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1 Adam O’Neill Leonid Reyzin Boston University A Unified Approach to Deterministic Encryption and a Connection to Computational Entropy Benjamin Fuller Boston University & MIT Lincoln Lab

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Public Key Encryption (PKE) 2 PK m Need randomness to achieve semantic security $ Enc c

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Public Key Encryption (PKE) 3 PK m $ What can be achieved without randomness? Enc

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Why deterministic PKE? The question of deterministic symmetric key encryption is well understood: Key: k Messages: m 1, …, m n Encryption: pad 1 || … || pad n = prg(k) c i = pad i m i Deterministic PKE is difficult but has important applications: –Supporting devices with limited/no randomness –Enabling encrypted search –E.g. spam filtering by keyword on encrypted 4 prg – pseudorandom generator Each bit appears random to bounded distinguisher

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Deterministic PKE PKE scheme where encryption is deterministic –Introduced by [BellareBoldyrevaO’Neill07] Need source of randomness messages are only hope Security defined w.r.t. high entropy message distribution M –H ∞ (M)≥μ for all m, Pr[M=m] ≤ (1/2) μ Even most likely message is hard to guess E.g.: Uniform with first bit 1, Network packet with fixed header –Message distribution must be independent of public key An approach: fake coins to chosen plaintext-secure (CPA) scheme [Bellare BoldyrevaO’Neill07, BelllareFischlinO’NeillRistenpart08] 5

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Results Deterministic PKE from: –General: Arbitrary TDF with enough hardcore bits –Efficient: Single application of TDF Framework yields constructions from Niederreiter RSA & Paillier –These TDF s have many hardcore bits under non-decisional (search) assumptions Tools of independent interest : –Improved Equivalence between Indistinguishability & Semantic Security –Conditional Computational Entropy First deterministic PKE for q arbitrarily correlated messages –Extension of LHL to correlated sources using 2q -wise indep. hash –Extension of crooked LHL to improve parameters 6

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Results Deterministic PKE from: –General: Arbitrary TDF with enough hardcore bits –Efficient: Single application of TDF Framework yields constructions from Niederreiter RSA & Paillier –These TDF s have many hardcore bits under non-decisional (search) assumptions Tools of independent interest : –Improved Equivalence between Indistinguishability & Semantic Security –Conditional Computational Entropy First deterministic PKE for q arbitrarily correlated messages –Extension of LHL to correlated sources using 2q -wise indep. hash –Extension of crooked LHL to improve parameters 7 Focus of the talk

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Our Scheme: Encrypt with hardcore Enc hc 8 $ PK m Enc

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Our Scheme −Enc hc 9 PK m Enc TDF – Trapdoor function hc – Hardcore function Ext – Randomness extractor Enc – Randomized Encrypt Alg. hc TDF Ext TDF : Easy to compute, hard to invert without key hc : Pseudorandom given output of TDF Ext : Converts high entropy distributions to uniform

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Our Scheme −Enc hc 10 PK m Enc TDF – Trapdoor function hc – Hardcore function Ext – Randomness extractor Enc – Randomized Encrypt Alg. hc TDF Ext Question: Why is this semantically secure?

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11 Indistinguishability Semantic SecurityFor a message distribution M Outline of Security Proof PK m Enc hc TDF c Ext General Definitional Equivalence

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Compute f from ciphertext Semantic Security for Deterministic PKE 12 AdversaryChallenger DetEnc b DetEnc(m b ), pk A M – message distribution f – test function

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Semantic Security for Deterministic PKE 13 AdversaryChallenger DetEnc b DetEnc(m b ), pk A M – message distribution f – test function Compute f from ciphertextCompute f from random ciphertext

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Indistinguishability for Deterministic PKE 14 b DetEnc(m), pk AdversaryChallenger A DetEnc M 0 – message distribution M 1 – message distribution

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15 Indistinguishability: Semantic Security:For a message distribution M Outline of Security Proof PK m Enc hc TDF c General Definitional Equivalence

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16 Indistinguishability: For all pairs M|e 0, M|e 1 e 0, e 1 are events s.t. Pr[e 0 ],Pr[e 1 ]≥1/4 Semantic Security:For a message distribution M Outline of Security Proof PK m Enc hc TDF c General Definitional Equivalence

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Our Scheme −Enc hc 17 PK m Enc TDF – Trapdoor function hc – Hardcore function Ext – Randomness extractor Enc – Randomized Encrypt Alg. hc TDF Ext Question: Why is this secure?

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Our Scheme −Enc hc 18 PK m Enc TDF – Trapdoor function hc – Hardcore function Ext – Randomness extractor Enc – Randomized Encrypt Alg. hc TDF Ext Question: Why is this secure indistinguishable? To gain intuition we will try removing the extractor.

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Toy Scheme −Enc hc Question: Is this scheme indistinguishable? NO: hc can reveal the first bit of m. Enc can reveal its first coin. 19 PK hc TDF m Enc

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Toy Scheme −Enc hc Question: Is this scheme indistinguishable? NO: hc can reveal the first bit of m. Enc can reveal its first coin. 20 PK hc TDF m Enc

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21 Indistinguishability: For all pairs M|e 0, M|e 1 e 0, e 1 are events s.t. Pr[e 0 ],Pr[e 1 ]≥1/4 Semantic Security:For a message distribution M Outline of Security Proof PK m Enc hc TDF c

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22 Robust hardcore function: hc is hardcore on M|e for all e, Pr[e] ≥ 1/4 Indistinguishability: For all pairs M|e 0, M|e 1 e 0, e 1 are events s.t. Pr[e 0 ],Pr[e 1 ]≥1/4 Semantic Security:For a message distribution M Outline of Security Proof PK m Enc hc TDF c

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23 Robust hardcore function: hc(M|e) is pseudorandom given TDF(M|e) for all e, Pr[e] ≥ 1/4 Indistinguishability: For all pairs M|e 0, M|e 1 e 0, e 1 are events s.t. Pr[e 0 ],Pr[e 1 ]≥1/4 Semantic Security:For a message distribution M Outline of Security Proof PK m Enc hc TDF c Q: Is any hc robust? A: NO! Define event e : fix first bit(previous example!)

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24 Robust hardcore function: hc(M|e) is pseudorandom given TDF(M|e) for all e, Pr[e] ≥ 1/4 Indistinguishability: For all pairs M|e 0, M|e 1 e 0, e 1 are events s.t. Pr[e 0 ],Pr[e 1 ]≥1/4 Semantic Security:For a message distribution M Outline of Security Proof PK m Enc hc TDF Q: Is any hc robust? A: NO! Define event e : fix first bit(previous example!)

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Robustness: Implicit in Prior Work 25 Iterated trapdoor permutation Lossy trapdoor function Arbitrary trapdoor function [GL89] hc bit at each iteration ([BM84] PRG) TDF Robust hc function [Belllare Fischlin O’Neill Ristenpart08] [Boldyreva Fehr O’Neill 08] This work Pairwise Independent Hash Function Any function with enough hc bits + extractor Ext

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Hardcore function: hc(M) is pseudorandom given TDF(M) Robust hardcore function: hc(M|e) is pseudorandom given TDF(M|e) for all e, Pr[e] ≥ 1/4 Indistinguishability: For all pairs M|e 0, M|e 1 e 0, e 1 are events s.t. Pr[e 0 ],Pr[e 1 ]≥1/4 Semantic Security:For a message distribution M 26 Outline of Security Proof PK m Enc hc TDF c Ext( )

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Hardcore function: hc(M) is pseudorandom given TDF(M) Robust hardcore function: hc(M|e) is pseudorandom given TDF(M|e) for all e, Pr[e] ≥ 1/4 Indistinguishability: For all pairs M|e 0, M|e 1 e 0, e 1 are events s.t. Pr[e 0 ],Pr[e 1 ]≥1/4 Semantic Security:For a message distribution M 27 Outline of Security Proof PK m Enc hc TDF c Ext Rest of the talk Ext( )

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Hardcore function Robust hardcore function Indistinguishability Semantic Security 28 Outline of Security Proof PK m Enc hc TDF c Ext

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29 Outline of Security Proof PK m Enc hc TDF c Ext 1.Hardcore function: hc(M) is pseudorandom given TDF(M) 2.Comp. Entropy: hc(M|e) high computational entropy 3.Uniform Ext Output: Ext( hc(M|e) ) pseudorandom 4.Robust hc function: Ext( hc(M|e) ) | TDF( M|e ) pseudorandom Hardcore function Robust hardcore function Indistinguishability Semantic Security

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(1) Hc function (2) Comp. Entropy 30 Know: hc produces pseudorandom bits on M Want: hc produces pseudorandom bits on M|e M hc(M)≈U hc

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31 Know: hc produces pseudorandom bits on M Want: hc produces pseudorandom bits on M|e hc(M)≈U Problem: hc(M|e) cannot be pseudorandom For example, event e can fix the first bit of hc(M) Solution: Use HILL entropy! M M|e (hc(M|e))≈U hc (1) Hc function (2) Comp. Entropy

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32 Know: hc produces pseudorandom bits on M Want: H HILL ( M | E ) is high M|e hc (1) Hc function (2) Comp. Entropy

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33 Know: hc produces pseudorandom bits on M Want: H HILL ( hc(M|e) ) is high M|e hc (1) Hc function (2) Comp. Entropy H HILL (X)≥μ if Y, H ∞ (Y)≥μ X≈ ε,s Y Distinguisher Advantage Distinguisher Size

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34 Know: hc produces pseudorandom bits on M Want: H HILL ( hc(M|e) ) is high M|e How is H HILL ( hc(M|e) ) related to H HILL ( hc(M) ) ? General question: How is H HILL ( X|E=e ) related to H HILL ( X ) ? hc (1) Hc function (2) Comp. Entropy H HILL (X)≥μ if Y, H ∞ (Y)≥μ X≈ ε,s Y ε,s Distinguisher Advantage Distinguisher Size

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Conditional Computational Entropy 35 Our Lemma: Info-Theoretic Case: Warning: this is not H HILL ! Different Y (that has true entropy) for each distinguisher (“metric*”) Notion used in [Barak Shaltiel Widgerson03] [DziembowskiPietrzak08]

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Conditional Computational Entropy 36 Our Lemma: Info-Theoretic Case: Warning: this is not H HILL ! Can be converted to HILL entropy with a loss in circuit size [BSW03, ReingoldTrevisanTulsianiVadhan08] Our Theorem:

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Tangent: Avg Case Cond. Entropy 37 Our Lemma: Info-Theoretic Case [Dodis Ostrovsky Reyzin Smith 04] : We can apply the lemma multiple times to measure H(M |E 1,E 2 ) Cannot measure entropy when original distribution is conditional Average case conditioning useful for leakage resilience Works on conditional computational entropy: [ReingoldTrevisanTulsianiVadhan08], [DziembowskiPietrzak08], [ChungKalaiLiuRaz11],[GentryWichs10] Distribution not a single event!

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38 M|e hc (1) Hc function (2) Comp. Entropy HILL entropy Our Theorem:

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39 Outline of Security Proof PK m Enc hc TDF c Ext 1.Hardcore function: hc(M) is pseudorandom given TDF(M) 2.Cond. Comp Entropy: hc(M|e) high computational entropy for e, Pr[e]≥1/4 3.Uniform Ext Output: Ext( hc(M|e) ) pseudorandom for e, Pr[e]≥1/4 4.Robust hc function: Ext( hc(M|e) ) | TDF(M|e) pseudorandom Hardcore function Robust hardcore function Indistinguishability Semantic Security

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40 M|e Ext HILL entropy pseudorandom Extractors convert distributions w/ min-entropy to uniform w/ H HILL to pseudorandom hc (2) Cond. Comp. Entropy (3) Unif. Ext Output

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41 Outline of Security Proof PK m Enc hc TDF c Ext 1.Hardcore function: hc(M) is pseudorandom given TDF(M) 2.Cond. Comp Entropy: hc(M|e) high computational entropy for e, Pr[e]≥1/4 3.Uniform Ext Output: Ext( hc(M|e) ) pseudorandom for e, Pr[e]≥1/4 4.Robust hc function: Ext( hc(M|e) ) | TDF(M|e) pseudorandom Hardcore function Robust hardcore function Indistinguishability Semantic Security

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42 (3) Unif. Ext Output (4) Robust hc function TDF M pseudorandom hc Know: hc(M) | TDF(M) is pseudorandom ( hc is hardcore)

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43 (3) Unif. Ext Output (4) Robust hc function TDF M pseudorandom hc Know: hc(M) | TDF(M) is pseudorandom ( hc is hardcore) Know: Ext( hc(M|e) ) is pseudorandom ((1) (3))

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M|e 44 (3) Unif. Ext Output (4) Robust hc function TDF pseudorandom hc Know: hc(M) | TDF(M) is pseudorandom ( hc is hardcore) Know: Ext( hc(M|e) ) is pseudorandom ((1) (3))

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45 (3) Unif. Ext Output (4) Robust hc function TDF hc Know: hc(M) | TDF(M) is pseudorandom ( hc is hardcore) Know: Ext( hc(M|e) ) is pseudorandom ((1) (3)) HILL entropy M|e

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46 (3) Unif. Ext Output (4) Robust hc function TDF Ext HILL entropy hc Know: hc(M) | TDF(M) is pseudorandom ( hc is hardcore) Know: Ext( hc(M|e) ) is pseudorandom ((1) (3)) Want: (Ext( hc(M|e) ) | TDF(M|e) ) is pseudorandom M|e pseudorandom

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(3) Unif. Ext Output (4) Robust hc function TDF Ext HILL entropy pseudorandom hc Know: hc(M) | TDF(M) is pseudorandom ( hc is hardcore) Know: Ext( hc(M|e) ) is pseudorandom ((1) (3)) Want: (Ext( hc(M|e) ) | TDF(M|e) ) is pseudorandom Unfortunately our entropy theorem does not work if the starting point is conditional Solution: Consider the joint distribution ( hc(M), TDF(M) ) Condition on e to measure entropy of ( hc(M|e), TDF(M|e) ) 47 M|e

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48 (3) Unif. Ext Output (4) Robust hc function TDF Ext HILL entropy pseudorandom hc Know: hc(M) | TDF(M) is pseudorandom ( hc is hardcore) Know: Ext( hc(M|e) ) is pseudorandom ((1) (3)) Lemma: (Ext( hc(M|e) ) | TDF(M|e) ) is pseudorandom Unfortunately our entropy theorem does not work if the starting point is conditional Solution: Consider the joint distribution ( hc(M), TDF(M) ) Condition on e to measure entropy of ( hc(M|e), TDF(M|e) ) M|e

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49 Outline of Security Proof PK m Enc hc TDF c Ext 1.Hardcore function: hc(M) is pseudorandom given TDF(M) 2.Cond. Comp Entropy: hc(M|e) high computational entropy for e, Pr[e]≥1/4 3.Uniform Ext Output: Ext( hc(M|e) ) pseudorandom for e, Pr[e]≥1/4 4.Robust hc function: Ext( hc(M|e) ) | TDF(M|e) pseudorandom Hardcore function Robust hardcore function Indistinguishability Semantic Security

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Our Scheme −Enc hc If hc is hardcore on M 50 PK m Enc Ext Enc hc is secure on M hc TDF

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Enc hc, deterministic PKE from: –General: Arbitrary TDF with enough hardcore bits –Efficient: Single application of TDF Framework yields constructions from Niederreiter RSA & Paillier –These TDF s have many hardcore bits under non-decisional (search) assumptions Tools of independent interest : –Improved Definitional Equivalence –Conditional Computational Entropy Allows encryption of messages from block sources –Each message has entropy conditioned on previous msgs: H ∞ (M i | M 1,…, M i-1 ) is high Results 51

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Results Enc hc, deterministic PKE from: –General: Arbitrary TDF with enough hardcore bits –Efficient: Single application of TDF Framework yields constructions from Niederreiter RSA & Paillier –These TDF s have many hardcore bits under non-decisional (search) assumptions Tools of independent interest : –Improved Definitional Equivalence –Conditional Computational Entropy First deterministic PKE for q arbitrarily correlated messages –Extension of LHL to correlated sources using 2q -wise indep. hash –Extension of crooked LHL to improve parameters 52 Briefly

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Extending to multiple messages 53 Enc hc does not extend when multiple arbitrarily correlated messages are encrypted We need an extractor that “decorrelates” messages: Use a 2 q -wise independent hash function

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Extending to multiple messages 54 Enc hc does not extend when multiple arbitrarily correlated messages are encrypted We need an extractor that “decorrelates” messages: Use a 2 q -wise independent hash function PK m Enc hc TDF c Ext

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Extending to multiple messages 55 Enc hc does not extend when multiple arbitrarily correlated messages are encrypted We need an extractor that “decorrelates” messages: Use a 2 q -wise independent hash function First scheme for q -arbitrarily correlated messages PK m Enc hc TDF c Hash

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Extending to multiple messages 56 Lemma (Extension of LHL): Let M 1,…, M q be high entropy, arbitrarily correlated random variables (M i ≠ M j ), Hash family of 2q -wise indep. hash functions (keyed by K ) K, Hash(K, M 1 ),…, Hash(K, M q ) ≈ K, U 1,…, U q

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Results Enc hc, deterministic PKE from: –General: Arbitrary TDF with enough hardcore bits –Efficient: Single application of TDF Framework yields constructions from Niederreiter RSA & Paillier –These TDF s have many hardcore bits under non-decisional (search) assumptions Tools of independent interest : –Improved Definitional Equivalence –Conditional Computational Entropy First deterministic PKE for q arbitrarily correlated messages –Extension of LHL to correlated sources using 2q -wise indep. hash –Extension of crooked LHL to improve parameters 57

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