Progress Report— 11/06 宗慶.

Slides:



Advertisements
Similar presentations
Notes 6.6 Fundamental Theorem of Algebra
Advertisements

Mathematics of Cryptography Part II: Algebraic Structures
Factoring of Large Numbers using Number Field Sieve Matrix Step Chandana Anand, Arman Gungor, and Kimberly A. Thomas ECE 646 Fall 2006.
Cryptography and Network Security
Rational Root Theorem.
ENGG2013 Unit 11 Row-Rank Feb,
A Creative Way of Breaking RSA Azeem Jiva. Overview ● What is RSA? – Public Key Algorithm – Is it secure? ● Ways to break RSA – Discover the Public Key.
Computing the Rational Univariate Reduction by Sparse Resultants Koji Ouchi, John Keyser, J. Maurice Rojas Department of Computer Science, Mathematics.
Forward Error Correction Steven Marx CSC45712/04/2001.
Monica Garika Chandana Guduru. METHODS TO SOLVE LINEAR SYSTEMS Direct methods Gaussian elimination method LU method for factorization Simplex method of.
Dirichlet’s Theorem for Polynomial Rings Lior Bary-Soroker, School of Mathematical Sciences, Sackler Faculty of Exact Sciences, Tel-Aviv University 1.
Dan Boneh Intro. Number Theory Modular e’th roots Online Cryptography Course Dan Boneh.
9.4 – Solving Quadratic Equations By Completing The Square
Section 8.3 – Systems of Linear Equations - Determinants Using Determinants to Solve Systems of Equations A determinant is a value that is obtained from.
Finite Mathematics Dr. Saeid Moloudzadeh Using Matrices to Solve Systems of Equations 1 Contents Algebra Review Functions and Linear Models.
Reduced Row Echelon Form Matrices and the Calculator.
An approach for solving the Helmholtz Equation on heterogeneous platforms An approach for solving the Helmholtz Equation on heterogeneous platforms G.
Mathematics of Cryptography Modular Arithmetic, Congruence,
Cryptography and Network Security
1.1.2 INTRODUCTION TO SYSTEMS OF LINEAR EQUATIONS Chapter 1: Systems of Linear Equations and Matrices SWBAT: Redefine algebraic operations as Elementary.
Pure and Applied Number Theory School Cheolmin Park NIMS.
Copyright, Yogesh Malhotra, PhD, 2013www.yogeshmalhotra.com SPECIAL PURPOSE FACTORING ALGORITHMS Special Purpose Factoring Algorithms For special class.
Finite Mathematics Dr. Saeid Moloudzadeh Solving Polynomial Equations 1 Contents Algebra Review Functions and Linear Models Systems of.
Monoids, Groups, Rings, Fields
Algebra 1 Notes: Lesson 8-5: Adding and Subtracting Polynomials.
Finite Mathematics Dr. Saeid Moloudzadeh Exponential Functions and Models 1 Contents Algebra Review Functions and Linear Models Systems.
Chapter 3 Polynomial and Rational Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc Zeros of Polynomial Functions.
6.6 The Fundamental Theorem of Algebra
7.5 Zeros of Polynomial Functions Objectives: Use the Rational Root Theorem and the Complex Conjugate Root Theorem. Use the Fundamental Theorem to write.
Factorization of a 768-bit RSA modulus Jung Daejin Lee Sangho.
5.6.1 – Square Root Method. Recall, we solved “quadratic equations” when we set a polynomial equation equal to 0 Example. x 2 + 5x + 6 = 0.
Data Security and Encryption (CSE348) 1. Lecture # 12 2.
Linear Algebra Libraries: BLAS, LAPACK, ScaLAPACK, PLASMA, MAGMA
Solving Polynomial Equations by Factoring Factoring by grouping Ex. 1. Solve:
Solve by factoring. x² = - 4 – 5x 2,. Solve by factoring. n² = -30 – 11n -4 and -1.
Find the exact value. 1.) √49 2.) - √ Use a calculator to approximate the value of √(82/16) to the nearest tenth.
Finite Mathematics Dr. Saeid Moloudzadeh Multiplying and Factoring Algebraic Expressions 1 Contents Algebra Review Functions and Linear.
Fundamental Theorem of Algebra Every polynomial function of positive degree with complex coefficients has at least one complex zero.
Section 1.7 Linear Independence and Nonsingular Matrices
Holt McDougal Algebra 2 Fundamental Theorem of Algebra How do we use the Fundamental Theorem of Algebra and its corollary to write a polynomial equation.
Lecture 2-3 Basic Number Theory and Algebra. In modern cryptographic systems, the messages are represented by numerical values prior to being encrypted.
Linear Algebra Libraries: BLAS, LAPACK, ScaLAPACK, PLASMA, MAGMA Shirley Moore CPS5401 Fall 2013 svmoore.pbworks.com November 12, 2012.
Polynomial P(x) Linear Factors Solutions of P(x)=0 Zeros of P(x) P(x) = 0.
Algebra Review. Systems of Equations Review: Substitution Linear Combination 2 Methods to Solve:
Do Now Determine which numbers in the set are natural, whole, integers, rational and irrational -9, -7/2, 5, 2/3, √2, 0, 1, -4, 2, -11 Evaluate |x + 2|,
Progress Report—11/13 宗慶. Problem Statement Find kernels of large and sparse linear systems over GF(2)
Kerimbekova M.S. MF-12 Equation. Equation is In mathematics, an equation is an equality containing one or more variables. The first use of an equals sign,
Page : 1 bfolieq.drw Technical University of Braunschweig IDA: Institute of Computer and Network Engineering  W. Adi 2011 Lecture-5 Mathematical Background:
S5.40. Module Structure 30% practical tests / 70% written exam 3h lectures / week (except reading week) 3 x 2h of computer labs (solving problems practicing.
A survey of Exascale Linear Algebra Libraries for Data Assimilation
Review Problems Matrices
Solving Quadratic Equations by the Complete the Square Method
FE Exam Tutorial
Warm - Up Perform the operation and write the result in standard form
An Introduction to Maple
Appendix A.5 Solving Equations.
Parallel Quadratic Sieve
P A R A L L E L C O M P U T I N G L A B O R A T O R Y
1. Use the quadratic formula to find all real zeros of the second-degree polynomial
Warm-up Complete this as a group on you Board. You have 15 minutes
University of South Florida and Eindhoven University of Technology
Rational Root Theorem.
Quadratic Equations.
Factoring RSA Moduli: Current State of the Art J
MA5242 Wavelets Lecture 1 Numbers and Vector Spaces
Mathematical Background for Cryptography
Lecture 2-3 Basic Number Theory and Algebra
Mathematical Background: Extension Finite Fields
Presentation transcript:

Progress Report— 11/06 宗慶

Major Steps of GNFS Polynomial Selection Sieving Matrix Reduction Find ‘good’ polynomials to speed up sieving Sieving Find sufficiently many relations to produce a matrix Matrix Reduction Find linear dependencies mod 2 among the rows of the matrix Square Root Calculate a square root in the number field for each dependency, until the factorization is found

Problem Statement Find kernels of large and sparse linear systems over GF(2)

Matrix Reduction A problem of solving a large sparse system of homogeneous linear equations over GF(2) Very large…... Block Wiedemann or Block Lanczos algorithms are popular For NFS, if there are k primes in the rational factor base, l first degree prime ideals of Z[θ] in the algebraic factor base, and m first degree prime ideals in the quadratic character base, then each “number” sieved will be comprised of (1+k+l+m) binary bits, determined by the sign and the respective bases.

Matrix Reduction (cont’d) According to “A Kilobit SNFS Factorization, ASIACRYPT’07” Did this step in parallel as four independent jobs on different clusters at various location by using block Wiedemann algorithm. 66718354X66718154 matrix were sought

Block Wiedemann Algorithm A Parallel GNFS Integrated with the Block Wiedemann’s Algorithm for Integer Factorization (DASC’06)

Why Cell High memory bandwidth SIMD DMA 陳君明老師的成果 IBM P595 with 64 cores & 256GB memory 但若使用 32 核心,由於某些硬體因素,執行效率甚差 SIMD Block edition + SIMD DMA Many TOP clusters are composed of Cell.

Resource GMP BLAS Prof. 陳君明’s openMP block Wiedemann source code Sage A free open-source mathematics software system licensed under the GPL. It combines the power of many existing open-source packages into a common Python-based interface. LinBox A C++ template library for exact, high-performance linear algebra computation with dense, sparse, and structured matrices over the integers and over finite fields. Givaro A C++ library for arithmetic and algebraic computations GMP BLAS Cell Blas library