Secret Securing with Multiple Protections and Minimum Costs

TL;DR

This paper addresses securing secrets in discrete-event systems by designing cost-effective protection policies that ensure secrets are sufficiently protected with minimal protection costs.

Contribution

It formulates a novel security problem with multiple protections and costs, providing necessary and sufficient conditions and an algorithm based on supervisory control theory.

Findings

The proposed algorithm guarantees minimum protection costs.

The security condition for problem solvability is clearly characterized.

Effectiveness demonstrated through a network security example.

Abstract

In this paper we study a security problem of protecting secrets with multiple protections and minimum costs. The target system is modeled as a discrete-event system (DES) in which a few states are secrets, and there are multiple subsets of protectable events with different cost levels. We formulate the problem as to ensure that every string that reaches a secret state (from the initial state) contains a specified number of protectable events and the highest cost level of these events is minimum. We first provide a necessary and sufficient condition under which this security problem is solvable, and then propose an algorithm to solve the problem based on the supervisory control theory of DES. The resulting solution is a protection policy which specifies at each state which events to protect and the highest cost level of protecting these events is minimum. Finally, we demonstrate the effectiveness of our solution with a network security example.

Equations

$$\mathbf{G}:=(Q,\Sigma ,\delta ,q_0)$$

$$[\forall s\in {\Sigma }^{\ast }]\delta (q_0,s)!\&\delta (q_0,s)\in Q_s⟹s\in {\Sigma }^{\ast }{\Sigma }_p^k{\Sigma }^{\ast }{\Sigma }_p^k{\Sigma }^{\ast }$$

Full text

I Introduction

Various security problems called cybersecurity issues have attracted much interest of researchers. In real systems, security problems can be caused by administrators’ mistakes or vulnerabilities of products. [1] introduces practical and technical methods relevant to security issues in the real world. In general, there are some secrets in the system which intruders want to steal without raising an alert, namely without being detected, and such secrets must be protected against malicious access of intruders. At the same time, the cost to protect secrets must be taken into account because infinite protection cost is infeasible in practice.

In this paper, we employ discrete-event systems (DES) to model real systems because it is suitable for describing dynamics and architectures of computer and network systems [2]. We also utilize the fundamental techniques from the supervisory control theory (SCT) of DES to compute solutions for problems we introduce. The SCT is the theory that Ramadge and Wonham originally proposed in [3]. For a comprehensive account of the SCT, the reader is referred to [4], and also see [5] for a historical overview of the theory.

One aspect of anonymity and secrecy that has been extensively studied in DES is opacity. This is a concept that intruders cannot identify secrets in the system because of their partial observability. For an overview of opacity, the reader is referred to [6], and also see [7] for historical remarks on opacity. In case that opacity is violated, several methods of enforcing opacity have been explored in literature [8, 9]. [8] investigates making languages that the system generates opaque, namely intruders cannot determine that the system has generated a secret language, by controlling the system with the SCT. [9] introduces inserting observable events into output languages from the system to make the language which intruders observe opaque. Opacity is based on the side of intruders, assuming that they have full knowledge of the target system’s structure but have only partial observability of the system’s behavior. By contrast, our work in this paper stands on the side of system administrators and focuses on the secret protection. We do not impose assumptions on the intruders’ knowledge and observability of the system. In particular, intruders may be able to observe all events, in which case opacity cannot hold. Instead, we study the problem of protecting the secrets as much as possible, while balanced by the cost of such protections.

For protecting secrets, we consider that there exist some operations or events which can be protected by administrators, e.g. connecting to a network or logging into a computer. In this paper, we represent an event to which system managers can apply a protection as a protectable event, and other events as unprotectable events. We also consider that there are multiple groups of protectable events, which have different levels of protection implementation costs. In addition, we represent secret information to be protected in the system as secret states. Secret information is a particular piece of information which should be available only to permitted users, for example, users’ credit card numbers, or system privileges like root in Unix operating systems. System administrators decide which protectable events to apply protections based on a protection policy that specifies which events to be protected at a given state. Our main objective is to solve the problem of finding an effective protection policy such that all secrets are protected with a predetermined number of protections, and the highest cost level to implement these protections is minimum. To compute a solution for this security problem, we convert the problem into a control problem and resort to the SCT. Our previous work [10] introduces a problem of secret securing with at least one protection and minimum protection cost, which is a special case of the problem considered in this paper.

The main contributions of this paper are fourfold. First, we formalize the security problem with DES as secret protection with multiple protections and minimum costs. Second, we present a necessary and sufficient condition under which the problem is solvable. This condition characterizes the situation where every string leading to the secret states in the system has at least a specified number of protectable events. Third, we introduce the concept conversion from security to control, and propose an algorithm to compute a solution for the converted and the original problem.

The remaining of this paper is organized as follows. Section II introduces a target system modeled by DES and formulates the problem of secret securing with multiple protections and minimum costs. In Section III, we first introduce a solvability condition such that the formulated original problem is solvable, and convert the security problem to a control problem, and then propose an algorithm to compute a solution for the converted problem. Section IV demonstrates our algorithm with an illustrating example.

II Problem Formulation

In this section, we formulate “Secret Securing with Multiple Protections and Minimum Costs Problem”. Its objective is to find a policy to protect all secret states with a prescribed number of protections and minimum protection cost. Consider a task to protect all secrets in the system, and assume that administrators want to use at least $m$ ($\geq 1$) protections. For this task, we need to find a protection policy to force intruders before reaching secrets to encounter $m$ protections. Meanwhile, the protection cost must be minimum. We consider that all secrets are protected with $m$ protections when every string reaching secrets from the initial state has at least $m$ protectable events.

We consider secret securing with minimum costs problem (SSMCP) in the framework of discrete-event systems (DES) modeled as finite-state automata

[TABLE]

where $Q$ is the set of states, $\Sigma$ is the set of all events, $\delta:Q\times\Sigma\to Q$ is the partial transition function, and $q_{0}\in Q$ is the initial state. We denote by $Q_{s}\subseteq Q$ the set of secret states in $\mathbf{G}$. $\delta$ is extended to $\delta:Q\times\Sigma^{\ast}\to Q$ in the standard manner [2]. $\delta(q,s)!$ denotes that string $s$ from state $q$ is defined. $\Sigma$ is a disjoint union of the protectable event set $\Sigma_{p}$ and the unprotectable event set $\Sigma_{up}$, namely $\Sigma=\Sigma_{p}\operatorname{\dot{\cup}}\Sigma_{up}$. In addition, $\Sigma_{p}$ is partitioned into $n$ disjoint subsets of protectable events $\Sigma_{i}$ where $i\in\set{0,1,\dots,n-1}$, namely $\Sigma_{p}=\operatorname*{\dot{\bigcup}}_{i=0}^{n-1}\Sigma_{i}$. The index $i$ of $\Sigma_{i}$ indicates the level of protection cost when the system administrator protects events in $\Sigma_{i}$. As the index $i$ increases, the protection cost becomes higher. We consider that the cost level of each subset is not comparable with other subsets. In other words, the cost to protect one event in $\Sigma_{i}$ is sufficiently higher than the cost to protect all events in $\Sigma_{i-1}$. For example, implementing a biometric protection is often more costly than setting up multiple password protections. We also denote the union of the subsets of protectable events until index $k$ by $\Sigma_{p}^{k}=\operatorname*{\dot{\bigcup}}_{i=0}^{k}\Sigma_{i}$.

In order to identify which transitions to protect, the system administrator needs a protection policy which specifies protectable events at suitable states. We define such a policy as a function $\mathcal{P}:Q\to\operatorname{Pwr}(\Sigma_{p})$ where $\operatorname{Pwr}(\Sigma_{p})$ is the power set of $\Sigma_{p}$. For example, $\mathcal{P}(q)=\set{\sigma_{i},\sigma_{j}}$ indicates that protectable events $\sigma_{i}$ and $\sigma_{j}$ are protected at state $q$.

For clarity of presentation, we henceforth focus on the case $m=2$. The case $m\geq 3$ can be addressed in the same fashion (but with more complicated notation). The case $m=1$ has been solved in [10], which is a special case of the problem addressed in this paper. We first define the following concept indicating that the secret states are protected with at least two protections.

Definition 1** (2-secure reachability).**

Consider a plant $\mathbf{G}$ in Eq. 1. The secret state $Q_{s}$ is securely reachable with at least two protectable events (2-securely reachable) w.r.t. $\mathbf{G}$ and $\Sigma_{p}^{k}=\operatorname*{\dot{\bigcup}}_{i=0}^{k}\Sigma_{i}$ if the following condition holds:

[TABLE]

Note that $s$ in Eq. 2 can contain two or more protectable events in $\Sigma_{p}^{k}$, which means that intruders have to penetrate at least two protections to reach secrets. When condition Eq. 2 does not hold, there exists a string which contains one or no protected event and reaches a secret state – this is the situation we try to avoid.

Next, we formulate the following security problem with Definition 1.

Problem 2** (Secret Securing with Two Protections and Minimum Costs Problem, or $2$-SSMCP).**

Consider a plant $\mathbf{G}$ in Eq. 1. Find a protection policy $\mathcal{P}:Q\to\operatorname{Pwr}(\Sigma_{p})$ s.t. $Q_{s}$ is 2-securely reachable w.r.t. $\Sigma_{p}^{k}=\operatorname*{\dot{\bigcup}}_{i=0}^{k}\Sigma_{i}$ and $k$ is the least index.

Let us explain Problem 2 with an illustrating example of a real system.

Example 3**.**

The plant $\mathbf{G}$ in Fig. 1 represents a computer network composed of two different local-area networks (LAN) and two servers. Consider a situation where a user uses a laptop and wants to browse his or her secret file in the server via Wi-Fi. $q_{0}$ means “the client is not connected”, and $q_{1}$ is a wireless router. Accordingly, $\sigma_{0}$ and $\sigma_{1}$ indicate connecting to and disconnecting from the router. $q_{2}$ and $q_{4}$ represent being in the respective LAN, that is, the client has been assigned an IP address. $\sigma_{2}$, $\sigma_{3}$, $\sigma_{8}$, and $\sigma_{9}$ are operations of connecting to and disconnecting from networks $q_{2}$ and $q_{4}$. $q_{5}$ is the server where the client’s secret file is stored. $q_{3}$ is a bastion server different from $q_{5}$, which allows clients to access to $q_{5}$ through $q_{3}$, namely $\sigma_{7}$. Accordingly, $\sigma_{4}$ and $\sigma_{5}$ are logging on to and logging out from $q_{3}$. Users in LANs $q_{2}$ and $q_{4}$ can access to $q_{5}$ directly. In order to protect the secret file against malicious access of intruders, the administrators must protect some suitable events in the plant, and they wish to do so with minimum cost.

Let $\Sigma_{p}=\Sigma_{0}\operatorname{\dot{\cup}}\Sigma_{1}\operatorname{\dot{\cup}}\Sigma_{2}$ be the set of protectable events, where $\Sigma_{0}=\set{\sigma_{0}}$, $\Sigma_{1}=\set{\sigma_{4},\sigma_{6},\sigma_{10}}$, $\Sigma_{2}=\set{\sigma_{2},\sigma_{8}}$. Also let $\Sigma_{up}=\set{\sigma_{1},\sigma_{3},\sigma_{5},\sigma_{7},\sigma_{9}}$ be the set of unprotectable events, and $\Sigma=\Sigma_{p}\operatorname{\dot{\cup}}\Sigma_{up}$. The secret state $q_{5}$ is depicted as a shaded state in Fig. 1; thus $Q_{s}=\set{q_{5}}$. The $2$-SSMCP is the problem of finding a protection policy which specifies at least two protectable events in every path from $q_{0}$ to $q_{5}$ with minimum protection cost. In other words, every string reaching $q_{5}$ from $q_{0}$ must have at least two protectable events, and the index $k\in\set{0,1,2}$ of $\Sigma_{p}^{k}$ of these protectable events must be the smallest.

III Main Results

In this section, we provide a necessary and sufficient condition for the solvability of Problem 2, and compute a solution by resorting to the SCT.

IV Illustrating Example

V Conclusions

We have introduced the problem of protecting secret states in the system with at least $m$ ($\geq 1$) protections and minimum protection costs ($m$-SSMCP). This problem has been formulated as finding a protection policy such that every string reaching secret states from the initial state has at least $m$ protectable events, and the protection cost is minimum. We have presented a solution algorithm for $m$-SSMCP which computes $m$ supervisors. Finally, we have demonstrated our solution with a network example.

In future work, we aim to investigate a situation where secrets have different importance and administrators are concerned with the balance between protection cost and secret importance.