Statistical Databases – Query Auditing

Li Xiong

CS573 Data Privacy and Anonymity

Partial slides credit: Vitaly Shmatikov, Univ Texas at Austin

slide 2

Query Audit Problem

Database

Q1

Auditor

Maintaining “privacy” of data

…

Q2 … Qn

A1 A2 … An

Q

Does answer to Q combined with answers to Q1,…,Qn reveal something?

A or “Denied”

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Variations of the Problem

Database

Q1

Auditor

…

Q2 … Qn

A1 A2 … An

List of real, integer, or Boolean values

Specifies subset of the variables

Min, max, median, sum, average, or count of specified subset

Wants to learn value of some variable

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Offline auditing Given a collection of queries and answers to them,

check whether anything “forbidden” was revealed Detects privacy breaches after the fact

Online auditing Queries are presented to auditor one at a time;

auditor checks if answering the current query (in combination with past answers) reveals “forbidden” information

Prevents privacy breaches on-the-fly

Offline vs. Online

Disclosure measures

Full disclosure (exact-value disclosure) – the exact value of a protected attribute is disclosed

Partial disclosure (interval-based disclosure) – the disclosed range (difference of the lower and upper bounds) of the protected attribute is smaller than a predefined threshold

Probability-based disclosure – the posterior distribution of the data after answering queries is significantly different from its prior distribution

Offline Auditors for Full Disclosure Sum queries Max and min queries Median and average queries

Audit Expert (Chin 1982)

Query auditing method for SUM queries A SUM query can be considered as a linear equation

where is whether record i belongs to the query set, xi is the sensitive value, and q is the query result

A set of SUM queries can be thought of as a system of linear equations

Maintains the binary matrix representing linearly independent queries and update it when a new query is issued

A row with all 0s except for ith column indicates disclosure

Offline Auditing for Full Disclosure Arbitrary combinations of aggregate queries It is unlikely there will be efficient on-line

application algorithms for SUM + MAX queries, SUM + MIN queries, or SUM + MAX + MIN queries.

Example hardness results:

Theorem. There is no polynomial time full-disclosure auditing algorithm for sum and max queries unless P=NP.

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Auditing Sum Queries on Booleans

Database: collection of secret Boolean variables Query: specifies subset S of variables Answer: sum of variables in S Privacy breach: after asking several queries, user

learns the value of some secret variable(s) Auditing problem: given a set of Boolean

equations, is there a variable that has the same value in all solutions?

Auditing Boolean attributes, Kleinberg, 2000

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Linear Diaphantine equations Query can be safe on real-valued, unbounded

data, but reveal information when the data are discrete, with known bounds

Hardness results: the auditing problem for Boolean values is coNP-complete.

x + y + w = 1y + z = 1x + z = 1

Real: multiple solutions, secureBoolean: unique solution, insecure (why?)

Why Is This Interesting?

Offline Auditor for Partial Disclosure

Partial disclosure – the disclosed range of the protected attribute is smaller than a predefined threshold

Sum queries

Interval-based disclosure: monitoring upper and lower bounds for all confidential attributes

Auditing interval-based inference. Li et al. 2002

Offline Auditor for Partial Disclosure New query A series of linear programming problems

Incremental evaluation of the LP

Treats the auditing problem as a series of updation problems and updates the bounds with certain rules

Horizontal updation – given the same set of queries, the bounds of one variable, how can the prior result be modified to get the bounds of other variables

Vertical updatation – given the same set of variables, and bounds under the previous queries, how can the prior result modified to get the bounds when a new query arrives

An efficient online auditing approach to limit private data disclosure, Lu, 2009

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Online Auditing

Database

Qi+1

Auditor

A or “Denied”

Previous queries Q1 … Qi

“Denied” if answering Qi+1 would cause a

privacy breach

Online Auditing

Given a sequence of queries and corresponding answers, and a new query, determine if the new query should be answered or denied in order to prevent a privacy breach.

Earliest approaches Query set size control Query set overlap control Limited data utility

Offline to Online?

Can an offline auditor directly solve the online auditing problem?

Denials leak information!

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“On the advice of my counsel I respectfully and regretfully decline to answer the question based on my constitutional rights.”

Colonel Oliver North, on the Iran-Contra arms deal

“Mr. Chairman, I would like to answer the committee's questions, but on the advice of my counsel I respectfully decline to answer the question based on the protection afforded me under the Constitution of the United States.”

David Duncan, former auditor for Enron and partner in Arthur Andersen

Sounds Familiar?[slide stolen from Kobbi Nissim]

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Variables di are real, privacy breached if adversary learns some di

Example: Sum/Max

Database

Gimme sum(d1,d2,d3)Auditor

Answer=15

Gimme max(d1,d2,d3)

“Denied”

Wait… there must be a reason why second

query was denied

Oh well

The only possible reason for

denial is if d1=d2=d3=5

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Online Audit

Denials reduce the search space

Possible assignments to {d1,…,dn}

Assignments consistent with (q1,…qi; a1,…,ai)

qi+1 denied

One workaround

Deny whenever the offline algorithm does, in addition, randomly deny queries.

Issues Leakage is not prevented Have to remember which queries were

randomly denied Semantically determine whether two queries

are equivalent

Simulatable Auditing

Observation: denials have the potential to leak information if the auditor uses information that is unavailable to the attacker (the answer to the new query)

Simlatable auditing: the attacker should be able to simulate or mimic the auditors decisions to answer or deny a query

Denials provably do not leak information

Simulatable Auditing, Kenthapadi, 2005

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Simulatable Auditing An auditor is a function of Q, A and X An auditor is simulatable if there exists a simulator

that is a function of only Q and A-ai+1 and whose output is always equal to the auditor.

Auditor

qi+1

Deny or answer

qi+1

Deny or answer

Simulator

q1,…,qi

a1,…,aiDatabase

q1,…,qi

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Possible assignments to {d1,…,dn}

Assignments consistent with (q1,…qi, a1,…,ai )

qi+1 denied/allowed

Simulatable Auditing

Simulatable Auditing

Query-set-size control Query-set-overlap control Audit expert for sum queries

Constructing Simulatable Auditor

General sufficient condition: the auditor should determine if there is any possible dataset, consistent with all past responses, in which the answer to the current query would cause some element to be fully disclosed

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Example revisited: Sum/Max

Database

Gimme sum(d1,d2,d3)Auditor

Answer=a1

Gimme max(d1,d2,d3)

“Denied”

Simulatable auditor would always deny the max query following a sum query

Lose some utility due to the requirement of simulatability

Challenges

Privacy definition Privacy of groups/families

Algorithmic limitations Simulatable algorithms computationally prohibitive Most work on sum queries, some on max, min,

median, hardness results on mixed queries Collusion

Reduced utility for legitimate users Large audit trail

Utility Percentage of denials may not be the best measure