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Codes and Pseudorandomness : A Survey

Codes and Pseudorandomness : A Survey. David Zuckerman University of Texas at Austin. Randomness and Computing. Randomness extremely useful in computing. Randomized algorithms Monte Carlo simulations Cryptography Distributed computing Problem: high-quality randomness expensive.

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Codes and Pseudorandomness : A Survey

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  1. Codes and Pseudorandomness:A Survey David Zuckerman University of Texas at Austin

  2. Randomness and Computing • Randomness extremely useful in computing. • Randomized algorithms • Monte Carlo simulations • Cryptography • Distributed computing • Problem: high-quality randomness expensive.

  3. What is minimal randomness requirement? • Can we eliminate randomness completely? • If not: • Can we minimize quantity of randomness? • Can we minimize quality of randomness? • What does this mean?

  4. What is minimal randomness requirement? • Can we eliminate randomness completely? • If not: • Can we minimize quantity of randomness? • Pseudorandom generator • Can we minimize quality of randomness? • Randomness extractor

  5. Outline • PRGs and Codes • Intro to PRGs. • Various connections. • Extractors and Codes • Intro to Extractors. • Connections with list decoding. • Non-Malleable Codes and Extractors. • Conclusions

  6. Pseudorandom Numbers • Computers rely on pseudorandom generators: PRG 141592653589793238 71294 long “random-enough” string short random string What does “random enough” mean?

  7. Modern Approach to PRGs[Blum-Micali 1982, Yao 1982] Require PRG to “fool” all efficient algorithms. Alg random ≈ same behavior Alg pseudorandom

  8. Which efficient algorithms? • Poly-time PRG fooling all polynomial-time circuits implies NP≠P. • So either: • Make unproven assumption. • Try to fool interesting subclasses of algorithms.

  9. Existence vs. Explicit Construction • Most functions are excellent PRGs. • Challenge: find explicit one. • Most codes have excellent properties. • Known: good explicit codes. • Can codes give good PRGs?

  10. Idea 1: PRG = Random Codeword • Choose a random codewordin [n,k] code. • n random variables, |sample space| = 2k. • But: linear code can’t fool all linear tests. • t-wise independence: • Dual distance > t  any t coordinates independ as rv’s, since any t columns of G linindepend. • t=2: Hadamard code, |Ω|=n+1 [Lancaster 1965] • t odd. Dual BCH code, |Ω| = 2(n+1)(t-1)/2 [ABI ‘86]

  11. Idea 2: PRG=Random Column of G Sample space Ω • k random variables, |sample space| = n. • If dual distance > 4, then Ω is a Sidon set: • All pairwise sums distinct. • Dual BCH code: |Ω|= 2(k-1)/2. • Don’t need row of 1’s; Ω={(x,x3)|x in F2k/2}. Generator matrix G k

  12. PRG=Random Column of G Sample space Ω X1 … Generator matrix G • Say all codewords≠0 have relative wt ½±ε. • corresponds to a codeword, S≠Φ. • ½-ε ≤ Pr[ =1] ≤ ½+ε: ε-biased space [NN ‘90] • RS concatHadamard: |Ω| = O(k2/ε2) [AGHP ‘90] • AG concatHadamard: |Ω| = O(k/ε3) • Degree < genus: |Ω| = O(k/ε2)5/4 [BT 2009] • Optimal, non-explicit: O(k/ε2). Ignored logs. Xk

  13. PRGs from Hard Functions [NW ‘88] • PRGs lower bounds. • Nisan-Wigderson: lower bounds  PRGs. • Suppose f is hard on average. Design (wt w code): |Si|=w f f Sj Si seed

  14. Worst Average Case Hardness [L,BF] • Given: worst-case hard f:{0,1}w{0,1}. • Encode f using RM code as g:FqwFq. • g=unique multilinear function s.t. g=f on {0,1}w. • g is avg-case hard. • Efficiently compute g on 1-1/(4m) fraction Efficiently compute g everywhere whp. • Pick random line with L(0)=x. • Degree(g(L(.))) ≤ w. • Interpolate g(L(0)) from g(L(1)),…,g(L(w+1)).

  15. Local Decodability • Compute any bit of message whp by querying at most r bits of encoding. • RM codes. • New family: Matching Vector Codes [Yekhanin, Efremenko,…]. Beats RM codes. • Stronger notion: Local correctability. • Can compute any bit of encoding. • RM codes.

  16. List Decoding [Elias 1957] • Output list of all close codewords. • Can sometimes decode beyond distance/2. • Efficient algorithms for: • Hadamard [Goldreich-Levin] • Reed-Solomon [Sudan, Guruswami-Sudan] • AG codes [Shokrollahi-Wasserman, GS] • Reed-Muller [large q: STV, q=2: GKZ] • Certain concatenated codes [GS, STV, GI, BM] • PV codes [Parvaresh-Vardy] • Folded RS [Guruswami-Rudra] • Multiplicity codes [Kopparty, Guruswami-Wang]

  17. PRGs and Hardcore Bits[Goldreich-Levin 1989, Impagliazzo 1997] • Given one-way function f: • Easy to compute, hard to invert. • E.g., f(x)=gx mod p (g generator, p prime). • Goal: computing bit b(x) hard given f(x). • Thm: Suppose C is (locally) list decodable. Then b(x,i) = C(x)i hard given f(x), i. • Pf idea: Suppose easy. List decode few candidates for x. Check if f(candidate)=f(x).

  18. PRG from Hardcore Bits • Given one-way permutation f, hardcore b. • PRG(x,i)=b(x,i),b(f(x),i),b(f2(x),i),…

  19. List Decoding Related to Randomness Extractors

  20. General Weak Random Source [Z ‘90] • Random variable X on {0,1}n. • General model: min-entropy • Flat source: • Uniform on A, |A| ≥ 2k. {0,1}n |A| ³ 2k

  21. 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.

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

  23. 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)

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

  25. Graph-Theoretic View: “Expansion” N=2n output  uniform  K=2k M=2m Ext(x,y) x y  (1-)M D=2d

  26. Alternate View M=2m N=2n D=2d S BADS x

  27. Extractor Codes via Alt-View[Ta-Shma-Z 2001] • List recovery – generalizes list decoding. S=(S1,…,SD), agreement = |{i|xi in Si}| |{Codewords with agreement ≥(μ(S) + ε)D}| ≤ |BADS|. • Can construct extractor codes with efficient decoding. • Give hardcore bits Ext(x,y) wrt 1-way (f(x),y).

  28. Leftover Hash Lemma Johnson Bound • Johnson bound: An [n,k,(½−ε2)n]-code has <L=1/ε2codewords within distance ½-ε of received word r. Alt pf [TZ ‘01]: • Let V=close codewords,D=distribution (i,vi), i in [n], v in V. |D-U|≥ε: Pr[(i,vi)= (i,ri)]. • If |V|≥L: • collision-pr(D)<(1/n)(1/|V| + 1-d/n)=(1+4ε2)/(2n) • Implies |D-U|< ε. Contradiction.

  29. Codes  Extractors • PRGs + Codes  Extractor [Trevisan 1999] • RM Codes Extractor [Ta-Shma, Z, Safra 2001; Shaltiel, Umans 2001] • Parvaresh-Vardy Codes Extractor [Guruswami, Umans, Vadhan 2007]

  30. 2-Stage Extractor  + O(log n) random bits Condense: .9 Extract: + O(log n) random bits  uniform

  31. Parvaresh-Vardy codes Condenser[Guruswami-Umans-Vadhan 2007] • Fqfinite field • parameter h ≤ q • deg. n polynomial E(Y) irreducible over Fq • source: degree n-1 univariate polynomial f • define fi(Y) = fhi(Y) mod E(Y) C(f, y 2Fq) = (y, f0(y), f1(y), f2(y), , fm-1(y))

  32. Independent Sources n/2 bits n/2 bits Ext m =Ω(k) bits statistical error 

  33. Bounds for 2 Independent Sources • Classical: H∞ (X) > n/2. • Lindsey Lemma: inner product. • Bourgain: H∞ (X) > .4999n. • Existence: H∞ (X) > 2 log n.

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

  35. 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.

  36. 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) • nmExt can’t 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.

  37. 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.

  38. A Simple 1-Bit Construction [Li] • Sidon set: set S with all s+t, s,t in S, distinct. • 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 (Lindsey’s 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)-1 = n/2 - 1.

  39. Non-Malleable Codes[Dziembowski, Pietrzak, Wichs 2010] • Adversary tampers with Enc(m) via f in F. • Ideally Dec(f(Enc(m)) = m or “error” • Impossible if f(x) = Enc(m’) allowed. • Dec(f(Enc(m)) = m or is independent of m. • Randomized encoding allowed. • Prob method: exist if |F| < 22αn, α<1. • Explicit? • Codes for f(x1,…,xn)=f1(x1),…,fn(xn).

  40. Split-State Tampering • f(x,y)=g(x),h(y) |x|=|y|=n/2. • 2-source ext for H(X)+H(Y)>2n/3 codes for 1-bit messages [Dz, Kazana, Obremski 2013] • Poly rate: n=k7+o(1) via additive combinatorics [Aggarwal, Dodis, Lovett 2013]. • Constant rate if can construct nonmalleable 2-source extractors for entropy rate .99. [Cheraghchi, Guruswami 2013].

  41. Non-Malleable 2-Source Extractor[Cheraghchi, Guruswami 2013] • X and Y independent weak sources. • Think of H∞(X)=H∞(Y)=.99(n/2). • For all A1, A2,x’=A1(x)≠x, y’=A2(y)≠y: • (nmExt(X,Y),nmExt(X,Y’)) ≈ (U,nmExt(X,Y’)) • (nmExt(X,Y),nmExt(X’,Y)) ≈ (U,nmExt(X’,Y)) • (nmExt(X,Y),nmExt(X’,Y’)) ≈ (U,nmExt(X’,Y’)) • Open question: explicit construction.

  42. Key Properties of Codes • Dual distance  k-wise independence, Sidon sets. • Relative distance ≈ ½  small-bias spaces. • Local decodability Amplifying hardness of functions for PRGs, extractors. • List decodability Cryptographic PRGs, extractors. • Non-malleability  Non-malleable 2-source extractors.

  43. Open Questions • Construct ε–biased spaces of size n=O(k/ε2). • [n=O(k/ε2),k,(½-ε)n] codes. • 2-source extractors for entropy rate α, any α>0. • Non-malleable extractors for H∞(X)=αn. • Non-malleable codes of constant rate. • Non-malleable 2-source extractors. • Other Applications & Connections.

  44. Thank you!

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