On Location Hiding in Distributed Systems

Abstract

We consider the following problem – a group of mobile agents perform some task on a terrain modeled as a graph. In a given moment of time an adversary gets access to the graph and agents’ positions. Shortly before adversary’s observation the devices have a chance to relocate themselves in order to hide their initial configuration, as the initial configuration may possibly reveal to the adversary some information about the task they performed. Clearly agents have to change their locations in possibly short time using minimal energy. In our paper we introduce a definition of a well hiding algorithm in which the starting and final configurations of the agents have small mutual information. Then we discuss the influence of various features of the model on running time of the optimal well hiding algorithm. We show that if the topology of the graph is known to the agents, then the number of steps proportional to the diameter of the graph is sufficient and necessary. In the unknown topology scenario we only consider a single agent case. We first show that the task is impossible in the deterministic case if the agent has no memory. Then we present a polynomial randomized algorithm. Finally in the model with memory we show that the number of steps proportional to the number of edges of the graph is sufficient and necessary. In some sense we investigate how complex is the problem of “losing” information about location (both physical and logical) for different settings.

The work of the second author was supported by Polish National Science Center grant 2013/09/B/ST6/02258. The work of the third author was supported by Polish National Science Center grant 2015/17/B/ST6/01897.

Appendices

Appendix 1: Information Theory

We recall some basic definitions and facts from Information Theory that can be found e.g. in [5]. In all cases below $\log$ will denote the base-2 logarithm.

Definition 3

(Entropy). For a discrete random variable $X :\mathcal {X}\rightarrow \mathbb {R}$ the entropy of X is defined as $H\left( X\right) = -\sum _{x \in \mathcal {X}} \Pr [X=x] \log \Pr [X=x]$.

Definition 4

(Conditional entropy). If $X :\mathcal {X}\rightarrow \mathbb {R}$ and $Y :\mathcal {Y}\rightarrow \mathbb {R}$ are two discrete random variables, we define the conditional entropy as

$$ H\left( X|Y\right) = -\sum _{y \in \mathcal {Y}} \Pr [Y=y] \sum _{x \in \mathcal {X}} \Pr [X=x|Y=y] \log \Pr [X=x|Y=y]. $$

Fact 1

For any random variables X and Y $H\left( X|Y\right) \le H\left( X\right)$ and the equality holds if and only if X and Y are independent.

Definition 5

(Relative entropy). Let X and Y be two discrete random variables defined on the common space $\mathcal {X}$ with pmf p(x) and q(x), respectively. The relative entropy (Kullback-Leibler distance) between p(x) and q(x) is

$$ D\left( p||q\right) = \sum _{x \in \mathcal {X}} p(x) \log \frac{p(x)}{q(x)}. $$

Fact 2

(Information inequality). Let p(x) and q(x) be probability mass functions of two discrete random variables $X, Y :\mathcal {X}\rightarrow \mathbb {R}$. Then $D\left( p||q\right) \ge 0$ with equality if and only if $\forall x \in \mathcal {X}$ $p(x) = q(x)$.

Fact 3

(Theorem 1 in [7]). Let p(x), $q(x) > 0$ be probability mass functions of two discrete random variables X and Y, respectively, defined on the space $\mathcal {X}$. Then

$$ D\left( p||q\right) \le \frac{1}{\ln 2} \left( \sum _{x \in \mathcal {X}} \frac{p^{2}(x)}{q(x)} - 1 \right) . $$

Definition 6

(Mutual information). If X and Y are two discrete random variables defined on the spaces $\mathcal {X}$ and $\mathcal {Y}$, respectively, then the mutual information of X and Y is defined as

$$\begin{aligned} I\left( X,Y\right) = \sum _{x \in \mathcal {X}} \sum _{y \in \mathcal {Y}} \Pr [X=x, Y=y] \log \left( \frac{\Pr [X=x, Y=y]}{\Pr [X=x] \Pr [Y=y]}\right) . \end{aligned}$$

(7)

Fact 4

For any discrete random variables X, Y

• $0 \le I\left( X,Y\right) \le \min \{H\left( X\right) , H\left( Y\right) \}$ and the first equality holds if and only if random variables X and Y are independent,

• $I\left( X,Y\right) = I\left( Y,X\right) = H\left( X\right) - H\left( X|Y\right) = H\left( Y\right) - H\left( Y|X\right)$.

• $I\left( X,Y\right) = D\left( p(x,y)||p(x)p(y)\right)$ where p(x, y) denotes the joint distribution, and p(x)p(y) the product distribution of X and Y.

Appendix 2: Markov Chains

We recall some definitions and facts from the theory of Markov chains. They can be found e.g. in [5, 13, 15]. Unless otherwise stated, we will consider only time-homogeneous chains, where transition probabilities do not change with time.

Definition 7

(Total variation distance). For probability distributions $\mu$ and $\nu$ on the space $\mathcal {X}$ we define the total variation distance between $\mu$ and $\nu$ as $d_{\mathrm {TV}}\left( \mu , \nu \right) = \max _{A \subseteq \mathcal {X}} |\mu (A) - \nu (A)|$.

Fact 5

Let $\mu$ and $\nu$ be two probability distributions on common space $\mathcal {X}$. Then we have $d_{\mathrm {TV}}\left( \mu , \nu \right) = \frac{1}{2} \sum _{x \in \mathcal {X}} |\mu (x) - \nu (x)|$.

Definition 8

Let $P^{t}(x_0, \cdot )$ denote the distribution of an ergodic Markov chain on finite space $\mathcal {X}$ in step t when starting in the state $x_0$. Let $\pi$ be the stationary distribution of M. We define $d(t) = \max _{x \in \mathcal {X}} d_{\mathrm {TV}}\left( P^{t}(x, \cdot ), \pi \right)$ and $\bar{d}(t) = \max _{x, y \in \mathcal {X}} d_{\mathrm {TV}}\left( P^{t}(x, \cdot ), P^{t}(y, \cdot )\right)$.

Fact 6

Let $\mathcal {P}$ be the family of all probability distributions on $\mathcal {X}$. Then

• $d(t) \le \bar{d}(t) \le 2 d(t)$,

• $d(t) = \sup _{\mu \in \mathcal {P}} d_{\mathrm {TV}}\left( \mu P^{t}, \pi \right) = \sup _{\mu , \nu \in \mathcal {P}} d_{\mathrm {TV}}\left( \mu P^{t}, \nu P^{t}\right)$.

Definition 9

(Mixing time). For an ergodic Markov chain M on finite space $\mathcal {X}$ we define the mixing time as $t_{\mathrm {mix}}\left( \varepsilon \right) = \min \{t :d(t) \le \varepsilon \}$ and $t_{\mathrm {mix}}= t_{\mathrm {mix}}\left( 1/4\right)$.

Fact 7

For any $\varepsilon > 0$, $t_{\mathrm {mix}}\left( \varepsilon \right) \le \left\lfloor \log \varepsilon ^{-1} \right\rfloor t_{\mathrm {mix}}$.

Definition 10

(Random walk). The random walk on a graph $G = (V,E)$ with n nodes and m edges is a Markov chain on V with transition probabilities

$$ p_{ij} = \Pr [X_{t+1} = v_j | X_t = v_i] = {\left\{ \begin{array}{ll} 1/\mathrm {deg}\left( v_i\right) , &{} if \, \{v_i,v_j\} \in E,\\ 0, &{} otherwise. \end{array}\right. } $$

The lazy random walk is the random walk which, in every time t, with probability 1/2 remains in current vertex or performs one step of a simple random walk.

The following Fact 8 gives an upper bound on the mixing time for random walks. It follows e.g. from Theorem 10.14 in [13] and the properties of cover time and its relation to mixing time (see [11]).

Fact 8

For a lazy random walk on an arbitrary connected graph G with n vertices $t_{\mathrm {mix}}= {{\mathrm{O}}}\left( n^3\right) .$

Fact 9

(cf. [4, 5, 22]). Let $M = (X_0, X_1, \ldots )$ be an ergodic Markov chain on finite space $\mathcal {X}$ with transition matrix P and stationary distribution $\pi$.

• For any two probability distributions $\mu$ and $\nu$ on space $\mathcal {X}$ the relative entropy $D\left( \mu P^{t}||\nu P^{t}\right)$ decreases with t, i.e. $D\left( \mu P^{t}||\nu P^{t}\right) \ge D\left( \mu P^{t+1}||\nu P^{t+1}\right)$.

• For any initial distribution $\mu$ the relative entropy $D\left( \mu P^{t}||\pi \right)$ decreases with t. Furthermore, $\lim _{t \rightarrow \infty } D\left( \mu P^{t}||\pi \right) = 0$.

• The conditional entropy $H\left( X_0|X_t\right)$ is increasing in t.