An Overview of Integer Factoring Algorithms. The Problem

Size: px
Start display at page:

Download "An Overview of Integer Factoring Algorithms. The Problem"

Transcription

1 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

2 A Difficult Problem No efficient algorithm (= taking time (log n) c ) is know for the problem. The fastest known algorithm takes time exp( c (log n) 1/3 (loglog n) 2/3 ) with c 1.9. With this, we can factor 140 digit numbers in reasonable time. It is believed that no efficient algorithm exists. Useful in Cryptography RSA cryptosystem s security is based on hardness of factoring. Several other cryptosystems rely on this problem as well. 2

3 We present an overview of the known factoring algorithms. #1: Trial Division Divide n with all primes up to n starting from 2 and collect all divisors. A very simple algorithm. Takes time exp(½ log n) = L(1, ½). Notation: Denote exp(c(log n) ε (loglog n) 1-ε ) as L(ε, c). 3

4 #2: Pollard s Rho Method 1. Randomly select x 0 {1, 2,, n-1}, and compute x i = x i (mod n) for i = 1, 2, 2. Compute gcd(x i x 2i, n) until a factor is found. Discovered by J. Pollard in Takes time L(1, ¼). Used to factorize eighth Fermat number , a 78 digit number. x t+2 x t+1 x t = x m x m-1 x m-2 x 2 x 1 x 0 Pollard s Rho Shape 4

5 Analysis Let p be the smallest prime factor of n, so p < n. Number sequence x 0, x 1, x 2, behaves randomly modulo p. So the probability that x t = x m (mod p) for t < m is roughly 1/ p. Notice that if x t = x m (mod p), then x t+k = x m+k (mod p) for all k > 0. Therefore, there exists a s < 2t with x s = x 2s (mod p). Again using randomness of the sequence, with probability at least ½, x s x 2s (mod n). Therefore, p gcd(x s x 2s, n) < n. For good probability of success, we need to generate roughly p = n 1/4 x i s. So the time complexity is exp(¼log n). 5

6 #3: Pollard s p-1 Method 1. Fix a factor base = set of all primes B. 2. Compute m = q prime, q B q log n. 3. Compute gcd(a m -1, n) for a random a. Discovered by J. Pollard in Takes time O(B (log n) 2 ). Works if prime p n and p-1 has no prime divisor greater than B. Fermat s Little Theorem If p is prime then for all a with gcd(a, p) = 1, a p-1 = 1 (mod p). In other words, the set of numbers { a 0 < a < p } forms a group of size p-1 under multiplication modulo p. 6

7 Analysis Suppose prime p n and p-1 has no factor greater than B. This implies that p-1 m. So, by Fermat s Little Theorem, p divides a m -1. So it might be found when computing gcd(a m -1, n). Useful only for a subset of numbers n. #4: Elliptic Curve Method Previous method works only for n s with a prime divisor p such that p-1 is a product of small primes. It is always true that a number m close to p will have this property. So if we can work with a group of size m, instead of p-1, the method will work for all numbers. 7

8 Elliptic Curves Elliptic curve E(a,b) has the following form: y 2 = 4x 3 - ax b; a 3 27 b 2 0 The set of points on an elliptic curve form a group under addition. We consider elliptic curves modulo n. The number of points on an elliptic curve modulo prime p (= #E p (a,b)) is between p+1-2 p and p+1+2 p. Curve y 2 = 4x 3-4x A F B -C E C Addition on curve: A + B = C; E + F = O, point at infinity 8

9 Algorithm 1. Fix a factor base = set of all primes B. 2. Compute m = q prime, q B q log n. 3. Choose a random a and b with a 3 27b 2 0 (mod n). 4. Choose a random point P on elliptic curve E n (a,b). 5. Attempt to compute a factor of n from mp O (the zero for addition ) Analysis Similar to Pollard s p-1 method. If prime p n and #E p (a,b) has no divisor > B, then n can be factored. This works for all the numbers since #E p (a,b) is randomly distributed between p+1-2 p and p+1+2 p. A careful analysis shows the running time to be L(½, 1) much better than earlier methods! 9

10 Used to factor tenth and eleventh Fermat numbers: (308 digits) and (610 digits). Fastest known algorithm for most of numbers. Discovered by H. Lenstra in #5: Fermat s Method 1. Compute m = [ n]. 2. For d = 1, 2, 3, do: i. Let x = m + d and test if x 2 -n is a perfect square. ii. If yes, let y 2 = x 2 -n and factor n using gcd(n,x+y). Discovered by P. Fermat in 17 th century. Works fast if n has two factors close to n. 10

11 Analysis Suppose n = k (k + t) with t small compared to k. Then m = [ n] k (1 + t/k) 1/2 k + ½t. Notice that with x = k + ½t, x 2 -n= k 2 + kt + ¼t 2 -k 2 -kt=(½t) 2 So the right x will be quickly found. #6: Dixon s Method Proposed by Dixon in 1970 s. Simple version of Morrison-Brillhart method. Based on Fermat s method. Aims to find x and y such that x 2 = y 2 (mod n). 11

12 Algorithm Data Collection Step: 1. Fix a factor base = set of primes B. 2. Randomly choose a number v and compute u = v 2 (mod n). 3. If u has all prime factors B, store the pair (v,u). Do this until about B pairs have been stored. Data Analysis Step: 1. Let p 1, p 2,, p t be primes B. 2. Let u i = p 1 e i,1 * p 2 e i,2 * * p t e i,t for every stored u i. 3. Let vector w i = [ e i,1 e i,2 e i,t ]. 4. Find a linear dependency amongst these vectors over F 2 : i β i w i = 0 (mod 2). 5. Compute x = Π i v i β i. 6. Compute y = (Π i u i β i ) ½. 7. Factor n as gcd(n, x+y). 12

13 Analysis Over integers, all numbers in i β i w i are multiples of 2. So, β Π i u i i = p i β i e i,1 1 * p i β i e i,2 2 * * p i β i e i,t t is a perfect square. Since v 2 i =u i (mod n), we get x 2 2β = Π i v i β i = Π i u i i = y 2 (mod n). Analysis How quickly can we find required number of pairs? Observation: If B is small, we need to find only a few pairs. But the chances of finding one pair are small. If B is large, we need to find many pairs. But chances of finding one pair are high. 13

14 Analysis What is the best value of B? It turns out to be L(½, 1/ 2) = exp(1/ 2(log n) 1/2 (loglog n) 1/2 ). With this value, the running time is L(½, 2) = exp( 2(log n) 1/2 (loglog n) 1/2 ). Not as good as Elliptic curve method. #7: Quadratic Sieve Proposed by C. Pomerance in A combination of Fermat s method and Dixon s method. Does the Data Collection step cleverly to reduce time. The best value of B becomes L(½, ½). The running time reduces to L(½, 1). Betters Elliptic curve method for large numbers that are used in cryptography. 14

15 Used to factor 129-digit RSA challenge in 1994: RSA-129 = The Sieving Idea The v i s to be tested are chosen from the range [ n, n+a]. For each v i, we check if v 2 i nhas all prime divisors B. For a prime q B, if q divides v 2 n, then it will also divide (v + kq) 2 n and (kq - v) 2 n for all integers k. 15

16 The Sieving Idea So, for each q B, do the following: Solve the equation x 2 = n (mod q) to obtain two solutions, say α and β. Divide all numbers in the range [ n, n+a] that are of the form α + kq or β + kq by q as many times as possible. Once all q s are finished, the numbers in the range that become 1 are the useful ones. The Time Complexity of Factoring A number of algorithms have time complexity L(½, c) for constants c. This led to the belief that the optimal complexity for factoring is L(½, c) for some c 1. And then the Number Field sieve appeared 16

17 #8: Number Field Sieve Proposed by J. Pollard in 1988 and improved by C. Pomerance, H. Lenstra and others. A generalization of Quadratic sieve to number fields. The running time is L(1/3, 1.923). Used to factor ninth Fermat number (153 digits) and RSA-130 (in 1996). Number Field Sieve Idea Select a small degree d. Find a polynomial f(x) and number m such that (1) m n 1/d and (2) n divides f(m). Let α be a root of f(x) over complex numbers. Consider ring Z[α], consisting of all complex numbers that can be written as: j c j α j with c j integers. 17

18 Define a map ψ from Z[α] to Z/nZ, the ring of residues modulo n as: Ψ( j c j α j ) = j c j m j (mod n). Clearly, 0 = Ψ(f(α)) = f(m) = 0 (mod n), and so Ψ is a ring homomorphism. Now find sequence of pairs (u i, v i ) and a sequence of exponents β i such that: 1. Π i (u i -mv i ) β i is a square in Z. 2. Π i (u i - αv i ) β i is a square in Z[α]. Then, x 2 = Π i (u i -mv i ) β i = Π i Ψ(u i - αv i ) β i (mod n) = Ψ(Π i (u i - αv i ) β i ) (mod n) = Ψ(g 2 (α)) (mod n) = [Ψ(g(α))] 2 (mod n) = g(m 2 ) = y 2 (mod n). 18

19 Pairs (u i, v i ) satisfying the first condition can be found as before. The second condition is more tricky. Using some additional ideas, it can be done. Time complexity reduces because numbers that we work with are now smaller ( n 1/d instead of n 1/2 ). #9: Shore s Algorithm Proposed by P. Shore in Works on Quantum computers only. The running time is L(0, 3) = n 3!! If one can build Quantum computers, factoring would become easy. As of now, we do not know 19

20 Remarks Number Field sieve is the fastest known general purpose algorithm. Is that the best possible? Perhaps not. Where does one find better algorithms?? 20

Factorization Methods: Very Quick Overview

Factorization 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 information

Factoring Algorithms

Factoring 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 information

Primality - Factorization

Primality - 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 information

Integer Factorization using the Quadratic Sieve

Integer 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 information

Primality Testing and Factorization Methods

Primality 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 information

RSA Question 2. Bob thinks that p and q are primes but p isn t. Then, Bob thinks Φ Bob :=(p-1)(q-1) = φ(n). Is this true?

RSA Question 2. Bob thinks that p and q are primes but p isn t. Then, Bob thinks Φ Bob :=(p-1)(q-1) = φ(n). Is this true? RSA Question 2 Bob thinks that p and q are primes but p isn t. Then, Bob thinks Φ Bob :=(p-1)(q-1) = φ(n). Is this true? Bob chooses a random e (1 < e < Φ Bob ) such that gcd(e,φ Bob )=1. Then, d = e -1

More information

Math 319 Problem Set #3 Solution 21 February 2002

Math 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 information

Breaking 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 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 information

FACTORING. n = 2 25 + 1. fall in the arithmetic sequence

FACTORING. n = 2 25 + 1. fall in the arithmetic sequence 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,

More information

MATH 168: FINAL PROJECT Troels Eriksen. 1 Introduction

MATH 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 information

Elements of Applied Cryptography Public key encryption

Elements 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 information

The Quadratic Sieve Factoring Algorithm

The 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 information

Notes on Factoring. MA 206 Kurt Bryan

Notes on Factoring. MA 206 Kurt Bryan The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor

More information

ALGEBRAIC APPROACH TO COMPOSITE INTEGER FACTORIZATION

ALGEBRAIC 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 553-50100, Kakamega, Kenya. Shem Aywa 2 Department of Mathematics,

More information

Arithmetic algorithms for cryptology 5 October 2015, Paris. Sieves. Razvan Barbulescu CNRS and IMJ-PRG. R. Barbulescu Sieves 0 / 28

Arithmetic algorithms for cryptology 5 October 2015, Paris. Sieves. Razvan Barbulescu CNRS and IMJ-PRG. R. Barbulescu Sieves 0 / 28 Arithmetic algorithms for cryptology 5 October 2015, Paris Sieves Razvan Barbulescu CNRS and IMJ-PRG R. Barbulescu Sieves 0 / 28 Starting point Notations q prime g a generator of (F q ) X a (secret) integer

More information

Elementary factoring algorithms

Elementary 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 information

Advanced Cryptography

Advanced 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 information

Is n a Prime Number? Manindra Agrawal. March 27, 2006, Delft. IIT Kanpur

Is 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 information

International Journal of Information Technology, Modeling and Computing (IJITMC) Vol.1, No.3,August 2013

International Journal of Information Technology, Modeling and Computing (IJITMC) Vol.1, No.3,August 2013 FACTORING CRYPTOSYSTEM MODULI WHEN THE CO-FACTORS DIFFERENCE IS BOUNDED Omar Akchiche 1 and Omar Khadir 2 1,2 Laboratory of Mathematics, Cryptography and Mechanics, Fstm, University of Hassan II Mohammedia-Casablanca,

More information

Elementary Number Theory We begin with a bit of elementary number theory, which is concerned

Elementary 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 information

FACTORING LARGE NUMBERS, A GREAT WAY TO SPEND A BIRTHDAY

FACTORING 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 information

TYPES Workshop, 12-13 june 2006 p. 1/22. The Elliptic Curve Factorization method

TYPES Workshop, 12-13 june 2006 p. 1/22. The Elliptic Curve Factorization method Ä ÙÖ ÒØ ÓÙ Ð ÙÖ ÒØ ÓÑ Ø ºÒ Ø TYPES Workshop, 12-13 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 information

Lecture 13 - Basic Number Theory.

Lecture 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 non-negative integers. We say that A divides B, denoted

More information

Factoring & Primality

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 information

The Sieve Re-Imagined: Integer Factorization Methods

The Sieve Re-Imagined: Integer Factorization Methods The Sieve Re-Imagined: Integer Factorization Methods by Jennifer Smith A research paper presented to the University of Waterloo in partial fulfillment of the requirement for the degree of Master of Mathematics

More information

U.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 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 information

STUDY 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. 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 information

Smooth numbers and the quadratic sieve

Smooth 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 information

ELEMENTARY THOUGHTS ON DISCRETE LOGARITHMS. Carl Pomerance

ELEMENTARY 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 information

Factoring Algorithms

Factoring 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 information

Factoring. Factoring 1

Factoring. 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 information

MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES

MA257: 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 information

(x + a) n = x n + a Z n [x]. Proof. If n is prime then the map

(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 information

Modern Factoring Algorithms

Modern 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 information

ELLIPTIC CURVES AND LENSTRA S FACTORIZATION ALGORITHM

ELLIPTIC 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 information

ABSTRACT ALGEBRA: A STUDY GUIDE FOR BEGINNERS

ABSTRACT ALGEBRA: A STUDY GUIDE FOR BEGINNERS ABSTRACT ALGEBRA: A STUDY GUIDE FOR BEGINNERS John A. Beachy Northern Illinois University 2014 ii J.A.Beachy This is a supplement to Abstract Algebra, Third Edition by John A. Beachy and William D. Blair

More information

minimal polyonomial Example

minimal 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 information

Study of algorithms for factoring integers and computing discrete logarithms

Study of algorithms for factoring integers and computing discrete logarithms Study of algorithms for factoring integers and computing discrete logarithms First Indo-French Workshop on Cryptography and Related Topics (IFW 2007) June 11 13, 2007 Paris, France Dr. Abhijit Das Department

More information

JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson

JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials

More information

On Generalized Fermat Numbers 3 2n +1

On 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 information

I. Introduction. MPRI Cours 2-12-2. Lecture IV: Integer factorization. What is the factorization of a random number? II. Smoothness testing. F.

I. Introduction. MPRI Cours 2-12-2. Lecture IV: Integer factorization. What is the factorization of a random number? II. Smoothness testing. F. F. Morain École polytechnique MPRI cours 2-12-2 2013-2014 3/22 F. Morain École polytechnique MPRI cours 2-12-2 2013-2014 4/22 MPRI Cours 2-12-2 I. Introduction Input: an integer N; logox F. Morain logocnrs

More information

I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I. GROUPS: BASIC DEFINITIONS AND EXAMPLES I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

More information

RSA and Primality Testing

RSA and Primality Testing and Primality Testing Joan Boyar, IMADA, University of Southern Denmark Studieretningsprojekter 2010 1 / 81 Correctness of cryptography cryptography Introduction to number theory Correctness of with 2

More information

Determining the Optimal Combination of Trial Division and Fermat s Factorization Method

Determining 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 information

Principles of Public Key Cryptography. Applications of Public Key Cryptography. Security in Public Key Algorithms

Principles of Public Key Cryptography. Applications of Public Key Cryptography. Security in Public Key Algorithms Principles of Public Key Cryptography Chapter : Security Techniques Background Secret Key Cryptography Public Key Cryptography Hash Functions Authentication Chapter : Security on Network and Transport

More information

Applications of Fermat s Little Theorem and Congruences

Applications of Fermat s Little Theorem and Congruences Applications of Fermat s Little Theorem and Congruences Definition: Let m be a positive integer. Then integers a and b are congruent modulo m, denoted by a b mod m, if m (a b). Example: 3 1 mod 2, 6 4

More information

Faster deterministic integer factorisation

Faster 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 information

Factoring integers, Producing primes and the RSA cryptosystem Harish-Chandra Research Institute

Factoring integers, Producing primes and the RSA cryptosystem Harish-Chandra Research Institute RSA cryptosystem HRI, Allahabad, February, 2005 0 Factoring integers, Producing primes and the RSA cryptosystem Harish-Chandra Research Institute Allahabad (UP), INDIA February, 2005 RSA cryptosystem HRI,

More information

Short Programs for functions on Curves

Short Programs for functions on Curves Short Programs for functions on Curves Victor S. Miller Exploratory Computer Science IBM, Thomas J. Watson Research Center Yorktown Heights, NY 10598 May 6, 1986 Abstract The problem of deducing a function

More information

Homework 5 Solutions

Homework 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 information

Factoring a semiprime n by estimating φ(n)

Factoring 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 Phi-Finder algorithm, is presented that factors a product of two primes, n = pq, by determining

More information

On the coefficients of the polynomial in the number field sieve

On the coefficients of the polynomial in the number field sieve On the coefficients of the polynomial in the number field sieve Yang Min a, Meng Qingshu b,, Wang Zhangyi b, Li Li a, Zhang Huanguo b a International School of Software, Wuhan University, Hubei, China,

More information

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization

More information

Quotient Rings and Field Extensions

Quotient Rings and Field Extensions Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.

More information

Discrete Mathematics, Chapter 4: Number Theory and Cryptography

Discrete 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 information

An Introduction to the General Number Field Sieve

An Introduction to the General Number Field Sieve An Introduction to the General Number Field Sieve Matthew E. Briggs Thesis submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements

More information

Lecture 13: Factoring Integers

Lecture 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 information

Mathematics of Cryptography Part I

Mathematics of Cryptography Part I CHAPTER 2 Mathematics of Cryptography Part I (Solution to Odd-Numbered Problems) Review Questions 1. The set of integers is Z. It contains all integral numbers from negative infinity to positive infinity.

More information

3. Applications of Number Theory

3. 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 information

Factoring pq 2 with Quadratic Forms: Nice Cryptanalyses

Factoring 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 information

ECE 842 Report Implementation of Elliptic Curve Cryptography

ECE 842 Report Implementation of Elliptic Curve Cryptography ECE 842 Report Implementation of Elliptic Curve Cryptography Wei-Yang Lin December 15, 2004 Abstract The aim of this report is to illustrate the issues in implementing a practical elliptic curve cryptographic

More information

Runtime and Implementation of Factoring Algorithms: A Comparison

Runtime 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 information

CONTINUED FRACTIONS AND FACTORING. Niels Lauritzen

CONTINUED FRACTIONS AND FACTORING. Niels Lauritzen CONTINUED FRACTIONS AND FACTORING Niels Lauritzen ii NIELS LAURITZEN DEPARTMENT OF MATHEMATICAL SCIENCES UNIVERSITY OF AARHUS, DENMARK EMAIL: niels@imf.au.dk URL: http://home.imf.au.dk/niels/ Contents

More information

a 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4)

a 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4) ROOTS OF POLYNOMIAL EQUATIONS In this unit we discuss polynomial equations. A polynomial in x of degree n, where n 0 is an integer, is an expression of the form P n (x) =a n x n + a n 1 x n 1 + + a 1 x

More information

Public-Key Cryptanalysis 1: Introduction and Factoring

Public-Key Cryptanalysis 1: Introduction and Factoring Public-Key Cryptanalysis 1: Introduction and Factoring Nadia Heninger University of Pennsylvania July 21, 2013 Adventures in Cryptanalysis Part 1: Introduction and Factoring. What is public-key crypto

More information

On prime-order elliptic curves with embedding degrees k = 3, 4 and 6

On prime-order elliptic curves with embedding degrees k = 3, 4 and 6 On prime-order elliptic curves with embedding degrees k = 3, 4 and 6 Koray Karabina and Edlyn Teske University of Waterloo ANTS VIII, Banff, May 20, 2008 K. Karabina and E. Teske (UW) Prime-order elliptic

More information

2 Primality and Compositeness Tests

2 Primality and Compositeness Tests Int. J. Contemp. Math. Sciences, Vol. 3, 2008, no. 33, 1635-1642 On Factoring R. A. Mollin Department of Mathematics and Statistics University of Calgary, Calgary, Alberta, Canada, T2N 1N4 http://www.math.ucalgary.ca/

More information

Optimization of the MPQS-factoring algorithm on the Cyber 205 and the NEC SX-2

Optimization of the MPQS-factoring algorithm on the Cyber 205 and the NEC SX-2 Optimization of the MPQS-factoring algorithm on the Cyber 205 and the NEC SX-2 Walter Lioen, Herman te Riele, Dik Winter CWI P.O. Box 94079, 1090 GB Amsterdam, The Netherlands ABSTRACT This paper describes

More information

RSA Attacks. By Abdulaziz Alrasheed and Fatima

RSA 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 information

3. Computational Complexity.

3. 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 information

Integer Factorisation

Integer Factorisation Integer Factorisation Vassilis Kostakos Department of Mathematical Sciences University of Bath vkostakos@yahoo.com http://www.geocities.com/vkostakos May 7, 2001 MATH0082 Double Unit Project Comparison

More information

THE NUMBER OF REPRESENTATIONS OF n OF THE FORM n = x 2 2 y, x > 0, y 0

THE NUMBER OF REPRESENTATIONS OF n OF THE FORM n = x 2 2 y, x > 0, y 0 THE NUMBER OF REPRESENTATIONS OF n OF THE FORM n = x 2 2 y, x > 0, y 0 RICHARD J. MATHAR Abstract. We count solutions to the Ramanujan-Nagell equation 2 y +n = x 2 for fixed positive n. The computational

More information

CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY

CHAPTER 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 information

CHAPTER 5. Number Theory. 1. Integers and Division. Discussion

CHAPTER 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 information

Continued Fractions. Darren C. Collins

Continued Fractions. Darren C. Collins Continued Fractions Darren C Collins Abstract In this paper, we discuss continued fractions First, we discuss the definition and notation Second, we discuss the development of the subject throughout history

More information

6.1 The Greatest Common Factor; Factoring by Grouping

6.1 The Greatest Common Factor; Factoring by Grouping 386 CHAPTER 6 Factoring and Applications 6.1 The Greatest Common Factor; Factoring by Grouping OBJECTIVES 1 Find the greatest common factor of a list of terms. 2 Factor out the greatest common factor.

More information

USING LUCAS SEQUENCES TO FACTOR LARGE INTEGERS NEAR GROUP ORDERS

USING LUCAS SEQUENCES TO FACTOR LARGE INTEGERS NEAR GROUP ORDERS USING LUCAS SEQUENCES TO FACTOR LARGE INTEGERS NEAR GROUP ORDERS Zhenxiang Zhang* Dept. of Math., Anhui Normal University, 241000 Wuhu, Anhui, P.R. China e-mail: zhangzhx@mail.ahwhptt.net.cn (Submitted

More information

Lecture Note 5 PUBLIC-KEY CRYPTOGRAPHY. Sourav Mukhopadhyay

Lecture Note 5 PUBLIC-KEY CRYPTOGRAPHY. Sourav Mukhopadhyay Lecture Note 5 PUBLIC-KEY CRYPTOGRAPHY Sourav Mukhopadhyay Cryptography and Network Security - MA61027 Modern/Public-key cryptography started in 1976 with the publication of the following paper. W. Diffie

More information

Homework until Test #2

Homework 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 information

Library (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. Library (versus Language) Based Parallelism in Factoring: Experiments in MPI Dr. Michael Alexander Dr. Sonja Sewera Talk 2007-10-19 Slide 1 of 20 Primes Definitions Prime: A whole number n is a prime number

More information

3 1. Note that all cubes solve it; therefore, there are no more

3 1. Note that all cubes solve it; therefore, there are no more Math 13 Problem set 5 Artin 11.4.7 Factor the following polynomials into irreducible factors in Q[x]: (a) x 3 3x (b) x 3 3x + (c) x 9 6x 6 + 9x 3 3 Solution: The first two polynomials are cubics, so if

More information

Cryptography and Network Security Number Theory

Cryptography and Network Security Number Theory Cryptography and Network Security Number Theory Xiang-Yang 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 information

Integer Factorization

Integer Factorization Integer Factorization Lecture given at the Joh. Gutenberg-Universität, Mainz, July 23, 1992 by ÖYSTEIN J. RÖDSETH University of Bergen, Department of Mathematics, Allégt. 55, N-5007 Bergen, Norway 1 Introduction

More information

FACTORING AFTER DEDEKIND

FACTORING AFTER DEDEKIND FACTORING AFTER DEDEKIND KEITH CONRAD Let K be a number field and p be a prime number. When we factor (p) = po K into prime ideals, say (p) = p e 1 1 peg g, we refer to the data of the e i s, the exponents

More information

APPLICATIONS OF THE ORDER FUNCTION

APPLICATIONS OF THE ORDER FUNCTION APPLICATIONS OF THE ORDER FUNCTION LECTURE NOTES: MATH 432, CSUSM, SPRING 2009. PROF. WAYNE AITKEN In this lecture we will explore several applications of order functions including formulas for GCDs and

More information

Basic Algorithms In Computer Algebra

Basic Algorithms In Computer Algebra Basic Algorithms In Computer Algebra Kaiserslautern SS 2011 Prof. Dr. Wolfram Decker 2. Mai 2011 References Cohen, H.: A Course in Computational Algebraic Number Theory. Springer, 1993. Cox, D.; Little,

More information

63. Graph y 1 2 x and y 2 THE FACTOR THEOREM. The Factor Theorem. Consider the polynomial function. P(x) x 2 2x 15.

63. Graph y 1 2 x and y 2 THE FACTOR THEOREM. The Factor Theorem. Consider the polynomial function. P(x) x 2 2x 15. 9.4 (9-27) 517 Gear ratio d) For a fixed wheel size and chain ring, does the gear ratio increase or decrease as the number of teeth on the cog increases? decreases 100 80 60 40 20 27-in. wheel, 44 teeth

More information

12 Greatest Common Divisors. The Euclidean Algorithm

12 Greatest Common Divisors. The Euclidean Algorithm Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 12 Greatest Common Divisors. The Euclidean Algorithm As mentioned at the end of the previous section, we would like to

More information

Zeros of Polynomial Functions

Zeros of Polynomial Functions Review: Synthetic Division Find (x 2-5x - 5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 3-5x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 3-5x 2 + x + 2. Zeros of Polynomial Functions Introduction

More information

March 29, 2011. 171S4.4 Theorems about Zeros of Polynomial Functions

March 29, 2011. 171S4.4 Theorems about Zeros of Polynomial Functions MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial

More information

Introduction to finite fields

Introduction to finite fields Introduction to finite fields Topics in Finite Fields (Fall 2013) Rutgers University Swastik Kopparty Last modified: Monday 16 th September, 2013 Welcome to the course on finite fields! This is aimed at

More information

Quantum Computing Lecture 7. Quantum Factoring. Anuj Dawar

Quantum Computing Lecture 7. Quantum Factoring. Anuj Dawar Quantum Computing Lecture 7 Quantum Factoring Anuj Dawar Quantum Factoring A polynomial time quantum algorithm for factoring numbers was published by Peter Shor in 1994. polynomial time here means that

More information

k, then n = p2α 1 1 pα k

k, then n = p2α 1 1 pα k Powers of Integers An integer n is a perfect square if n = m for some integer m. Taking into account the prime factorization, if m = p α 1 1 pα k k, then n = pα 1 1 p α k k. That is, n is a perfect square

More information

Cubic Polynomials in the Number Field Sieve. Ronnie Scott Williams, Jr., B.S. A Thesis. Mathematics and Statistics

Cubic Polynomials in the Number Field Sieve. Ronnie Scott Williams, Jr., B.S. A Thesis. Mathematics and Statistics Cubic Polynomials in the Number Field Sieve by Ronnie Scott Williams, Jr., B.S. A Thesis In Mathematics and Statistics Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment

More information

Revised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m)

Revised 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 information

Computer and Network Security

Computer 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 information

Computing Cubic Fields in Quasi-Linear Time

Computing Cubic Fields in Quasi-Linear Time Computing Cubic Fields in Quasi-Linear Time K. Belabas Département de mathématiques (A2X) Université Bordeaux I 351, cours de la Libération, 33405 Talence (France) belabas@math.u-bordeaux.fr Cubic fields

More information

Modern Algebra Lecture Notes: Rings and fields set 4 (Revision 2)

Modern Algebra Lecture Notes: Rings and fields set 4 (Revision 2) Modern Algebra Lecture Notes: Rings and fields set 4 (Revision 2) Kevin Broughan University of Waikato, Hamilton, New Zealand May 13, 2010 Remainder and Factor Theorem 15 Definition of factor If f (x)

More information

Index Calculation Attacks on RSA Signature and Encryption

Index Calculation Attacks on RSA Signature and Encryption Index Calculation Attacks on RSA Signature and Encryption Jean-Sébastien Coron 1, Yvo Desmedt 2, David Naccache 1, Andrew Odlyzko 3, and Julien P. Stern 4 1 Gemplus Card International {jean-sebastien.coron,david.naccache}@gemplus.com

More information

Recent Breakthrough in Primality Testing

Recent 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 information