Lecture 11: Number Theoretic Assumptions
|
|
- Garry Horn
- 5 years ago
- Views:
Transcription
1 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. Due: 5/1/2008 at 5:30pm, in class. For problem 8 in the homework, we are to read a paper by M. Bellare. For this paper, focus primarily on understanding the significance of the results, rather than the specific details of the proofs. Everyone should have received a response for the project updates. Anyone who has not submitted a project update or received a response should contact the professor. Presentations will be in the week of the finals, and they should be minutes in length. 1.2 Review of Last Week Definitions and constructions of NMAC and HMAC Definitions of groups Introduction of modular operations and analysis of costs ( a represents the length of a in digits) Mod: a mod N : Running time O( N 2 ) Mod Addition: (a + b) mod N : Running time O( N ) if a, b < N Mod Multiplication: (ab) mod N : Running time O( N 2 ) Mod Exponentiation: a n mod N : Running time O( n N 2 ) 11-1
2 2 Modular Arithmetic (Continued) 2.1 Greatest Common Denominator Given a, b N, gcd(a, b) represents the greatest common denominator that these two numbers share. The naive solution to this problem would be to simply try all values from 1 to max( a, b). For numbers that may be hundreds of digits long, this is not feasible. Therefore, we will improve upon this algorithm by using the Euclidean Algorithm. Before we introduce this algorithm, we must prove a related fact. Claim: gcd(a, b) = gcd(a b, b) Proof: To show this, we will break the proof into 2 parts. We show that (1)gcd(a, b) gcd(a b, b) (2)gcd(a, b) gcd(a b, b) Proof of (1) We define CD a,b = {k N : k a and k b}, this is the set of all common divisors of a and b. Similarly, CD a b,b = {k N : k (a b) and k b}. Consider some k in CD a,b. We know that: a = ka b = kb for some A, B N. We therefore know that: a b = ka kb = k(a B) We can therefore conclude that k CD a b,b. Therefore, CD a,b CD a b,b. We therefore see that the largest elements of CD a,b must be at least as large as the largest element of CD a b,b. We therefore conclude gcd(a, b) gcd(a b, b) Proof of (2) We now consider some element k CD a b,b. We know that a b = ka Therefore we can see that: b = kb a = ka + kb = k(a + B ) We therefore know that if some element k is in CD a b,b, then it is also in CD a,b, meaning CD a b,b CD a,b. We can therefore conclude that: gcd(a, b) gcd(a b, b) 11-2
3 From (1) and (2), we know that CD a,b = CD a b,b and we also see that: gcd(a, b) = gcd(a b, b) We see a corollary of this result is: gcd(a, b) = gcd(a mod b, b). (We can say that a > b without loss of generality). In order to find the gcd of two arbitrary numbers, we will continually use this fact until one of the two numbers is a multiple of the other one. Example: We will use the Euclidean algorithm to find gcd(37, 15). We see that 37 mod 15 = 7, so, gcd(37, 15) = gcd(7, 15) 1 = 7 mod 15 gcd(7, 15) = gcd(7, 1) = 1 We can see this algorithm another way using the Euclidean remainder. Recall that every pair of natural numbers, a, b, (with a b) can be written uniquely as a = kb + r. gcd(37, 15) 37 = = = gcd(37, 15) = 1 The running time of this algorithm will be O( N 2 ). We do not provide a complete analysis here, but we can see that we will be recursively cutting the size of the input in half at every step (one can easily show that a (mod b) a/2). We recall the geometric series: 1 i = 1 i=2 And we see that we are doing O( N 2 ) work at every step, because each step requires a gcd computation. 2.2 Modular Multiplicative Inverse We are given a N, and we want to find a 1 mod N. Recall that the inverse of a group element is the group element such that aa 1 = 1 = a 1 a, even in non-abelian groups. Because we are trying to find x = a 1 mod N, this is equivalent to saying xa = 1 mod N. Therefore xa = yn + 1 for some y N, we can also say xa yn = 1. Note: Such an inverse only occurs when a and N are relatively prime, so gcd(a, N) = 1. To find the inverse, we use the extended Euclidean algorithm, as we will demonstrate in the following example. 11-3
4 Example: 15 1 mod 37 We begin as we would with the standard Euclidean algorithm for gcd: gcd(37, 15) = = This is the step we are interested in, because the remainder is 1 15 = = 1 7 = (from step 1) 15 2( ) = = 1 This is the form we described before (xa yn = 1), so we conclude that x = 15 1 mod 37 = 5. The running time of this algorithm is O( N 2 ). 3 Number Theory Definitions Order of a group: The order of group G, denoted G, represents the number of elements in G. e.g. Z 11 = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} Z 11 = 10 Order of an element: Given group element a in G, the order of a, denoted o(a), is the smallest i N such that a i = 1. Note that only the identity element of the group is of order 1. e.g. The element 2 Z = 1, so o(2) = 2. Lagrange s Theorem: The order of an element divides the order of the group. e.g. In Z 11, the order of any element (or in fact any subgroup) can only be 1, 2, 5, or
5 Euler s Theorem: For some group G, this theorem states that a G a G = 1. Proof: Let i be the order of a, so a i = 1. From Lagrange s theorem, G = k i for some k. a G = a i k = (a i ) k = 1 k = 1 Fermat s Little Theorem: If p is a prime number, then a p 1 = 1 mod p. Generator of a group: For group G, g G is said to be a generator if: {g 0, g 1, g 2...g G 1 } = G Intuitively, this means that we can find every other element of G by continuing to multiply g to itself. An element is a generator if and only if the order of the element equals the order of the whole group. A group is said to be cyclic if it contains a generator element. e.g. We consider the group Z 11, and we see that 2 is a generator of this group: 2 1 = = = = = = = = = = 1 We do not need to check all G exponents of every element to find a group generator, there is a faster algorithm that is made possible by Lagrange s theorem. Begin by consider the unique prime factorization of the order of the group: G = We will say that m i = G p i. Now, because of Lagrange, some element g will be a generator if and only if g mi 1 for all i [1, n]. e.g. We will once again consider Z 11. The order is 10, and the prime factorization is 10 = 2 5. Therefore: m 1 = 10 2 = 5 m 2 = 10 5 = 2 All we need to do is find some g such that g 2 1 and g 5 1. We see that 2 satisfies these conditions as expected n i=1 p αi i
6 4 Number Theoretic/Computational Assumptions 4.1 Discrete Logarithmic Setting We will consider the group G, such that G = m and G is cyclic (has a generator). If the generator is g, then: G = {g 0, g 1...g m 1 } Now we will consider some x Z m, such that y = g x. Given g and y, it should be difficult to find x. The brute force approach will simply try every element in G, giving a running time of O(m) = O(2 m ) which will be infeasible if m 128. DL Assumption: We will now define an experiment Exp DL (A) for adversary A. We will describe the experiment with the following image: A g, y, m x b = 1 if x = x Otherwise b = 0 Chall Challenger Picks a group ( G = m) with a generator g $ x Z m y g x The DL assumption is said the hold in a group G, if A Adv DL (A) ɛ, meaning P r(exp DL (A) = 1) ɛ. 4.2 Computational Diffie-Hellman Assumption (CDH) In this case, the challenger is going to pick 2 group elements at random, x and y. The challenger will then send the adversary g x and g y, and the adversary will have to try and compute g xy. The image of the experiment Exp CDH (A) is as follows: 11-6
7 A m, g, X, Y Z Chall $ x, y Z m X = g x Y = g y Return 1 if Z = g xy, else return 0 The CDH assumption is said to hold on a group if A Adv CDH (A) ɛ. 4.3 Decisional Diffie-Hellman Assumption (DDH) In this problem, the adversary will simply be tasked with differentiating two different scenarios. In scenario 1, the challenger sends over a group element g xy for some x and y, while in scenario 2 the challenger sends over a random group element. The experiment Exp 0/1 DDH (A) is shown below: m, g, X, Y, Z A b Chall $ x, y Z m X = g x Y = g y If world 1: Z = g xy $ If world 0: z Z m and Z = g z Return b The adversary will return a guess b {0, 1}, for which world the experiment took place in. We see that the advantage of the adversary is: Adv(A) = P r(exp 1 DDH(A) = 1) P r(exp 0 DDH(A) = 1) The DDH assumption holds on group G if this advantage is negligible. 11-7
8 4.4 Relations Between the Assumptions We can state the relationship between the assumptions with the following implication series: DDH CDH DL We first show that DDH CDH, using the contrapositive CDH DDH. We will show that if some adversary A can defeat CDH, we can construct another adversary B that can defeat DDH. The construction works as follows: CDH DDH X, Y, m X, Y, Z, m A B Chall (g xy ) b b = 1 if (g xy ) = Z We see that B succeeds whenever A does, unless we are in world 0, but g z = g xy by chance. Therefore: Adv DDH (B) = Adv CDH (A) 1 G We can therefore conclude that if the advantage of A is non-negligible, then so is the advantage of B. The proof that DL CDH is fairly trivial, and it is omitted here. The converse is not true: Now we show that this implication is uni-directional. Specifically: DL DDH. We will show this by counter example, using the group Z p, where p is a prime. It has been shown previously that the DL assumption holds with this group. Now we will show that the DDH assumption does not hold here. We begin by proving the following lemma: Lemma: For group generator g and prime p: g p 1 2 = 1 mod p Proof: We will say that y = g p 1 2. We know that y 2 = g p 1 = 1 (by Little Fermat), so y {1, 1}. However, g is a generator, so the only exponents i [0, p] that make g i = 1 are 0 and p. But p 1 falls strictly into this range, so g p Therefore g p 1 2 = 1. 2 Now for DDH, we are given X = g x, Y = g y, and Z {g z, g xy }. We begin by computing X p 1 2 = g x p 1 p 1 2k 2. If x is even (x = 2k), then g 2 = (g p 1 ) k = 1. Otherwise, if x is odd, then g x p 1 2 = ( 1) x = 1. Using these facts, we can determine the parity of x. Following the same procedure, we can determine the polarities of y and the exponent of Z. We now use the following table of elementary multiplications: 11-8
9 x y xy even even even even odd even odd even even odd odd odd So, we check if the parity of xy matches the parity of the exponent of Z. If it does not, we declare that we are in world 0. When the parities do not match, we are completely certain with our outcome. The only error comes when Z = g z, and z happens to have the same polarity as xy. This will happen around 50% of the time. Therefore: Adv DDH (A) = =
Lecture 10: HMAC and Number Theory
CS 6903 Modern Cryptography April 15, 2010 Lecture 10: HMAC and Number Theory Instructor: Nitesh Saxena Scribes: Anand Bidla, Samiksha Saxena,Varun Sanghvi 1 HMAC A Hash-based Message Authentication Code
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 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 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 informationLecture 4 Chiu Yuen Koo Nikolai Yakovenko. 1 Summary. 2 Hybrid Encryption. CMSC 858K Advanced Topics in Cryptography February 5, 2004
CMSC 858K Advanced Topics in Cryptography February 5, 2004 Lecturer: Jonathan Katz Lecture 4 Scribe(s): Chiu Yuen Koo Nikolai Yakovenko Jeffrey Blank 1 Summary The focus of this lecture is efficient public-key
More informationINTEGERS. In this section we aim to show the following: Goal. Every natural number can be written uniquely as a product of primes.
INTEGERS PETER MAYR (MATH 2001, CU BOULDER) In this section we aim to show the following: Goal. Every natural number can be written uniquely as a product of primes. 1. Divisibility Definition. Let a, b
More informationComputational Number Theory. Adam O Neill Based on
Computational Number Theory Adam O Neill Based on http://cseweb.ucsd.edu/~mihir/cse207/ Secret Key Exchange - * Is Alice Ka Public Network Ka = KB O KB 0^1 Eve should have a hard time getting information
More information4 Powers of an Element; Cyclic Groups
4 Powers of an Element; Cyclic Groups Notation When considering an abstract group (G, ), we will often simplify notation as follows x y will be expressed as xy (x y) z will be expressed as xyz x (y z)
More informationLecture 17: Constructions of Public-Key Encryption
COM S 687 Introduction to Cryptography October 24, 2006 Lecture 17: Constructions of Public-Key Encryption Instructor: Rafael Pass Scribe: Muthu 1 Secure Public-Key Encryption In the previous lecture,
More informationIntroduction to Modern Cryptography Recitation 3. Orit Moskovich Tel Aviv University November 16, 2016
Introduction to Modern Cryptography Recitation 3 Orit Moskovich Tel Aviv University November 16, 2016 The group: Z N Let N 2 be an integer The set Z N = a 1,, N 1 gcd a, N = 1 with respect to multiplication
More informationCSC 5930/9010 Modern Cryptography: Number Theory
CSC 5930/9010 Modern Cryptography: Number Theory Professor Henry Carter Fall 2018 Recap Hash functions map arbitrary-length strings to fixedlength outputs Cryptographic hashes should be collision-resistant
More informationIntroduction to Cybersecurity Cryptography (Part 4)
Introduction to Cybersecurity Cryptography (Part 4) Review of Last Lecture Blockciphers Review of DES Attacks on Blockciphers Advanced Encryption Standard (AES) Modes of Operation MACs and Hashes Message
More informationIntroduction to Cryptography. Lecture 8
Introduction to Cryptography Lecture 8 Benny Pinkas page 1 1 Groups we will use Multiplication modulo a prime number p (G, ) = ({1,2,,p-1}, ) E.g., Z 7* = ( {1,2,3,4,5,6}, ) Z p * Z N * Multiplication
More informationIntroduction to Cybersecurity Cryptography (Part 4)
Introduction to Cybersecurity Cryptography (Part 4) Review of Last Lecture Blockciphers Review of DES Attacks on Blockciphers Advanced Encryption Standard (AES) Modes of Operation MACs and Hashes Message
More informationPractice Number Theory Problems
Massachusetts Institute of Technology Handout 9 6.857: Network and Computer Security March 21, 2013 Professor Ron Rivest Due: N/A Problem 3-1. GCD Practice Number Theory Problems (a) Compute gcd(85, 289)
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 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 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 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 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 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 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 10: NMAC, HMAC and Number Theory
CS 6903 Modern Cryptography April 10, 2008 Lecture 10: NMAC, HMAC and Number Theory Instructor: Nitesh Saxena Scribes: Jonathan Voris, Md. Borhan Uddin 1 Recap 1.1 MACs A message authentication code (MAC)
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 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 informationPublic Key Cryptography
Public Key Cryptography Spotlight on Science J. Robert Buchanan Department of Mathematics 2011 What is Cryptography? cryptography: study of methods for sending messages in a form that only be understood
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 information14 Diffie-Hellman Key Agreement
14 Diffie-Hellman Key Agreement 14.1 Cyclic Groups Definition 14.1 Example Let д Z n. Define д n = {д i % n i Z}, the set of all powers of д reduced mod n. Then д is called a generator of д n, and д n
More informationCOMS W4995 Introduction to Cryptography September 29, Lecture 8: Number Theory
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.
More information9 Modular Exponentiation and Square-Roots
9 Modular Exponentiation and Square-Roots Modular arithmetic is used in cryptography. In particular, modular exponentiation is the cornerstone of what is called the RSA system. 9. Modular Exponentiation
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 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 informationMATH 25 CLASS 21 NOTES, NOV Contents. 2. Subgroups 2 3. Isomorphisms 4
MATH 25 CLASS 21 NOTES, NOV 7 2011 Contents 1. Groups: definition 1 2. Subgroups 2 3. Isomorphisms 4 1. Groups: definition Even though we have been learning number theory without using any other parts
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 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 informationDiscrete Mathematics GCD, LCM, RSA Algorithm
Discrete Mathematics GCD, LCM, RSA Algorithm Abdul Hameed http://informationtechnology.pk/pucit abdul.hameed@pucit.edu.pk Lecture 16 Greatest Common Divisor 2 Greatest common divisor The greatest common
More informationCSC 373: Algorithm Design and Analysis Lecture 30
CSC 373: Algorithm Design and Analysis Lecture 30 Allan Borodin April 5, 2013 1 / 12 Announcements and Outline Announcements Two misstated questions on term test Grading scheme for term test 3: 1 Test
More informationLecture Notes, Week 6
YALE UNIVERSITY DEPARTMENT OF COMPUTER SCIENCE CPSC 467b: Cryptography and Computer Security Week 6 (rev. 3) Professor M. J. Fischer February 15 & 17, 2005 1 RSA Security Lecture Notes, Week 6 Several
More informationDiscrete Mathematics and Probability Theory Summer 2014 James Cook Note 5
CS 70 Discrete Mathematics and Probability Theory Summer 2014 James Cook Note 5 Modular Arithmetic In several settings, such as error-correcting codes and cryptography, we sometimes wish to work over a
More information2 Arithmetic. 2.1 Greatest common divisors. This chapter is about properties of the integers Z = {..., 2, 1, 0, 1, 2,...}.
2 Arithmetic This chapter is about properties of the integers Z = {..., 2, 1, 0, 1, 2,...}. (See [Houston, Chapters 27 & 28]) 2.1 Greatest common divisors Definition 2.16. If a, b are integers, we say
More informationLecture 9 Julie Staub Avi Dalal Abheek Anand Gelareh Taban. 1 Introduction. 2 Background. CMSC 858K Advanced Topics in Cryptography February 24, 2004
CMSC 858K Advanced Topics in Cryptography February 24, 2004 Lecturer: Jonathan Katz Lecture 9 Scribe(s): Julie Staub Avi Dalal Abheek Anand Gelareh Taban 1 Introduction In previous lectures, we constructed
More informationCS 5319 Advanced Discrete Structure. Lecture 9: Introduction to Number Theory II
CS 5319 Advanced Discrete Structure Lecture 9: Introduction to Number Theory II Divisibility Outline Greatest Common Divisor Fundamental Theorem of Arithmetic Modular Arithmetic Euler Phi Function RSA
More informationIntroduction to Number Theory. The study of the integers
Introduction to Number Theory The study of the integers of Integers, The set of integers = {... 3, 2, 1, 0, 1, 2, 3,...}. In this lecture, if nothing is said about a variable, it is an integer. Def. We
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 informationLecture 1: Introduction to Public key cryptography
Lecture 1: Introduction to Public key cryptography Thomas Johansson T. Johansson (Lund University) 1 / 44 Key distribution Symmetric key cryptography: Alice and Bob share a common secret key. Some means
More informationLecture 10: NMAC, HMAC and Number Theory
CS 6903 Modern Cryptography April 13, 2011 Lecture 10: NMAC, HMAC and Number Theory Instructor: Nitesh Saxena Scribes: Anand Desai,Manav Singh Dahiya,Amol Bhavekar 1 Recap 1.1 MACs A Message Authentication
More informationMa/CS 6a Class 2: Congruences
Ma/CS 6a Class 2: Congruences 1 + 1 5 (mod 3) By Adam Sheffer Reminder: Public Key Cryptography Idea. Use a public key which is used for encryption and a private key used for decryption. Alice encrypts
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 informationSecurity II: Cryptography exercises
Security II: Cryptography exercises Markus Kuhn Lent 2015 Part II Some of the exercises require the implementation of short programs. The model answers use Perl (see Part IB Unix Tools course), but you
More informationCourse MA2C02, Hilary Term 2013 Section 9: Introduction to Number Theory and Cryptography
Course MA2C02, Hilary Term 2013 Section 9: Introduction to Number Theory and Cryptography David R. Wilkins Copyright c David R. Wilkins 2000 2013 Contents 9 Introduction to Number Theory 63 9.1 Subgroups
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 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 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 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 informationKatz, Lindell Introduction to Modern Cryptrography
Katz, Lindell Introduction to Modern Cryptrography Slides Chapter 8 Markus Bläser, Saarland University Weak factoring experiment The weak factoring experiment 1. Choose two n-bit integers x 1, x 2 uniformly.
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 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 informationMATH 145 Algebra, Solutions to Assignment 4
MATH 145 Algebra, Solutions to Assignment 4 1: a) Find the inverse of 178 in Z 365. Solution: We find s and t so that 178s + 365t = 1, and then 178 1 = s. The Euclidean Algorithm gives 365 = 178 + 9 178
More information1 Structure of Finite Fields
T-79.5501 Cryptology Additional material September 27, 2005 1 Structure of Finite Fields This section contains complementary material to Section 5.2.3 of the text-book. It is not entirely self-contained
More informationA Few Primality Testing Algorithms
A Few Primality Testing Algorithms Donald Brower April 2, 2006 0.1 Introduction These notes will cover a few primality testing algorithms. There are many such, some prove that a number is prime, others
More informationFor your quiz in recitation this week, refer to these exercise generators:
Monday, Oct 29 Today we will talk about inverses in modular arithmetic, and the use of inverses to solve linear congruences. For your quiz in recitation this week, refer to these exercise generators: GCD
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 informationALGEBRA. 1. Some elementary number theory 1.1. Primes and divisibility. We denote the collection of integers
ALGEBRA CHRISTIAN REMLING 1. Some elementary number theory 1.1. Primes and divisibility. We denote the collection of integers by Z = {..., 2, 1, 0, 1,...}. Given a, b Z, we write a b if b = ac for some
More informationOWO Lecture: Modular Arithmetic with Algorithmic Applications
OWO Lecture: Modular Arithmetic with Algorithmic Applications Martin Otto Winter Term 2008/09 Contents 1 Basic ingredients 1 2 Modular arithmetic 2 2.1 Going in circles.......................... 2 2.2
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 informationDiscrete Logarithm Problem
Discrete Logarithm Problem Çetin Kaya Koç koc@cs.ucsb.edu (http://cs.ucsb.edu/~koc/ecc) Elliptic Curve Cryptography lect08 discrete log 1 / 46 Exponentiation and Logarithms in a General Group In a multiplicative
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 informationIntroduction to Cybersecurity Cryptography (Part 5)
Introduction to Cybersecurity Cryptography (Part 5) Prof. Dr. Michael Backes 13.01.2017 February 17 th Special Lecture! 45 Minutes Your Choice 1. Automotive Security 2. Smartphone Security 3. Side Channel
More informationIntroduction to Elliptic Curve Cryptography. Anupam Datta
Introduction to Elliptic Curve Cryptography Anupam Datta 18-733 Elliptic Curve Cryptography Public Key Cryptosystem Duality between Elliptic Curve Cryptography and Discrete Log Based Cryptography Groups
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 informationCourse 2BA1: Trinity 2006 Section 9: Introduction to Number Theory and Cryptography
Course 2BA1: Trinity 2006 Section 9: Introduction to Number Theory and Cryptography David R. Wilkins Copyright c David R. Wilkins 2006 Contents 9 Introduction to Number Theory and Cryptography 1 9.1 Subgroups
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 informationCh 4.2 Divisibility Properties
Ch 4.2 Divisibility Properties - Prime numbers and composite numbers - Procedure for determining whether or not a positive integer is a prime - GCF: procedure for finding gcf (Euclidean Algorithm) - Definition:
More informationNUMBER THEORY AND CODES. Álvaro Pelayo WUSTL
NUMBER THEORY AND CODES Álvaro Pelayo WUSTL Talk Goal To develop codes of the sort can tell the world how to put messages in code (public key cryptography) only you can decode them Structure of Talk Part
More informationNotes for Lecture 17
U.C. Berkeley CS276: Cryptography Handout N17 Luca Trevisan March 17, 2009 Notes for Lecture 17 Scribed by Matt Finifter, posted April 8, 2009 Summary Today we begin to talk about public-key cryptography,
More informationLecture 6: Cryptanalysis of public-key algorithms.,
T-79.159 Cryptography and Data Security Lecture 6: Cryptanalysis of public-key algorithms. Helsinki University of Technology mjos@tcs.hut.fi 1 Outline Computational complexity Reminder about basic number
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 informationFERMAT S TEST KEITH CONRAD
FERMAT S TEST KEITH CONRAD 1. Introduction Fermat s little theorem says for prime p that a p 1 1 mod p for all a 0 mod p. A naive extension of this to a composite modulus n 2 would be: for all a 0 mod
More informationLECTURE 5: APPLICATIONS TO CRYPTOGRAPHY AND COMPUTATIONS
LECTURE 5: APPLICATIONS TO CRYPTOGRAPHY AND COMPUTATIONS Modular arithmetics that we have discussed in the previous lectures is very useful in Cryptography and Computer Science. Here we discuss several
More informationIntroduction to Cryptography k. Lecture 5. Benny Pinkas k. Requirements. Data Integrity, Message Authentication
Common Usage of MACs for message authentication Introduction to Cryptography k Alice α m, MAC k (m) Isα= MAC k (m)? Bob k Lecture 5 Benny Pinkas k Alice m, MAC k (m) m,α Got you! α MAC k (m )! Bob k Eve
More informationNOTES ON SIMPLE NUMBER THEORY
NOTES ON SIMPLE NUMBER THEORY DAMIEN PITMAN 1. Definitions & Theorems Definition: We say d divides m iff d is positive integer and m is an integer and there is an integer q such that m = dq. In this case,
More informationNUMBER SYSTEMS. Number theory is the study of the integers. We denote the set of integers by Z:
NUMBER SYSTEMS Number theory is the study of the integers. We denote the set of integers by Z: Z = {..., 3, 2, 1, 0, 1, 2, 3,... }. The integers have two operations defined on them, addition and multiplication,
More informationLecture 7: ElGamal and Discrete Logarithms
Lecture 7: ElGamal and Discrete Logarithms Johan Håstad, transcribed by Johan Linde 2006-02-07 1 The discrete logarithm problem Recall that a generator g of a group G is an element of order n such that
More informationLecture Note 3 Date:
P.Lafourcade Lecture Note 3 Date: 28.09.2009 Security models 1st Semester 2007/2008 ROUAULT Boris GABIAM Amanda ARNEDO Pedro 1 Contents 1 Perfect Encryption 3 1.1 Notations....................................
More informationChapter 5. Number Theory. 5.1 Base b representations
Chapter 5 Number Theory The material in this chapter offers a small glimpse of why a lot of facts that you ve probably nown and used for a long time are true. It also offers some exposure to generalization,
More informationSlides by Christopher M. Bourke Instructor: Berthe Y. Choueiry. Spring 2006
Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry Spring 2006 1 / 1 Computer Science & Engineering 235 Introduction to Discrete Mathematics Sections 2.4 2.6 of Rosen Introduction I When talking
More informationLecture 17 - Diffie-Hellman key exchange, pairing, Identity-Based Encryption and Forward Security
Lecture 17 - Diffie-Hellman key exchange, pairing, Identity-Based Encryption and Forward Security Boaz Barak November 21, 2007 Cyclic groups and discrete log A group G is cyclic if there exists a generator
More informationMa/CS 6a Class 2: Congruences
Ma/CS 6a Class 2: Congruences 1 + 1 5 (mod 3) By Adam Sheffer Reminder: Public Key Cryptography Idea. Use a public key which is used for encryption and a private key used for decryption. Alice encrypts
More informationPUTNAM TRAINING NUMBER THEORY. Exercises 1. Show that the sum of two consecutive primes is never twice a prime.
PUTNAM TRAINING NUMBER THEORY (Last updated: December 11, 2017) Remark. This is a list of exercises on Number Theory. Miguel A. Lerma Exercises 1. Show that the sum of two consecutive primes is never twice
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 informationPublic-Key Cryptosystems CHAPTER 4
Public-Key Cryptosystems CHAPTER 4 Introduction How to distribute the cryptographic keys? Naïve Solution Naïve Solution Give every user P i a separate random key K ij to communicate with every P j. Disadvantage:
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 information2WF15 - Discrete Mathematics 2 - Part 1. Algorithmic Number Theory
1 2WF15 - Discrete Mathematics 2 - Part 1 Algorithmic Number Theory Benne de Weger version 0.54, March 6, 2012 version 0.54, March 6, 2012 2WF15 - Discrete Mathematics 2 - Part 1 2 2WF15 - Discrete Mathematics
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 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 informationMath 299 Supplement: Modular Arithmetic Nov 8, 2013
Math 299 Supplement: Modular Arithmetic Nov 8, 2013 Numbers modulo n. We have previously seen examples of clock arithmetic, an algebraic system with only finitely many numbers. In this lecture, we make
More informationPublic Key Algorithms
Public Key Algorithms Raj Jain Washington University in Saint Louis Saint Louis, MO 63130 Jain@cse.wustl.edu Audio/Video recordings of this lecture are available at: http://www.cse.wustl.edu/~jain/cse571-09/
More information