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Computational Entropy

Computational Entropy. Salil Vadhan Harvard University (on sabbatical at Microsoft Research SVC and Stanford). Joint works with Iftach Haitner (Tel Aviv), Thomas Holenstein (ETH Zurich ), Omer Reingold (MSR-SVC), Hoeteck Wee (George Washington U.),

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Computational Entropy

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  1. Computational Entropy SalilVadhanHarvard University (on sabbatical at Microsoft Research SVC and Stanford) Joint works with IftachHaitner (Tel Aviv), Thomas Holenstein (ETH Zurich), Omer Reingold (MSR-SVC), Hoeteck Wee (George Washington U.), and Colin Jia Zheng (Harvard) TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAA

  2. Complexity-Based Cryptography • Shannon `49: Information-theoretic security is infeasible. • |Key| ¸ |All Encrypted Data| • On a standard, insecure communication channel. • Diffie& Hellman `76: Complexity-based cryptography • Assume adversary has limited computational resources • Base cryptography on hard computational problems • Enables public-key crypto, digital signatures, …

  3. One-Way Functions [DH76] easy • Candidate: f(x,y) = x¢y Formally, a OWF is f : {0,1}n! {0,1}n s.t. • f poly-time computable • 8 poly-time A Pr[A(f(X))2f-1(f(X))] = 1/n!(1) for XÃ{0,1}n x f(x) hard

  4. OWFs & Cryptography secure protocols & applications digitalsignatures zero-knowledgeproofs private-key encryption statistical ZKarguments MACs [GMW86] statistically bindingcommitments pseudorandom functions [NY89] [BCC86] [GGM86] [N89] pseudorandomgenerators statistically hiding commitments target-collision-resistant hash functions (UOWHFs) [R90] [HILL90] [HNORV07] one-way functions

  5. OWFs & Cryptography secure protocols & applications digitalsignatures zero-knowledgeproofs private-key encryption statistical ZKarguments MACs [GMW86] statistically bindingcommitments pseudorandom functions [NY89] [BCC86] [GGM86] [N89] pseudorandomgenerators statistically hiding commitments target-collision-resistant hash functions (UOWHFs) [R90] [HILL90] [HNORV07] one-way functions

  6. Computational Entropy[Y82,HILL90,BSW03] Question: How can we use the “raw hardness” of a OWF to build useful crypto primitives? Answer (today’s talk): • Every crypto primitive amounts to some form of “computational entropy”. • One-way functions already have a little bit of “computational entropy”.

  7. Entropy Def: The Shannon entropyof r.v. X is H(X) = ExÃX[log(1/Pr[X=x)] • H(X) = “Bits of randomness in X (on avg)” • 0 · H(X) · log|Supp(X)| • Conditional Entropy: H(X|Y) = EyÃY[H(X|Y=y)] X uniform onSupp(X) X concentratedon single point

  8. Worst-Case Entropy Measures • Min-Entropy: H1(X) = minx log(1/Pr[X=x]) • Max-Entropy: H0(X) = log |Supp(X)| H1(X) · H(X) · H0(X)

  9. Computational Entropy A poly-time algorithm may “perceive” the entropy of X to be very different from H(X). • Example:a pseudorandom generator [BM82,Y82] G: {0,1}m!{0,1}n • G(Um) “computationally indistinguishable” from Un • But H(G(Um))·m. e.g. G(N,x) = (lsb(x),lsb(x2 mod N), lsb(x4 mod N),…)for N=pq, x2ZN* is a PRG if factoring is hard [BBS82,ACGS82]. • Def[GM82]: X ´c Yiff8 poly-time TPr[T(X)=1] ¼Pr[T(Y)=1]

  10. Pseudoentropy Def [HILL90]: X has pseudoentropy¸ k iff there exists a random variable Y s.t. • Y ´c X • H(Y) ¸ k Interesting when k > H(X), i.e. Pseudoentropy > Real Entropy

  11. OWFs & Cryptography secure protocols & applications digitalsignatures zero-knowledgeproofs private-key encryption statistical ZKarguments MACs [GMW86] statistically bindingcommitments pseudorandom functions [NY89] [BCC86] [GGM86] [N89] pseudorandomgenerators statistically hiding commitments target-collision-resistant hash functions (UOWHFs) [R90] [HILL90] pseudoentropy [HNORV07] one-way functions

  12. Application of Pseudoentropy Thm [HILL90]:9 OWF )9 PRG Proof idea: OWF to discuss X with pseudoentropy ¸ H(X)+1/poly(n) repetitions X with pseudo-min-entropy ¸ H0(X)+poly(n) hashing PRG

  13. Pseudoentropy from OWF: Intuition Computational Setting: Information Theory: For jointly distributed (X,Y): For 1-1 OWF f, XÃ{0,1}n: H(X|f(X))=0, but • 8functionA Pr[A(Y)=X] ·p • 8 poly-time A Pr[A(f(X))=X] · 1/n!(1) • def [DORS04] • def[HLR07] • X has “average min-entropy” ¸log(1/p) given Y • X has “unpredictability entropy”¸!(log n) given f(X) ? • X has pseudoentropy!(log n) given f(X) • H(X|Y) ¸ log(1/p)

  14. Pseudoentropy from OWF: Intuition Computational Setting: Challenges: How to convert unpredictability intopseudoentropy? When f not 1-1, unpredictability can be trivial. For 1-1 OWF f, XÃ{0,1}n: H(X|f(X))=0, but • 8 poly-time A Pr[A(f(X))=X] · 1/n!(1) • def[HLR07] • X has “unpredictability entropy”¸!(log n) given f(X) FALSE! T(x,y) = [[f(x)=?y]]distinguishes (X,f(X)) from every (Z,f(X)) with H(Z|f(X))>0 • X has pseudoentropy!(log n) given f(X)

  15. Pseudoentropy from OWF • Thm[HILL90]: W=(f(X),H,H(X)1,…H(X)J)has pseudoentropy¸ H(W)+!(log n)/n • H : {0,1}n! {0,1}n a certain kind of hash func. • XÃ{0,1}n, JÃ{1,…,n}. • Thm[HRV10,VZ11]: (f(X),X1,…,Xn) has “next-bit pseudoentropy” ¸n+!(log n). • No hashing! • Total amount of pseudoentropy known & > n. • Get full !(log n) bits of pseudoentropy.

  16. Next-bit Pseudoentropy • Thm[HRV10,VZ11]: (f(X),X1,…,Xn) has “next-bit pseudoentropy” ¸ n+!(log n). • Note: (f(X),X) easily distinguishable from every random variable of entropy > n. • Next-bit pseudoentropy: 9 (Y1,…,Yn) s.t. • (f(X),X1,…,Xi) ´c (f(X),X1,…,Xi-1,Yi) • H(f(X))+iH(Yi|f(X),X1,…,Xi-1) = n+!(log n).

  17. Consequences • Simpler and more efficient construction of pseudorandom generators from one-way functions. • [HILL90,H06]: OWF f of input length n ) PRG G of seed length O(n8). • [HRV10,VZ11]: OWF f of input length n ) PRG G of seed length O~(n3).

  18. Pseudoentropy,Unpredictability wrt KL Divergence • Thm[VZ11]: Let (Y,Z) 2 {0,1}n£ {0,1}O(log n). The pseudoentropy of Z given Y is ¸ H(Z|Y)+±mThere is no probabilistic poly-time A s.t.D((Y,Z)||(Y,A(Y)) ·±. [D = Kullback-Liebler Divergence] • Special case: H(Z|Y)=0 • “Z has pseudoentropy¸±” iff“hard to predict Z with divergence ·±” • Can’t take Z=f-1(Y) for 1-1 OWF f since |f-1(Y)|=n.

  19. Pseudoentropy,Unpredictability wrt KL Divergence • Thm[VZ11]: Let (Y,Z) 2 {0,1}n£ {0,1}O(log n). The pseudoentropy of Z given Y is ¸ H(Z|Y)+±mThere is no probabilistic poly-time A s.t.D((Y,Z)||(Y,A(Y)) ·±. [D = Kullback-Liebler Divergence] • Analogue of Impagliazzo’s Hardcore Thm [I95,N95,H05,BHK09] for Shannon entropy rather than min-entropy.

  20. Next-Bit Pseudoentropy from OWF: Proof Sketch f a one-way function Given f(X), hard to achieve divergence O(log n) from X Given (f(X),X1,…,XJ), hard to achieve divergence O(log n)/n from XJ+1 thm Given (f(X),X1,…,XJ), XJ+1 has pseudoentropy¸entropy+!(log n)/n (f(X),X1,…,Xn) has next-bit pseudoentropy¸ n+!(log n)

  21. OWFs & Cryptography secure protocols & applications digitalsignatures zero-knowledgeproofs private-key encryption statistical ZKarguments MACs [GMW86] statistically bindingcommitments pseudorandom functions [NY89] [BCC86] [GGM86] [N89] pseudorandomgenerators statistically hiding commitments target-collision-resistant hash functions (UOWHFs) next-bitpseudoentropy [HRV10,VZ11] [R90] [HNORV07] one-way functions

  22. Application of Next-Bit Pseudoentropy Thm [HILL90]:9 OWF )9 PRG Proof outline [HRV10]: OWF done Z=(f(Un),Un) with next-bit pseudoentropy¸n+!(log n) repetitions Z’ with next-block pseudo-min-entropy ¸|seed|+poly(n) hashing PRG

  23. OWFs & Cryptography secure protocols & applications digitalsignatures zero-knowledgeproofs private-key encryption statistical ZKarguments MACs [GMW86] statistically bindingcommitments pseudorandom functions [NY89] [BCC86] [GGM86] [N89] pseudorandomgenerators statistically hiding commitments target-collision-resistant hash functions (UOWHFs) next-bitpseudoentropy [HRV10,VZ11] [R90] [HNORV07] one-way functions

  24. OWFs & Cryptography secure protocols & applications digitalsignatures zero-knowledgeproofs private-key encryption statistical ZKarguments MACs [GMW86] statistically bindingcommitments pseudorandom functions [NY89] [BCC86] [GGM86] [N89] pseudorandomgenerators statistically hiding commitments target-collision-resistant hash functions (UOWHFs) next-bitpseudoentropy [HRV10,VZ11] [HRVW09,HHRVW10] inaccessible entropy one-way functions

  25. Inaccessible Entropy [HRVW09,HHRVW10] • Example: if h : {0,1}n! {0,1}n-k is collision-resistant and XÃ {0,1}n, then • H(X|h(X)) ¸ k, but • To an efficient algorithm, once it produces h(X), X is determined ) “accessible entropy” 0. • Accessible entropy ¿ Real Entropy! • Thm[HRVW09]: f a OWF ) (f(X)1,…,f(X)n,X) has accessible entropy n-!(log n). • Cf. (f(X),X1,…,Xn) has pseudoentropy n+!(log n).

  26. Conclusion Complexity-based cryptography is possible because of gaps between real & computational entropy. “Secrecy”pseudoentropy > real entropy “Unforgeability”accessible entropy < real entropy

  27. Research Directions • Formally unify inaccessible entropy and pseudoentropy. • OWF f : {0,1}n! {0,1}n) Pseudorandom generators of seed length O(n)? • More applications of inaccessible entropy in crypto or complexity (or mathematics?)

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