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List decoding and pseudorandom constructions: lossless expanders and extractors from Parvaresh-Vardy codes Venkatesan Guruswami Carnegie Mellon University --- CMI Pseudorandomness Workshop, Aug 23,

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[GW94,WZ95, TUZ01,RVW00, CRVW02] Connections in Pseudorandomness Randomness Extractors Expander Graphs Error-Correcting Codes Pseudorandom Generators [STV99,SU01,Uma02] [Tre99,TZ01, TZS01,SU01] Algebraic list decoding [SS96,Spi96, GI02,GI03, GR06,GUV07] [Tre99,RRV99, ISW99,SU01,Uma02] Euclidean Sections, Compressed sensing [GLR08,GLW08] Expander codes

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[GW94,WZ95, TUZ01,RVW00, CRVW02] Connections in Pseudorandomness Randomness Extractors Expander Graphs List-Decodable Error-Correcting Codes Pseudorandom Generators [STV99,SU01,U02] [Tre99,TZ01, TZS01,SU01] This talk [PV05,GR06] [GI02,GI03] [Tre99,RRV99, ISW99,SU01,U02] This talk

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List Decodable codes Code C D with N codewords, alphabet size | | = Q (e,L)-list-decodable: Every Hamming ball of radius e has at most L codewords of C –Combinatorial packing condition –Balls of radius e around codewords cover each point L times. –List error correction of e errors with worst-case list size L

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List Decoding Centric View of Pseudorandom Objects

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List decoding, in different notation Encoding function E : [N] [Q] D View as map (bipartite graph) : [N] x [D] [D] x [Q] – (x, y) = (y, E(x) y ) List decoding property: For all r [Q] D, if T = { (y, r y ) : y [D] } then |LIST(T)| L where we define LIST(T) = { x : (x, y) T for at least D - e values of y } N D D x Q x

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Bipartite expanders For all K’ ≤ K, and T [M] with |T| < AK’, LIST(T) < K’ where LIST(T) = { x [N] : for all y [D], (x, y) T } | (S)| A ¢ |S| ( vertex expansion A = expansion factor ) M S, |S| K “ (K,A) expander” D N : [N] x [D] [M]

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Extractors : [N] x [D] [M] is a (k, )-extractor if for all T [M], |LIST(T)| < 2 k where LIST(T) = { x [N] : Pr y [ (x,y) T ] ≥ |T|/M + } d random bits “seed” E XT unknown source of length n with k bits of “min-entropy” m almost-uniform bits M = 2 m Would like m k N = 2 n D = 2 d

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Condensers (weaker object en route extractors) Output not close to uniform but is close to source with good min-entropy –Ideally k’ k (don’t lose entropy), m k (good entropy “rate”) Can also be captured by list decoding type definition –LIST(T) small for all small subsets T [M], where LIST(T) = { x : Pr y [ (x,y) T ] ≥ } d random bits seed C OND k - source of length n ~ k’-source of length m

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The common framework Definitions of various useful objects : [N] x [D] [M] captured as: “For all subsets T [M] that obey certain property, a suitably defined list decoding of T, LIST(T), has small size” –List decodable codes: T arising out of received words –Expanders, condensers: T of small size Also case for “list recoverable codes” –Extractors: arbitrary T The framework gives not just unified abstractions, but also a proof method that leads to the best constructions and analysis.

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Parameters of interest Map : [N] x [D] [M] What we care about varies for different objects Extractors: small seed length D (= poly(log N)); large output length M Codes: want small alphabet size M, small D (= O(log N)) –Small |LIST(T)|, plus efficient algorithm to recover LIST(T) Tight analysis of size of LIST(T) : –exact value not too crucial for codes; –for lossless expanders it is crucial (factor 2 worse bound implies factor 2 worse expansion)

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The abstraction in action Unbalanced expanders Expander Construction from Parvaresh-Vardy codes View as condensers and application to extractors Conclusions

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Unbalanced Expander Graphs Goals: Minimize D Maximize A ( lossless expansion: A close to D ) Minimize M (not much larger than O(KD)) | (S)| A ¢ |S| ( vertex expansion) M S, |S| K “ (K,A) expander” N D

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Expanders have many uses … Fault-tolerant networks (e.g., [Pin73,Chu78,GG81]) Sorting in parallel [AKS83] Derandomization [AKS87,IZ89,INW94,IW97,Rei05,…] PCP theorem [Din06] Randomness Extractors [CW89,GW94,TUZ01,RVW00,GUV07] Error-correcting codes [SS96,Spi96,LMSS01,GI01-04] Distributed routing in networks [ PU89,ALM96,BFU99 ]. Data structures [ BMRV00 ]. Hard tautologies in proof complexity [BW99,ABRW00,AR01 ]. Pseudorandom matrices, Almost Euclidean sections of L 1 N [GLR’08,GLW’08] …. Need explicit constructions (deterministic, time poly(log N)).

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(Bipartite) Expander Graphs Goals: Minimize D Maximize A Minimize M | (S)| A ¢ |S| M S, |S| K Optimal (Non-constructive): D = O(log (N/M) / ) A = (1- ) ¢ D M = O(KD/ “ (K,A) expander” N D

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Explicit Constructions Optimal O(log (N/M)) (1- ) ¢ D O(KD Ramanujan graphs O(1) ¼ D/2N Zig-zag CRVW02] O(1) (1- ) ¢ D N Ta-Shma, Umans, Zuckerman[TUZ01] polylog(N) exp(poly(log log N)) (1- ) ¢ D exp(poly(log KD) poly(KD) G., Umans, Vadhan polylog(N) (1- ) ¢ Dpoly(KD) degree D expansion A |right-side| M arbitrary positive constant.

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Utility of Expansion Utility of Expansion (1- ) ¢ D At least (1-2 ) D |S| elements of (S) are unique neighbors: touch exactly one edge from S | (S)| (1- ) D |S| D N M S, |S| K x Set membership in bit-probe model [BMRV’00] Fault tolerance: Even if an adversary removes say ¾ edges from each vertex, lossless expansion maintained (with =4 ) Useful in Expander codes [SS’96]

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The Result Theorem [GUV]: N, K, >0, 9 explicit (K,A) expander with degree D = poly(log N, 1/ ) expansion A = (1- ) ¢ D #right vertices M = D 2 ¢ K 1.01 | (S)| A ¢ |S| M S, |S| K “ (K,A) expander” N D

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Parvaresh-Vardy codes Variant of Reed-Solomon codes Parameters of construction: n, F q, m, h, an irreducible polynomial E(Y) of degree n over F q Encoding: Given message f F q n or polynomial f(Y) F q [Y] of degree (n-1), –PV(f) y = (f 0 (y), f 1 (y), …, f m-1 (y)) for y F q where f i (Y) = (f(Y)) h^i mod E(Y) Define (f, y) = (y, PV(f) y ) –Consider bipartite expander with neighborhood given by

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Expander theorem Left vertices = polynomials of degree · n-1 over F q (N = q n ) Degree D = q Right vertices = F q m+1 (M = q m+1 ) ( f,y ) = y ’th neighbor of f = (y, f(y), (f h mod E)(y), (f h 2 mod E)(y), …, (f h m-1 mod E)(y)) where E(Y) = irreducible * poly of degree n over F q h = a parameter Thm [GUV’07] : This is a (K,A) expander for K = h m, A = q-hnm. * can be found deterministically in poly(n, log q, char( F q )) time

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Close relation to list decoding Proof of expansion based on list decoding of Parvaresh- Vardy codes –Need a tight analysis of list size –For “list recovery” version S1S1 S2S2 SqSq y 1 y 2 y q K Possible values for each position

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Recall list decoding view For T µ [M], define LIST(T) = {x 2 [N] : (x) µ T} Lemma: G is a (=K,A) expander if and only if for all T µ [M] of size AK-1, we have |LIST(T)| · K-1 | (S)| A ¢ K “ (=K,A) expander” M S, |S|=K N D

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Expansion analysis ( f,y ) = (y, f(y), (f h mod E)(y), …, (f h m-1 mod E)(y)) f = poly of degree · n-1, y F q, E = irreducible of degree n Theorem: For A = q - nmh and any K · h m, we have T µ F q m+1 of size AK-1 ) |LIST(T)| · K-1 Proof outline, following [S97,GS99,PV05]: 1.Find a nonzero low-degree multivariate polynomial Q vanishing on T. 2.Show that every f 2 LIST(T) is a root of a related univariate polynomial Q*. 3.Show that Q * is nonzero and deg(Q * ) · K-1 =

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Proof of Expansion: Step 1 Thm: For A=q-nmh, K= h m, |T| · AK-1 ) |LIST(T)| · K-1. Step 1: Find a low-degree poly Q vanishing on T µ F q m+1 Take Q(Y,Z 1,…,Z m ) to be of degree · A-1 in Y, degree · h-1 in each Z i. # coefficients = A K > |T| = # homogeneous constraints, so a nonzero solution exists Wlog E(Y) doesn’t divide Q(Y,Z 1,…,Z m ).

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Proof of Expansion: Step 2 ( f,y ) = (y, f(y), (f h mod E)(y), …, (f h m-1 mod E)(y)) Step 1: 9 Q(Y,Z 1,…,Z m ) vanishing on T, deg · A-1 in Y, h-1 in Z i, E - Q Step 2: Every f 2 LIST(T) is a “root” of a related Q * Polynomial f 2 LIST(T) ) 8 y 2 F q Q(y, f(y), (f h mod E)(y), …, (f h m-1 mod E)(y)) = 0 ) Q(Y, f(Y), (f h mod E)(Y), …, (f h m-1 mod E)(Y)) 0 ) Q(Y, f(Y), f(Y) h, …, f(Y) h m-1 ) 0 (mod E(Y)) ) Q * (f) = 0 in extension field U= F q [Y]/(E(Y)), where Q* U[Z] is given by Q * (Z) = Q(Y,Z,Z h,…,Z h m-1 ) mod E(Y) Degree ≤ A-1+nmh < q ≤ # roots

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Proof of Expansion: Step 3 Step 2: 8 f 2 LIST(T) Q * (f) = 0 where Q * (Z) = Q(Y,Z,Z h,…,Z h m-1 ) mod E(Y) Step 3: Show that Q * is nonzero and deg(Q * ) · K-1 Q * (Z) nonzero because –Q(Y,Z 1,….,Z m ) mod E(Y) is nonzero –Q is of deg · h-1 in Z i so distinct monomals get mapped to distinct powers of Z when we set Z i = Z h i deg(Q * ) · h-1+(h-1) ¢ h+ +(h-1) ¢ h m-1 = h m -1 = K-1

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Proof of Expansion: Wrap-Up ( f,y ) = (y, f(y), (f h mod E)(y), …, (f h m-1 mod E)(y)) LIST(T) = { x 2 [N] : (x) µ T } Theorem: For A = q - nmh, K= h m, |T| · AK-1 ) |LIST(T)| · K-1. There is a nonzero polynomial Q * over U= F q [Y]/(E(Y)) with deg(Q * ) · K - 1 such that every f LIST(T) satisfies Q * (f) = 0. Hence |LIST(T)| · deg(Q * ) · K - 1. ¥

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Parameter Choices LHS = F q n, degree D = q, RHS = F q m+1 We have a (K,A) expander with K = h m, A = q - nmh To make A (1- ) ¢ D, pick q nmh/ . To make M ¼ KD, need q m+1 ¼ q h m, so take q ¼ h 1+ Set h ¼ ( nm / ) 1/ q ¼ h 1+ . Then: A = q - nmh (1- q = (1- ) ¢ D M = q m+1 ¼ q ¢ h ( 1+ m ¼ D ¢ K 1+ D = ( nm / ) 1+1/ ¼ ((log N)(log K)/ ) 1+1/

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Our Expander Result Thm: For every N, K, >0, 9 explicit (K,A) expander with degree D = O((log N) ¢ (log K)/ ) 1+1/ expansion A = (1- ) ¢ D #right vertices M = (D ¢ K) 1+ | (S)| A ¢ |S| M S, |S| K “ (K,A) expander” N D

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Outline Unbalanced expanders Expander Construction from Parvaresh-Vardy codes View as condensers and application to Extractors Conclusions

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Extractors [NZ’93] Goal: Output -close to uniform on {0,1} m (for large m and small d) Optimal (nonconstructive): d = log n + 2 log(1/ ) + O(1) m = (k+d) - 2 log(1/ ) - O(1) d random bits “seed” E XT Uniform sample from unknown subset X {0,1} n of size 2 k m almost-uniform bits

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Extractors: Original Motivation Randomization is pervasive in CS –Algorithm design, cryptography, distributed computing, … Typically assume perfect random source. –Unbiased, independent random bits –Unrealistic? Can we use a “weak” random source? –Source of biased & correlated bits. –More realistic model of physical sources. (Randomness) Extractors: convert a weak random source into an almost-perfect random source. Dozens of constructions over 15+ years

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Extractors: many “extraneous” uses… Derandomization of (poly-time/log-space) algorithms [Sip88,NZ93,INW94, GZ97,RR99, MV99,STV99,GW02] Distributed & Network Algorithms [WZ95,Zuc97,RZ98,Ind02]. Hardness of Approximation [Zuc93,Uma99,MU01,Zuc06] Data Structures [Ta02] Cryptography [BBR85,HILL89,CDHKS00,Lu02,DRS04,NV04] List decodable codes [TZ01,Gur04] Metric Embeddings [Ind06] Compressed sensing [Ind07]

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[GUV] Result on Extractors Thm: For every n, k, >0, 9 explicit (k, ) extractor with seed length d=O(log n + log (1/ )) and output length m=.99k. Previously achieved by [LRVW03] –Only worked for ¸ 1/n o(1) –Complicated recursive construction Optimal up to constant factors

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2k2k Expanders & Lossless Condensers Lemma [TUZ01]: : {0,1} n £ {0,1} d ! {0,1} m is a lossless ((n, k) ! (m,k +d )) condenser if graph is a (2 k,(1- ) ¢ 2 d ) expander. Proof: Expansion ) can make 1-1 by moving fraction of edges {0,1} n {0,1} m 2d2d ¸ (1- ) 2 d ¢ 2 k n - bit source with entropy k m ¼ 1.01k bit source with entropy ( k+d) d -bit seed C OND x (x,y)(x,y) y

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Extractor Using PV code, we have compressed the n bit source to 1.01k bits while retaining all the entropy (using O(log n) bit seed) –Cond ( f,y ) = (y, f(y), (f h mod E)(y), …, (f h m-1 mod E)(y)) Now extract 0.99k bits from the 1.01k bit source with entropy k –Easier, specialized task (due to high entropy percentage) –Good constructions already known For constant error , can use a simple random walk based extractor –Compose with our condenser to get final extractor

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Extractor for high min-entropy Extractor for min-entropy rate 99% that extracts 99% of the input min-entropy with constant error : Ext(x,y) = y’th vertex on expander walk specified by x ( n bit source: specify walk of length n/c) 2 c -degree expander on 2 (1- )n nodes Extraction follows from Chernoff bound for expander walks [Gil98]

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Variation on the Condenser Cond ( f,y ) = (y, f(y), (f h mod E)(y), …, (f h m-1 mod E)(y)) Use E(Y) = Y q-1 - , for generator of F q * [G.-Rudra’06] ) (f q i mod E)(y) = f ( i y) Cond(f,y) = (y, f(y), f (γy), f(γ 2 y)…, f(γ m-1 y)) Condenser from Folded Reed-Solomon code [ GR06 ] –Loses small constant fraction of min-entropy Okay for the extractor application –Univariate analogue of Shaltiel-Umans extractor f(Y) q = f(Y q ) f( Y) mod E(Y)

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Conclusions List decoding view + an algebraic code construction ) best known constructions of –Highly unbalanced expanders –Lossless condensers –Randomness extractors Future directions? –Constant degree lossless expanders (alternative to zig-zag) Non-bipartite expanders? –Direct construction of a simple, algebraic extractor –Extractors with better (or even optimal) entropy loss? Suffices to achieve this for entropy rate –Other pseudorandom objects: multi-source extractors?

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