebeek Scribe: Mark Wellons In this lecture, we review order finding and use this to develop a method
More informationIntroduction. Digital Signature
Introduction Electronic transactions and activities taken place over Internet need to be protected against all kinds of interference, accidental or malicious. The general task of the information technology
More information3. Applications of Number Theory
3. APPLICATIONS OF NUMBER THEORY 163 3. Applications of Number Theory 3.1. Representation of Integers. Theorem 3.1.1. Given an integer b > 1, every positive integer n can be expresses uniquely as n = a
More informationTwogenerator numerical semigroups and Fermat and Mersenne numbers
Twogenerator numerical semigroups and Fermat and Mersenne numbers Shalom Eliahou and Jorge Ramírez Alfonsín Abstract Given g N, what is the number of numerical semigroups S = a,b in N of genus N\S = g?
More informationThe RSA Algorithm: A Mathematical History of the Ubiquitous Cryptological Algorithm
The RSA Algorithm: A Mathematical History of the Ubiquitous Cryptological Algorithm Maria D. Kelly December 7, 2009 Abstract The RSA algorithm, developed in 1977 by Rivest, Shamir, and Adlemen, is an algorithm
More informationARITHMETICAL FUNCTIONS II: CONVOLUTION AND INVERSION
ARITHMETICAL FUNCTIONS II: CONVOLUTION AND INVERSION PETE L. CLARK 1. Sums over divisors, convolution and Möbius Inversion The proof of the multiplicativity of the functions σ k, easy though it was, actually
More informationIs n a Prime Number? Manindra Agrawal. March 27, 2006, Delft. IIT Kanpur
Is n a Prime Number? Manindra Agrawal IIT Kanpur March 27, 2006, Delft Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 1 / 47 Overview 1 The Problem 2 Two Simple, and Slow, Methods
More informationRSA Attacks. By Abdulaziz Alrasheed and Fatima
RSA Attacks By Abdulaziz Alrasheed and Fatima 1 Introduction Invented by Ron Rivest, Adi Shamir, and Len Adleman [1], the RSA cryptosystem was first revealed in the August 1977 issue of Scientific American.
More informationPRIME FACTORS OF CONSECUTIVE INTEGERS
PRIME FACTORS OF CONSECUTIVE INTEGERS MARK BAUER AND MICHAEL A. BENNETT Abstract. This note contains a new algorithm for computing a function f(k) introduced by Erdős to measure the minimal gap size in
More information3. Computational Complexity.
3. Computational Complexity. (A) Introduction. As we will see, most cryptographic systems derive their supposed security from the presumed inability of any adversary to crack certain (number theoretic)
More informationNumber Theory. Proof. Suppose otherwise. Then there would be a finite number n of primes, which we may
Number Theory Divisibility and Primes Definition. If a and b are integers and there is some integer c such that a = b c, then we say that b divides a or is a factor or divisor of a and write b a. Definition
More informationCHAPTER 5. Number Theory. 1. Integers and Division. Discussion
CHAPTER 5 Number Theory 1. Integers and Division 1.1. Divisibility. Definition 1.1.1. Given two integers a and b we say a divides b if there is an integer c such that b = ac. If a divides b, we write a
More information11 Ideals. 11.1 Revisiting Z
11 Ideals The presentation here is somewhat different than the text. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e.g., Z[ 5] = O Q(
More informationMA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES
MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES 2016 47 4. Diophantine Equations A Diophantine Equation is simply an equation in one or more variables for which integer (or sometimes rational) solutions
More informationDetermining the Optimal Combination of Trial Division and Fermat s Factorization Method
Determining the Optimal Combination of Trial Division and Fermat s Factorization Method Joseph C. Woodson Home School P. O. Box 55005 Tulsa, OK 74155 Abstract The process of finding the prime factorization
More informationSTUDY ON ELLIPTIC AND HYPERELLIPTIC CURVE METHODS FOR INTEGER FACTORIZATION. Takayuki Yato. A Senior Thesis. Submitted to
STUDY ON ELLIPTIC AND HYPERELLIPTIC CURVE METHODS FOR INTEGER FACTORIZATION by Takayuki Yato A Senior Thesis Submitted to Department of Information Science Faculty of Science The University of Tokyo on
More informationHomework until Test #2
MATH31: Number Theory Homework until Test # Philipp BRAUN Section 3.1 page 43, 1. It has been conjectured that there are infinitely many primes of the form n. Exhibit five such primes. Solution. Five such
More informationCryptography and Network Security Department of Computer Science and Engineering Indian Institute of Technology Kharagpur
Cryptography and Network Security Department of Computer Science and Engineering Indian Institute of Technology Kharagpur Module No. # 01 Lecture No. # 05 Classic Cryptosystems (Refer Slide Time: 00:42)
More informationA New Generic Digital Signature Algorithm
Groups Complex. Cryptol.? (????), 1 16 DOI 10.1515/GCC.????.??? de Gruyter???? A New Generic Digital Signature Algorithm Jennifer Seberry, Vinhbuu To and Dongvu Tonien Abstract. In this paper, we study
More informationModule MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013
Module MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013 D. R. Wilkins Copyright c David R. Wilkins 1997 2013 Contents A Cyclotomic Polynomials 79 A.1 Minimum Polynomials of Roots of
More informationTextbook: Introduction to Cryptography 2nd ed. By J.A. Buchmann Chap 12 Digital Signatures
Textbook: Introduction to Cryptography 2nd ed. By J.A. Buchmann Chap 12 Digital Signatures Department of Computer Science and Information Engineering, Chaoyang University of Technology 朝 陽 科 技 大 學 資 工
More informationPYTHAGOREAN TRIPLES PETE L. CLARK
PYTHAGOREAN TRIPLES PETE L. CLARK 1. Parameterization of Pythagorean Triples 1.1. Introduction to Pythagorean triples. By a Pythagorean triple we mean an ordered triple (x, y, z) Z 3 such that x + y =
More informationCryptographic hash functions and MACs Solved Exercises for Cryptographic Hash Functions and MACs
Cryptographic hash functions and MACs Solved Exercises for Cryptographic Hash Functions and MACs Enes Pasalic University of Primorska Koper, 2014 Contents 1 Preface 3 2 Problems 4 2 1 Preface This is a
More informationminimal polyonomial Example
Minimal Polynomials Definition Let α be an element in GF(p e ). We call the monic polynomial of smallest degree which has coefficients in GF(p) and α as a root, the minimal polyonomial of α. Example: We
More informationFactoring a semiprime n by estimating φ(n)
Factoring a semiprime n by estimating φ(n) Kyle Kloster May 7, 2010 Abstract A factoring algorithm, called the PhiFinder algorithm, is presented that factors a product of two primes, n = pq, by determining
More informationOn the largest prime factor of x 2 1
On the largest prime factor of x 2 1 Florian Luca and Filip Najman Abstract In this paper, we find all integers x such that x 2 1 has only prime factors smaller than 100. This gives some interesting numerical
More informationDoug Ravenel. October 15, 2008
Doug Ravenel University of Rochester October 15, 2008 s about Euclid s Some s about primes that every mathematician should know (Euclid, 300 BC) There are infinitely numbers. is very elementary, and we
More informationA Comparison Of Integer Factoring Algorithms. Keyur Anilkumar Kanabar
A Comparison Of Integer Factoring Algorithms Keyur Anilkumar Kanabar Batchelor of Science in Computer Science with Honours The University of Bath May 2007 This dissertation may be made available for consultation
More informationOverview of PublicKey Cryptography
CS 361S Overview of PublicKey Cryptography Vitaly Shmatikov slide 1 Reading Assignment Kaufman 6.16 slide 2 PublicKey Cryptography public key public key? private key Alice Bob Given: Everybody knows
More informationBreaking The Code. Ryan Lowe. Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and
Breaking The Code Ryan Lowe Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and a minor in Applied Physics. As a sophomore, he took an independent study
More informationComputing Cubic Fields in QuasiLinear Time
Computing Cubic Fields in QuasiLinear Time K. Belabas Département de mathématiques (A2X) Université Bordeaux I 351, cours de la Libération, 33405 Talence (France) belabas@math.ubordeaux.fr Cubic fields
More informationInteger Factorization
Integer Factorization Lecture given at the Joh. GutenbergUniversität, Mainz, July 23, 1992 by ÖYSTEIN J. RÖDSETH University of Bergen, Department of Mathematics, Allégt. 55, N5007 Bergen, Norway 1 Introduction
More informationTYPES Workshop, 1213 june 2006 p. 1/22. The Elliptic Curve Factorization method
Ä ÙÖ ÒØ ÓÙ Ð ÙÖ ÒØ ÓÑ Ø ºÒ Ø TYPES Workshop, 1213 june 2006 p. 1/22 ÄÇÊÁ ÍÒ Ú Ö Ø À ÒÖ ÈÓ Ò Ö Æ ÒÝÁ. The Elliptic Curve Factorization method Outline 1. Introduction 2. Factorization method principle 3.
More informationComputing exponents modulo a number: Repeated squaring
Computing exponents modulo a number: Repeated squaring How do you compute (1415) 13 mod 2537 = 2182 using just a calculator? Or how do you check that 2 340 mod 341 = 1? You can do this using the method
More informationComputational Number Theory
Computational Number Theory C. Pomerance 1 Introduction Historically, computation has been a driving force in the development of mathematics. To help measure the sizes of their fields, the Egyptians invented
More informationThe Mathematics of the RSA PublicKey Cryptosystem
The Mathematics of the RSA PublicKey Cryptosystem Burt Kaliski RSA Laboratories ABOUT THE AUTHOR: Dr Burt Kaliski is a computer scientist whose involvement with the security industry has been through
More information