Lecture Introduction. 2 Formal Definition. CS CTT Current Topics in Theoretical CS Oct 30, 2012
|
|
- Wendy McDonald
- 5 years ago
- Views:
Transcription
1 CS CTT Current Topics in Theoretical CS Oct 30, 0 Lecturer: Elena Grigorescu Lecture 9 Scribe: Vivek Patel Introduction In this lecture we study locally decodable codes. Locally decodable codes are error correcting codes that allow very efficient access to encoded data, and in addition they are highly resilient to noise. As a motivating example suppose that we have a list of songs say s, s,... s n that we would like to store so as to ensure both efficient decoding and error resilience. One way to encode this library would be to use an error-correcting code and encode each song separately and store the concatenation of these encodings E(s ), E(s ),..., E(s n ). To recover a song one would only need to access the corresponding block in the database. If a hacker takes out one song from this list then there will be no way of getting that song back. To deal with this possibility one might want instead to concatenate all songs into a string s = s, s, s 3,..., s n and then encode them altogether in a string E(s). However, the drawback now is that the time to recover/decode one song will be proportional to the length of all songs as we now have to decode the whole library. Locally decodable codes help us overcome such scenario by allowing both resilience to errors and fast access to the data. Formal Definition Let F q be a a field of size q (for simplicity you can just think of the more familiar field Z q, where q is a prime and all operations are mod q). Informally, an r-query locally decodable code (LDC) encodes k-symbol messages x into n-symbol messages C(x) in such a way that one can probabilistically recover any symbol x i of the message by querying only r symbols of the (possibly corrupted) codeword C(x), where r is a very small number. Definition (Locally decodable codes) A code C : F k q F n q is a (r, δ, ɛ)- locally decodable if there exists a randomized algorithm A such that. x F k q, iɛ[k], y F n q with d(y, c(x)) δ (where d(y, c(x)) denotes the relative Hamming distance between y and c(x)) we have P r[a(y, i) = x i ] ɛ (that is, A recovers the ith bit of x w.p. ɛ over its random coins), and. A reads at most r coordinates of the received vector y The ideal settings are n = O(k), r = o(n). Some particularly interesting settings for cryptographic applications and private information retrieval schemes is when r =, 3, 4. Building LDCs of small rate is interesting for all ranges of r up to O(n). For binary codes δ < /4. The running time of the decoder is poly(r, log n) which is asymptotically much smaller then poly(n), the time to decode the entire received word. A related and stronger notion of local decoding is that of local correction. Here we disregard the message that we started with and only focus on its actual encoding, and we want to
2 recover each bit of the encoding from a possibly corrupted received word. Recall that by definition a codeword of a systematic code contains the actual message itself (together with some redundancy), so for systematic codes local correction implies local decoding. Also recall that every linear code is systematic. In this lecture we only look at locally correctable linear codes and so we ll use the terms locally decodable/correctable interchangeably. Definition (Locally correctable codes (LCCs)) A code C F n q is a (r, δ, ɛ)- locally correctable if there exists a randomized algorithm A such that. c C i [n], and vector y F n q such that d(y, c) δ (again, d(y, c) is the relative Hamming distance between y and c) we have Pr[A(y, i) = c i ] ɛ.. A makes only r queries into y 3 A local decoder for the Hadamard Code Recall that the Hadamard code is Had = {l a : {0, } k {0, }, l a (x) = a x mod a {0, } k }. In other words, a binary message a = (a,..., a k ) is encoded as (a x 0, a x,..., a x k ) = l a (x) x F k. To locally decode/correct Had means to provide an algorithm that makes a constant number of queries and is able to output l a (b) for any value of b {0, } k, when we have access to a received word that has a δ fraction of error compared to l a (x) x F k. Theorem 3 Had is (, δ, δ)-locally decodable. Proof Recall that since l a is a linear function we have l a (b) = l a (b + c) + l a (c) c F k. The intuition is that if not too many values of l a (x) got corrupted, we can count the votes of each c F k for the value of l a(b) = l a (b + c) + l a (c) and output the majority. This idea gives us the basic decoder below. Algorithm Local decoding of the Hadamard Code Input: A function f, such that d(l a, f) δ, and b F k. Goal: output l a(b).. Pick c uniformly at random from F k. Output: f(b + c) + f(c). We can now analyse this decoder. Since the distance between f and l a is δ we have that Pr c [f(c) l a (c)] δ. For any fixed b, if c is chosen uniformly from F k we have that b + c is uniformly distributed in F k so, we also have Pr[f(c + b) l a(c + b)] δ. From the Union Bound Pr[l a (b)] = f(b + c) + f(c)] δ. So we have shown an example of a code for which there is a local decoder with optimal query complexity ( bits) but which has terrible rate (the codeword length is exponential in the message length). We will next see an important family of LDCs with much better rate (only a poly blowup in message length) yet the query complexity is a constant.
3 4 Reed Muller Codes Reed-Muller codes (RM) are multivariate extensions of Reed-Solomon codes, which we ve seen in a previous lecture. Most of the new families of LDCs are generalizations of RM codes. Informally, RM(m, l) is the code consisting of evaluations of m-variate polynomials of degree l over F q. Definition 4 Let m, l be positive integers and F q a finite field, and l q. RM(m, l) = { p(α) α F m q p F q [x,..., x m ], deg(p) l}. For example the polynomials p(x, x ) = x 3 + 3x is a polynomial in variables over the field F 5 = Z 5 and has total degree of 3. The codeword corresponding to it can also be thought as the evaluation of p at every point in Z Parameters of RM Dimension (message length) We can think of an RM code as a generalization of RS in the following sense. Recall that a RS codeword was an encoding of a message (m 0,..., m k ) F k q into the codeword m(α) α Fq, where m(x) = k i=0 m ix i. Similarly, a RM codeword encodes a message whose coordinates are viewed as the the coefficients of a degree l polynomial in m variables. Hence the dimension( of the) code is the number of possible monomials of such m + l polynomials, which turns out to be. To see this, note that a monomial of degree l is x d l x d xd m m with m d i = l and so we are asking about the size of the set {(d, d,..., d m ) d i = l}, which ( is the ) number ( of) ways one can place m delimiters between l units, which is easily seen m + l m + l to be =. m l Block length By definition this is q m. Distance ( l/q)q m i.e. relative distance l/q. This is obtained from the following useful lemma about the number of roots of a multivariate polynomials over a finite field. Lemma 5 (Schwartz-Zippel) Let p F q [x,..., x m ] be a poly of total degree l. Then the number of x F m q s.t. p(x) = 0 is at most lq m. Since RM is a linear code, its minimum distance is the weight of a minimum weight codeword (say given by a polynomial p). Therefore this is q m {x p(x) = 0} = q m ( l/q). ( ) m + l Useful settings for the parameters So RM(l, m) is a [n, k, d] code where k =, l n = q m, d = ( l/q)q m. Expressing everything in terms of k, we may choose m = log k/ log log k, q = log k, and so n = k, and it follows that l < log k log log k << q, and so d > ( log log k )n. 3
4 5 A local decoder for RM codes Suppose f : F m q F q is the function to which the algorithm has oracle access and g RM(l, m) is s.t. d(f, g) = δ (so, x : g(x) f(x)] = δq m ). A local decoder for RM is required that, on input a F m q it outputs g(a) in time poly(m, l, q) (w.h.p.). The idea is to query points that are structured in a way that is specific to RM codes. Recall that for the Hadamard code, any tuple of points (b, c, b + c) was always satisfying the pattern h(b) = h(c) + h(b + c) (where h Had). It turns out that a similar pattern characterizes higher degree polynomials, but this time the pattern is more complicated and the points that give us the useful structure form a so-called line in F m q. Definition 6 A line in F m q is defined by a F m q and b F m q 0 and is given by the collection of points L a,b = {a + bt t F q }. We will sometimes view the line as a function L a,b : F m q F m q, L a,b (t) = a + bt. Before proceeding with the decoder we state some useful facts. Proposition 7 For any a F m q, t F q if b is uniform over F m q then L a,b (t) is uniform in F m q. Definition 8 If f : F m q F q then the restriction of f to line L = L a,b is the function f L : F q F q, f L (t) = f(l(t)). Proposition 9 If p F q [x,..., x m ] is a polynomial of degree l then p L = p L (i.e. p L (t) = p(l(t)) t) is a polynomial in one variable of degree l. The following proposition is the main tool in the proof of the correctness of the local decoder that we will present. It indirectly gives a way of characterizing degree l polynomials by l + points on a line, which will be the query complexity of a decoder for RM codes. Proposition 0 If g F q [x] is a degree l polynomial and for l+ distinct values α 0, α,..., α l F q we know g(α i ) = β i, then g can be recovered exactly at any point α F q. Theorem RM q (l, m) is (l +, 3(l+), 3 )-locally decodable. Proof Algorithm A basic local decoder for RM Code Input: A function f, such that d(g, f) δ, for some g RM(l, m) and a F m. Goal: output g(a).. Pick random direction b F m q {0} uniformly at random from F m and let L a,b(t) = a + bt.. Let α 0,..., α l be distinct elements in F q and let β i = f(l a,b (α i )) = f(a + α i b). 3. Interpolate (using Proposition 0) to find the unique polynomial of degree l s.t. g(α i ) = β i, i. 4. Output g(0) (which should be the corrected value of f(a) = f(a + 0 b)) By Proposition 7 a, if b is chosen uniformly at random a + bt is a uniform point in F m q for any t. So Pr[f(a + α i b) g(a + α i b)] 3(l+) by assumption, for every i {0,,..., l}. Again, 4
5 by a union bound Pr[ some α i s.t. f(a+α i b] g(a+α i b)] (l +) 3(l+) = /3, and so all the queried points are correct w.p. /3. When that is the case the interpolation step successfully finds the unique polynomials g that agrees with f on the queried points. Notice that the amount of error that the above decodes can recover from (i.e. 3(l+) ) degrades with the degree l. More sophisticated analyses can however show that RM codes are locally decodable from a constant fraction of error (unambiguously, even from /4-fraction of error). 5
Lecture Introduction. 2 Linear codes. CS CTT Current Topics in Theoretical CS Oct 4, 2012
CS 59000 CTT Current Topics in Theoretical CS Oct 4, 01 Lecturer: Elena Grigorescu Lecture 14 Scribe: Selvakumaran Vadivelmurugan 1 Introduction We introduced error-correcting codes and linear codes in
More informationLocally Decodable Codes
Foundations and Trends R in sample Vol. xx, No xx (xxxx) 1 114 c xxxx xxxxxxxxx DOI: xxxxxx Locally Decodable Codes Sergey Yekhanin 1 1 Microsoft Research Silicon Valley, 1065 La Avenida, Mountain View,
More informationLecture 19 : Reed-Muller, Concatenation Codes & Decoding problem
IITM-CS6845: Theory Toolkit February 08, 2012 Lecture 19 : Reed-Muller, Concatenation Codes & Decoding problem Lecturer: Jayalal Sarma Scribe: Dinesh K Theme: Error correcting codes In the previous lecture,
More informationLecture 3: Error Correcting Codes
CS 880: Pseudorandomness and Derandomization 1/30/2013 Lecture 3: Error Correcting Codes Instructors: Holger Dell and Dieter van Melkebeek Scribe: Xi Wu In this lecture we review some background on error
More informationLocally Decodable Codes
Foundations and Trends R in sample Vol. xx, No xx (xxxx) 1 98 c xxxx xxxxxxxxx DOI: xxxxxx Locally Decodable Codes Sergey Yekhanin 1 1 Microsoft Research Silicon Valley, 1065 La Avenida, Mountain View,
More informationCS151 Complexity Theory. Lecture 9 May 1, 2017
CS151 Complexity Theory Lecture 9 Hardness vs. randomness We have shown: If one-way permutations exist then BPP δ>0 TIME(2 nδ ) ( EXP simulation is better than brute force, but just barely stronger assumptions
More informationTutorial: Locally decodable codes. UT Austin
Tutorial: Locally decodable codes Anna Gál UT Austin Locally decodable codes Error correcting codes with extra property: Recover (any) one message bit, by reading only a small number of codeword bits.
More informationLecture 12: November 6, 2017
Information and Coding Theory Autumn 017 Lecturer: Madhur Tulsiani Lecture 1: November 6, 017 Recall: We were looking at codes of the form C : F k p F n p, where p is prime, k is the message length, and
More informationDecoding Reed-Muller codes over product sets
Rutgers University May 30, 2016 Overview Error-correcting codes 1 Error-correcting codes Motivation 2 Reed-Solomon codes Reed-Muller codes 3 Error-correcting codes Motivation Goal: Send a message Don t
More information6.895 PCP and Hardness of Approximation MIT, Fall Lecture 3: Coding Theory
6895 PCP and Hardness of Approximation MIT, Fall 2010 Lecture 3: Coding Theory Lecturer: Dana Moshkovitz Scribe: Michael Forbes and Dana Moshkovitz 1 Motivation In the course we will make heavy use of
More informationLecture 4: Codes based on Concatenation
Lecture 4: Codes based on Concatenation Error-Correcting Codes (Spring 206) Rutgers University Swastik Kopparty Scribe: Aditya Potukuchi and Meng-Tsung Tsai Overview In the last lecture, we studied codes
More informationLecture 03: Polynomial Based Codes
Lecture 03: Polynomial Based Codes Error-Correcting Codes (Spring 016) Rutgers University Swastik Kopparty Scribes: Ross Berkowitz & Amey Bhangale 1 Reed-Solomon Codes Reed Solomon codes are large alphabet
More informationError Correcting Codes Questions Pool
Error Correcting Codes Questions Pool Amnon Ta-Shma and Dean Doron January 3, 018 General guidelines The questions fall into several categories: (Know). (Mandatory). (Bonus). Make sure you know how to
More informationError Correcting Codes: Combinatorics, Algorithms and Applications Spring Homework Due Monday March 23, 2009 in class
Error Correcting Codes: Combinatorics, Algorithms and Applications Spring 2009 Homework Due Monday March 23, 2009 in class You can collaborate in groups of up to 3. However, the write-ups must be done
More information1 Randomized Computation
CS 6743 Lecture 17 1 Fall 2007 1 Randomized Computation Why is randomness useful? Imagine you have a stack of bank notes, with very few counterfeit ones. You want to choose a genuine bank note to pay at
More informationLecture B04 : Linear codes and singleton bound
IITM-CS6845: Theory Toolkit February 1, 2012 Lecture B04 : Linear codes and singleton bound Lecturer: Jayalal Sarma Scribe: T Devanathan We start by proving a generalization of Hamming Bound, which we
More informationNotes 10: List Decoding Reed-Solomon Codes and Concatenated codes
Introduction to Coding Theory CMU: Spring 010 Notes 10: List Decoding Reed-Solomon Codes and Concatenated codes April 010 Lecturer: Venkatesan Guruswami Scribe: Venkat Guruswami & Ali Kemal Sinop DRAFT
More information: Error Correcting Codes. October 2017 Lecture 1
03683072: Error Correcting Codes. October 2017 Lecture 1 First Definitions and Basic Codes Amnon Ta-Shma and Dean Doron 1 Error Correcting Codes Basics Definition 1. An (n, K, d) q code is a subset of
More informationError Detection and Correction: Hamming Code; Reed-Muller Code
Error Detection and Correction: Hamming Code; Reed-Muller Code Greg Plaxton Theory in Programming Practice, Spring 2005 Department of Computer Science University of Texas at Austin Hamming Code: Motivation
More informationBasic Probabilistic Checking 3
CS294: Probabilistically Checkable and Interactive Proofs February 21, 2017 Basic Probabilistic Checking 3 Instructor: Alessandro Chiesa & Igor Shinkar Scribe: Izaak Meckler Today we prove the following
More informationHigh-rate codes with sublinear-time decoding
High-rate codes with sublinear-time decoding Swastik Kopparty Shubhangi Saraf Sergey Yekhanin September 23, 2010 Abstract Locally decodable codes are error-correcting codes that admit efficient decoding
More informationLocality in Coding Theory
Locality in Coding Theory Madhu Sudan Harvard April 9, 2016 Skoltech: Locality in Coding Theory 1 Error-Correcting Codes (Linear) Code CC FF qq nn. FF qq : Finite field with qq elements. nn block length
More informationLecture 8 (Notes) 1. The book Computational Complexity: A Modern Approach by Sanjeev Arora and Boaz Barak;
Topics in Theoretical Computer Science April 18, 2016 Lecturer: Ola Svensson Lecture 8 (Notes) Scribes: Ola Svensson Disclaimer: These notes were written for the lecturer only and may contain inconsistent
More informationHardness Amplification
Hardness Amplification Synopsis 1. Yao s XOR Lemma 2. Error Correcting Code 3. Local Decoding 4. Hardness Amplification Using Local Decoding 5. List Decoding 6. Local List Decoding 7. Hardness Amplification
More information1 The Low-Degree Testing Assumption
Advanced Complexity Theory Spring 2016 Lecture 17: PCP with Polylogarithmic Queries and Sum Check Prof. Dana Moshkovitz Scribes: Dana Moshkovitz & Michael Forbes Scribe Date: Fall 2010 In this lecture
More informationLecture 21: P vs BPP 2
Advanced Complexity Theory Spring 206 Prof. Dana Moshkovitz Lecture 2: P vs BPP 2 Overview In the previous lecture, we began our discussion of pseudorandomness. We presented the Blum- Micali definition
More informationGreat Theoretical Ideas in Computer Science
15-251 Great Theoretical Ideas in Computer Science Polynomials, Lagrange, and Error-correction Lecture 23 (November 10, 2009) P(X) = X 3 X 2 + + X 1 + Definition: Recall: Fields A field F is a set together
More informationQuantum algorithms (CO 781/CS 867/QIC 823, Winter 2013) Andrew Childs, University of Waterloo LECTURE 13: Query complexity and the polynomial method
Quantum algorithms (CO 781/CS 867/QIC 823, Winter 2013) Andrew Childs, University of Waterloo LECTURE 13: Query complexity and the polynomial method So far, we have discussed several different kinds of
More informationLecture 7 September 24
EECS 11: Coding for Digital Communication and Beyond Fall 013 Lecture 7 September 4 Lecturer: Anant Sahai Scribe: Ankush Gupta 7.1 Overview This lecture introduces affine and linear codes. Orthogonal signalling
More informationLecture 6. k+1 n, wherein n =, is defined for a given
(67611) Advanced Topics in Complexity: PCP Theory November 24, 2004 Lecturer: Irit Dinur Lecture 6 Scribe: Sharon Peri Abstract In this lecture we continue our discussion of locally testable and locally
More informationLecture 9: List decoding Reed-Solomon and Folded Reed-Solomon codes
Lecture 9: List decoding Reed-Solomon and Folded Reed-Solomon codes Error-Correcting Codes (Spring 2016) Rutgers University Swastik Kopparty Scribes: John Kim and Pat Devlin 1 List decoding review Definition
More informationCSCI-B609: A Theorist s Toolkit, Fall 2016 Oct 4. Theorem 1. A non-zero, univariate polynomial with degree d has at most d roots.
CSCI-B609: A Theorist s Toolkit, Fall 2016 Oct 4 Lecture 14: Schwartz-Zippel Lemma and Intro to Coding Theory Lecturer: Yuan Zhou Scribe: Haoyu Zhang 1 Roots of polynomials Theorem 1. A non-zero, univariate
More information2 Completing the Hardness of approximation of Set Cover
CSE 533: The PCP Theorem and Hardness of Approximation (Autumn 2005) Lecture 15: Set Cover hardness and testing Long Codes Nov. 21, 2005 Lecturer: Venkat Guruswami Scribe: Atri Rudra 1 Recap We will first
More informationLecture 9 - One Way Permutations
Lecture 9 - One Way Permutations Boaz Barak October 17, 2007 From time immemorial, humanity has gotten frequent, often cruel, reminders that many things are easier to do than to reverse. Leonid Levin Quick
More informationList Decoding of Reed Solomon Codes
List Decoding of Reed Solomon Codes p. 1/30 List Decoding of Reed Solomon Codes Madhu Sudan MIT CSAIL Background: Reliable Transmission of Information List Decoding of Reed Solomon Codes p. 2/30 List Decoding
More informationEE 229B ERROR CONTROL CODING Spring 2005
EE 229B ERROR CONTROL CODING Spring 2005 Solutions for Homework 1 1. Is there room? Prove or disprove : There is a (12,7) binary linear code with d min = 5. If there were a (12,7) binary linear code with
More informationHigh-Rate Codes with Sublinear-Time Decoding
High-Rate Codes with Sublinear-Time Decoding Swastik Kopparty Institute for Advanced Study swastik@ias.edu Shubhangi Saraf MIT shibs@mit.edu Sergey Yekhanin Microsoft Research yekhanin@microsoft.com ABSTRACT
More informationTwo Query PCP with Sub-Constant Error
Electronic Colloquium on Computational Complexity, Report No 71 (2008) Two Query PCP with Sub-Constant Error Dana Moshkovitz Ran Raz July 28, 2008 Abstract We show that the N P-Complete language 3SAT has
More informationLecture 11: Quantum Information III - Source Coding
CSCI5370 Quantum Computing November 25, 203 Lecture : Quantum Information III - Source Coding Lecturer: Shengyu Zhang Scribe: Hing Yin Tsang. Holevo s bound Suppose Alice has an information source X that
More informationQuestions Pool. Amnon Ta-Shma and Dean Doron. January 2, Make sure you know how to solve. Do not submit.
Questions Pool Amnon Ta-Shma and Dean Doron January 2, 2017 General guidelines The questions fall into several categories: (Know). (Mandatory). (Bonus). Make sure you know how to solve. Do not submit.
More informationNotes for Lecture 18
U.C. Berkeley Handout N18 CS294: Pseudorandomness and Combinatorial Constructions November 1, 2005 Professor Luca Trevisan Scribe: Constantinos Daskalakis Notes for Lecture 18 1 Basic Definitions In the
More informationLecture 24: Goldreich-Levin Hardcore Predicate. Goldreich-Levin Hardcore Predicate
Lecture 24: : Intuition A One-way Function: A function that is easy to compute but hard to invert (efficiently) Hardcore-Predicate: A secret bit that is hard to compute Theorem (Goldreich-Levin) If f :
More informationLecture 12: Reed-Solomon Codes
Error Correcting Codes: Combinatorics, Algorithms and Applications (Fall 007) Lecture 1: Reed-Solomon Codes September 8, 007 Lecturer: Atri Rudra Scribe: Michel Kulhandjian Last lecture we saw the proof
More informationNotes 3: Stochastic channels and noisy coding theorem bound. 1 Model of information communication and noisy channel
Introduction to Coding Theory CMU: Spring 2010 Notes 3: Stochastic channels and noisy coding theorem bound January 2010 Lecturer: Venkatesan Guruswami Scribe: Venkatesan Guruswami We now turn to the basic
More informationNotes for the Hong Kong Lectures on Algorithmic Coding Theory. Luca Trevisan. January 7, 2007
Notes for the Hong Kong Lectures on Algorithmic Coding Theory Luca Trevisan January 7, 2007 These notes are excerpted from the paper Some Applications of Coding Theory in Computational Complexity [Tre04].
More information6.842 Randomness and Computation Lecture 5
6.842 Randomness and Computation 2012-02-22 Lecture 5 Lecturer: Ronitt Rubinfeld Scribe: Michael Forbes 1 Overview Today we will define the notion of a pairwise independent hash function, and discuss its
More informationLecture 29: Computational Learning Theory
CS 710: Complexity Theory 5/4/2010 Lecture 29: Computational Learning Theory Instructor: Dieter van Melkebeek Scribe: Dmitri Svetlov and Jake Rosin Today we will provide a brief introduction to computational
More informationEfficiently decodable codes for the binary deletion channel
Efficiently decodable codes for the binary deletion channel Venkatesan Guruswami (venkatg@cs.cmu.edu) Ray Li * (rayyli@stanford.edu) Carnegie Mellon University August 18, 2017 V. Guruswami and R. Li (CMU)
More informationIP = PSPACE using Error Correcting Codes
Electronic Colloquium on Computational Complexity, Report No. 137 (2010 IP = PSPACE using Error Correcting Codes Or Meir Abstract The IP theorem, which asserts that IP = PSPACE (Lund et. al., and Shamir,
More informationDiscrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Discussion 6A Solution
CS 70 Discrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Discussion 6A Solution 1. Polynomial intersections Find (and prove) an upper-bound on the number of times two distinct degree
More informationComputational Complexity: A Modern Approach
i Computational Complexity: A Modern Approach Sanjeev Arora and Boaz Barak Princeton University http://www.cs.princeton.edu/theory/complexity/ complexitybook@gmail.com Not to be reproduced or distributed
More informationThree Query Locally Decodable Codes with Higher Correctness Require Exponential Length
Three Query Locally Decodable Codes with Higher Correctness Require Exponential Length Anna Gál UT Austin panni@cs.utexas.edu Andrew Mills UT Austin amills@cs.utexas.edu March 8, 20 Abstract Locally decodable
More informationLecture 24: Randomized Complexity, Course Summary
6.045 Lecture 24: Randomized Complexity, Course Summary 1 1/4 1/16 1/4 1/4 1/32 1/16 1/32 Probabilistic TMs 1/16 A probabilistic TM M is a nondeterministic TM where: Each nondeterministic step is called
More informationLecture Examples of problems which have randomized algorithms
6.841 Advanced Complexity Theory March 9, 2009 Lecture 10 Lecturer: Madhu Sudan Scribe: Asilata Bapat Meeting to talk about final projects on Wednesday, 11 March 2009, from 5pm to 7pm. Location: TBA. Includes
More informationLocally testable and Locally correctable Codes Approaching the Gilbert-Varshamov Bound
Electronic Colloquium on Computational Complexity, Report No. 1 (016 Locally testable and Locally correctable Codes Approaching the Gilbert-Varshamov Bound Sivakanth Gopi Swastik Kopparty Rafael Oliveira
More informationChapter 6 Reed-Solomon Codes. 6.1 Finite Field Algebra 6.2 Reed-Solomon Codes 6.3 Syndrome Based Decoding 6.4 Curve-Fitting Based Decoding
Chapter 6 Reed-Solomon Codes 6. Finite Field Algebra 6. Reed-Solomon Codes 6.3 Syndrome Based Decoding 6.4 Curve-Fitting Based Decoding 6. Finite Field Algebra Nonbinary codes: message and codeword symbols
More informationA list-decodable code with local encoding and decoding
A list-decodable code with local encoding and decoding Marius Zimand Towson University Department of Computer and Information Sciences Baltimore, MD http://triton.towson.edu/ mzimand Abstract For arbitrary
More informationLinear Algebra. F n = {all vectors of dimension n over field F} Linear algebra is about vectors. Concretely, vectors look like this:
15-251: Great Theoretical Ideas in Computer Science Lecture 23 Linear Algebra Linear algebra is about vectors. Concretely, vectors look like this: They are arrays of numbers. fig. by Peter Dodds # of numbers,
More informationCS294: Pseudorandomness and Combinatorial Constructions September 13, Notes for Lecture 5
UC Berkeley Handout N5 CS94: Pseudorandomness and Combinatorial Constructions September 3, 005 Professor Luca Trevisan Scribe: Gatis Midrijanis Notes for Lecture 5 In the few lectures we are going to look
More information1 Vandermonde matrices
ECE 771 Lecture 6 BCH and RS codes: Designer cyclic codes Objective: We will begin with a result from linear algebra regarding Vandermonde matrices This result is used to prove the BCH distance properties,
More informationReed-Muller Codes. Sebastian Raaphorst Carleton University
Reed-Muller Codes Sebastian Raaphorst Carleton University May 9, 2003 Abstract This paper examines the family of codes known as Reed-Muller codes. We begin by briefly introducing the codes and their history
More informationCS151 Complexity Theory. Lecture 14 May 17, 2017
CS151 Complexity Theory Lecture 14 May 17, 2017 IP = PSPACE Theorem: (Shamir) IP = PSPACE Note: IP PSPACE enumerate all possible interactions, explicitly calculate acceptance probability interaction extremely
More informationLocal correctability of expander codes
Local correctability of expander codes Brett Hemenway Rafail Ostrovsky Mary Wootters IAS April 4, 24 The point(s) of this talk Locally decodable codes are codes which admit sublinear time decoding of small
More informationMATH3302 Coding Theory Problem Set The following ISBN was received with a smudge. What is the missing digit? x9139 9
Problem Set 1 These questions are based on the material in Section 1: Introduction to coding theory. You do not need to submit your answers to any of these questions. 1. The following ISBN was received
More informationThe Tensor Product of Two Codes is Not Necessarily Robustly Testable
The Tensor Product of Two Codes is Not Necessarily Robustly Testable Paul Valiant Massachusetts Institute of Technology pvaliant@mit.edu Abstract. There has been significant interest lately in the task
More informationLecture 4 : Quest for Structure in Counting Problems
CS6840: Advanced Complexity Theory Jan 10, 2012 Lecture 4 : Quest for Structure in Counting Problems Lecturer: Jayalal Sarma M.N. Scribe: Dinesh K. Theme: Between P and PSPACE. Lecture Plan:Counting problems
More informationIMPROVING THE ALPHABET-SIZE IN EXPANDER BASED CODE CONSTRUCTIONS
IMPROVING THE ALPHABET-SIZE IN EXPANDER BASED CODE CONSTRUCTIONS 1 Abstract Various code constructions use expander graphs to improve the error resilience. Often the use of expanding graphs comes at the
More informationPCP Theorem and Hardness of Approximation
PCP Theorem and Hardness of Approximation An Introduction Lee Carraher and Ryan McGovern Department of Computer Science University of Cincinnati October 27, 2003 Introduction Assuming NP P, there are many
More informationList decoding of binary Goppa codes and key reduction for McEliece s cryptosystem
List decoding of binary Goppa codes and key reduction for McEliece s cryptosystem Morgan Barbier morgan.barbier@lix.polytechnique.fr École Polytechnique INRIA Saclay - Île de France 14 April 2011 University
More informationLecture 11: Polar codes construction
15-859: Information Theory and Applications in TCS CMU: Spring 2013 Lecturer: Venkatesan Guruswami Lecture 11: Polar codes construction February 26, 2013 Scribe: Dan Stahlke 1 Polar codes: recap of last
More informationfor some error exponent E( R) as a function R,
. Capacity-achieving codes via Forney concatenation Shannon s Noisy Channel Theorem assures us the existence of capacity-achieving codes. However, exhaustive search for the code has double-exponential
More informationList and local error-correction
List and local error-correction Venkatesan Guruswami Carnegie Mellon University 8th North American Summer School of Information Theory (NASIT) San Diego, CA August 11, 2015 Venkat Guruswami (CMU) List
More informationLecture 8: Shannon s Noise Models
Error Correcting Codes: Combinatorics, Algorithms and Applications (Fall 2007) Lecture 8: Shannon s Noise Models September 14, 2007 Lecturer: Atri Rudra Scribe: Sandipan Kundu& Atri Rudra Till now we have
More informationLecture 2 Linear Codes
Lecture 2 Linear Codes 2.1. Linear Codes From now on we want to identify the alphabet Σ with a finite field F q. For general codes, introduced in the last section, the description is hard. For a code of
More informationLecture 12: Interactive Proofs
princeton university cos 522: computational complexity Lecture 12: Interactive Proofs Lecturer: Sanjeev Arora Scribe:Carl Kingsford Recall the certificate definition of NP. We can think of this characterization
More informationThe BCH Bound. Background. Parity Check Matrix for BCH Code. Minimum Distance of Cyclic Codes
S-723410 BCH and Reed-Solomon Codes 1 S-723410 BCH and Reed-Solomon Codes 3 Background The algebraic structure of linear codes and, in particular, cyclic linear codes, enables efficient encoding and decoding
More informationLecture 11: Key Agreement
Introduction to Cryptography 02/22/2018 Lecture 11: Key Agreement Instructor: Vipul Goyal Scribe: Francisco Maturana 1 Hardness Assumptions In order to prove the security of cryptographic primitives, we
More informationNotes for Lecture 11
Stanford University CS254: Computational Complexity Notes 11 Luca Trevisan 2/11/2014 Notes for Lecture 11 Circuit Lower Bounds for Parity Using Polynomials In this lecture we prove a lower bound on the
More informationLecture 15: Conditional and Joint Typicaility
EE376A Information Theory Lecture 1-02/26/2015 Lecture 15: Conditional and Joint Typicaility Lecturer: Kartik Venkat Scribe: Max Zimet, Brian Wai, Sepehr Nezami 1 Notation We always write a sequence of
More informationCS Communication Complexity: Applications and New Directions
CS 2429 - Communication Complexity: Applications and New Directions Lecturer: Toniann Pitassi 1 Introduction In this course we will define the basic two-party model of communication, as introduced in the
More informationGoldreich-Levin Hardcore Predicate. Lecture 28: List Decoding Hadamard Code and Goldreich-L
Lecture 28: List Decoding Hadamard Code and Goldreich-Levin Hardcore Predicate Recall Let H : {0, 1} n {+1, 1} Let: L ε = {S : χ S agrees with H at (1/2 + ε) fraction of points} Given oracle access to
More informationArrangements, matroids and codes
Arrangements, matroids and codes first lecture Ruud Pellikaan joint work with Relinde Jurrius ACAGM summer school Leuven Belgium, 18 July 2011 References 2/43 1. Codes, arrangements and matroids by Relinde
More informationLecture 14: Cryptographic Hash Functions
CSE 599b: Cryptography (Winter 2006) Lecture 14: Cryptographic Hash Functions 17 February 2006 Lecturer: Paul Beame Scribe: Paul Beame 1 Hash Function Properties A hash function family H = {H K } K K is
More informationLecture 19: Elias-Bassalygo Bound
Error Correcting Codes: Combinatorics, Algorithms and Applications (Fall 2007) Lecturer: Atri Rudra Lecture 19: Elias-Bassalygo Bound October 10, 2007 Scribe: Michael Pfetsch & Atri Rudra In the last lecture,
More informationArthur-Merlin Streaming Complexity
Weizmann Institute of Science Joint work with Ran Raz Data Streams The data stream model is an abstraction commonly used for algorithms that process network traffic using sublinear space. A data stream
More informationLecture 8: Channel and source-channel coding theorems; BEC & linear codes. 1 Intuitive justification for upper bound on channel capacity
5-859: Information Theory and Applications in TCS CMU: Spring 23 Lecture 8: Channel and source-channel coding theorems; BEC & linear codes February 7, 23 Lecturer: Venkatesan Guruswami Scribe: Dan Stahlke
More information20.1 2SAT. CS125 Lecture 20 Fall 2016
CS125 Lecture 20 Fall 2016 20.1 2SAT We show yet another possible way to solve the 2SAT problem. Recall that the input to 2SAT is a logical expression that is the conunction (AND) of a set of clauses,
More informationImproving the Alphabet Size in Expander Based Code Constructions
Tel Aviv University Raymond and Beverly Sackler Faculty of Exact Sciences School of Computer Sciences Improving the Alphabet Size in Expander Based Code Constructions Submitted as a partial fulfillment
More informationBounding the number of affine roots
with applications in reliable and secure communication Inaugural Lecture, Aalborg University, August 11110, 11111100000 with applications in reliable and secure communication Polynomials: F (X ) = 2X 2
More informationThe number of message symbols encoded into a
L.R.Welch THE ORIGINAL VIEW OF REED-SOLOMON CODES THE ORIGINAL VIEW [Polynomial Codes over Certain Finite Fields, I.S.Reed and G. Solomon, Journal of SIAM, June 1960] Parameters: Let GF(2 n ) be the eld
More informationComputing Error Distance of Reed-Solomon Codes
Computing Error Distance of Reed-Solomon Codes Guizhen Zhu Institute For Advanced Study Tsinghua University, Beijing, 100084, PR China Email:zhugz08@mailstsinghuaeducn Daqing Wan Department of Mathematics
More informationLecture 10 Oct. 3, 2017
CS 388R: Randomized Algorithms Fall 2017 Lecture 10 Oct. 3, 2017 Prof. Eric Price Scribe: Jianwei Chen, Josh Vekhter NOTE: THESE NOTES HAVE NOT BEEN EDITED OR CHECKED FOR CORRECTNESS 1 Overview In the
More informationDiscrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Note 8
CS 70 Discrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Note 8 Polynomials Polynomials constitute a rich class of functions which are both easy to describe and widely applicable in
More informationOn the NP-Hardness of Bounded Distance Decoding of Reed-Solomon Codes
On the NP-Hardness of Bounded Distance Decoding of Reed-Solomon Codes Venkata Gandikota Purdue University vgandiko@purdue.edu Badih Ghazi MIT badih@mit.edu Elena Grigorescu Purdue University elena-g@purdue.edu
More informationGuess & Check Codes for Deletions, Insertions, and Synchronization
Guess & Check Codes for Deletions, Insertions, and Synchronization Serge Kas Hanna, Salim El Rouayheb ECE Department, Rutgers University sergekhanna@rutgersedu, salimelrouayheb@rutgersedu arxiv:759569v3
More informationB(w, z, v 1, v 2, v 3, A(v 1 ), A(v 2 ), A(v 3 )).
Lecture 13 PCP Continued Last time we began the proof of the theorem that PCP(poly, poly) = NEXP. May 13, 2004 Lecturer: Paul Beame Notes: Tian Sang We showed that IMPLICIT-3SAT is NEXP-complete where
More informationNotes for Lecture 3... x 4
Stanford University CS254: Computational Complexity Notes 3 Luca Trevisan January 18, 2012 Notes for Lecture 3 In this lecture we introduce the computational model of boolean circuits and prove that polynomial
More informationAnd for polynomials with coefficients in F 2 = Z/2 Euclidean algorithm for gcd s Concept of equality mod M(x) Extended Euclid for inverses mod M(x)
Outline Recall: For integers Euclidean algorithm for finding gcd s Extended Euclid for finding multiplicative inverses Extended Euclid for computing Sun-Ze Test for primitive roots And for polynomials
More informationReport on PIR with Low Storage Overhead
Report on PIR with Low Storage Overhead Ehsan Ebrahimi Targhi University of Tartu December 15, 2015 Abstract Private information retrieval (PIR) protocol, introduced in 1995 by Chor, Goldreich, Kushilevitz
More informationA Public Key Encryption Scheme Based on the Polynomial Reconstruction Problem
A Public Key Encryption Scheme Based on the Polynomial Reconstruction Problem Daniel Augot and Matthieu Finiasz INRIA, Domaine de Voluceau F-78153 Le Chesnay CEDEX Abstract. The Polynomial Reconstruction
More information