Computational Analogues of Entropy

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Computational Analogues of Entropy

• Boaz BarakRonen ShaltielAvi Wigderson

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Statistical Min-Entropy
Definition H(X)k iff maxx Pr Xx lt2-k
( X r.v. over 0,1n )
Properties

• H(X) Shannon-Ent(X)

• H(X)n iff XUn

• H(X,Y) H(X) (concatenation)

• If H(X)k then 9 (efficient) f s.t.
  f(X)?Uk/2 (extraction)

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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

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Review - Pseudorandomness
Def X is pseudorandom if
maxD2C biasD(X,Un) lt ?
i.e., X is computationally indistinguishable from
Un
C class of efficient algorithms (e.g. s-sized
circuits)
biasD(X,Y) EXD(X) - EYD(Y)
? parameter (in this talk some constant gt 0)
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Defining Pseudoentropy
Def 1 HILL HHILL(X)k if
9Y s.t. H(Y) k and maxD2C biasD(X,Y) lt ?
minH(Y) K maxD2C biasD(X,Y) lt ?
Def 2 HMet(X)k if
maxD2C minH(Y) K biasD(X,Y) lt ?
Def 3 Yao HYao(X)k if X cannot be efficiently
compressed to k-1 bits.
maxD2C biasD(X,Un) lt ?
X is pseudorandom if
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Defining Pseudoentropy
HHILL(X)k if minH(Y) K maxD2C biasD(X,Y) lt ?
HMet(X)k if maxD2C minH(Y) K biasD(X,Y) lt ?
HYao(X)k if X cant be efficiently compressed to
k-1 bits.
Claim 1 H(X) HHILL(X) HMet(X) HYao(X)
Claim 2 For kn all 3 defs equivalent to
pseudorandomness.
Claim 3 All 3 defs satisfy extraction
property.Tre
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HILL Metric Def are Equivalent
(For C poly-sized circuits, any ?)
Thm 1 HHILL(X) HMet(X)
Proof Suppose HHILL(X)ltk
Player 2 D
Player 1 D
Y
Y
biasD(X,Y)?
Player 1
HHILL(X)k if minH(Y)K maxD2C biasD(X,Y) lt ?
HMet(X)k if maxD2C minH(Y)K biasD(X,Y) lt ?
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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)
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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.
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Thm 2 If X is unpredictable by SAT-gate circuits
with const. adv. then HMet(X)?(n)
Proof Suppose that HMet(X)lt?n Well
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) lt 2?n 2) PrX D(X)1
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Construct P from D
1) D-1(1) lt 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 gt ½
Note that P does not depend on X and can be
constructed w/ NP oracle. (approx counting JVV)
Claim 8x2D, P predicts at least (1-?)n indices
of x
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Claim 8x2D, P predicts at least (1-?)n indices
of x
Proof Suppose P fails to predictx in m indices.
2m
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Well show that Dgt2m,obtaining a contradiction.
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2
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P(x1,,xi)0 iff Pr D(x1,,xi,0,Un-i-1)1 gt ½
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Open Problems
More results for poly-time computation

• Analog of Thm 2 (unpredictability?entropy)?
• Meaningful concatenation property?
• Separate Yao Metric pseudoentropy.

Prove that RLL