Abstract channels, gain functions
and the information order
B Espinoza1 , AK McIver2 , LA Meinicke3 , CC Morgan4 and G Smith1
1
3
School of Computing and Inf. Sciences, Florida International University, USA
2
Dept. Computer Science, Macquarie University, Australia
School of Inf. Technology and Electrical Engineering, U. Queensland, Australia
4
School of Computer Science and Engineering, UNSW, Australia
A probabilistic channel produces observable outputs from secret inputs; given
a prior distribution of inputs, the channel’s output defines a posterior distribution of inputs which, compared via some entropy with the prior, measures
information leakage caused by transmission through the channel.
Elsewhere [2, 4] it is shown that defining a generalised test (g-test or gain
function) determines a natural order on channels in terms of the information
they leak. A “spin-off” advantage of this is that the conventional presentation
of a channel, as a stochastic matrix, is easily seen to contain information that
is redundant with respect to leakage alone: examples are duplication, scaling or
permutation of columns.
In this paper we introduce abstract channels, the space obtained by quotienting over the equivalences derived from gain-function testing. Within this space
we show how [3] resolved the Coriaceous Conjecture of [2], namely that the
gain-function testing order is equivalent to a separately formulated information
order based on matrix structure alone. As proposed in [2], these orders together
generalise the Lattice of Information.
In addition we derive a number of interesting mathematical properties of
information leakage, both additive and multiplicative. Additive leakage [1] measures leakage as the difference between the posterior and prior vulnerabilities,
rather than the (log of) their ratio.
In the Appendix, we illustrate some of these ideas with an example.
References
1. C. Braun, K. Chatzikokolakis, and C. Palamidessi. Quantitative notions of leakage
for one-try attacks. In Proc. MFPS, volume 249 of ENTCS, pages 75–91. Elsevier,
2009.
2. Mário S. Alvim, Kostas Chatzikokolakis, Catuscia Palamidessi, and Geoffrey Smith.
Measuring information leakage using generalized gain functions. In Proc. 25th IEEE
CSF, pages 265–279, 2012.
3. Annabelle McIver, Larissa Meinicke, and Carroll Morgan. Compositional closure
for Bayes Risk in probabilistic noninterference. In ICALP, volume 6199 of LNCS,
pages 223–235, 2010.
4. Annabelle McIver, Larissa Meinicke, and Carroll Morgan. A kantorovich-monadic
powerdomain for information hiding, with probability and nondeterminism. In
LICS, pages 461–470, 2012.
A
An Example
Given space of secrets X = {x1 , x2 , x3 }, consider the following two channels:




2/5 0 3/5
1 0 0
C1 =  1/4 1/2 1/4 
C2 =  1/10 3/4 3/20 
1/2 1/3 1/6
1/5 1/2 3/10
While these channels may appear superficially to be very different, in fact they
are semantically the same channel as far as leakage of X is concerned. Indeed
both map a prior distribution π = (p1 , p2 , p3 ) on X to the same hyperdistribution
(i.e. distribution of posterior distributions on X ):
p2
2p3
1
4p1
,
,
with prob. (4p1 + p2 + 2p3 )
4p1 + p2 + 2p3 4p1 + p2 + 2p3 4p1 + p2 + 2p3
4
3p2
2p3
1
0,
,
with prob. (3p2 + 2p3 )
3p2 + 2p3 3p2 + 2p3
4
To understand this, note that the second and third columns of C1 are similar
(indeed column 2 is two times column 3); hence both columns 2 and 3 result
in the same posterior distribution. In the same way, columns 1 and 3 of C2 are
similar (indeed column 1 is two-thirds times column 3). Merging these similar
columns and ordering the resulting columns lexicographically gives the same
reduced matrix for both C1 and C2 :


1 0
C1r = C2r =  1/4 3/4 
1/2 1/2
The reduced matrix can be seen as a canonical representation of an abstract
channel.
Finally, we consider C1 and C2 with respect to the composition refinement
preorder v◦ of [2], defined by C1 v◦ C2 iff C1 can be expressed as C2 followed
by some “post-processing”; that is, C1 = C2 C3 for some channel matrix C3 . We
find that C1 v◦ C2 and also C2 v◦ C1 :


1 0 0
C1 = C2  0 2/3 1/3 
1 0 0
and


0 3/5
C2 = C1  0 1 0 
0 1 0
2/5
We hence say that C1 is composition equivalent to C2 . It turns out that composition equivalence coincides with semantic equivalence, which means that v◦ is
a partial order on abstract channels; it moreover coincides with the information
leakage order, by the resolution of the Coriaceous Conjecture.