COMS W4995 Introduction to Cryptography September 29, Lecture 8: Number Theory
|
|
- Allison Pearson
- 5 years ago
- Views:
Transcription
1 COMS W4995 Introduction to Cryptography September 29, 2005 Lecture 8: Number Theory Lecturer: Tal Malkin Scribes: Elli Androulaki, Mohit Vazirani Summary This lecture focuses on some basic Number Theory. After proving basic theorems related to prime numbers, this chapter is referred to the extended Euclid s algorithm, Modular Arithmetic, and the groups Z n, Z n and their properties. Recall from the previous lecture... In the last lecture we defined (weak) one-way functions (OWFs) - functions easy to compute and difficult to invert - and collections of One-Way Functions. We also introduced the Factoring Assumption and showed a OWF based on it. 1 Prime-Number Basic Theorems In the last lecture we have mentioned the following theorem: Theorem 1 If the factoring assumption holds, then F = {f i : D i {0, 1} } i I where I = {n N : n is even} and D i = {p, q : p = q = i/2 and p, q are primes} and f i (p, q) = p q is a collection of OWF. In the previous lecture we proved that three of the four conditions that are required to show that F is a collection of OWFs are satisfied. Now we have to prove that condition (2) is also satisfied. Namely, we have to prove that it is feasible in polynomial time to sample random pairs of i 2-bit primes. This is easily proven using the following claim. Claim 2 Choosing a k-bit random integer that is a prime can be done in polynomial time in k. 1
2 Proof: In order to prove the above claim, we state the following facts and corollary and later come up with an algorithm that chooses a k-bit prime integer at random in polynomial time in k. Fact 3 Primality testing can be done in polynomial time. Fact 4 (Prime Number Theorem, Hadamard, de la Vallée-P. 1896) π(x) = O( x x i.e. the number π(x) of primes p x is roughly logx as x Corollary 5 Since the biggest k-bit number is (2 k 1) there are about Algorithm 1 A PPT algorithm to find a random k-bit prime. 1. Choose random k-bit number P {2 k 1,..., 2 k 1} 2. Run a primality test on P. 3. If P is prime then output P, else goto 1. 2 k kln2 k-bit primes. What we need to prove is that the above algorithm is PPT. Because of Corollary 4, the test in step 2 succeeds with probability Ω( 1 k ). Therefore, the expected number of iterations in the algorithm is O(k), namely its expected running time is polynomial. 1 logx ) 2 Review of Number Theory Fact 6 (Extended Euclid s Algorithm) For every two integers x, y two integers a, b such that ax+by = gcd(x, y). Moreover, there is a polynomial time algorithm that computes a, b. This algorithm is called the Extended Euclid s Algorithm (EEA). Following is an example of the application of the EEA. Example 1 gcd(72, 15) 72 = = = = = 15 ( ) = ( 1) 72 gcd = 3 a = 1, b = 5 1 In order to get an algorithm that always runs in polynomial time, we can fix a polynomial bound on the number of iterations, after which the algorithm will halt and output fail. The polynomial can be set large enough so that this happens with negligible probability. 2
3 Modular Arithmetic x y (mod n) if n (x y) Example (mod 12) 3 27 (mod 15) Definition 1 x mod n Unique x {0, 1, 2, 3,..., n 1} s.t. x x (mod n) 2.1 Operations (addition, subtraction, multiplication) x y (mod n) a (x + a y + a (mod n)) x y (mod n) a (x a y a (mod n)) x y (mod n) = a (a x a y (mod n)) Note: The following example demonstrates that we cannot divide in general. Example mod mod Basic Definitions Definition 2 Z n = {0, 1, 2,..., n 1} Example 4 Z 6 = {0, 1, 2, 3, 4, 5} Z 15 = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14} Definition 3 Z n = {x Z n gcd(x, n) = 1} 3
4 Example 5 Z 6 = {1, 5} = {1, 2, 4, 7, 8, 11, 13, 14} Z 15 Fact 7 Z n is the set of all elements in Z n that have a multiplicative inverse. Namely, Z n = {x Z n a : x a = 1 (mod n)} Example = 1 mod = 1 mod 6 Thus, 1, 5 Z n Because of Fact 6, if a Z n, then we can divide by a, namely multiply both sides by a 1 mod n. Definition 4 A group is a set G with a binary operation such that G is closed under the operation : a, b G = a b G. associativity: a (b c) = (a b) c identity: I G : a I = I a = a a G inverse: a G a 1 G : a a 1 = I All of the groups we consider will also be commutative, namely a,b G a b = b a. Example 7 The integers are a group under addition ( + ). The rationals are a group under multiplication ( * ). Fact 8 Z n is a group under + (mod n). The identity element of this group for this operation is element 0. Z n is a group under * (mod n). The identity element of this group for this operation is element 1. Fact 9 For any finite group G, and any x G, if we apply operation in x G times, then we get the Identity element of. x x x x... x = I (Identity element) G times 4
5 Example 8 (Z 15, +(mod n)) ( Z 15 = 15) (mod 15) In general, 15 X 0 mod 15 X (Z 15, (mod n)) ( Z 15 = 8) 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 1 (mod 15) In general, X 8 1 mod 15 X Note: From this we can simplify many tedious exponential calculations, as we know for example that Z 15 = 8, we can immediately calculate = 1 mod 15. Definition 5 Euler s Totient Function Φ(n) = Z n Fact 10 if p is prime, then Φ(p) = p 1 Fact 11 if n = p α i i for distinct primes p i : Φ(n) = p α i 1 i (p i 1) Example 9 15 = 5 * 3 Φ(15) = Φ(5) Φ(3) = (5 1) (3 1) = 4 2 = 8 Fact 9 implies the following theorem. Theorem 12 x Z n, x Φ(n) 1 (mod n) Special case for primes (Fermat s little theorem): If p is prime then x Z n, x p 1 1 mod p 5
1 Rabin Squaring Function and the Factoring Assumption
COMS W461 Introduction to Cryptography October 11, 005 Lecture 11: Introduction to Cryptography Lecturer: Tal Malkin Scribes: Kate McCarthy, Adam Vartanian Summary In this lecture we will prove that Rabin
More informationNumbers. Çetin Kaya Koç Winter / 18
Çetin Kaya Koç http://koclab.cs.ucsb.edu Winter 2016 1 / 18 Number Systems and Sets We represent the set of integers as Z = {..., 3, 2, 1,0,1,2,3,...} We denote the set of positive integers modulo n as
More informationCPSC 467b: Cryptography and Computer Security
CPSC 467b: Cryptography and Computer Security Michael J. Fischer Lecture 8 February 1, 2012 CPSC 467b, Lecture 8 1/42 Number Theory Needed for RSA Z n : The integers mod n Modular arithmetic GCD Relatively
More informationTopics in Cryptography. Lecture 5: Basic Number Theory
Topics in Cryptography Lecture 5: Basic Number Theory Benny Pinkas page 1 1 Classical symmetric ciphers Alice and Bob share a private key k. System is secure as long as k is secret. Major problem: generating
More informationCPSC 467: Cryptography and Computer Security
CPSC 467: Cryptography and Computer Security Michael J. Fischer Lecture 9 September 30, 2015 CPSC 467, Lecture 9 1/47 Fast Exponentiation Algorithms Number Theory Needed for RSA Elementary Number Theory
More informationApplied Cryptography and Computer Security CSE 664 Spring 2018
Applied Cryptography and Computer Security Lecture 12: Introduction to Number Theory II Department of Computer Science and Engineering University at Buffalo 1 Lecture Outline This time we ll finish the
More informationNumber Theory. CSS322: Security and Cryptography. Sirindhorn International Institute of Technology Thammasat University CSS322. Number Theory.
CSS322: Security and Cryptography Sirindhorn International Institute of Technology Thammasat University Prepared by Steven Gordon on 29 December 2011 CSS322Y11S2L06, Steve/Courses/2011/S2/CSS322/Lectures/number.tex,
More informationECEN 5022 Cryptography
Elementary Algebra and Number Theory University of Colorado Spring 2008 Divisibility, Primes Definition. N denotes the set {1, 2, 3,...} of natural numbers and Z denotes the set of integers {..., 2, 1,
More informationLecture 11 - Basic Number Theory.
Lecture 11 - Basic Number Theory. Boaz Barak October 20, 2005 Divisibility and primes Unless mentioned otherwise throughout this lecture all numbers are non-negative integers. We say that a divides b,
More informationCS 290G (Fall 2014) Introduction to Cryptography Oct 21st, Lecture 5: RSA OWFs
CS 290G (Fall 2014) Introduction to Cryptography Oct 21st, 2014 Instructor: Rachel Lin 1 Recap Lecture 5: RSA OWFs Scribe: Harichandan Pulagam In the last class, we continued the discussion of One-Way
More informationOutline. Number Theory and Modular Arithmetic. p-1. Definition: Modular equivalence a b [mod n] (a mod n) = (b mod n) n (a-b)
Great Theoretical Ideas In CS Victor Adamchik CS - Lecture Carnegie Mellon University Outline Number Theory and Modular Arithmetic p- p Working modulo integer n Definitions of Z n, Z n Fundamental lemmas
More informationBasic Algorithms in Number Theory
Basic Algorithms in Number Theory Algorithmic Complexity... 1 Basic Algorithms in Number Theory Francesco Pappalardi Discrete Logs, Modular Square Roots & Euclidean Algorithm. July 20 th 2010 Basic Algorithms
More informationNumber Theory and Group Theoryfor Public-Key Cryptography
Number Theory and Group Theory for Public-Key Cryptography TDA352, DIT250 Wissam Aoudi Chalmers University of Technology November 21, 2017 Wissam Aoudi Number Theory and Group Theoryfor Public-Key Cryptography
More informationQ 2.0.2: If it s 5:30pm now, what time will it be in 4753 hours? Q 2.0.3: Today is Wednesday. What day of the week will it be in one year from today?
2 Mod math Modular arithmetic is the math you do when you talk about time on a clock. For example, if it s 9 o clock right now, then it ll be 1 o clock in 4 hours. Clearly, 9 + 4 1 in general. But on a
More informationIntroduction to Cryptology. Lecture 19
Introduction to Cryptology Lecture 19 Announcements HW6 due today HW7 due Thursday 4/20 Remember to sign up for Extra Credit Agenda Last time More details on AES/DES (K/L 6.2) Practical Constructions of
More informationCIS 551 / TCOM 401 Computer and Network Security
CIS 551 / TCOM 401 Computer and Network Security Spring 2008 Lecture 15 3/20/08 CIS/TCOM 551 1 Announcements Project 3 available on the web. Get the handout in class today. Project 3 is due April 4th It
More informationOutline. Some Review: Divisors. Common Divisors. Primes and Factors. b divides a (or b is a divisor of a) if a = mb for some m
Outline GCD and Euclid s Algorithm AIT 682: Network and Systems Security Topic 5.1 Basic Number Theory -- Foundation of Public Key Cryptography Modulo Arithmetic Modular Exponentiation Discrete Logarithms
More informationOutline. AIT 682: Network and Systems Security. GCD and Euclid s Algorithm Modulo Arithmetic Modular Exponentiation Discrete Logarithms
AIT 682: Network and Systems Security Topic 5.1 Basic Number Theory -- Foundation of Public Key Cryptography Instructor: Dr. Kun Sun Outline GCD and Euclid s Algorithm Modulo Arithmetic Modular Exponentiation
More informationCIS 6930/4930 Computer and Network Security. Topic 5.1 Basic Number Theory -- Foundation of Public Key Cryptography
CIS 6930/4930 Computer and Network Security Topic 5.1 Basic Number Theory -- Foundation of Public Key Cryptography 1 Review of Modular Arithmetic 2 Remainders and Congruency For any integer a and any positive
More informationIntroduction to Cryptography. Lecture 6
Introduction to Cryptography Lecture 6 Benny Pinkas page 1 Public Key Encryption page 2 Classical symmetric ciphers Alice and Bob share a private key k. System is secure as long as k is secret. Major problem:
More informationAn integer p is prime if p > 1 and p has exactly two positive divisors, 1 and p.
Chapter 6 Prime Numbers Part VI of PJE. Definition and Fundamental Results Definition. (PJE definition 23.1.1) An integer p is prime if p > 1 and p has exactly two positive divisors, 1 and p. If n > 1
More informationLecture notes: Algorithms for integers, polynomials (Thorsten Theobald)
Lecture notes: Algorithms for integers, polynomials (Thorsten Theobald) 1 Euclid s Algorithm Euclid s Algorithm for computing the greatest common divisor belongs to the oldest known computing procedures
More informationCSC 474 Network Security. Outline. GCD and Euclid s Algorithm. GCD and Euclid s Algorithm Modulo Arithmetic Modular Exponentiation Discrete Logarithms
Computer Science CSC 474 Network Security Topic 5.1 Basic Number Theory -- Foundation of Public Key Cryptography CSC 474 Dr. Peng Ning 1 Outline GCD and Euclid s Algorithm Modulo Arithmetic Modular Exponentiation
More informationMa/CS 6a Class 4: Primality Testing
Ma/CS 6a Class 4: Primality Testing By Adam Sheffer Reminder: Euler s Totient Function Euler s totient φ(n) is defined as follows: Given n N, then φ n = x 1 x < n and GCD x, n = 1. In more words: φ n is
More informationBasic elements of number theory
Cryptography Basic elements of number theory Marius Zimand By default all the variables, such as a, b, k, etc., denote integer numbers. Divisibility a 0 divides b if b = a k for some integer k. Notation
More informationBasic elements of number theory
Cryptography Basic elements of number theory Marius Zimand 1 Divisibility, prime numbers By default all the variables, such as a, b, k, etc., denote integer numbers. Divisibility a 0 divides b if b = a
More informationSome Facts from Number Theory
Computer Science 52 Some Facts from Number Theory Fall Semester, 2014 These notes are adapted from a document that was prepared for a different course several years ago. They may be helpful as a summary
More informationChapter 6 Randomization Algorithm Theory WS 2012/13 Fabian Kuhn
Chapter 6 Randomization Algorithm Theory WS 2012/13 Fabian Kuhn Randomization Randomized Algorithm: An algorithm that uses (or can use) random coin flips in order to make decisions We will see: randomization
More information2.3 In modular arithmetic, all arithmetic operations are performed modulo some integer.
CHAPTER 2 INTRODUCTION TO NUMBER THEORY ANSWERS TO QUESTIONS 2.1 A nonzero b is a divisor of a if a = mb for some m, where a, b, and m are integers. That is, b is a divisor of a if there is no remainder
More informationCHAPTER 6. Prime Numbers. Definition and Fundamental Results
CHAPTER 6 Prime Numbers Part VI of PJE. Definition and Fundamental Results 6.1. Definition. (PJE definition 23.1.1) An integer p is prime if p > 1 and the only positive divisors of p are 1 and p. If n
More informationCS483 Design and Analysis of Algorithms
CS483 Design and Analysis of Algorithms Lectures 2-3 Algorithms with Numbers Instructor: Fei Li lifei@cs.gmu.edu with subject: CS483 Office hours: STII, Room 443, Friday 4:00pm - 6:00pm or by appointments
More informationCongruence Classes. Number Theory Essentials. Modular Arithmetic Systems
Cryptography Introduction to Number Theory 1 Preview Integers Prime Numbers Modular Arithmetic Totient Function Euler's Theorem Fermat's Little Theorem Euclid's Algorithm 2 Introduction to Number Theory
More informationBasic Algorithms in Number Theory
Basic Algorithms in Number Theory Algorithmic Complexity... 1 Basic Algorithms in Number Theory Francesco Pappalardi #2-b - Euclidean Algorithm. September 2 nd 2015 SEAMS School 2015 Number Theory and
More informationNumber Theory. Modular Arithmetic
Number Theory The branch of mathematics that is important in IT security especially in cryptography. Deals only in integer numbers and the process can be done in a very fast manner. Modular Arithmetic
More informationLecture 14: Hardness Assumptions
CSE 594 : Modern Cryptography 03/23/2017 Lecture 14: Hardness Assumptions Instructor: Omkant Pandey Scribe: Hyungjoon Koo, Parkavi Sundaresan 1 Modular Arithmetic Let N and R be set of natural and real
More informationNumber Theory A focused introduction
Number Theory A focused introduction This is an explanation of RSA public key cryptography. We will start from first principles, but only the results that are needed to understand RSA are given. We begin
More informationCSC 474 Information Systems Security
CSC Information Systems Security Topic. Basic Number Theory CSC Dr. Peng Ning Basic Number Theory We are talking about integers! Divisor We say that b divides a if a = mb for some m, denoted b a. b is
More informationComputational Hardness
Chapter 2 Computational Hardness 2.1 Efficient Computation and Efficient Adversaries We start by formalizing what it means to compute a function. Definition 19.1 (Algorithm). An algorithm is a deterministic
More informationcse 311: foundations of computing Fall 2015 Lecture 12: Primes, GCD, applications
cse 311: foundations of computing Fall 2015 Lecture 12: Primes, GCD, applications n-bit unsigned integer representation Represent integer x as sum of powers of 2: If x = n 1 i=0 b i 2 i where each b i
More informationCS 4770: Cryptography. CS 6750: Cryptography and Communication Security. Alina Oprea Associate Professor, CCIS Northeastern University
CS 4770: Cryptography CS 6750: Cryptography and Communication Security Alina Oprea Associate Professor, CCIS Northeastern University March 15 2018 Review Hash functions Collision resistance Merkle-Damgaard
More informationIntroduction to Public-Key Cryptosystems:
Introduction to Public-Key Cryptosystems: Technical Underpinnings: RSA and Primality Testing Modes of Encryption for RSA Digital Signatures for RSA 1 RSA Block Encryption / Decryption and Signing Each
More informationLecture 11: Hash Functions, Merkle-Damgaard, Random Oracle
CS 7880 Graduate Cryptography October 20, 2015 Lecture 11: Hash Functions, Merkle-Damgaard, Random Oracle Lecturer: Daniel Wichs Scribe: Tanay Mehta 1 Topics Covered Review Collision-Resistant Hash Functions
More informationIntroduction to Cryptology. Lecture 20
Introduction to Cryptology Lecture 20 Announcements HW9 due today HW10 posted, due on Thursday 4/30 HW7, HW8 grades are now up on Canvas. Agenda More Number Theory! Our focus today will be on computational
More informationChapter 7 Randomization Algorithm Theory WS 2017/18 Fabian Kuhn
Chapter 7 Randomization Algorithm Theory WS 2017/18 Fabian Kuhn Randomization Randomized Algorithm: An algorithm that uses (or can use) random coin flips in order to make decisions We will see: randomization
More informationThis is a recursive algorithm. The procedure is guaranteed to terminate, since the second argument decreases each time.
8 Modular Arithmetic We introduce an operator mod. Let d be a positive integer. For c a nonnegative integer, the value c mod d is the remainder when c is divided by d. For example, c mod d = 0 if and only
More informationChapter 4 Finite Fields
Chapter 4 Finite Fields Introduction will now introduce finite fields of increasing importance in cryptography AES, Elliptic Curve, IDEA, Public Key concern operations on numbers what constitutes a number
More informationMathematical Foundations of Cryptography
Mathematical Foundations of Cryptography Cryptography is based on mathematics In this chapter we study finite fields, the basis of the Advanced Encryption Standard (AES) and elliptical curve cryptography
More informationDiscrete Mathematics with Applications MATH236
Discrete Mathematics with Applications MATH236 Dr. Hung P. Tong-Viet School of Mathematics, Statistics and Computer Science University of KwaZulu-Natal Pietermaritzburg Campus Semester 1, 2013 Tong-Viet
More informationCPSC 467b: Cryptography and Computer Security
CPSC 467b: Cryptography and Computer Security Michael J. Fischer Lecture 9 February 14, 2013 CPSC 467b, Lecture 9 1/42 Integer Division (cont.) Relatively prime numbers, Z n, and φ(n) Computing in Z n
More informationCOMS W4995 Introduction to Cryptography October 12, Lecture 12: RSA, and a summary of One Way Function Candidates.
COMS W4995 Introduction to Cryptography October 12, 2005 Lecture 12: RSA, and a summary of One Way Function Candidates. Lecturer: Tal Malkin Scribes: Justin Cranshaw and Mike Verbalis 1 Introduction In
More informationLecture Notes. Advanced Discrete Structures COT S
Lecture Notes Advanced Discrete Structures COT 4115.001 S15 2015-01-13 Recap Divisibility Prime Number Theorem Euclid s Lemma Fundamental Theorem of Arithmetic Euclidean Algorithm Basic Notions - Section
More informationLemma 1.2. (1) If p is prime, then ϕ(p) = p 1. (2) If p q are two primes, then ϕ(pq) = (p 1)(q 1).
1 Background 1.1 The group of units MAT 3343, APPLIED ALGEBRA, FALL 2003 Handout 3: The RSA Cryptosystem Peter Selinger Let (R, +, ) be a ring. Then R forms an abelian group under addition. R does not
More informationAlgebra for error control codes
Algebra for error control codes EE 387, Notes 5, Handout #7 EE 387 concentrates on block codes that are linear: Codewords components are linear combinations of message symbols. g 11 g 12 g 1n g 21 g 22
More informationLecture 11: Number Theoretic Assumptions
CS 6903 Modern Cryptography April 24, 2008 Lecture 11: Number Theoretic Assumptions Instructor: Nitesh Saxena Scribe: Robert W.H. Fisher 1 General 1.1 Administrative Homework 3 now posted on course website.
More informationMa/CS 6a Class 4: Primality Testing
Ma/CS 6a Class 4: Primality Testing By Adam Sheffer Send anonymous suggestions and complaints from here. Email: adamcandobetter@gmail.com Password: anonymous2 There aren t enough crocodiles in the presentations
More informationRSA Key Generation. Required Reading. W. Stallings, "Cryptography and Network-Security, Chapter 8.3 Testing for Primality
ECE646 Lecture RSA Key Generation Required Reading W. Stallings, "Cryptography and Network-Security, Chapter 8.3 Testing for Primality A.Menezes, P. van Oorschot, and S. Vanstone, Handbook of Applied Cryptography
More informationECE646 Lecture 11 Required Reading Chapter 8.3 Testing for Primality RSA Key Generation
ECE646 Lecture equired eading W. Stallings, "Cryptography and Network-Security, Chapter 8. Testing for Primality SA Key Generation A.Menezes, P. van Oorschot, and S. Vanstone, Handbook of Applied Cryptography
More informationPublic Key Encryption
Public Key Encryption 3/13/2012 Cryptography 1 Facts About Numbers Prime number p: p is an integer p 2 The only divisors of p are 1 and p s 2, 7, 19 are primes -3, 0, 1, 6 are not primes Prime decomposition
More informationFinite Fields. Mike Reiter
1 Finite Fields Mike Reiter reiter@cs.unc.edu Based on Chapter 4 of: W. Stallings. Cryptography and Network Security, Principles and Practices. 3 rd Edition, 2003. Groups 2 A group G, is a set G of elements
More informationChapter 5. Modular arithmetic. 5.1 The modular ring
Chapter 5 Modular arithmetic 5.1 The modular ring Definition 5.1. Suppose n N and x, y Z. Then we say that x, y are equivalent modulo n, and we write x y mod n if n x y. It is evident that equivalence
More informationApplied Cryptography and Computer Security CSE 664 Spring 2017
Applied Cryptography and Computer Security Lecture 11: Introduction to Number Theory Department of Computer Science and Engineering University at Buffalo 1 Lecture Outline What we ve covered so far: symmetric
More informationcse 311: foundations of computing Spring 2015 Lecture 12: Primes, GCD, applications
cse 311: foundations of computing Spring 2015 Lecture 12: Primes, GCD, applications casting out 3s Theorem: A positive integer n is divisible by 3 if and only if the sum of its decimal digits is divisible
More informationThe next sequence of lectures in on the topic of Arithmetic Algorithms. We shall build up to an understanding of the RSA public-key cryptosystem.
CS 70 Discrete Mathematics for CS Fall 2003 Wagner Lecture 10 The next sequence of lectures in on the topic of Arithmetic Algorithms. We shall build up to an understanding of the RSA public-key cryptosystem.
More informationLECTURE NOTES IN CRYPTOGRAPHY
1 LECTURE NOTES IN CRYPTOGRAPHY Thomas Johansson 2005/2006 c Thomas Johansson 2006 2 Chapter 1 Abstract algebra and Number theory Before we start the treatment of cryptography we need to review some basic
More informationCPSC 467b: Cryptography and Computer Security
CPSC 467b: Cryptography and Computer Security Michael J. Fischer Lecture 10 February 19, 2013 CPSC 467b, Lecture 10 1/45 Primality Tests Strong primality tests Weak tests of compositeness Reformulation
More informationModular Arithmetic. Examples: 17 mod 5 = 2. 5 mod 17 = 5. 8 mod 3 = 1. Some interesting properties of modular arithmetic:
Modular Arithmetic If a mod n = b, then a = c n + b. When you reduce a number a modulo n you usually want 0 b < n. Division Principle [Bar02, pg. 61]: Let n be a positive integer and let a be any integer.
More informationBasic Algorithms in Number Theory
Basic Algorithms in Number Theory Algorithmic Complexity... 1 Basic Algorithms in Number Theory Francesco Pappalardi #2 - Discrete Logs, Modular Square Roots, Polynomials, Hensel s Lemma & Chinese Remainder
More informationPrimes. Rational, Gaussian, Industrial Strength, etc. Robert Campbell 11/29/2010 1
Primes Rational, Gaussian, Industrial Strength, etc Robert Campbell 11/29/2010 1 Primes and Theory Number Theory to Abstract Algebra History Euclid to Wiles Computation pencil to supercomputer Practical
More informationTheoretical Cryptography, Lecture 13
Theoretical Cryptography, Lecture 13 Instructor: Manuel Blum Scribe: Ryan Williams March 1, 2006 1 Today Proof that Z p has a generator Overview of Integer Factoring Discrete Logarithm and Quadratic Residues
More informationIntroduction to Information Security
Introduction to Information Security Lecture 5: Number Theory 007. 6. Prof. Byoungcheon Lee sultan (at) joongbu. ac. kr Information and Communications University Contents 1. Number Theory Divisibility
More informationTheoretical Cryptography, Lectures 18-20
Theoretical Cryptography, Lectures 18-20 Instructor: Manuel Blum Scribes: Ryan Williams and Yinmeng Zhang March 29, 2006 1 Content of the Lectures These lectures will cover how someone can prove in zero-knowledge
More informationAddition. Ch1 - Algorithms with numbers. Multiplication. al-khwārizmī. al-khwārizmī. Division 53+35=88. Cost? (n number of bits) 13x11=143. Cost?
Ch - Algorithms with numbers Addition Basic arithmetic Addition ultiplication Division odular arithmetic factoring is hard Primality testing 53+35=88 Cost? (n number of bits) O(n) ultiplication al-khwārizmī
More informationAll variables a, b, n, etc are integers unless otherwise stated. Each part of a problem is worth 5 points.
Math 152, Problem Set 2 solutions (2018-01-24) All variables a, b, n, etc are integers unless otherwise stated. Each part of a problem is worth 5 points. 1. Let us look at the following equation: x 5 1
More informationChapter 8. Introduction to Number Theory
Chapter 8 Introduction to Number Theory CRYPTOGRAPHY AND NETWORK SECURITY 1 Index 1. Prime Numbers 2. Fermat`s and Euler`s Theorems 3. Testing for Primality 4. Discrete Logarithms 2 Prime Numbers 3 Prime
More informationFactoring Algorithms Pollard s p 1 Method. This method discovers a prime factor p of an integer n whenever p 1 has only small prime factors.
Factoring Algorithms Pollard s p 1 Method This method discovers a prime factor p of an integer n whenever p 1 has only small prime factors. Input: n (to factor) and a limit B Output: a proper factor of
More information1 Recommended Reading 1. 2 Public Key/Private Key Cryptography Overview RSA Algorithm... 2
Contents 1 Recommended Reading 1 2 Public Key/Private Key Cryptography 1 2.1 Overview............................................. 1 2.2 RSA Algorithm.......................................... 2 3 A Number
More informationA. Algebra and Number Theory
A. Algebra and Number Theory Public-key cryptosystems are based on modular arithmetic. In this section, we summarize the concepts and results from algebra and number theory which are necessary for an understanding
More informationCPE 776:DATA SECURITY & CRYPTOGRAPHY. Some Number Theory and Classical Crypto Systems
CPE 776:DATA SECURITY & CRYPTOGRAPHY Some Number Theory and Classical Crypto Systems Dr. Lo ai Tawalbeh Computer Engineering Department Jordan University of Science and Technology Jordan Some Number Theory
More informationLecture 5: Arithmetic Modulo m, Primes and Greatest Common Divisors Lecturer: Lale Özkahya
BBM 205 Discrete Mathematics Hacettepe University http://web.cs.hacettepe.edu.tr/ bbm205 Lecture 5: Arithmetic Modulo m, Primes and Greatest Common Divisors Lecturer: Lale Özkahya Resources: Kenneth Rosen,
More informationDiscrete Mathematics and Probability Theory Fall 2013 Vazirani Note 3
CS 70 Discrete Mathematics and Probability Theory Fall 2013 Vazirani Note 3 Modular Arithmetic In several settings, such as error-correcting codes and cryptography, we sometimes wish to work over a smaller
More informationAlgorithms (II) Yu Yu. Shanghai Jiaotong University
Algorithms (II) Yu Yu Shanghai Jiaotong University Chapter 1. Algorithms with Numbers Two seemingly similar problems Factoring: Given a number N, express it as a product of its prime factors. Primality:
More informationbasics of security/cryptography
RSA Cryptography basics of security/cryptography Bob encrypts message M into ciphertext C=P(M) using a public key; Bob sends C to Alice Alice decrypts ciphertext back into M using a private key (secret)
More informationCPSC 467b: Cryptography and Computer Security
CPSC 467b: Cryptography and Computer Security Michael J. Fischer Lecture 9 February 6, 2012 CPSC 467b, Lecture 9 1/53 Euler s Theorem Generating RSA Modulus Finding primes by guess and check Density of
More informationECE596C: Handout #11
ECE596C: Handout #11 Public Key Cryptosystems Electrical and Computer Engineering, University of Arizona, Loukas Lazos Abstract In this lecture we introduce necessary mathematical background for studying
More informationCSc 466/566. Computer Security. 5 : Cryptography Basics
1/84 CSc 466/566 Computer Security 5 : Cryptography Basics Version: 2012/03/03 10:44:26 Department of Computer Science University of Arizona collberg@gmail.com Copyright c 2012 Christian Collberg Christian
More informationDiscrete Math, Fourteenth Problem Set (July 18)
Discrete Math, Fourteenth Problem Set (July 18) REU 2003 Instructor: László Babai Scribe: Ivona Bezakova 0.1 Repeated Squaring For the primality test we need to compute a X 1 (mod X). There are two problems
More informationNumber theory (Chapter 4)
EECS 203 Spring 2016 Lecture 10 Page 1 of 8 Number theory (Chapter 4) Review Questions: 1. Does 5 1? Does 1 5? 2. Does (129+63) mod 10 = (129 mod 10)+(63 mod 10)? 3. Does (129+63) mod 10 = ((129 mod 10)+(63
More informationLecture 4: One Way Functions - II
CSE 594 : Modern Cryptography 05/02/2017 Lecture 4: One Way Functions - II Instructor: Omkant Pandey Scribe: Bharathkrishna G Murali, Swarnima Shrivastava 1 Weak to Strong OWFs Theorem 1 The multiplication
More informationDiscrete Mathematics and Probability Theory Fall 2018 Alistair Sinclair and Yun Song Note 6
CS 70 Discrete Mathematics and Probability Theory Fall 2018 Alistair Sinclair and Yun Song Note 6 1 Modular Arithmetic In several settings, such as error-correcting codes and cryptography, we sometimes
More informationArithmetic Algorithms, Part 1
Arithmetic Algorithms, Part 1 DPV Chapter 1 Jim Royer EECS January 18, 2019 Royer Arithmetic Algorithms, Part 1 1/ 15 Multiplication à la Français function multiply(a, b) // input: two n-bit integers a
More informationCSE 521: Design and Analysis of Algorithms I
CSE 521: Design and Analysis of Algorithms I Randomized Algorithms: Primality Testing Paul Beame 1 Randomized Algorithms QuickSelect and Quicksort Algorithms random choices make them fast and simple but
More informationMathematics of Cryptography
UNIT - III Mathematics of Cryptography Part III: Primes and Related Congruence Equations 1 Objectives To introduce prime numbers and their applications in cryptography. To discuss some primality test algorithms
More informationUniversity of Tokyo: Advanced Algorithms Summer Lecture 6 27 May. Let s keep in mind definitions from the previous lecture:
University of Tokyo: Advanced Algorithms Summer 2010 Lecture 6 27 May Lecturer: François Le Gall Scribe: Baljak Valentina As opposed to prime factorization, primality testing is determining whether a given
More informationPublic-key Cryptography: Theory and Practice
Public-key Cryptography Theory and Practice Department of Computer Science and Engineering Indian Institute of Technology Kharagpur Chapter 2: Mathematical Concepts Divisibility Congruence Quadratic Residues
More informationLecture 3.1: Public Key Cryptography I
Lecture 3.1: Public Key Cryptography I CS 436/636/736 Spring 2015 Nitesh Saxena Today s Informative/Fun Bit Acoustic Emanations http://www.google.com/search?source=ig&hl=en&rlz=&q=keyboard+acoustic+em
More informationChapter 9 Mathematics of Cryptography Part III: Primes and Related Congruence Equations
Chapter 9 Mathematics of Cryptography Part III: Primes and Related Congruence Equations Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 9.1 Chapter 9 Objectives
More informationMathematical Foundations of Public-Key Cryptography
Mathematical Foundations of Public-Key Cryptography Adam C. Champion and Dong Xuan CSE 4471: Information Security Material based on (Stallings, 2006) and (Paar and Pelzl, 2010) Outline Review: Basic Mathematical
More informationSOLUTIONS Math 345 Homework 6 10/11/2017. Exercise 23. (a) Solve the following congruences: (i) x (mod 12) Answer. We have
Exercise 23. (a) Solve the following congruences: (i) x 101 7 (mod 12) Answer. We have φ(12) = #{1, 5, 7, 11}. Since gcd(7, 12) = 1, we must have gcd(x, 12) = 1. So 1 12 x φ(12) = x 4. Therefore 7 12 x
More informationCryptography IV: Asymmetric Ciphers
Cryptography IV: Asymmetric Ciphers Computer Security Lecture 7 David Aspinall School of Informatics University of Edinburgh 31st January 2011 Outline Background RSA Diffie-Hellman ElGamal Summary Outline
More information