Computational Analogues of Entropy

1. Computational Analogues of Entropy Boaz BarakRonen ShaltielAvi Wigderson

2. Our Objectives: 1. Investigate possible defs forcomputationalMin-Entropy. 2. Check whether computational defs satisfy analogs of statistical properties. Statistical Min-Entropy Definition:H(X)¸k iff maxx Pr[ X=x ]<2-k ( X r.v. over {0,1}n ) Properties: • H(X) · Shannon-Ent(X) • H(X)=n iff X~Un • H(X,Y) ¸ H(X) (concatenation) • If H(X)¸k then 9(efficient)fs.t.f(X)~Uk/2(extraction)

3. Our Contributions • Study 3 variants (1 new) of pseudoentropy. • Equivalence & separation results for several computational model. • Study analogues of IT results. In this talk: • Present the 3 variants. • Show 2 results + proof sketches

4. Review - Pseudorandomness Def:X is pseudorandom if maxD2C biasD(X,Un) <  i.e., X is computationally indistinguishable from Un C– class of efficient algorithms (e.g. s-sized circuits) biasD(X,Y) =| EX[D(X)] - EY[D(Y)] |  – parameter (in this talk: some constant > 0)

5. i.e., X is computationally indist. from someY with ¸k statistical min-entropy. i.e., 8 efficient D, X is computationally indist. by D from someY=Y(D) with ¸k statistical min-entropy. Defining Pseudoentropy Def 1 [HILL]: HHILL(X)¸k if 9Y s.t. H(Y)¸ k and maxD2C biasD(X,Y) <  minH(Y)¸ K maxD2C biasD(X,Y) <  Def 2: HMet(X)¸k if maxD2C minH(Y)¸ K biasD(X,Y) <  Def 3 [Yao]: HYao(X)¸k if X cannot be efficiently compressed to k-1 bits. maxD2C biasD(X,Un) <  *X is pseudorandom if

6. Defining Pseudoentropy HHILL(X)¸k if minH(Y)¸ K maxD2C biasD(X,Y) <  HMet(X)¸k if maxD2C minH(Y)¸ K biasD(X,Y) <  HYao(X)¸k if X can’t be efficiently compressed tok-1bits. Claim 1: H(X) · HHILL(X) · HMet(X) · HYao(X) Claim 2: Fork=n all 3 defs equivalent to pseudorandomness. Claim 3: All 3 defs satisfy extraction property.[Tre]

7. 2: Use the “Min-Max” theorem. [vN28] HILL & Metric Def are Equivalent (For C = poly-sized circuits, any ) Thm 1: HHILL(X) = HMet(X) Proof: SupposeHHILL(X)<k Player 2: D Player 1:D Y Y Player 1: biasD(X,Y)¸ HHILL(X)¸k if minH(Y)¸K maxD2C biasD(X,Y) <  HMet(X)¸k if maxD2C minH(Y)¸K biasD(X,Y) < 

8. Can we do better? Unpredictability & Entropy Thm [Yao]: If X is unpredicatble with adv.  then X is pseudorandom w/ param ’=n¢ Loss of factor of n due to hybrid argument –useless for constant advantage  This loss can be crucial for some applications (e.g., extractors, derandomizing small-space algs)

9. Unpredictability & Entropy IT Fact [TZS]: If X is IT-unpredictable with const. adv. then H(X)=(n) We obtain the following imperfect analog: Thm 2: If X is unpredictable by SAT-gate circuits with const. adv. then HMet(X)=(n) In paper: A variant of Thm 2 for nonuniform online logspace.

10. {0,1}n D X Thm 2: If X is unpredictable by SAT-gate circuits with const. adv. then HMet(X)=(n) Proof: Suppose thatHMet(X)<n We’ll construct a SAT-gate predictor P s.t. Pri,X[ P(X1,…,Xi-1)=Xi] = 1 –  We have that maxD2CminH(Y)¸n biasD(X,Y)¸ i.e., 9D s.t.8Y If H(Y)¸n then biasD(X,Y)¸ Assume: 1) |D-1(1)| < 2n*2) PrX[ D(X)=1 ] = 1

11. {0,1}n D X Construct P from D 1) |D-1(1)| < 2n2) PrX[ D(X)=1 ] = 1 Define predictor P as follows:P(x1,…,xi)=0 iff Pr[ D(x1,…,xi,0,Un-i-1)=1] > ½ Note that P does not depend on X and can be constructed w/ NP oracle. (approx counting [JVV]) Claim: 8x2D, Ppredicts at least (1-)n indices ofx

12. Claim: 8x2D, Ppredicts at least(1-)n indices of x Proof: SupposePfails to predictx in m indices. ¸2m ¸8 We’ll show that |D|>2m,obtaining a contradiction. ¸4 ¸4 ¸2 ¸2 1 P(x1,…,xi)=0 iff Pr[ D(x1,…,xi,0,Un-i-1)=1] > ½

13. Open Problems More results for poly-time computation: • Analog of Thm 2 (unpredictabilityentropy)? • Meaningful concatenation property? • Separate Yao & Metric pseudoentropy. Prove that RL=L