Lecture 11: The Goldreich-Levin Theorem

Save this PDF as:

Size: px
Start display at page:

Transcription

1 COM S 687 Introduction to Cryptography September 28, 2006 Lecture 11: The Goldreich-Levin Theorem Instructor: Rafael Pass Scribe: Krishnaprasad Vikram Hard-Core Bits Definition: A predicate b : {0, 1} γ {0, 1} is hardcore for a function f if (a) b is efficiently computable (b) p.p.t. A, a negligible polynomial ǫ s.t. k Pr[X {0, 1} k : A(1 k, f(x) = b(x))] ǫ(k) Intuitively, a hardcore bit (described as a function b) is efficiently computable given an input x, but is hard to compute given only f(x). In other words, f hides the bit b. This definition can be trivially extended to collections of one-way functions. Construction of a PRG[1] Using the idea of hardcore bits, and assuming the existence of a one-way permutation f we constructed in the previous lecture a PRG G : {0, 1} n {0, 1} n+1 given by G(s) = f(s) b(s) where f is a one-way permutation and b is a hard core bit for f, and is the string concatenation operator. Intuitively, this is a PRG given that it passes the next-bit test since it would be hard to compute the n+1 th bit b(s) given the first n bits f(s). However, this directly does not hold true of a OWF (Why? We might be proving that in an upcoming homework). However, there is a theorem that says of a OWF of a Pseudo-random number generator [3]. We ll prove the (supposedly easy) direction in one of the homeworks. Now, we ll show in class that of a OWP of a PRG. Note that it is an open problem to prove that any OWF or a OWP has a hard core bit. What we ll show is that every OWF (or OWP) can be transformed into a new OWF (respectively OWP) that has a hard-core bit. One possibility of a hardcore bit is a parity function, but that might be easy to compute given f(x). We ll try something more sophisticated. 11-1

2 Theorem Let f be a OWF. Then f (X, r) = f(x), r (where X = r ) is a OWF and b(x, r) = X, r 2 = ΣX i r i mod 2 (inner product mod 2) is a hardcore predicate for f. Here r essentially tells us which bits to take parity of. Note that f is a OWP if f is a OWP. Proof. (by reductio ad absurdum) We show that if A, given f (X, r) can compute b(x, r) w.p. significantly better than 1/2 p.p.t. B that inverts f. We ll do the proof in three steps. In the first step, we consider a very oversimplified case and prove the theorem for that case. In the next step, we take a less simplified case and finally we take the general case. In the very oversimplified case, we assume A always computes b correctly. And so, we can construct a f with an r such that the first bit is 1 and the other bits are 0. In such a case A would return the first bit of X. Similarly we can set the second bits of r to be 1 to obtain the second bit of X. Thus, we have B given by B(y): Let X i = A(y, e i ) where e i = where the 1 is on the i th position. -Output X 1, X 2,...,X n This works, since X, e i 2 = X i Now, in the less simplified case, we assume that A, when given random y = f(x) and random r, computes b(x, r) w.p ǫ, (ǫ =, n is the length of X). 4 poly(n) Intuition: we want the attacker to compute b with a fixed X and a varying r so that given enough observations, X can be computed eventually. The trick is to find the set of good X, for which this will work. As an attempt to find such X, let S = {X Pr[A(f(X), r) = b(x, r)] > 3 + ǫ }. It can be 4 2 shown that S > ǫ/2. A simple attack with various e i might not work here. More rerandomization is required. Idea: Use linearity of a, b. Useful relevant fact: a, b c = a, b a, c mod 2 Proof. a, b c = Σa i (b i + c i ) = Σa i b i + Σa i c i = a, b + a, c mod 2 Attacker asks: X, r, X, r + e

3 and then XOR both to get X, e 1 without ever asking for e 1. And so, B inverts f as follows: B(y) : For i = 1 to n 1. Pick random r in {0, 1} n 2. Let r = e i r 3. Compute guess for X i as A(y, r) A(y, r ) 4. Repeat poly(1/ǫ) times and let X i be majority of guesses. Finally output X 1,...,X n. If we assume e 1 and r + e 1 as independent, the proof works fine. However, they are not independent. The proof is still OK though, as can be seen using the union bound: The proof works because: w.p. 1 4 ǫ 2 A(y, r) b(x, r) w.p. 1 4 ǫ 2 A(y, r ) b(x, r) by union bound w.p. 1 2 both answers of A are OK. Since y, r + y, r = y, r r = y, e i, each guess is correct w.p ǫ Since samples are independent, using Chernoff Bound it can be shown that every bit is OK w.h.p. Now, to the general case. Here, we assume that A, given random y = f(x), random r computes b(x, r) w.p ǫ (ǫ = 1 poly(n) ) Let S = {X Pr[A(f(X), r) = b(x, r)] > ǫ 2 }. It again follows that S > ǫ 2. Assume set access to oracle C that given f(x) gives us samples X, r 1, r 1. (where r 1,..., r n are independent and random) X, r n, r n We now recall Homework 1, where given an algorithm that computes a correct bit value w.p. greater than 1 +ǫ, we can run it multiple times and take the majority result, thereby 2 computing the bit w.p. as close to 1 as desired. 11-3

4 From here on, the idea is to eliminate C from the constructed machine step by step, so that we don t need an oracle in the final machine B. Consider the following B(y): For i = 1 to n 1. C(y) (b 1, r 1 ),..., (b m, r m ) 2. Let r j = e i r j 3. Compute g j = b j A(y, r ) 4. Let X i = majority(g 1,...,g m ) Output X 1,...,X m 1 Each guess g i is correct w.p. + ǫ = ǫ. As in HW1, by Chernoff bound, an x i is wrong w.p. 2 ǫ 2m (was 2 4ǫ2m in the HW). If m >> 1, we are OK. ǫ 2 Now, we assume that C gives us samples X, r 1, r 1 ;...; X, r n, r n which are random but only pairwise independent. Again, using results from HW1, by Chebyshev s theorem, each X i is wrong w.p. 1 4ǫ 2 1 (ignoring constants). 4mǫ 2 mǫ 2 By union bound, any of the X i is wrong w.p. n 1 2n, when m. Therefore, as mǫ 2 2 ǫ 2 long as we have polynomially many samples (precisely 2n pairwise independent samples), ǫ 2 we d be done. The question now is: How do we get pairwise independent samples? So, our initial attempt to remove C would be to pick r 1,...,r m on random and guess b 1,...,b m randomly. However, b i would be correct only w.p. 2 m. A better attempt is to pick log(m) samples s 1,...,s log(m) and guessing b 1,..., b log(m) randomly. Here the guess is correct with probability 1/m. Now, generate r 1, r 2,...,r m 1 as all possible sums (mod 2) of subsets of s 1,...,s log(m), and b 1, b 2,...,b m as the corresponding subsets of b i. Mathematically r i = j I i s j j I iff i j = 1 b i = j I i b j In HW1, we showed that these r i are pairwise independent samples. Yet w.p. 1/m, all guesses for b 1,..., b log(m) are correct, which means that b 1,...,b m 1 are also correct. 11-4

5 Thus, for a fraction of ǫ of X it holds that w.p. 1/m we invert w.p. 1/2. That is B(y) inverts w.p. ǫ 2m = ǫ 3 4n = (ǫ/2)3 4n which contradicts the (strong) one-way-ness of f. (m = 2n ǫ 2 ) Yao proved that if OWF exists, then there exists OWF with hard core bits. But this construction is due to Goldreich and Levin[2] and by Charles Rackoff[?]. References [1] Manuel Blum and Silvio Micali. How to generate cryptographically strong sequences of pseudo-random bits. SIAM J. Comput., 13(4): , [2] O. Goldreich and L. A. Levin. A hard-core predicate for all one-way functions. In STOC 89: Proceedings of the twenty-first annual ACM symposium on Theory of computing, pages 25 32, New York, NY, USA, ACM Press. [3] Johan Hastad, Russell Impagliazzo, Leonid A. Levin, and Michael Luby. A pseudorandom generator from any one-way function. SIAM J. Comput., 28(4): ,

Pseudorandom Generators from One-Way Functions: A Simple Construction for Any Hardness

Pseudorandom Generators from One-Way Functions: A Simple Construction for Any Hardness Thomas Holenstein ETH Zurich, Department of Computer Science, CH-8092 Zurich thomahol@inf.ethz.ch Abstract. In a seminal

Lecture 2: Complexity Theory Review and Interactive Proofs

600.641 Special Topics in Theoretical Cryptography January 23, 2007 Lecture 2: Complexity Theory Review and Interactive Proofs Instructor: Susan Hohenberger Scribe: Karyn Benson 1 Introduction to Cryptography

Lecture 5 - CPA security, Pseudorandom functions

Lecture 5 - CPA security, Pseudorandom functions Boaz Barak October 2, 2007 Reading Pages 82 93 and 221 225 of KL (sections 3.5, 3.6.1, 3.6.2 and 6.5). See also Goldreich (Vol I) for proof of PRF construction.

Concrete Security of the Blum-Blum-Shub Pseudorandom Generator

Appears in Cryptography and Coding: 10th IMA International Conference, Lecture Notes in Computer Science 3796 (2005) 355 375. Springer-Verlag. Concrete Security of the Blum-Blum-Shub Pseudorandom Generator

Lecture 15 - Digital Signatures

Lecture 15 - Digital Signatures Boaz Barak March 29, 2010 Reading KL Book Chapter 12. Review Trapdoor permutations - easy to compute, hard to invert, easy to invert with trapdoor. RSA and Rabin signatures.

Lecture 10: CPA Encryption, MACs, Hash Functions. 2 Recap of last lecture - PRGs for one time pads

CS 7880 Graduate Cryptography October 15, 2015 Lecture 10: CPA Encryption, MACs, Hash Functions Lecturer: Daniel Wichs Scribe: Matthew Dippel 1 Topic Covered Chosen plaintext attack model of security MACs

How to Prove a Theorem So No One Else Can Claim It

Proceedings of the International Congress of Mathematicians Berkeley, California, USA, 1986 How to Prove a Theorem So No One Else Can Claim It MANUEL BLUM Goldwasser, Micali, and Rackoff [GMR] define for

Lecture 4: AC 0 lower bounds and pseudorandomness

Lecture 4: AC 0 lower bounds and pseudorandomness Topics in Complexity Theory and Pseudorandomness (Spring 2013) Rutgers University Swastik Kopparty Scribes: Jason Perry and Brian Garnett In this lecture,

The Challenger-Solver game: Variations on the theme of P =?NP

The Challenger-Solver game: Variations on the theme of P =?NP Yuri Gurevich Electrical Engineering and Computer Science Department University of Michigan, Ann Arbor, MI 48109, U.S.A. Logic in Computer

A Secure Protocol for the Oblivious Transfer (Extended Abstract) M. J. Fischer. Yale University. S. Micali Massachusetts Institute of Technology

J, Cryptoiogy (1996) 9:191-195 Joumol of CRYPTOLOGY O 1996 International Association for Cryptologic Research A Secure Protocol for the Oblivious Transfer (Extended Abstract) M. J. Fischer Yale University

The Bit Security of Paillier s Encryption Scheme and its Applications

The Bit Security of Paillier s Encryption Scheme and its Applications Dario Catalano 1, Rosario Gennaro and Nick Howgrave-Graham 1 Dipartimento di Matematica e Informatica Università di Catania. Viale

Lecture 1: Course overview, circuits, and formulas

Lecture 1: Course overview, circuits, and formulas Topics in Complexity Theory and Pseudorandomness (Spring 2013) Rutgers University Swastik Kopparty Scribes: John Kim, Ben Lund 1 Course Information Swastik

Yale University Department of Computer Science

Yale University Department of Computer Science On Backtracking Resistance in Pseudorandom Bit Generation (preliminary version) Michael J. Fischer Michael S. Paterson Ewa Syta YALEU/DCS/TR-1466 October

Ch.9 Cryptography. The Graduate Center, CUNY.! CSc 75010 Theoretical Computer Science Konstantinos Vamvourellis

Ch.9 Cryptography The Graduate Center, CUNY! CSc 75010 Theoretical Computer Science Konstantinos Vamvourellis Why is Modern Cryptography part of a Complexity course? Short answer:! Because Modern Cryptography

Victor Shoup Avi Rubin. fshoup,rubing@bellcore.com. Abstract

Session Key Distribution Using Smart Cards Victor Shoup Avi Rubin Bellcore, 445 South St., Morristown, NJ 07960 fshoup,rubing@bellcore.com Abstract In this paper, we investigate a method by which smart

Computational Soundness of Symbolic Security and Implicit Complexity

Computational Soundness of Symbolic Security and Implicit Complexity Bruce Kapron Computer Science Department University of Victoria Victoria, British Columbia NII Shonan Meeting, November 3-7, 2013 Overview

2.1 Complexity Classes

15-859(M): Randomized Algorithms Lecturer: Shuchi Chawla Topic: Complexity classes, Identity checking Date: September 15, 2004 Scribe: Andrew Gilpin 2.1 Complexity Classes In this lecture we will look

Computational Complexity: A Modern Approach

i Computational Complexity: A Modern Approach Draft of a book: Dated January 2007 Comments welcome! Sanjeev Arora and Boaz Barak Princeton University complexitybook@gmail.com Not to be reproduced or distributed

P versus NP, and More

1 P versus NP, and More Great Ideas in Theoretical Computer Science Saarland University, Summer 2014 If you have tried to solve a crossword puzzle, you know that it is much harder to solve it than to verify

Lecture 3: One-Way Encryption, RSA Example

ICS 180: Introduction to Cryptography April 13, 2004 Lecturer: Stanislaw Jarecki Lecture 3: One-Way Encryption, RSA Example 1 LECTURE SUMMARY We look at a different security property one might require

T Cryptology Spring 2009

T-79.5501 Cryptology Spring 2009 Homework 2 Tutor : Joo Y. Cho joo.cho@tkk.fi 5th February 2009 Q1. Let us consider a cryptosystem where P = {a, b, c} and C = {1, 2, 3, 4}, K = {K 1, K 2, K 3 }, and the

Breaking Generalized Diffie-Hellman Modulo a Composite is no Easier than Factoring

Breaking Generalized Diffie-Hellman Modulo a Composite is no Easier than Factoring Eli Biham Dan Boneh Omer Reingold Abstract The Diffie-Hellman key-exchange protocol may naturally be extended to k > 2

a Course in Cryptography

a Course in Cryptography rafael pass abhi shelat c 2010 Pass/shelat All rights reserved Printed online 11 11 11 11 11 15 14 13 12 11 10 9 First edition: June 2007 Second edition: September 2008 Third edition:

Lecture 13: Factoring Integers

CS 880: Quantum Information Processing 0/4/0 Lecture 3: Factoring Integers Instructor: Dieter van Melkebeek Scribe: Mark Wellons In this lecture, we review order finding and use this to develop a method

1 Formulating The Low Degree Testing Problem

6.895 PCP and Hardness of Approximation MIT, Fall 2010 Lecture 5: Linearity Testing Lecturer: Dana Moshkovitz Scribe: Gregory Minton and Dana Moshkovitz In the last lecture, we proved a weak PCP Theorem,

1 Message Authentication

Theoretical Foundations of Cryptography Lecture Georgia Tech, Spring 200 Message Authentication Message Authentication Instructor: Chris Peikert Scribe: Daniel Dadush We start with some simple questions

1 Signatures vs. MACs

CS 120/ E-177: Introduction to Cryptography Salil Vadhan and Alon Rosen Nov. 22, 2006 Lecture Notes 17: Digital Signatures Recommended Reading. Katz-Lindell 10 1 Signatures vs. MACs Digital signatures

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory

Lecture 2: Universality

CS 710: Complexity Theory 1/21/2010 Lecture 2: Universality Instructor: Dieter van Melkebeek Scribe: Tyson Williams In this lecture, we introduce the notion of a universal machine, develop efficient universal

1 Construction of CCA-secure encryption

CSCI 5440: Cryptography Lecture 5 The Chinese University of Hong Kong 10 October 2012 1 Construction of -secure encryption We now show how the MAC can be applied to obtain a -secure encryption scheme.

Computer Science and Engineering, UCSD October 7, 1999 Goldreic-Levin Teorem Autor: Bellare Te Goldreic-Levin Teorem 1 Te problem We æx a an integer n for te lengt of te strings involved. If a is an n-bit

COMP 250 Fall 2012 lecture 2 binary representations Sept. 11, 2012

Binary numbers The reason humans represent numbers using decimal (the ten digits from 0,1,... 9) is that we have ten fingers. There is no other reason than that. There is nothing special otherwise about

On-Line/Off-Line Digital Signatures

J. Cryptology (996) 9: 35 67 996 International Association for Cryptologic Research On-Line/Off-Line Digital Signatures Shimon Even Computer Science Department, Technion Israel Institute of Technology,

Factoring & Primality

Factoring & Primality Lecturer: Dimitris Papadopoulos In this lecture we will discuss the problem of integer factorization and primality testing, two problems that have been the focus of a great amount

Guaranteed Slowdown, Generalized Encryption Scheme, and Function Sharing

Guaranteed Slowdown, Generalized Encryption Scheme, and Function Sharing Yury Lifshits July 10, 2005 Abstract The goal of the paper is to construct mathematical abstractions of different aspects of real

Linearity and Group Homomorphism Testing / Testing Hadamard Codes

Title: Linearity and Group Homomorphism Testing / Testing Hadamard Codes Name: Sofya Raskhodnikova 1, Ronitt Rubinfeld 2 Affil./Addr. 1: Pennsylvania State University Affil./Addr. 2: Keywords: SumOriWork:

Lecture 9 - Message Authentication Codes

Lecture 9 - Message Authentication Codes Boaz Barak March 1, 2010 Reading: Boneh-Shoup chapter 6, Sections 9.1 9.3. Data integrity Until now we ve only been interested in protecting secrecy of data. However,

princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora

princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora Scribe: One of the running themes in this course is the notion of

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 11

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note Conditional Probability A pharmaceutical company is marketing a new test for a certain medical condition. According

Introduction. Digital Signature

Introduction Electronic transactions and activities taken place over Internet need to be protected against all kinds of interference, accidental or malicious. The general task of the information technology

The sum of d small-bias generators fools polynomials of degree d

The sum of d small-bias generators fools polynomials of degree d Emanuele Viola April 15, 2009 Abstract We prove that the sum of d small-bias generators L : F s F n fools degree-d polynomials in n variables

Non-Black-Box Techniques In Crytpography. Thesis for the Ph.D degree Boaz Barak

Non-Black-Box Techniques In Crytpography Introduction Thesis for the Ph.D degree Boaz Barak A computer program (or equivalently, an algorithm) is a list of symbols a finite string. When we interpret a

MACs Message authentication and integrity Foundations of Cryptography Computer Science Department Wellesley College Table of contents Introduction MACs Constructing Secure MACs Secure communication and

FACTORING LARGE NUMBERS, A GREAT WAY TO SPEND A BIRTHDAY

FACTORING LARGE NUMBERS, A GREAT WAY TO SPEND A BIRTHDAY LINDSEY R. BOSKO I would like to acknowledge the assistance of Dr. Michael Singer. His guidance and feedback were instrumental in completing this

Security Aspects of. Database Outsourcing. Vahid Khodabakhshi Hadi Halvachi. Dec, 2012

Security Aspects of Database Outsourcing Dec, 2012 Vahid Khodabakhshi Hadi Halvachi Security Aspects of Database Outsourcing Security Aspects of Database Outsourcing 2 Outline Introduction to Database

The Goldberg Rao Algorithm for the Maximum Flow Problem

The Goldberg Rao Algorithm for the Maximum Flow Problem COS 528 class notes October 18, 2006 Scribe: Dávid Papp Main idea: use of the blocking flow paradigm to achieve essentially O(min{m 2/3, n 1/2 }

Network Security. Chapter 6 Random Number Generation

Network Security Chapter 6 Random Number Generation 1 Tasks of Key Management (1)! Generation:! It is crucial to security, that keys are generated with a truly random or at least a pseudo-random generation

Influences in low-degree polynomials

Influences in low-degree polynomials Artūrs Bačkurs December 12, 2012 1 Introduction In 3] it is conjectured that every bounded real polynomial has a highly influential variable The conjecture is known

Network Security. Chapter 6 Random Number Generation. Prof. Dr.-Ing. Georg Carle

Network Security Chapter 6 Random Number Generation Prof. Dr.-Ing. Georg Carle Chair for Computer Networks & Internet Wilhelm-Schickard-Institute for Computer Science University of Tübingen http://net.informatik.uni-tuebingen.de/

General Hardness Amplification of Predicates and Puzzles

General Hardness Amplification of Predicates and Puzzles (extended abstract) Thomas Holenstein 1 and Grant Schoenebeck 2 1 Department of Computer Science, ETH Zurich, Switzerland, thomas.holenstein@inf.ethz.ch,

Lecture 13. Lecturer: Yevgeniy Dodis Spring 2012

CSCI-GA.3210-001 MATH-GA.2170-001 Introduction to Cryptography April 18, 2012 Lecture 13 Lecturer: Yevgeniy Dodis Spring 2012 This lecture is dedicated to constructions of digital signature schemes. Assuming

Family Name:... First Name:... Section:... Advanced Cryptography Final Exam July 18 th, 2006 Start at 9:15, End at 12:00 This document consists of 12 pages. Instructions Electronic devices are not allowed.

1 Domain Extension for MACs

CS 127/CSCI E-127: Introduction to Cryptography Prof. Salil Vadhan Fall 2013 Reading. Lecture Notes 17: MAC Domain Extension & Digital Signatures Katz-Lindell Ÿ4.34.4 (2nd ed) and Ÿ12.0-12.3 (1st ed).

Security of Blind Digital Signatures

Security of Blind Digital Signatures (Revised Extended Abstract) Ari Juels 1 Michael Luby 2 Rafail Ostrovsky 3 1 RSA Laboratories. Email: ari@rsa.com. 2 Digital Fountain 3 UCLA, Email: rafail@cs.ucla.edu.

Limits of Computational Differential Privacy in the Client/Server Setting

Limits of Computational Differential Privacy in the Client/Server Setting Adam Groce, Jonathan Katz, and Arkady Yerukhimovich Dept. of Computer Science University of Maryland {agroce, jkatz, arkady}@cs.umd.edu

8.1 Makespan Scheduling

600.469 / 600.669 Approximation Algorithms Lecturer: Michael Dinitz Topic: Dynamic Programing: Min-Makespan and Bin Packing Date: 2/19/15 Scribe: Gabriel Kaptchuk 8.1 Makespan Scheduling Consider an instance

Improved Online/Offline Signature Schemes

Improved Online/Offline Signature Schemes Adi Shamir and Yael Tauman Applied Math. Dept. The Weizmann Institute of Science Rehovot 76100, Israel {shamir,tauman}@wisdom.weizmann.ac.il Abstract. The notion

Polynomial Invariants

Polynomial Invariants Dylan Wilson October 9, 2014 (1) Today we will be interested in the following Question 1.1. What are all the possible polynomials in two variables f(x, y) such that f(x, y) = f(y,

Universal Hash Proofs and a Paradigm for Adaptive Chosen Ciphertext Secure Public-Key Encryption

Universal Hash Proofs and a Paradigm for Adaptive Chosen Ciphertext Secure Public-Key Encryption Ronald Cramer Victor Shoup December 12, 2001 Abstract We present several new and fairly practical public-key

Talk announcement please consider attending! Where: Maurer School of Law, Room 335 When: Thursday, Feb 5, 12PM 1:30PM Speaker: Rafael Pass, Associate Professor, Cornell University, Topic: Reasoning Cryptographically

Lecture 7: Approximation via Randomized Rounding

Lecture 7: Approximation via Randomized Rounding Often LPs return a fractional solution where the solution x, which is supposed to be in {0, } n, is in [0, ] n instead. There is a generic way of obtaining

Lecture 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

(67902) Topics in Theory and Complexity Nov 2, 2006. Lecture 7

(67902) Topics in Theory and Complexity Nov 2, 2006 Lecturer: Irit Dinur Lecture 7 Scribe: Rani Lekach 1 Lecture overview This Lecture consists of two parts In the first part we will refresh the definition

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2 Proofs Intuitively, the concept of proof should already be familiar We all like to assert things, and few of us

Notes on Complexity Theory Last updated: August, 2011. Lecture 1

Notes on Complexity Theory Last updated: August, 2011 Jonathan Katz Lecture 1 1 Turing Machines I assume that most students have encountered Turing machines before. (Students who have not may want to look

MESSAGE AUTHENTICATION IN AN IDENTITY-BASED ENCRYPTION SCHEME: 1-KEY-ENCRYPT-THEN-MAC

MESSAGE AUTHENTICATION IN AN IDENTITY-BASED ENCRYPTION SCHEME: 1-KEY-ENCRYPT-THEN-MAC by Brittanney Jaclyn Amento A Thesis Submitted to the Faculty of The Charles E. Schmidt College of Science in Partial

CIS 6930 Emerging Topics in Network Security. Topic 2. Network Security Primitives

CIS 6930 Emerging Topics in Network Security Topic 2. Network Security Primitives 1 Outline Absolute basics Encryption/Decryption; Digital signatures; D-H key exchange; Hash functions; Application of hash

Breaking The Code. Ryan Lowe. Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and

Breaking The Code Ryan Lowe Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and a minor in Applied Physics. As a sophomore, he took an independent study

9 Digital Signatures: Definition and First Constructions. Hashing.

Leo Reyzin. Notes for BU CAS CS 538. 1 9 Digital Signatures: Definition and First Constructions. Hashing. 9.1 Definition First note that encryption provides no guarantee that a message is authentic. For

I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

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

minimal polyonomial Example

Minimal Polynomials Definition Let α be an element in GF(p e ). We call the monic polynomial of smallest degree which has coefficients in GF(p) and α as a root, the minimal polyonomial of α. Example: We

CIS 5371 Cryptography. 8. Encryption --

CIS 5371 Cryptography p y 8. Encryption -- Asymmetric Techniques Textbook encryption algorithms In this chapter, security (confidentiality) is considered in the following sense: All-or-nothing secrecy.

Definition 11.1. Given a graph G on n vertices, we define the following quantities:

Lecture 11 The Lovász ϑ Function 11.1 Perfect graphs We begin with some background on perfect graphs. graphs. First, we define some quantities on Definition 11.1. Given a graph G on n vertices, we define

6.080 / 6.089 Great Ideas in Theoretical Computer Science Spring 2008

MIT OpenCourseWare http://ocw.mit.edu 6.080 / 6.089 Great Ideas in Theoretical Computer Science Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.

NP-Completeness I. Lecture 19. 19.1 Overview. 19.2 Introduction: Reduction and Expressiveness

Lecture 19 NP-Completeness I 19.1 Overview In the past few lectures we have looked at increasingly more expressive problems that we were able to solve using efficient algorithms. In this lecture we introduce

Proofs in Cryptography

Proofs in Cryptography Ananth Raghunathan Abstract We give a brief overview of proofs in cryptography at a beginners level. We briefly cover a general way to look at proofs in cryptography and briefly

An improved LPN algorithm

An improved LPN algorithm Éric Levieil and Pierre-Alain Fouque École normale supérieure, 45 rue d Ulm, 75230 Paris Cedex 05, France {Eric.Levieil,Pierre-Alain.Fouque}@ens.fr Abstract. HB + is a shared-key

List-Decoding Reed-Muller Codes over Small Fields

List-Decoding Reed-Muller Codes over Small Fields Parikshit Gopalan University of Washington parik@cs.washington.edu Adam R. Klivans UT Austin klivans@cs.utexas.edu David Zuckerman UT Austin diz@cs.utexas.edu

Lecture 1: Introduction. CS 6903: Modern Cryptography Spring 2009. Nitesh Saxena Polytechnic University

Lecture 1: Introduction CS 6903: Modern Cryptography Spring 2009 Nitesh Saxena Polytechnic University Outline Administrative Stuff Introductory Technical Stuff Some Pointers Course Web Page http://isis.poly.edu/courses/cs6903-s10

The Complexity of Online Memory Checking

The Complexity of Online Memory Checking Moni Naor Guy N. Rothblum Abstract We consider the problem of storing a large file on a remote and unreliable server. To verify that the file has not been corrupted,

1 Problem Statement. 2 Overview

Lecture Notes on an FPRAS for Network Unreliability. Eric Vigoda Georgia Institute of Technology Last updated for 7530 - Randomized Algorithms, Spring 200. In this lecture, we will consider the problem

CSC 373: Algorithm Design and Analysis Lecture 16

CSC 373: Algorithm Design and Analysis Lecture 16 Allan Borodin February 25, 2013 Some materials are from Stephen Cook s IIT talk and Keven Wayne s slides. 1 / 17 Announcements and Outline Announcements

CSC2420 Fall 2012: Algorithm Design, Analysis and Theory

CSC2420 Fall 2012: Algorithm Design, Analysis and Theory Allan Borodin November 15, 2012; Lecture 10 1 / 27 Randomized online bipartite matching and the adwords problem. We briefly return to online algorithms

Identity-Based Encryption from the Weil Pairing

Appears in SIAM J. of Computing, Vol. 32, No. 3, pp. 586-615, 2003. An extended abstract of this paper appears in the Proceedings of Crypto 2001, volume 2139 of Lecture Notes in Computer Science, pages

Testing Low-Degree Polynomials over

Testing Low-Degree Polynomials over Noga Alon, Tali Kaufman, Michael Krivelevich, Simon Litsyn, and Dana Ron Department of Mathematics, Tel Aviv University, Tel Aviv 69978, Israel and Institute for Advanced

SECURITY EVALUATION OF EMAIL ENCRYPTION USING RANDOM NOISE GENERATED BY LCG

SECURITY EVALUATION OF EMAIL ENCRYPTION USING RANDOM NOISE GENERATED BY LCG Chung-Chih Li, Hema Sagar R. Kandati, Bo Sun Dept. of Computer Science, Lamar University, Beaumont, Texas, USA 409-880-8748,

Lecture 13 - Basic Number Theory.

Lecture 13 - Basic Number Theory. Boaz Barak March 22, 2010 Divisibility and primes Unless mentioned otherwise throughout this lecture all numbers are non-negative integers. We say that A divides B, denoted

A Sublinear Bipartiteness Tester for Bounded Degree Graphs

A Sublinear Bipartiteness Tester for Bounded Degree Graphs Oded Goldreich Dana Ron February 5, 1998 Abstract We present a sublinear-time algorithm for testing whether a bounded degree graph is bipartite

A New Generic Digital Signature Algorithm

Groups Complex. Cryptol.? (????), 1 16 DOI 10.1515/GCC.????.??? de Gruyter???? A New Generic Digital Signature Algorithm Jennifer Seberry, Vinhbuu To and Dongvu Tonien Abstract. In this paper, we study

THE TURING DEGREES AND THEIR LACK OF LINEAR ORDER

THE TURING DEGREES AND THEIR LACK OF LINEAR ORDER JASPER DEANTONIO Abstract. This paper is a study of the Turing Degrees, which are levels of incomputability naturally arising from sets of natural numbers.

(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

Exponential time algorithms for graph coloring

Exponential time algorithms for graph coloring Uriel Feige Lecture notes, March 14, 2011 1 Introduction Let [n] denote the set {1,..., k}. A k-labeling of vertices of a graph G(V, E) is a function V [k].

Question: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit?

ECS20 Discrete Mathematics Quarter: Spring 2007 Instructor: John Steinberger Assistant: Sophie Engle (prepared by Sophie Engle) Homework 8 Hints Due Wednesday June 6 th 2007 Section 6.1 #16 What is the

Blazing Fast 2PC in the Offline/Online Setting with Security for Malicious Adversaries

Blazing Fast 2PC in the Offline/Online Setting with Security for Malicious Adversaries Yehuda Lindell Ben Riva October 12, 2015 Abstract Recently, several new techniques were presented to dramatically

Message Authentication Code

Message Authentication Code Ali El Kaafarani Mathematical Institute Oxford University 1 of 44 Outline 1 CBC-MAC 2 Authenticated Encryption 3 Padding Oracle Attacks 4 Information Theoretic MACs 2 of 44

How Efficient can Memory Checking be?

How Efficient can Memory Checking be? Cynthia Dwork 1, Moni Naor 2,, Guy N. Rothblum 3,, and Vinod Vaikuntanathan 4, 1 Microsoft Research 2 The Weizmann Institute of Science 3 MIT 4 IBM Research Abstract.

Theorem (informal statement): There are no extendible methods in David Chalmers s sense unless P = NP.

Theorem (informal statement): There are no extendible methods in David Chalmers s sense unless P = NP. Explication: In his paper, The Singularity: A philosophical analysis, David Chalmers defines an extendible

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

CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,