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


 Abigayle Anthony
 3 years ago
 Views:
Transcription
1 F. Morain École polytechnique MPRI cours /22 F. Morain École polytechnique MPRI cours /22 MPRI Cours I. Introduction Input: an integer N; logox F. Morain logocnrs logoinria Output: N = k i=1 pα i i with p i (proven) prime. Lecture IV: Integer factorization Major impact: estimate the security of RSA cryptosystems. Also: primitive for a lot of number theory problems. I. Introduction. II. Smoothness testing. III. Pollard s RHO method. IV. Pollard s p 1 method. 2013/10/14 How do we test and compare algorithms? Cunningham project, RSA Security (partitions, RSA keys) though abandoned? Decimals of π. F. Morain École polytechnique MPRI cours /22 What is the factorization of a random number? F. Morain École polytechnique MPRI cours /22 II. Smoothness testing N = N 1 N 2 N r with N i prime, N i N i+1. Prop. r log 2 N; r = log log N. Size of the factors: D k = lim N + log N k / log N exists and On average k D k N 1 N 0.62, N 2 N 0.21, N 3 N an integer has one large factor, a medium size one and a bunch of small ones. Def. a Bsmooth number has all its prime factors B. Bsmooth numbers are the heart of all efficient factorization or discrete logarithm algorithms. De Bruijn s function: ψ(x, y) = #{z x, z is y smooth}. Thm. (Candfield, Erdős, Pomerance) ε > 0, uniformly in y (log x) 1+ε, as x with u = log x/ log y. ψ(x, y) = x u u(1+o(1))
2 Bsmooth numbers (cont d) A) Trial division Prop. Let L(x) = exp ( log x log log x ). For all real α > 0, β > 0, as x ψ(x α, L(x) β x α ) = L(x). α 2β +o(1) Ordinary interpretation: a number x α is L(x) β smooth with probability ψ(x α, L(x) β ) x α = L(x) α 2β +o(1). Algorithm: divide x X by all p B, say {p 1, p 2,..., p m }, m = π(b) = O(B). Cost: m divisions or p B T(x, p) = O(m lg X lg B). Implementation: use any method to compute and store all primes 2 32 (one char per (p i+1 p i )/2; see Brent). Useful generalization: given x 1, x 2,..., x n X, can we find the Bsmooth part of the x i s more rapidly than repeating the above in O(nm lg B lg X)? F. Morain École polytechnique MPRI cours /22 B) Product trees Algorithm: Franke/Kleinjung/FM/Wirth improved by Bernstein 1. [Product tree] Compute z = p 1 p m. p 1 p 2 p 3 p 4 p 1 p 2 p 3 p 4 p 1 p 2 p 3 p 4 2. [Remainder tree] Compute z mod x 1,..., z mod x n. z mod (x 1 x 2 x 3 x 4 ) z mod (x 1 x 2 ) z mod (x 3 x 4 ) z mod x 1 z mod x 2 z mod x 3 z mod x 4 3. [explode valuation] For each k {1,..., n}, compute y k = z 2e mod x k with e s.t. 2 2e x k ; print gcd(x k, y k ). F. Morain École polytechnique MPRI cours /22 F. Morain École polytechnique MPRI cours /22 Validity and analysis Validity: let y k = z 2e mod x k. Suppose p x k. Then ν p (x k ) 2 e, since 2 ν p ν 2 2e. Therefore ν p (y k ) 2 e ν and the gcd will contain the right valuation. Division: If A has r + s digits and B has s digits, then plain division requires D(r + s, s) = O(rs) word operations. In case r s, break into r/s divisions of complexity M(s). Step 1: M((m/2) lg B). Step 2: D(m lg B, n lg X) (m/n)m(n) lg B lg X. Ex. B = 2 32, m , X = Rem. If space is an issue, do this by blocks. Rem. For more information see Bernstein s web page. F. Morain École polytechnique MPRI cours /22
3 F. Morain École polytechnique MPRI cours /22 F. Morain École polytechnique MPRI cours /22 III. Pollard s RHO method Epact Prop. Let f : E E, #E = m; X n+1 = f (X n ) with X 0 E. X 0 X 1 X 2 X µ 1 X µ X µ+1 X µ+λ 1 Thm. (Flajolet, Odlyzko, 1990) When m πm λ µ m. Prop. There exists a unique e > 0 (epact) s.t. µ e < λ + µ and X 2e = X e. It is the smallest nonzero multiple of λ that is µ: if µ = 0, e = λ and if µ > 0, e = µ λ λ. Floyd s algorithm: X < X0; Y < X0; e < 0; repeat X < f(x); Y < f(f(y)); e < e+1; until X = Y; Thm. e π 5 m m. F. Morain École polytechnique MPRI cours /22 Application to the factorization of N F. Morain École polytechnique MPRI cours /22 Practice Idea: suppose p N and we have a random f mod N s.t. f mod p is random. function f(x, N) return (x 2 + 1) mod N; end. function rho(n) 1. [initialization] x:=1; y:=1; g:=1; 2. [loop] while (g = 1) do x:=f(x, N); y:=f(f(y, N), N); g:=gcd(xy, N); endwhile; 3. return g; Choosing f : some choices are bad, as x x 2 et x x 2 2. Tables exist for given f s. Trick: compute gcd( i (x 2i x i ), N), using backtrack whenever needed. Improvements: reducing the number of evaluations of f, the number of comparisons (see Brent, Montgomery). Conjecture. RHO finds p N using O( p) iterations.
4 F. Morain École polytechnique MPRI cours /22 F. Morain École polytechnique MPRI cours /22 Theoretical results (1/4) Bach (2/4) Thm. (Bach, 1991) Proba RHO with f (x) = x finding p N after k iterations is at least ( k 2) when p goes to infinity. Sketch of the proof: define p + O(p 3/2 ) f 0 (X, Y) = X, f i+1 (X, Y) = f 2 i + Y, f 1 (X, Y) = X 2 + Y, f 2 (X, Y) = (X 2 + Y 2 ) + Y,... Divisor of N found by inspecting gcd(f 2i+1 (x, y) f i (x, y), N) for i = 0, 1, 2,.... Prop. (a) deg X (f i ) = 2 i ; (b) f i X 2i mod Y; (c) f i Y 2i Y mod X; (d) f i+j (X, Y) = f i (f j (X, Y), Y). (e) f i is absolutely irreducible (Eisentein s criterion). (f) f j f i = fj 1 2 f i 1 2 = (f j 1 + f i 1 )(f j 2 + f i 2 ) (f j i + X)(f j i X). (g) For all k 1 f l+n ± f l f l+kn ± f l. F. Morain École polytechnique MPRI cours /22 Bach (3/4) For i < j, ρ i,j is the unique poly in Z[X, Y] s.t. (a) ρ i,j is a monic (in Y) irreducible divisor of f j f i. (b) Let ω i,j denote a primitive (2 j 2 i ) root of unity. Then ρ i,j (ω i,j, 0) = 0. Proof: and and for i 1: f j f i X 2i (X 2j 2 i 1) mod Y, X 2j 2 i 1 = µ 2 j 2 i Φ µ (X). f j f 0 ρ 0,j d j,d j ρ 0,d f j 1 + f i 1 ρ i,j d j i,d j i ρ. i,i+d Conj. we have in fact equalities instead of divisibility., F. Morain École polytechnique MPRI cours /22 Bach (4/4) Thm1. f k f l factor over Z[X, Y]. Moreover, they are squarefree. Proof uses projectivization of f. Weil s thm: if f Z[X, Y] is absolutely irreducible of degree d, then N p the number of projective zeroes of f is s.t. ( ) d 1 p. N p (p + 1) 2 2 Thm2. Fix k 1. Choose x and y at random s.t. 0 x, y < p. Then, proba for some i, j < k, i j, f i (x, y) = f j (x, y) mod p is at least ( k ) 2 /p + O(1/p 3/2 ) as p tends to infinity. Proof: same as i < j < k and ρ i,j (x, y) 0 mod p. Use inclusionexclusion, Weil s inequality and Bézout s theorems. Same result for RHO to find p N.
5 F. Morain École polytechnique MPRI cours /22 F. Morain École polytechnique MPRI cours /22 IV. Pollard s p 1 method First phase Idea: assume p N and a is prime to p. Then Invented by Pollard in Williams: p + 1. Bach and Shallit: Φ k factoring methods. Shanks, Schnorr, Lenstra, etc.: quadratic forms. Lenstra (1985): ECM. Rem. Almost all the ideas invented for the classical p 1 can be transposed to the other methods. (p a p 1 1 and p N) p gcd(a p 1 1, N). Generalization: if R is known s.t. p 1 R, will yield a factor. gcd((a R mod N) 1, N) How do we find R? Only reasonable hope is that p 1 B! for some (small) B. In other words, p 1 is Bsmooth. Algorithm: R = p α B 1 p α = lcm(2,..., B 1 ). Rem. (usual trick) we compute gcd( k ((ar k 1) mod N), N). F. Morain École polytechnique MPRI cours /22 Second phase: the classical one F. Morain École polytechnique MPRI cours /22 Second phase: BSGS Let b = a R mod N and gcd(b, N) = 1. Hyp. p 1 = Qs with Q R and s prime, B 1 < s B 2. Test: is gcd(b s 1, N) > 1 for some s. s j = jth prime. In practice all s j+1 s j are small (Cramer s conjecture implies s j+1 s j (log B 2 ) 2 ). Precompute c δ b δ mod N for all possible δ (small); Compute next value with one multiplication b s j+1 = b s j c sj+1 s j mod N. Cost: O((log B 2 ) 2 ) + O(log s 1 ) + (π(b 2 ) π(b 1 )) multiplications +(π(b 2 ) π(b 1 )) gcd s. When B 2 B 1, π(b 2 ) dominates. Rem. We need a table of all primes < B 2 ; memory is O(B 2 ). Record. Nohara (66dd of , 2006; see Select w B 2, v 1 = B 1 /w, v 2 = B 2 /w. Write our prime s as s = vw u, with 0 u < w, v 1 v v 2. Lem. gcd(b s 1, N) > 1 iff gcd(b wv b u, N) > 1. Algorithm: 1. Precompute b u mod N for all 0 u < w. 2. Precompute all(b w ) v for all v 1 v v For all u and all v evaluate gcd(b vw b u, N). Number of multiplications: w + (v 2 v 1 ) + O(log 2 w) = O( B 2 ) Memory: O( B 2 ). Number of gcd: π(b 2 ) π(b 1 ).
6 Second phase: using fast polynomial arithmetic Algorithm: 1. Compute h(x) = 0 u<w (X bu ) Z/NZ[X] 2. Evaluate all h((b w ) v ) for all v 1 v v Evaluate all gcd(h(b wv ), N). Analysis: Step 1: O((log w)m pol (w)) operations (using a product tree). Step 2: O((log w)m int (log N)) for b w ; v 2 v 1 for (b w ) v ; multipoint evaluation on w points takes O((log w)m pol (w)). Rem. Evaluating h(x) along a geometric progression of length w takes O(w log w) operations (see MontgomerySilverman). ( ) Total cost: O((log w)m pol (w)) = O. B 0.5+o(1) 2 Trick: use gcd(u, w) = 1 and w = F. Morain École polytechnique MPRI cours /22 Second phase: using the birthday paradox Consider B = b mod p ; s := #B. If we draw s elements at random in B, then we have a collision (birthday paradox). Algorithm: build (b i ) with b 0 = b, and { b 2 b i+1 = i mod N with proba 1/2 b 2 i b mod N with proba 1/2. We gather r s values and compute where r i=1 j i (b i b j ) = Disc(P(X)) = i P(X) = r (X b i ). i=1 use fast polynomial operations again. P (b i ) F. Morain École polytechnique MPRI cours /22
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 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 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 informationFactoring & 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 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 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 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 informationModern 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 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 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 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 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 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 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 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 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 informationLecture 10: Distinct Degree Factoring
CS681 Computational Number Theory Lecture 10: Distinct Degree Factoring Instructor: Piyush P Kurur Scribe: Ramprasad Saptharishi Overview Last class we left of with a glimpse into distant degree factorization.
More informationThe Division Algorithm for Polynomials Handout Monday March 5, 2012
The Division Algorithm for Polynomials Handout Monday March 5, 0 Let F be a field (such as R, Q, C, or F p for some prime p. This will allow us to divide by any nonzero scalar. (For some of the following,
More informationPROBLEM SET 6: POLYNOMIALS
PROBLEM SET 6: POLYNOMIALS 1. introduction In this problem set we will consider polynomials with coefficients in K, where K is the real numbers R, the complex numbers C, the rational numbers Q or any other
More informationCHAPTER 3 THE NEW MMP CRYPTO SYSTEM. mathematical problems Hidden Root Problem, Discrete Logarithm Problem and
79 CHAPTER 3 THE NEW MMP CRYPTO SYSTEM In this chapter an overview of the new Mixed Mode Paired cipher text Cryptographic System (MMPCS) is given, its three hard mathematical problems are explained, and
More informationFactorization Algorithms for Polynomials over Finite Fields
Degree Project Factorization Algorithms for Polynomials over Finite Fields Sajid Hanif, Muhammad Imran 20110503 Subject: Mathematics Level: Master Course code: 4MA11E Abstract Integer factorization is
More informationChapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.
Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize
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 informationWinter Camp 2011 Polynomials Alexander Remorov. Polynomials. Alexander Remorov alexanderrem@gmail.com
Polynomials Alexander Remorov alexanderrem@gmail.com Warmup Problem 1: Let f(x) be a quadratic polynomial. Prove that there exist quadratic polynomials g(x) and h(x) such that f(x)f(x + 1) = g(h(x)).
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 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 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 informationIs n a prime number? Nitin Saxena. Turku, May Centrum voor Wiskunde en Informatica Amsterdam
Is n a prime number? Nitin Saxena Centrum voor Wiskunde en Informatica Amsterdam Turku, May 2007 Nitin Saxena (CWI, Amsterdam) Is n a prime number? Turku, May 2007 1 / 36 Outline 1 The Problem 2 The High
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 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 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 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 informationAlgebra for Digital Communication
EPFL  Section de Mathématiques Algebra for Digital Communication Fall semester 2008 Solutions for exercise sheet 1 Exercise 1. i) We will do a proof by contradiction. Suppose 2 a 2 but 2 a. We will obtain
More informationThe Laws of Cryptography Cryptographers Favorite Algorithms
2 The Laws of Cryptography Cryptographers Favorite Algorithms 2.1 The Extended Euclidean Algorithm. The previous section introduced the field known as the integers mod p, denoted or. Most of the field
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 informationMOP 2007 Black Group Integer Polynomials Yufei Zhao. Integer Polynomials. June 29, 2007 Yufei Zhao yufeiz@mit.edu
Integer Polynomials June 9, 007 Yufei Zhao yufeiz@mit.edu We will use Z[x] to denote the ring of polynomials with integer coefficients. We begin by summarizing some of the common approaches used in dealing
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 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 informationMathematical induction & Recursion
CS 441 Discrete Mathematics for CS Lecture 15 Mathematical induction & Recursion Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Proofs Basic proof methods: Direct, Indirect, Contradiction, By Cases,
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 informationHYPERELLIPTIC CURVE METHOD FOR FACTORING INTEGERS. 1. Thoery and Algorithm
HYPERELLIPTIC CURVE METHOD FOR FACTORING INTEGERS WENHAN WANG 1. Thoery and Algorithm The idea of the method using hyperelliptic curves to factor integers is similar to the elliptic curve factoring method.
More informationThe Mixed Binary Euclid Algorithm
Electronic Notes in Discrete Mathematics 35 (009) 169 176 www.elsevier.com/locate/endm The Mixed Binary Euclid Algorithm Sidi Mohamed Sedjelmaci LIPN CNRS UMR 7030 Université ParisNord Av. J.B. Clément,
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 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 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 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 informationLecture 8: Applications of Quantum Fourier transform
Department of Physical Sciences, University of Helsinki http://theory.physics.helsinki.fi/ quantumgas/ p. 1/25 Quantum information and computing Lecture 8: Applications of Quantum Fourier transform JaniPetri
More informationFactoring Polynomials
Factoring Polynomials Sue Geller June 19, 2006 Factoring polynomials over the rational numbers, real numbers, and complex numbers has long been a standard topic of high school algebra. With the advent
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 informationUnique Factorization
Unique Factorization Waffle Mathcamp 2010 Throughout these notes, all rings will be assumed to be commutative. 1 Factorization in domains: definitions and examples In this class, we will study the phenomenon
More informationUSING 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 email: zhangzhx@mail.ahwhptt.net.cn (Submitted
More informationFACTORING SPARSE POLYNOMIALS
FACTORING SPARSE POLYNOMIALS Theorem 1 (Schinzel): Let r be a positive integer, and fix nonzero integers a 0,..., a r. Let F (x 1,..., x r ) = a r x r + + a 1 x 1 + a 0. Then there exist finite sets S
More informationTheorem (The division theorem) Suppose that a and b are integers with b > 0. There exist unique integers q and r so that. a = bq + r and 0 r < b.
Theorem (The division theorem) Suppose that a and b are integers with b > 0. There exist unique integers q and r so that a = bq + r and 0 r < b. We re dividing a by b: q is the quotient and r is the remainder,
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 informationPolynomials Classwork
Polynomials Classwork What Is a Polynomial Function? Numerical, Analytical and Graphical Approaches Anatomy of an n th degree polynomial function Def.: A polynomial function of degree n in the vaiable
More informationThe van Hoeij Algorithm for Factoring Polynomials
The van Hoeij Algorithm for Factoring Polynomials Jürgen Klüners Abstract In this survey we report about a new algorithm for factoring polynomials due to Mark van Hoeij. The main idea is that the combinatorial
More informationH/wk 13, Solutions to selected problems
H/wk 13, Solutions to selected problems Ch. 4.1, Problem 5 (a) Find the number of roots of x x in Z 4, Z Z, any integral domain, Z 6. (b) Find a commutative ring in which x x has infinitely many roots.
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 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 informationThe Elliptic Curve Factorization method. Rencontres Arithmétique de l Informatique Mathématique, 23 jan p. 1/28
. The Elliptic Curve Factorization method Ä ÙÖ ÒØ ÓÙ Ð ÙÖ ÒØ ÓÑ Ø ºÒ Ø ÄÇÊÁ ÍÒ Ú Ö Ø À ÒÖ ÈÓ Ò Ö Æ ÒÝ Á Rencontres Arithmétique de l Informatique Mathématique, 23 jan. 2007 p. 1/28 Outline 1. Introduction
More informationPrime Numbers and Irreducible Polynomials
Prime Numbers and Irreducible Polynomials M. Ram Murty The similarity between prime numbers and irreducible polynomials has been a dominant theme in the development of number theory and algebraic geometry.
More informationQuotient 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 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 information7. Some irreducible polynomials
7. Some irreducible polynomials 7.1 Irreducibles over a finite field 7.2 Worked examples Linear factors x α of a polynomial P (x) with coefficients in a field k correspond precisely to roots α k [1] of
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 information4. Number Theory (Part 2)
4. Number Theory (Part 2) Terence Sim Mathematics is the queen of the sciences and number theory is the queen of mathematics. Reading Sections 4.8, 5.2 5.4 of Epp. Carl Friedrich Gauss, 1777 1855 4.3.
More informationCLASS 3, GIVEN ON 9/27/2010, FOR MATH 25, FALL 2010
CLASS 3, GIVEN ON 9/27/2010, FOR MATH 25, FALL 2010 1. Greatest common divisor Suppose a, b are two integers. If another integer d satisfies d a, d b, we call d a common divisor of a, b. Notice that as
More informationAPPLICATIONS 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 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 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 informationNUMBER THEORY AMIN WITNO
NUMBER THEORY AMIN WITNO ii Number Theory Amin Witno Department of Basic Sciences Philadelphia University JORDAN 19392 Originally written for Math 313 students at Philadelphia University in Jordan, this
More informationIndex Calculation Attacks on RSA Signature and Encryption
Index Calculation Attacks on RSA Signature and Encryption JeanSébastien Coron 1, Yvo Desmedt 2, David Naccache 1, Andrew Odlyzko 3, and Julien P. Stern 4 1 Gemplus Card International {jeansebastien.coron,david.naccache}@gemplus.com
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 informationOn 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 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 informationIs this number prime? Berkeley Math Circle Kiran Kedlaya
Is this number prime? Berkeley Math Circle 2002 2003 Kiran Kedlaya Given a positive integer, how do you check whether it is prime (has itself and 1 as its only two positive divisors) or composite (not
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 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 informationcalculating the result modulo 3, as follows: p(0) = 0 3 + 0 + 1 = 1 0,
Homework #02, due 1/27/10 = 9.4.1, 9.4.2, 9.4.5, 9.4.6, 9.4.7. Additional problems recommended for study: (9.4.3), 9.4.4, 9.4.9, 9.4.11, 9.4.13, (9.4.14), 9.4.17 9.4.1 Determine whether the following polynomials
More informationFurther linear algebra. Chapter I. Integers.
Further linear algebra. Chapter I. Integers. Andrei Yafaev Number theory is the theory of Z = {0, ±1, ±2,...}. 1 Euclid s algorithm, Bézout s identity and the greatest common divisor. We say that a Z divides
More informationThe Chebotarev Density Theorem
The Chebotarev Density Theorem Hendrik Lenstra 1. Introduction Consider the polynomial f(x) = X 4 X 2 1 Z[X]. Suppose that we want to show that this polynomial is irreducible. We reduce f modulo a prime
More information1 Die hard, once and for all
ENGG 2440A: Discrete Mathematics for Engineers Lecture 4 The Chinese University of Hong Kong, Fall 2014 6 and 7 October 2014 Number theory is the branch of mathematics that studies properties of the integers.
More informationElementary Number Theory and Methods of Proof. CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.
Elementary Number Theory and Methods of Proof CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.edu/~cse215 1 Number theory Properties: 2 Properties of integers (whole
More informationALGEBRAIC SYSTEMS, FALL Notation
ALGEBRAIC SYSTEMS, FALL 2012 TODD COCHRANE 1. Notation N = {1, 2, 3, 4, 5,... } = Natural numbers Z = {0, ±1, ±, 2, ±3,... } = Integers E = {0, ±2, ±4, ±6,... } = Even integers O = {±1, ±3, ±5,... } =
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 informationSolutions to Practice Problems
Solutions to Practice Problems March 205. Given n = pq and φ(n = (p (q, we find p and q as the roots of the quadratic equation (x p(x q = x 2 (n φ(n + x + n = 0. The roots are p, q = 2[ n φ(n+ ± (n φ(n+2
More informationData Structure. Lecture 3
Data Structure Lecture 3 Data Structure Formally define Data structure as: DS describes not only set of objects but the ways they are related, the set of operations which may be applied to the elements
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 informationThe finite field with 2 elements The simplest finite field is
The finite field with 2 elements The simplest finite field is GF (2) = F 2 = {0, 1} = Z/2 It has addition and multiplication + and defined to be 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 0 0 0 = 0 0 1 = 0
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 informationChapter Objectives. Chapter 9. Sequential Search. Search Algorithms. Search Algorithms. Binary Search
Chapter Objectives Chapter 9 Search Algorithms Data Structures Using C++ 1 Learn the various search algorithms Explore how to implement the sequential and binary search algorithms Discover how the sequential
More informationSection 4.2: The Division Algorithm and Greatest Common Divisors
Section 4.2: The Division Algorithm and Greatest Common Divisors The Division Algorithm The Division Algorithm is merely long division restated as an equation. For example, the division 29 r. 20 32 948
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 informationFPGA and ASIC Implementation of Rho and P1 Methods of Factoring. Master s Thesis Presentation Ramakrishna Bachimanchi Director: Dr.
FPGA and ASIC Implementation of Rho and P1 Methods of Factoring Master s Thesis Presentation Ramakrishna Bachimanchi Director: Dr. Kris Gaj Contents Introduction Background Hardware Architecture FPGA
More informationExercises with solutions (1)
Exercises with solutions (). Investigate the relationship between independence and correlation. (a) Two random variables X and Y are said to be correlated if and only if their covariance C XY is not equal
More informationChapter 9. Computational Number Theory. 9.1 The basic groups Integers mod N Groups
Chapter 9 Computational Number Theory 9.1 The basic groups We let Z = {..., 2, 1,0,1,2,...} denote the set of integers. We let Z + = {1,2,...} denote the set of positive integers and N = {0,1,2,...} the
More informationLecture 3: Finding integer solutions to systems of linear equations
Lecture 3: Finding integer solutions to systems of linear equations Algorithmic Number Theory (Fall 2014) Rutgers University Swastik Kopparty Scribe: Abhishek Bhrushundi 1 Overview The goal of this lecture
More informationPrime Numbers. Chapter Primes and Composites
Chapter 2 Prime Numbers The term factoring or factorization refers to the process of expressing an integer as the product of two or more integers in a nontrivial way, e.g., 42 = 6 7. Prime numbers are
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 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 information