1. Security in Outsourcing of Association Rule Mining

2. Agenda
Introduction
Item mapping
Algorithms and experiments
Summary

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

4. Association rules Possible solutions: Complexity of exponential order
Apriori, FPclose, …
Complexity of exponential order
Required a long time for simple mining algorithm
Motivated to outsource mining responsibility

5. Concerns in outsourcing
Correctness
Security
Privacy of records
Information of the company
Company DB
Data Miner

6. Item mapping

7. Item mapping
Substitution cipher has been widely used in encryption in text
Each letter is replaced by another letter
In encryption of transactions
Each item is replaced by another item

8. Example bread -> 54 chocolate -> 165
What is <8, 69, 153, 756>?
<cheese, fork, ice-cream, clock>

9. Security Substitution cipher is not secure Frequency analysis
Frequency analysis can be done to recover the original plain text
Frequency analysis
We know the relative frequencies of each letter in dictionary
Find frequencies of letters in the cipher text
Perform matching using both frequencies

10. Security in item substitution
Item substitution is more secure than substitution in text
The number of possible mappings is much larger (|I|! where I is the entire set of items)
Relative frequencies of itemset is not known
There is no ways to determine whether a guess on the mappings is correct

11. Architecture Association Rules Association Rules Mappings DB DB
Transformer

12. Simple scheme (one-to-one item mapping)
T: the database
I: a set of items in T
D: another set of items with |D| = |I|
T’: the transformed database
m: I -> D : one-to-one item mapping
Encryption
For each transaction t in T = <x1, x2, …, xn>
t’ = <m(x1), m(x2), …, m(xn)>
Decryption of association rules
For every association rule in T’ <a1, .., ak> => <b1, …bn>
<m-1(a1), …, m-1(ak)> => <m-1(b1), …, m-1(bn)> is a rule in T

13. Privacy breach
Statistical information is known by the third party miner
Counts of some itemsets
Number of rules in the database
Number of large itemsets

14. Fake items m: I -> D : one-to-one item mapping
F: a set of items not in D
For each transaction t, a subset of F is added to t’

15. Example I = {a, b}, D = {1, 2} m(a) = 1, m(b) = 2 F = {3}
T = {<a>, <b>}
T’ = {<1, 3>, <2>}

16. Functions of fake items
Increase the difficulty to get a correct guess on a mapping of item (from 1/|I| to 1/(|I| + |F|))
Protect statistical information of the data owner
The miner do not know the exact information

17. Better way to utilize “fake” items
A one-to-n item mapping
B: a set of items
m: I -> 2B
Example
m(a) = {1, 4, 5}
m(b) = {2}
m(c) = {3, 5}

18. Advantage
In an ideal situation, it is even more difficulty to get a correct guess on a mapping of item
One-to-one item mapping (no fake items): 1/|I|
With fake items :1/(|I| + |F|)
One-to-n item mapping: 1 / (2|B| - 1)
It does not introduce additional overheads compared to adding fake items

19. Itemset mapping m: I -> 2B : one-to-n item mapping
M: 2I -> 2B : itemset mapping
X: an itemset with (x1, x2,…, xn)
M(X) = Ux in X m(x)
M-1(Y) = {x | x in I and m(x) is subset of Y}

20. Example itemset mapping
m(b) = {2}
m(c) = {3, 5}
M(<a, b>) = <1, 2, 4, 5>
M(<a, c>) = <1, 3, 4, 5>
M(<b, c>) = <2, 3, 5>

21. Example inverse itemset mapping
m(b) = {2}
m(c) = {3, 5}
M-1(<1, 2, 4, 5>) = <a, b>
M-1(<1, 3, 4, 5>) = <a, c>
M-1(<1, 2, 3, 5>) is not well-defined

22. Correctness – restrictions on one-to-n mapping
m(b) = {2, 3}
m(c) = {1, 3}
M(<a, b>) = <1, 2, 3>
M(<a, b, c>) = <1, 2, 3>
Each mapping of an item must contain unique stuff

23. Admissible item mapping
m: I -> 2B
m is admissible iff
For all x in I
m(x) – U y in I – {x} m(y) is not empty
Guarantee
M-1(M(X)) = X (correct decryption)

24. Security of one-to-n item mapping
Function cover
M1: 2I -> 2D1
M2: 2I -> 2D2
M1 covers M2 iff
for all X is a subset of I
Y = M2(X)
M2-1(Y ) = M1-1(Y intersect D1)

25. Function cover If M1 covers M2 Our results
Any transaction encrypted by M2 can be decrypted by using the inverse of M1
Our results
A one-to-n itemset mapping built on an admissible one-to-n item mapping is covered by some one-to-one itemset mapping

26. Example transformation
{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} M:
a -> {1, 4, 5}
b -> {2}
c -> {3, 5}
M’:
a -> {1}
b -> {2}
c -> {3}

27. Function cover
Suppose a data owner encrypted his own database using a one-to-n item mapping m
An adversary do not need to find m, but instead he look for a one-to-one item mapping m’
Conclusion
One-to-n item mapping is not more secure than a one-to-one item mapping

28. Transaction transformation
M: 2I -> 2B : One-to-n itemset mapping
F: a set of items not in B
To hide B
N: transaction transformation
Maps from 2I to 2BUF
N(t) = M(t) U E
E is a random subset of B U F
N-1(t’) = {x | m(x) in t’}

29. Transaction transformation
Valid
N-1(N(t)) = t
Complete (based on valid)
s is a program to generate E
outs(t) is the set of all possible outputs of s with transaction t
If for all t
For any E such that t’ = M(t) U E and N-1(t’) = t
E in outs(t)

30. Valid and complete transaction transformation
N: valid and complete transaction transformation
There does not exist a one-to-one itemset mappint M’ which covers N
Cannot analyze it as a one-to-one mapping
Increased difficulty in cryptanalysis

31. Example transformation
{b}
{c}
{a, b}
{a, c}
{b, c}
{a, b, c}
T’ =
{1, 4, 5}
{2, 1, 3}
{3, 5, 4}
{1, 2, 4, 5}
{1, 3, 5}
{2, 3, 5, 1}
{1, 2, 3, 4, 5} M:
a -> {1, 4, 5}
b -> {2}
c -> {3, 5}
M’:
a -> ?

32. Conclusion on transaction transformation
Correct results
More difficult to perform cryptanalysis
Overhead comparable to one-to-one item mapping with fake items
Transaction transformation is a non-deterministic approach

33. Recovering association rules
Given an encrypted rule in T’
X => Y
Find A = M-1(X) and B = M-1(X U Y)
If both are well-defined
The rule in T is
A => B - A

34. Example Given 1 => 4 (rejected) 2 => 1, 5 (rejected)
b  2
c  3, 5
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

35. Algorithms

36. Algorithms Developed Generate an admissible one-to-n item mapping
Perform a transaction transformation given a transaction t and an admissible one-to-n item mapping
Valid and complete

37. Experiments

38. Frequency analysis
One-to-one item mapping vs one-to-n transaction transformation
One-to-n mapping
Limited to size 2
Each item can be mapped to at most 2 items
Adversary: Genetic algorithm
Automated frequency analysis tools using a random approach

39. Parameters One-to-one: 5 fake items
One-to-n: |I| items maps to |I|+5 items
Two initialization strategies
Good
Each mapping of item has a chance to be initiated to be a correct one
Random
Each mapping of item is initiated totally random

41. 20 items
35 items

42. Observation One-to-n requires more information to be prepared
Takes a longer time to calculate the goodness of current state
Long time spent in each iteration
Shows slow progress when near the optimal solution
Similar progress but in fact a poorer result

43. Background knowledge
Assume the adversary has an approximation from a global result of mining
alpha: knows alpha% of large itemsets in original database
beta: the support in the knowledge is in the range
real support * (1 beta)

44. beta
alpha
beta
alpha

45. Overhead measure
Measure the overhead introduced by transaction transformation
Parameters
Em: expected size of a mapping of an item
NE: expected size of B intersect E
NF: expected size of F intersect E

46. Em = 1.6,
Em = 1.6,
Em = 1.4,

47. Summary
The idea of substitution cipher is used in the problem of encryption of transaction database
One-to-n item mapping cannot be directly applied since it is effectively a one-to-one item mapping
Transaction transformation is proposed
Experiments show that it is suitable for outsourcing

48. The End