Security in Outsourced Association Rule Mining

Agenda  Introduction  Approximate randomized technique  Encryption  Summary and future work

Introduction  Data mining in company know about the past activities of their customers make strategic decisions  Types of data mining Association rules mining Clustering Classification

Association rules  “X => Y” If a transaction contains itemset X, the transaction will probably contain itemset Y Support: number of supporting transactions Confidence: proportion of transactions containing X which also contains Y

Performing data mining  Build application Development cost? Time?  Buy software Fit requirements? Maintenance?  Outsource

Concerns in outsourcing  Output Execution Assurance Correctness  Security Privacy of records Information of the company Company DB Data Miner

Approximate randomized technique

Approximate solution  Privacy Preserving Mining of Association Rules SIGKDD 2002 Authors: Alexandre Evfimievski, Ramakrishnan Srikant, Rakesh Agrawal, Johannes Gehrke

Problem formulation  Let the set of transactions be T = {t 1, t 2, … t N }  Transform T to T’ = {t’ 1, t’ 2, … t’ N }  Mine in T’  Privacy breaches Itemset A cause a privacy breach of level p if for some item a in A  P[a in t i |A in t’ i ] >= p

Select-a-size randomization  For each transaction t i in T m = length of t i Select (non-uniformly) randomly an integer j from [0, m] Copy uniformly at random j items in t i to t’ i Consider every item a not in t i, add a to t’ i with a given probability p m

Run on real data  Privacy breach of level <= 50% P[a in t i |A in t’ i ] <= 50%  Accuracy = # true positive / (# found itemsets)  Set 1 Itemset Size True Itemset True Positive False Drops False Positive Accuracy % % %

Accuracy  Set 2: Itemset Size True Itemset True Positive False Drops False Positive Accuracy % % %

Problems  Estimated counts of large itemsets varies Lower accuracy of association rules  "beer and diaper" story customers who buy diapers tend also to buy beer hard to believe some strange rules  Expensive to make wrong decision Supermarket: layout design Health center: identify new disease

Security concerns  Individual transaction is protected  Private association rules can be estimated by other parties Adversary actions may be based on found association rules

Encryption

Problem formulation  Let the set of transactions be T = {t 1, t 2, … t N }  I is the entire set of items All t i is a subset of I  Transform T to T’ = {t’ 1, t’ 2, … t’ N }  A third party mines in T’ and gets AR’  Transform AR’ to AR

Architecture DB Transformer Association Rules Association Rules Mappings

Encryption  To protect a message, simple encryption can be applied “GOOD DOG” can be encrypted as “PLLX XLP”  Association rule encryption 752 => 891? Milk => Bread  Transaction encryption ?

Simple scheme  Encryption  For every transaction t i For every item x in t i  Add f(x) to t’ i where f is a bi-jective function  Decryption For every association rule r i  For every item y in r Replace y by f -1 (y)

Problems with simple encryption  They are easy to crack “PLLX XLP”  26 P 3 combinations, with at least one vowel Association rules  # Bread > # Car  # association rules, # large itemsets are disclosed  Solution Use a more complex scheme

Fake items  Probability to make a correct guess of a single mapping = 1 / |I|  Randomly add some fake items to each transaction Decrease the above probability to 1 / (|I| + |F|)

One-to-n Mapping  Originally, we are “one-to-one” mapping One item  One item A  1 B  2 C  3  We form “one-to-n” mapping A  1, 4, 5 B  2 C  3, 5 Greatly increase the number of possible mapping of an item  |I|+|F| C 1 + |I|+|F| C 2 + … |I|+|F| C |F|

Example transformation  T = {A} {B} {C} {A, B} {A, C} {B, C} {A, B, C}  T’ = {1, 4, 5} {2} {3, 5} {1, 2, 4, 5} {1, 3, 5} {2, 3, 5} {1, 2, 3, 4, 5} A  1, 4, 5 B  2 C  3, 5

Limitation on the mapping f  For any item x, there does not exist items y 1, y 2, …, y k (x ≠ y 1 ≠ … ≠ y k ) Such that f(x) subset in f(y 1 ) U f(y 2 ) U…f(y k )  Consider an example A  1, 2 B  2, 3 C  3, 4 AC  1, 2, 3, 4 ABC  1, 2, 3, 4

Limitation on the mapping f  For any item x f(x) – U i != x, i in I f(i) != empty  Every item must map to something unique

Mapping generation – Item Extend  Initialize every item to map to something unique I’  For every item x in IE Randomly pick some mappings Extend each mapping by x

Example run  A  1  B  2  C  3  IE = {4, 5}

Considering item 4  A  1  B  2  C  3  A  1, 4  B  2  C  3 Pick A

Considering item 5  A  1  B  2  C  3  A  1, 4, 5  B  2  C  3, 5 Pick A, C

Item Extend  Every item must map to something unique Say 1 is unique to f(A)  supp T (A) = supp T’ (1)  For a transaction t without item A Add a subset of unique mapping set to t’ with some probabilitysome probability {1, 4} is unique mapping set in f(A)  {}, {1}, {4}, {1, 4} may be added A  1, 4, 5 B  2 C  3, 5

Fake items again  Now, every item in t’ i must be in some mappings  Randomly add some fake items in |F| to each transaction  Mapping f: I -> |I’| U |IE| U |F| |I’|: core “unique” items |IE|: expanding items |F|: fake items

Basic transformation framework  For each transaction t For each item x in t  Add f(x) to t’ For item i in I - t  Add randomly subset of unique mapping set of f(i) to t’ For item f in F  Toss a biased coin for each item, add f to t’ if head (probability should be difference)

Recovering association rules  Given an encrypted rule in AR’ r’: X => Y  If there exists i 1, i 2, …, i m in I U k=1 m f(i k ) = X  And there exists j 1, j 2, …, j n in I U k=1 n f(j k ) = XUY  r: {i 1, i 2, … i m } => {j 1, j 2, …, j n } – {i 1, i 2, … i m } is a rule in AR  Otherwise, the rule is not correct

Example  Given 1 => 4 (rejected) 2 => 1, 5 (rejected) 2 => 1, 3, 5 (rejected) 2 => 1, 3, 4, 5 (B => AC) 2, 3, 5 => 1, 4 (BC => A)  2, 3, 5 => BC  1, 2, 3, 4, 5 => ABC Mapping f A  1, 4, 5 B  2 C  3, 5

Correctness  Proposition For any item x, y, f is transformation mapping  supp T (x) = supp T’ (f(x))  supp T (xUy) = supp T’ (f(x) U f(y)) For any itemset X, Y, F is the transformation mapping  supp T (X) = supp T’ (F(X))  supp T (XUY) = supp T’ (F(X) U F(Y))  No false drops and false positives

Summary  Generation of mappings One-to-n mappings Item Extend  Transformation of transactions Mapping f(x) Subsets of unique mapping set Fake items  Recovering association rules Reverse mappings and filtering

Test run  # Items = 1k, |T| = 1k  Without transformation One rule Time: 8s  Item Extend 147 rules Total times: 26s Mappings generation and transformation: 219ms

Future Work  Define parameters to the problem Size of |IE| Size of |F|  Give a clear measure of security  Give a clear measure of overhead  Correctness of association rules Query execution proof Result verification

The End

Choosing probability  Uniform distribution or any fixed distribution give patterns which may be easily identified  Random probability distribution {}: 70%, {1}: 5%, {4}: 15%, {1, 4}: 20% Storage: need additional storage Back

Algorithm for transformation  Transformation is the most costly process  Execution time linear to database size |T|  Should be as fast as possible

Optimization  Mapping Retrieval For an item x, use a hash table to retrieve the mapping, h(x)  Adding fake items First randomly (according to the probability of adding items) determine the number of items to add Randomly pick in the set (non-uniform distribution) Gives a much shorter runtime in average

Choice of mapped items 12…|I|+|IE|+|F| * (1+ δ)  Acceptable as long as it is not easy to identify I’, IE, F  One way is to use random permutation of first |I| + |IE| + |F| natural numbers  First |I| numbers are mapped to |I’|  Next |IE| numbers are IE

Cut and paste randomization  One case of select-a-size randomization  The way to perform selection of j Given an integer K m > 0 Randomly choose j in [0, K m ] If (j > m)  Set j = m  Overall input parameters K m p m

Effects on support  Support of A in T’ A in t, without replaced A’ in t, randomly add A  Support of AB in T’ AB in t, without replaced A and B AB’ in t, randomly add B A’B in t, randomly add A A’B’ in t, randomly add A and B

Estimating original support  Support of A in T, x Support of A in T’, y x * P(A remains in original transaction) + (|DB| - x) * p m = y  Support of AB in T Support of AB in T’ Support of AB’, A’B in T’ Support of A’B’ in T’

Apriori property  Suppose m = 2 for all t in T  |T| = 10, |I| = {A, B}  p m = 0, j = 1,  Support of B in T’ supp T’ (B)= 0 E(supp T (B)) = 0  supp T’ (A)= 10  supp T’ (AB)= 0  E(supp T (AB)) = supp T’ (A) * 1 = 10

Apriori property  An expected large itemset may have an expected small sub-set  But generally the support of subsets are not too small  Instead of using the support threshold to filter all small candidates, use a smaller value

Apriori algorithm  Generate candidate sets  Scan database for counts  Recover the predicted support  Discard candidates with support smaller than <= candidate limit  Save for output candidates with support >= support threshold  Apriori_gen(remaining candidate)

Candidate limit  A high value Increase numbers of false drops Poor correctness  A small value Increase number of candidate sets High running time  Experiment Support threshold: s min estimated s.d.: δ s min – δ is found to be a good value

Other applications  Outsourced transaction database (secure) storage  Outsourced association rule mining using data stream  Secure distributed association rule mining with third party miner

Outsourced database with association rule mining service DB Transformer Association Rules Association Rules Mappings Transactions Query