Hide and Seek in Digital Communication: The Steganography Game

Two players, Alice and Bob play the following game. A list of positive numbers d1, d2, . . . , dn and an integer 1  k  n are given. Alice chooses a k-element subset S of {1, 2, . . . , n} and, simultaneously, Bob chooses an integer i 2 {1, 2, . . . , n}. If i / 2 S then there is no payo↵. If, on the other hand, i 2 S then Alice pays Bob the amount of di. This two-player, zero-sum game was introduced in [6] as a means of analyzing the security (or detectability) of contentadaptive steganography. A formula for computing a pair of Nash-equilibrium strategies was also given in [6], but this formula was shown in [10] to be incorrect for certain choices of the parameters. In [10], the game was also generalized to allow for costs, and this more general version was solved in the sense that finding a Nash-equilibrium was shown to be possible in polynomial time by solving an appropriate linear program. In this paper, we solve the (generalized version of the) game in a stronger sense: we provide (correct) formulas that give a pair of Nash-equilibrium strategies and thus show that these are possible to compute in strongly-polynomial time.

[1]  Jessica Fridrich,et al.  Steganography in Digital Media: References , 2009 .

[2]  Rainer Böhme,et al.  Where to Hide the Bits ? , 2012, GameSec.

[3]  Bernd Gärtner,et al.  Understanding and using linear programming , 2007, Universitext.

[4]  Rainer Böhme,et al.  A Game-Theoretic Approach to Content-Adaptive Steganography , 2012, Information Hiding.

[5]  Mark F. Bocko,et al.  Steganalysis aware steganography: statistical indistinguishability despite high distortion , 2008, Electronic Imaging.

[6]  Aron Laszka,et al.  Modeling Content-Adaptive Steganography with Detection Costs as a Quasi-Zero-Sum Game , 2013 .

[7]  Jessica J. Fridrich,et al.  Detection of content adaptive LSB matching: a game theory approach , 2014, Electronic Imaging.

[8]  Bruce Bueno de Mesquita,et al.  An Introduction to Game Theory , 2014 .

[9]  Rainer Böhme,et al.  Advanced Statistical Steganalysis , 2010, Information Security and Cryptography.

[10]  Rainer Böhme,et al.  Bitspotting: Detecting Optimal Adaptive Steganography , 2013, IWDW.

[11]  Yoav Shoham,et al.  Essentials of Game Theory: A Concise Multidisciplinary Introduction , 2008, Essentials of Game Theory: A Concise Multidisciplinary Introduction.

[12]  Rainer Böhme,et al.  A game-theoretic analysis of content-adaptive steganography with independent embedding , 2013, 21st European Signal Processing Conference (EUSIPCO 2013).

[13]  Jack Edmonds,et al.  Matroids and the greedy algorithm , 1971, Math. Program..

[14]  Andrew D. Ker.  Batch steganography and the threshold game , 2007, Electronic Imaging.

[15]  Mark Ettinger,et al.  Steganalysis and Game Equilibria , 1998, Information Hiding.