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Randomness Extraction: A Survey David Zuckerman University of Texas at Austin

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Randomness in Computer Science Many uses of randomness in CS. – Randomized algorithms – Cryptography – Distributed computing But: high-quality randomness expensive. Can low-quality (weak) randomness suffice?

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Models for Weak Randomness Independent bits with same, unknown bias – [von Neumann 51] Semirandom sources [Santha-Vazirani 84] – δ < Pr[X i |X 1 =x 1,…,X i-1 =x i-1 ] < 1-δ – Block sources [Chor-Goldreich 85] Bit-fixing sources [CFGHRS 85,…] – k uniform bits; others set by adversary.

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General Weak Random Source [Z 90] Random variable X on {0,1} n. General model: min-entropy Flat source: – Uniform on A, |A| 2 k. |A| 2 k {0,1} n

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General Weak Random Source [Z 90] Can arise in different ways: – Physical source of randomness. – Cryptography: condition on adversary s information, e.g. bounded storage model. – Pseudorandom generators (for space s machines): condition on TM configuration.

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Goal: Extract Randomness Ext n bits m bits statistical error Problem: Impossible, even for k=n-1, m=1, ε<1/2.

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Impossibility Proof Suppose f:{0,1} n {0,1} satisfies sources X with H (X) n-1, f(X) U. f -1 (0) f -1 (1) Take X=f -1 (0)

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Randomness Extractor: short seed [Nisan-Z 93,…, Guruswami-Umans-Vadhan 07] Ext n bits m =.99k bits statistical error d=O(log (n/ε)) random bit seed Y Strong extractor: (Ext(X,Y),Y) Uniform

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Outline Seeded Extractors – Basic applications – Alternate view with applications – Sketch of two constructions Seedless Extractors for Structured Sources – Algebraic sources: independent, affine, … Applications in cryptography – Complexity-theoretic sources Crypto-tailored Extractors

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Simulating Randomized Algorithms Randomized algorithm R using m random bits. Assume only random bits X have H (X)k>m. – No high-quality randomness available. Given Ext for H (X)k – seed length d, output length m. Simulation with factor 2 d blowup: – Run R with random string Ext(x,y 1 ),…,Ext(x,y 2 d ). – Take majority vote or median.

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Use in Privacy Amplification [Bennett, Brassard, Robert 1985] Goal: convert weak shared secret X to uniform secret. Unbounded passive adversary. public Pick Y Shared secret = Ext(X,Y). Correct by strong extractor definition.

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PRGs for Space-Bounded Machines Basic PRG: G(x,y) = (x,Ext(x,y))[Nisan-Z] Condition on configuration v after read x. Whp Hence whp Ext(X,Y) close to uniform. G:{0,1} O(s) {0,1} poly(s) fools space s TMs [NisanZ] Sometimes can avoid union bound! – O(log n log log n) bit seed fools read-once polylog- width regular BPs [BRRY 10,BV 10] – O(log n) bit seed fools read-once O(1)-width permutation BPs [KNP].

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PRGs from Shrinkage Hardness vs. Randomness paradigm: – Lower bounds give PRGs [Nisan-Wigderson,…]. But: need superpolynomial lower bounds. Known: polynomial lower bounds for restricted models. – E.g., formulas Ω(n 3 /polylog n) [Andreev, Hastad]. [Impagliazzo, Meka, Z 2012]: polynomial lower bounds proved via shrinkage give PRGs. – E.g., seed length s 1/3+o(1) fools size s formulas.

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Graph-Theoretic View: Expansion (1- )M K=2 k D=2 d N=2 n M=2 m Can use this to construct expanders beating eigenvalue bound [WZ] xy Ext(x,y) output uniform

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K-Expanding Graphs K N K |A|K |Γ(A)|>N-K Goal: minimize degree D D>N/K Random graphs: D=O((N/K) log (N/K)) 2 nd Eigenvalue: D(N/K) 2 /2 Extractors: D=N 1+o(1) /K [Wigderson-Z 93] Useful for sorting, networks

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Extractors K-Expanding Graphs (1- )M K N M K K-Expanding Graph: V=[N] E=Paths of length 2 in Ext

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Alternate View S BAD S D=2 d N=2 n M=2 m x Other direction: Error S |BAD S |2 -k + ε

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Averaging Sampler via Alternate View [Z 96] Goal: Estimate mean μ of – Black box access to f. Algorithm: Pick x randomly in {0,1} n. Sample f at Γ(x) = {x 1,…,x D }. Output μ f. Pr[error] = |BAD f |/2 n. Can use 1.01m random bits with Pr[error]=2 -Ω(m).

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Extractor Perspective Helps Proposition: Sampler using O(m) random bits implies sampler using 1.01m random bits. Equivalent Statement: Extractor outputting Ω(k) bits implies extractor outputting.99k bits. Ext(x,(y 1,y 2 )) = Ext(x,y 1 )Ext(x,y 2 ) [Wigderson-Z] – Conditioned on Ext(X,y 1 ) of length m, still k-m bits of entropy in X.

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Extractor Codes via Alt-View [Ta-Shma-Z 2001] List recovery – generalizes list decoding. S=(S 1,…,S D ), agreement = |{i|x i in S i }| |{Codewords with agreement (μ(S) + ε)D}| |BAD S |. Extractor codes with efficient decoding give hardcore bits Ext(x,y) wrt 1-way (f(x),y). Codes Extractors [Tre,TZS, SU, GUV].

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Max Clique and Chromatic Number [FGLSS,…,Hastad]: Max Clique inapproximable to n 1-, any >0, assuming NP ZPP. [LY,…,FK]: Same for Chromatic Number. Derandomize with linear degree extractors: Thm [Z]: Both inapproximable to n 1-, any >0, assuming NP P.

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Constructions of Strong Extractors RestrictionsDegree D=2 d Output Length m ExistenceNone(n-k)/ε 2 k – 2lg(1/ε) Leftover Hash Lemma [ILL] None2n2n k – 2lg(1/ε) GUV 2007None(n/ε) O(1) (1-α)k GUV 2007Nonen O(log(k/ε)) k – 2lg(1/ε)-O(1) DKSS 2009ε1/log c nn O(1) (1-1/log c n)k Z 2006k=Ω(n) ε=Ω(1) O(n)(1-α)k

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Pseudorandom Generators Cryptographically secure PRGs: – Run in time less than adversary. – Exist iff one-way functions exist [HILL]. PRGs for derandomization: – Can take slightly more time than adversary. – Exist iff hard functions exist [Nisan-Wigderson...] PRG pseudorandomrandom seed

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PRGs from Hard Functions [Nisan-Wigderson 1988 …] PRG comp. error εrandom seed hard function

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NW-Style PRGs Give Extractors [Trevisan 1999] View x as hard function f:{0,1} lg n {0,1} – Most functions hard Set Ext(x,y) = NW-PRG(f,y) Better: Ext(x,y) = NW-PRG(Code(f),y) Ext n bits statistical error seed

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Linear Degree Extractor [Z] (Sketch) Condense: Extract:.9 uniform + lg n+O(1) random bits + O(1) random bits

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Condensing via Incidence Graph 1-Bit Somewhere Condenser: – Input: edge – Output: random endpoint Condenses rate to rate (1+ ), some > 0. Proof uses bound on incidences [BKT]+ probabilistic lemma. Combine with technique of [Raz] to get actual condenser. lines points = F q 2 L P (L,P) an edge iff P on L |P| 3/2 edges

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High Entropy Extractor Chernoff bound for random walks on expanders [Gillman,Kahale] Implies Sampler Implies Extractor.

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Seeded Extractor Techniques/History Hashing based: Z 90-91, Nisan-Z 93, Wigderson-Z 93, Srinivasan-Z 94, Z 96, Ta-Shma 96, Raz-Reingold-Vadhan 99, Reingold-Shaltiel-Wigderson 00, NW-PRG based: Trevisan 99, Raz-Reingold-Vadhan 99, Impagliazzo-Shaltiel-Wigderson 99-00, Ta-Shma-Umans-Z 01 Algebraic/coding theory based: Ta-Shma-Z-Safra 01, Shaltiel- Umans 01, Lu-Reingold-Vadhan-Wigderson 03, Gurusmami- Umans-Vadhan 07, Ta-Shma-Umans 12 Additive combinatorics based: Barak-Kindler-Shaltiel- Sudakov-Wigderson 05, Raz 05, Z 07, Dvir-Wigderson 08, Dvir-Kopparty-Sharaf-Sudan 09

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Seedless (Deterministic) Extractors for Structured Sources Probabilistic Method: If sources of min-entropy k: Can deterministically extract m=(1-α)k bits with error 2 -αk/3. Algebraic sources: – Bit-fixing, affine. Independent sources. Complexity-theoretic sources: – AC 0 sources, small-space sources.

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Oblivious Bit-Fixing Sources Example: ?0010?111??11. – ? = uniform on {0,1}. – (n-k) bits fixed by adversary; k uniform bits. – Parity extracts 1 bit. For klog c n, can extract k-o(k) bits [GRS, Rao]. Application: Exposure Resilient Cryptography. – Adversary learns many bits of secret key. – Can still do cryptography.

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Affine Extractors X = random element from affine subspace. Generalizes bit-fixing sources. Extractor for min-entropy αn, any α>0 [Bourgain]. 1-bit disperser for min-entropy exp(log.9 n) [Shaltiel]. Large fields: any k>0 [Gabizon-Raz].

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Independent Sources n bits Ext m =Ω(k) bits statistical error

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Classical: entropy rate > 1/2 Lindsey Lemma: H (X) + H (Y) > n+t implies X. Y U, error 2 -t/2.

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Independent Sources # sourcesk=H (X)Restrictions Existence2k 2log nNone Bourgain2k.499nNone BRSW2k n α Disperser Li3k n 1/2+α None Rao-Z3k n α Uneven lengths LiO(1)k log 3 nNone

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Cryptography with Weak Sources Players have independent weak sources. Allow Byzantine faults. For 2 players, impossible [DOPS]. For more players, possible!

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Network Extractor Protocol [Goldwasser-Sudan-Vaikunthanatan05, Dodis-Oliveira03] Input: x 1,…,x p 2 {0,1} n from independent weak random sources Output: z 1,…,z p 2 {0,1} m private nearly-uniform random strings (for honest parties) Byzantine faults: can send arbitrary messages

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Network Extractor Protocols After running network extractor protocol, run standard protocol, e.g., Byzantine Agreement. Naïve idea to design protocol: – A few players broadcast sources. – Remaining players apply independent-source extractor to those sources and own source. – Problem: what if only malicious players broadcast?

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Network Extractor Constructions Information-theoretic setting [Kalai-Li-Rao-Z]: – For k exp(log α n), can still tolerate linear number of faults in BA and leader election, any α>0. Computational setting [Kalai-Li-Rao]: – Under certain crypto assumptions, for k = αn, secure multiparty computation if 2 honest players. – Under certain crypto assumptions, 2-source extractors for k = αn, any α>0.

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Complexity-Theoretic Sources X=f(U), complexity(f) small. Deterministic extraction possible under assumptions [Trevisan-Vadhan 00]. No assumptions: – NC 0 [De-Watson 11, Viola 11] – AC 0 [Viola 11] – Proofs reduce to low-weight affine extractors [Rao 09].

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Small Space Sources Space s source: min-entropy k source generated by width 2 s branching program. n+1 layers /, 0 1-1/, 0 1,1 0.1,0 0.8,1 0.1,0 0.3,0 0.5,1 0.1,1 0.1,0 1 width 2 s

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Bit Fixing Sources can be modelled by Space 0 sources ? 1 ? ? ,1 0.5,0 1,11,01,1

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Extractors for Small Space Sources For k αn, any α>0, space αβn, β>0 sufficiently small, can extract k-o(k) bits [Kamp-Rao-Vadhan-Z 06]. Proof reduces to variants of independent sources by conditioning on intermediate states.

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Crypto-Tailored Extractors Fuzzy extractors – Noise tolerant [Dodis-Ostrovsky-Reyzin-Smith 04] Correlation extractors – [Ishai-Kushilevitz-Ostrovsky-Sahai 09]. Non-malleable extractors [Dodis-Wichs 09]

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Privacy Amplification With Active Adversary Problem: Active adversary could change Y to Y. public Pick Y Shared secret = Ext(X,Y).

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Active Adversary Can arbitrarily insert, delete, modify, and reorder messages. E.g., can run several rounds with one party before resuming execution with other party.

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Non-Malleable Extractor [Dodis-Wichs 2009] Strong extractor: (Ext(X,Y),Y) (U,Y). nmExt is a non-malleable extractor if for arbitrary A:{0,1} d {0,1} d with y = A(y) y. (nmExt(X,Y),nmExt(X,Y),Y) (U,nmExt(X,Y),Y) Cant ignore a bit of the seed. Existence: k > log log n + c, d = log n + O(1), m = (k-log d)/2.01. Gives privacy amplification with active adversary in 2 rounds with optimal entropy loss.

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Explicit Non-Malleable Extractor Even k=n-1, m=1 nontrivial. – E.g., Ext(x,y) = x. y. X=0??...?, y=A(y) flips first bit, x. y= x. y. Dodis-Li-Wooley-Z 2011: H (X) > n/2. Cohen-Raz-Segev 2012: Seed length O(log n). Li 2012: H (X) >.499n. – Connection with 2-source extractors.

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A Simple 1-Bit Construction [Li] Sidon set: set S with all s+t, s,t in S, distinct. Example: S={(x,x 3 )|x in F 2 n/2 }. Thm [Li]: f(x,y) = x. y, y uniform from S, nonmalleable extractor for H (X) > n/2. Proof: H (Y) = n/2, so X. Y U (Lindseys lemma). Suffices to show X. Y+X. A(Y) U (XOR lemma). X. Y+X. A(Y) = X.( Y+A(Y)). H (Y+A(Y)) = H (Y) = n/2.

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Proof Via Character Sum Estimate For m=1, we show For larger m, consider (χ,χ) with χ nontrivial. Give non-uniform XOR lemma. – nmExt(x,A(y)) need not be uniform.

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Conclusions Interesting mathematics used in constructions: additive combinatorics, coding theory, random walks on expander graphs, hashing, … Crypto Expanders Coding Theory Extractors PRGs Inapproximability

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Open Questions Seeded Extractors – O(n) degree for all min-entropy. – O(log n) seed to extract k - 2log(1/ε) – O(1). Seedless Extractors – 2-source extractors for min-entropy αn, any α>0. – Affine extractors for min-entropy n α. – Other general models. Crypto-Tailored Extractors – Non-malleable extractors for min-entropy αn. Other Applications & Connections.

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