Weakly Private Information Retrieval Under the Maximal Leakage Metric Ruida Zhou, Tao Guo, Chao Tian Texas A&M University ISIT 2020 R. Zhou, T. Guo, C. Tian Weakly PIR Under the Maximal Leakage Metric 1 / 20
Private Information Retrieval (PIR) ( N , K ) non-colluding PIR: server 0 server 1 server 2 server N-1 ...... N servers: server 0, server 1, . . . , server N-1; K messages: W 0 , W 1 , . . . , W K − 1 ; R. Zhou, T. Guo, C. Tian Weakly PIR Under the Maximal Leakage Metric 2 / 20
Private Information Retrieval (PIR) User wants the message W M , where M is a random variable in { 0 , 1 , . . . , K − 1 } . R. Zhou, T. Guo, C. Tian Weakly PIR Under the Maximal Leakage Metric 3 / 20
Private Information Retrieval (PIR) User wants the message W M , where M is a random variable in { 0 , 1 , . . . , K − 1 } . When M = k , server 0 server 1 server 2 server N-1 ...... R. Zhou, T. Guo, C. Tian Weakly PIR Under the Maximal Leakage Metric 3 / 20
Private Information Retrieval (PIR) User wants the message W M , where M is a random variable in { 0 , 1 , . . . , K − 1 } . When M = k , server 0 server 1 server 2 server N-1 ...... User: uses a random key F to generate queries Q [ k ] n ; R. Zhou, T. Guo, C. Tian Weakly PIR Under the Maximal Leakage Metric 3 / 20
Private Information Retrieval (PIR) User wants the message W M , where M is a random variable in { 0 , 1 , . . . , K − 1 } . When M = k , server 0 server 1 server 2 server N-1 ...... User: uses a random key F to generate queries Q [ k ] n ; Server n : return answer A [ k ] after receiving the query; n R. Zhou, T. Guo, C. Tian Weakly PIR Under the Maximal Leakage Metric 3 / 20
Private Information Retrieval (PIR) User wants the message W M , where M is a random variable in { 0 , 1 , . . . , K − 1 } . When M = k , server 0 server 1 server 2 server N-1 ...... User: uses a random key F to generate queries Q [ k ] n ; Server n : return answer A [ k ] after receiving the query; n W k = ψ ( A [ k ] User: recovers ˆ 0: N − 1 , k , F ). R. Zhou, T. Guo, C. Tian Weakly PIR Under the Maximal Leakage Metric 3 / 20
Private Information Retrieval (PIR) User wants the message W M , where M is a random variable in { 0 , 1 , . . . , K − 1 } . When M = k , server 0 server 1 server 2 server N-1 ...... User: uses a random key F to generate queries Q [ k ] n ; Server n : return answer A [ k ] after receiving the query; n W k = ψ ( A [ k ] User: recovers ˆ 0: N − 1 , k , F ). Requirements: Correctness: ˆ W k = W k ; Privacy: the query distribution Pr ( Q [ k ] = q ) = Pr ( Q [ k ′ ] = q ). n n R. Zhou, T. Guo, C. Tian Weakly PIR Under the Maximal Leakage Metric 3 / 20
Private Information Retrieval (PIR) User wants the message W M , where M is a random variable in { 0 , 1 , . . . , K − 1 } . When M = k , server 0 server 1 server 2 server N-1 ...... Requirements: Correctness: ˆ W k = W k ; Privacy: for any n , Q [ M ] does not leak the privacy of M . n R. Zhou, T. Guo, C. Tian Weakly PIR Under the Maximal Leakage Metric 4 / 20
How do we measure privacy leakage? Privacy: for any n , Q [ M ] does not leak any privacy of M . n R. Zhou, T. Guo, C. Tian Weakly PIR Under the Maximal Leakage Metric 5 / 20
How do we measure privacy leakage? Privacy: for any n , Q [ M ] does not leak any privacy of M . n Mutual information; Differential privacy; Maximal leakage metric; etc. R. Zhou, T. Guo, C. Tian Weakly PIR Under the Maximal Leakage Metric 5 / 20
How do we measure privacy leakage? Privacy: for any n , Q [ M ] does not leak any privacy of M . n Mutual information; Differential privacy; Maximal leakage metric; etc. Use a metric to quantify the leakage of identity privacy; If allow certain amount of the leakage, we have the weakly private information retrieval. R. Zhou, T. Guo, C. Tian Weakly PIR Under the Maximal Leakage Metric 5 / 20
Weakly Private Information Retrieval (WPIR) User wants the message W M , where M is a random variable in { 0 , 1 , . . . , K − 1 } . When M = k , server 0 server 1 server 2 server N-1 ...... Requirements: Correctness: ˆ W k = W k ; ρ -Privacy: Leak ( M → Q [ M ] ) ≤ ρ n R. Zhou, T. Guo, C. Tian Weakly PIR Under the Maximal Leakage Metric 6 / 20
Performance measure – download cost Message size L – length of the message L := H ( W 0 ) = H ( W 1 ) = · · · = H ( W K − 1 ); Answer length ℓ n – number of symbols downloaded from server n Download cost D � N − 1 � D := 1 � E ( ℓ n ) . L n =0 We aim to have lower download cost. R. Zhou, T. Guo, C. Tian Weakly PIR Under the Maximal Leakage Metric 7 / 20
Review of WPIR By the type of leakage metric Leak ( M → Q [ M ] ) n Lin et al (ISIT 2019) : mutual information is used to measure leakage Samy-Tandon-Lazos (ISIT 2019): differential privacy is used to measure leakage Jia (MS Thesis 2019): conditional entropy is used to measure leakage R. Zhou, T. Guo, C. Tian Weakly PIR Under the Maximal Leakage Metric 8 / 20
Review of WPIR By the type of leakage metric Leak ( M → Q [ M ] ) n Lin et al (ISIT 2019) : mutual information is used to measure leakage Samy-Tandon-Lazos (ISIT 2019): differential privacy is used to measure leakage Jia (MS Thesis 2019): conditional entropy is used to measure leakage In this work, we use the maximal leakage metric L . R. Zhou, T. Guo, C. Tian Weakly PIR Under the Maximal Leakage Metric 8 / 20
Maximal Leakage Metric Operational meaning of L ( X → Y ): When guessing a function of X upon observing Y , the leakage is the logarithm of the ratio of the probability of a correct guess when Y is observed, to the probability of a correct guess when Y is not observed. log Pr( U = ˆ U ) L ( X → Y ) = sup max u P U ( u ) U − X − Y − ˆ U R. Zhou, T. Guo, C. Tian Weakly PIR Under the Maximal Leakage Metric 9 / 20
Maximal Leakage Metric Operational meaning of L ( X → Y ): When guessing a function of X upon observing Y , the leakage is the logarithm of the ratio of the probability of a correct guess when Y is observed, to the probability of a correct guess when Y is not observed. log Pr( U = ˆ U ) L ( X → Y ) = sup max u P U ( u ) U − X − Y − ˆ U Math formula: � L ( X → Y ) = log P Y | X ( y | x ) . max x ∈ X : y ∈ Y P X ( x ) > 0 R. Zhou, T. Guo, C. Tian Weakly PIR Under the Maximal Leakage Metric 9 / 20
Maximal Leakage Metric Operational meaning of L ( X → Y ): When guessing a function of X upon observing Y , the leakage is the logarithm of the ratio of the probability of a correct guess when Y is observed, to the probability of a correct guess when Y is not observed. log Pr( U = ˆ U ) L ( X → Y ) = sup max u P U ( u ) U − X − Y − ˆ U Math formula: � L ( X → Y ) = log P Y | X ( y | x ) . max x ∈ X : y ∈ Y P X ( x ) > 0 For WPIR system L ( M → Q [ M ] � ) = log max P Q [ M ] | M ( q | M = k ) n k ∈ [0: K − 1]: n q ∈ Q n P M ( k ) > 0 The distribution of M is unknown, and also not required for calculating L ( M → Q [ M ] ). n R. Zhou, T. Guo, C. Tian Weakly PIR Under the Maximal Leakage Metric 9 / 20
Extreme cases Define the worst case maximal leakage – maximum privacy leakage of all servers n ∈ [0: N − 1] L ( M → Q [ M ] ρ = max ) . n R. Zhou, T. Guo, C. Tian Weakly PIR Under the Maximal Leakage Metric 10 / 20
Extreme cases Define the worst case maximal leakage – maximum privacy leakage of all servers n ∈ [0: N − 1] L ( M → Q [ M ] ρ = max ) . n Two extreme cases ρ ? D Perfect privacy: No privacy: Sun-Jafar’s scheme[TIT17]; Tian-Sun-Chen’s scheme[TIT19] “direct download” D pir = 1 + 1 1 1 N + N 2 + · · · + N K − 1 , ρ = 0 . D = 1 , ρ = log K . R. Zhou, T. Guo, C. Tian Weakly PIR Under the Maximal Leakage Metric 10 / 20
Gentle Start: WPIR ( N , K , ρ ) = (2 , 2 , 0) Review of TSC’s scheme server 0 server 1 a 1 a 1 b 1 b 1 Coding scheme: �� �� �� �� � � � � 0 1 server 0 server 1 0 1 server 0 server 1 F ∈ F ∈ , ; , 0 1 ∅ a 1 0 1 a 1 + b 1 b 1 Use random code structure to maintain privacy; R. Zhou, T. Guo, C. Tian Weakly PIR Under the Maximal Leakage Metric 11 / 20
Gentle Start: WPIR ( N , K , ρ ) = (2 , 2 , 0) Review of TSC’s scheme server 0 server 1 a 1 a 1 b 1 b 1 Coding scheme: �� �� �� �� � � � � 0 1 server 0 server 1 0 1 server 0 server 1 F ∈ , ; F ∈ , 0 1 ∅ a 1 0 1 a 1 + b 1 b 1 Some choice of F may lead to “direct download” code structure; R. Zhou, T. Guo, C. Tian Weakly PIR Under the Maximal Leakage Metric 12 / 20
From no privacy to perfect privacy p F p F p F ? F · · · F F · · · · · · (a) (b) (c) Figure: Between the extreme case of (a) where download cost is minimized and the other extreme case of (c) where the privacy is prefect, we can adjust the probability distribution to achieve the tradeoff between ρ and D . What is the optimal behavior of TSC’ code in between the two extreme cases? R. Zhou, T. Guo, C. Tian Weakly PIR Under the Maximal Leakage Metric 13 / 20
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