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Exploring Pseudorandomness and Shrinkage: Understanding Minimal Randomness Requirements

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  1. Pseudorandomness from Shrinkage David Zuckerman University of Texas at Austin Joint with Russell Impagliazzo and Raghu Meka

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

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

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

  8. Two Major Challenges • Prove circuit lower bounds. • EXP does not have poly-size circuits. • Derandomize algorithms. • Hardness vs. Randomness paradigm • (1) implies (2) [Nisan-Wigderson, BFNW,…] • Almost equivalent [Kabanets-Impagliazzo …]

  9. Pseudorandom Generators random seed pseudorandom PRG • PRG fools class F of functions if |Pr[f(Un)=1] - Pr[f(PRG(Ud))=1]| ≤ ε. • Cryptography: e.g., F=BPTIME(nlog n). • Equivalent to one-way functions [HILL]. • Derandomizing BPP: F=nc-size circuits. • Need unproven lower bound assumptions. • What F, d without unproven assumptions? d n

  10. Pseudorandom Generators random seed pseudorandom PRG • PRG fools class F of functions if |Pr[f(Un)=1] - Pr[f(PRG(Ud))=1]| ≤ ε. • PRG fooling{f | sizeM(f)≤s} with seed length s1/c implies g in NP with sizeM(g)≥≈nc. • Can we achieve converse: does g in P with sizeM(g)≥nc imply PRG with seed of length ≈ s1/c? • Previous work gives nothing in this case. d n

  11. New Results • Construct such near optimal PRGs if lower bound is proved via “shrinkage.” • Obtain following seed lengths to fool size s, error = 1/poly. • Formulas over {∨,∧,NOT}: s1/3+o(1) • Formulas over arbitrary basis: s1/2+o(1) • Read-once formulas over {∨,∧,NOT}: s.234… • Branching programs: s1/2+o(1)

  12. Previous Work • Seed length (1-α)n fooling read-once formulas and read-once branching programs of width 2αn, α>0 small enough constant. [Bogdanov, Papakonstantinou, Wan]. • For ROBPs reading bits in known order, seed length O(log2 n) [Nisan,…].

  13. Random Restrictions • Choose random restriction ρ, fraction p unset. • E[size(f|ρ)] ≤ p size(f), size(formula)= # leaves. • Whpsize(f|ρ) ≤ 2p size(f). • Holds even if ρ chosen k-wise independently.

  14. Shrinkage Exponent • Random ρ, fraction p unset. Shrinkage Γ: E[size(f|ρ)] = O(pΓ s). • Example: Formulas. • Formulas over arbitrary basis: Γ = 1. • Formulas over DM={∨,∧,NOT}: Γ = 2 [Subbotovskaya ‘61, …., Hastad ‘93] • Read-once formulas over DM: Γ = 3.27… [Paterson-Zwick ‘91, Hastad-Razborov-Yao ‘95] • General circuits: Γ = 0.

  15. Branching Programs n+1 layers • Layered, ordered, read-once BPs needed for PRG for Space • Size = # edges ≤ 2wn. • Γ = 1: size of shrunken BP proportionally to |{unfixed var’s}|. • |{layered, ordered ROBPs}| ≤ w2wn. • We consider arbitrary BPs, reading bits in arbitrary order. 0 1 x2 width w 1 x1 acc 0 rej

  16. PRGs from Shrinkage • Random ρ, fraction p unset. Shrinkage Γ: E[size(f|ρ)] = O(pΓ s). • Shrinkage Γ nΓ+1/polylog(n) lower bounds [Andreev]. • Main theorem: High probability shrinkage Γwrt pseudorandom restrictions gives PRG with seed length s1/(Γ+1) + o(1). • Showing shrinkage wrt pseudorandom restrictions is nontrivial when Γ ≠ 1.

  17. Outline • Background on Randomness Extractors • New Theorem about Old PRG • New PRG • Correctness Proof • Pseudorandom Restrictions • Conclusions

  18. Weak Random Source […CG ‘85 Z ‘90] • Random variable X on {0,1}r. • General model: min-entropy • Flat source: • Uniform on A, |A| ≥ 2k. {0,1}r |A| ³ 2k

  19. How Arise in PRGs • Condition on information • E.g., TM configuration • Uniform X in {0,1}r, f:{0,1}r{0,1}b. • f regular: H∞(X|f(X) = a) = r - b. • Any f: Pra=f(X’)[H∞(X|f(X) = a) ≥ r – b – Δ] ≥ 1-2-Δ.

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

  21. Impossibility Proof • Suppose f:{0,1}r{0,1} satisfies ∀sources X with H∞(X) ≥ r-1, f(X) ≈ U. f-1(0) f-1(1) Take X=f-1(0)

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

  23. Extractor-Based PRG for Read-Once Branching Programs [Nisan-Z ‘93] • Basic PRG: G(x, y1,…,yt)=Ext(x,y1)…Ext(x,yt) • Parameters: r = |x| = 2√n d = |yi| = O(log n) t = m = |Ext(x,yi)| = √n

  24. PRG for Ordered Read-Once BPs n+1 layers • G(x, y1,…, yt)=Ext(x,y1)…Ext(x,yt) • Condition on v reached after reading up to Ext(X,Yi-1). • Whp H∞(X|reach v) ≥ |x| – log w - Δ. • Hence (Ext(X,Yi)|reach v) ≈ uniform. 0 1 z2 width w v 1 z1 acc 0 rej

  25. New: Same PRG works if bits read in any order n+1 layers • z1,z2,…,zm can appear anywhere. • Still, after fixing all zi, i>m, restricted function is a ROBP on z1,z2,…,zmread in the same order as original ROBP. 0 1 z26 width w 1 z41 acc 0 rej

  26. New: Same PRG works if bits read in any order n+1 layers • Still, after fixing all zi, i>m, restricted function is a ROBP on z1,z2,…,zmread in the same order as original ROBP. • Information = lg(# restricted functions) = lg(w2wm) 0 1 z26 width w 1 z41 acc 0 rej

  27. New: Works if bits read in any order • PRG: G(x, y1,…,yt)=Ext(x,y1)…Ext(x,yt)=z1…zn • BP could read in order z12z7z8… • D=distribution of PRG output, U=Unif({0,1}n). • Suppose |Pr[f(D)=1] – Pr[f(U)=1]| > δ. • Let Zi=Ext(X,Yi), Ui =Unif({0,1}m) • Z1=z1z2…zm,Z2=zm+1…z2m,… • Bits in Zican appear anywhere.

  28. New: Works if bits read in any order • PRG: G(x, y1,…,yt)=Ext(x,y1)…Ext(x,yt). • D=distribution of PRG output, U=Unif({0,1}n). • Suppose |Pr[f(D)=1] – Pr[f(U)=1]| > δ. • Let Zi=Ext(X,Yi), Ui =Unif({0,1}m). • Hybrid argument. • Let Di = (U1,…,Ui,Zi+1,…,Zt). D0=D, Dt=U. • Exists i: |Pr[f(Di)=1] – Pr[f(Di-1=1)]| > δ/t. • Changing Zi=Ext(X,Yi) to Ui changes Pr[accept].

  29. New: Works if bits read in any order • Exists i: |Pr[f(Di)=1] – Pr[f(Di-1=1)]| > δ/t. • Changing Zi=Ext(X,Yi) to Ui changes Pr[accept]. • Consider ρ = (Z1,…,Zi-1,**…*,Ui+1,…,Ut) • Then g = f|ρ is a ROBP on m bits. • f(Di)=g(Zi), f(Di-1)=g(Ui). Goal: whp g(Zi) ≈ g(Ui). • Only w2wm possibilities for g. • Whp, H∞(X|G=g) ≥ r – 2mw log w - Δ. • Whp, conditioned G=g, Ext(X,Yi) ≈ Ui.

  30. General Branching Programs • Even PRG for unordered ROBPs is new • Our seed length is O(√(wn) log n) • Previous was (1-α)n [Bogdanov, Papakonstantinou, Wan] • Known order: O(log2 n) [Nisan,…]. • What if not read once? • Some variables could be read many times. • Pseudorandomly permute variables before construction. • Gives seed length size(f)½+o(1). • What about formulas? General reduction?

  31. General PRG Construction • Assume have pseudorandom restrictions which give shrinkage Γwhp. ρ1 = 0 1 * 1 1 0 1 1 * 0 0 1 0 * 0 1 0 0 1 1 1 ρ2 = 0 0 1 0 1 0 * 0 1 1 0 1 * 0 1 1 0 * * 1 0 … ρt = * 0 1 0 1 1 * 1 * 0 0 1 0 0 0 1 * 0 1 1 1 • Set t=c(log n)/p so whp all columns have *.

  32. General PRG Construction ρ1 = 0 1 * 1 1 0 1 1 * 0 0 1 0 * 0 1 0 0 1 1 1 ρ2 = 0 0 1 0 1 0 * 0 1 1 0 1 * 0 1 1 0 * * 1 0 … ρt = * 0 1 0 1 1 * 1 * 0 0 1 0 0 0 1 * 0 1 1 1 • Choose X, Y1,…,Yt randomly. • Replace *’s in ith row with Ext(X,Yi). • PRG output = XOR of resulting strings.

  33. Correctness Proof • D=distribution of PRG output, U=uniform. • Suppose |Pr[f(D)=1] – Pr[f(U=1)]| > δ. • Let Zi=Ext(X,Yi). Hybrid argument. • Change Z1,…,Zi to U1,…,Ui to get Di. • Dt ≈ U: Whp *’s cover all columns. • Exists i: |Pr[f(Di)=1] – Pr[f(Di-1=1)]| > δ/t. • Changing Zito Uichanges Pr[f accepts].

  34. Correctness Proof • Exists i: changing Zi=Ext(X,Yi) to Ui changes Pr[f accepts]. • Fix everything but ρ=ρi, Zi, Ui. Let v = ithrow. • Let fi(v) = f(v+w), w = XOR of rows except ith. • Let g = fi|ρ, so g(v|A) = fi (v) , A = *’s of ρ. • f(Di)=g(Zi), f(Di-1)=g(Ui). Goal: whp g(Zi) ≈ g(Ui). • E=event that size(g) ≤ s=cpΓ size(fi). Pr[E] ≥ 1-ε. • Conditioned on E, gdescribable by b ≈ s log s bits. • Whp, H∞(X|E,G=g) ≥ r – b - Δ. • Whp conditioned on Eand G=g, Ext(X,Yi) ≈ Ui.

  35. Improving the PRG • To get nearly optimal output length for Γ > 1, replace *’s with Gk-wise(Ext(X,Yi)).

  36. Pseudorandom Restrictions • Need pseudorandom restrictions that yield shrinkage. • BPs and formulas over arbitrary basis: • clog n wise independence suffices. • Deal with heavy variables separately. • Formulas over {∧,∨,NOT}, incl. read-once: • More work. • Hastad and Hastad-Razborov-Yao as black boxes. • They only guarantee shrinkage in expectation for truly random restrictions.

  37. Proof Idea Decompose formula: O(n/k) subformulas of size ≤k=no(1). Use k2-wise independence. Goal: p ≈ n-1/(Γ+1). Too small here. Instead, shrink by q≈ k-.1 and iterate.

  38. Unrestrictable inputs • Many subformulas have inputs that must = *. • Does shrinkage for random restrictions imply shrinkage when some inputs must = *? • Further decomposition: each subformula has ≤ 2 such inputs. • h such inputs increase size by ≤ 2h. • For each setting of variables have subformula. • Combine with selector formula.

  39. Read-Once Formulas • Need different trick for read-once formula. • g small but unlikely to shrink to nothing. g g * *

  40. Dependencies • Read-once case: k-wise independence. • Read-t case: Consider independent sets in dependency graph on subformulas. • General case: tricky dependencies.

  41. Conclusions • New, extractor-based PRG based on shrinkage. • Without improving lower bounds, essentially best possible PRGs for: • Formulas over {∨,∧,NOT}: s1/3+o(1) seed length. • Formulas over arbitrary basis: s1/2+o(1) • Read-once formulas over {∨,∧,NOT}: s.234… • Branching programs: s1/2+o(1)

  42. Open Questions • Better PRGs for unordered ROBPs? • Can we recurse somehow? • Subsequent work: Reingold-Steinke-Vadhan give O(log2 n) seed for unordered permutation ROBPs. • PRGs from other lower bound techniques? • Subsequent work: Trevisan-Xue on PRGs for AC0. • Improve lower bounds? • Our PRG gives alternate function f: formula-size(f) ≥ n3-o(1), matching Hastad/Andreev. • Subsequent: average-case lower bound of n3-o(1) [Komargodski-Raz-Tal] (improving [Komargodski-Raz])

  43. Thank you!

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