Security in Data Warehouses and OLAP Systems - Database Security: Applications and Trends - page 198

Table 1. Modeling A Core Cuboid

Q 1 Q 2 Q 3 Q 4 ALL

Bob x 1 x 2 x 3 8500

Alice x 4 x 5 10000

Jim x 6 x 7 6100

Mallory x 8 x 9 12400

ALL 10000 6000 11000 9000 36000

Table 2. Modeling the Aggregation Cuboids

⎛
⎝

⎞
⎠

⎛
⎝

⎞
⎠

⎛
⎝

⎞
⎠

111000000
000110000
000001100
000000011
100001010
010100000
001010000
000000101
111111111

x 1
x 2
x 3
x 4
x 5
x 6
x 7
x 8
x 9

8500
10000
6100
12400
10000
6000
11000
9000
36000

×

=

in Table 3. From M rref it can be observed that the system of linear equations
in Table 2 has infinitely many solutions. This means that an adversary cannot
infer the entire core cuboid from the given aggregation cuboids. However, the
first row vector of M rref being a unit vector (that is, it has a single 1) indi-
cates that the value of x 1 must remain the same among all the solutions to the
system of equations. Consequently, the adversary can infer Bob's commission
in Q 1 .

Table 3. The Reduced Row Echelon Form M rref

⎛
⎝

⎞
⎠

1000 0 000 0
0100 − 1000 0
0010 1 000 0
0001 1 000 0
0000 0 100 − 1
0000 0 010 1
0000 0 001 1
0000 0 000 0
0000 0 000 0

The existence of at least one unit row vectors in M rref is indeed the neces-
sary and sucient condition for any unknown variable to have the same value