Self-ad-MCNHA-SLOS

A Self-adaptive Minimum-Cost Network Hardening Algorithm based on Stochastic

Loose Optimize Strategy

Yonglin Sun, Yongjun Wang and Yi Zhang

College of Computer Science, National University of Defense Technology (NUDT), Changsha, P.R. China

Keywords: Minimum-cost Network Hardening, Stochastic Loose Optimize, Self-adaptive, Network Vulnerability,

Attack Graph.

Abstract: Given a network, it inevitable contains various vulnerabilities, which could be exploited by malicious

attackers. It is an effective way to harden a network by searching and remedying those critical

vulnerabilities. That is the so-called Minimum-Cost Network Hardening (MCNH) problem, but there

haven’t any effective enough method to address this problem yet, especially, when facing large-scale

network. We proposed Self-ad-MCNHA-SLOS, an algorithm using Stochastic Loose Optimize Strategy

(SLOS) and self-adaptive parameter adjustment method ingeniously, to meet the problem. Experiment

results show that it has the merits of high-efficiency, controllable, asymptotically optimal, and suitable for

large-scale network.

1 INTRODUCTION

Networks changed and are changing people’s life,

anyone and anything could be tied together by the

Internet to share the benefits of interconnected and

information sharing. However, the neglect of

security in the history of network techniques

development is becoming a nightmare. Drived by

economic interests, hacker industry chains create

uncountable malwares and attack techniques

threatening to legitimate users continuously by

finding and exploiting vulnerabilities in networks,

network security is becoming more and more serious.

Network vulnerability means those exploitable

defects existing in network environment, which

could be exploited by adversaries to do harm to the

network and its users, such as software bugs,

protocol defects, security policy conflicts,

unreasonable network structure, etc. MCNH is based

on network vulnerability association analysis and

target for finding minimum-cost hardening plan for

those critical resources. (Jha, 2002) proposed the

most original concept of MCNH based on state

attack graph. Since then, several teams addressed

this problem and made significant improvements but

still unsolved.

The MCNH problem contains 3H questions: how

to determine the whole space of possible hardening

plans, how to determine the cost of a hardening plan,

and how to find the effective minimum-cost

hardening plan. Attack graph can be used to

determine the whole plan space and the validity

criteria of plan, since it can reflect the exploit-

dependence relations among all of the vulnerabilities

existing in given network. Usually, the costs of plans

are assumed to be known for the complexity and

subjectivity of plan cost evaluation. As to the third H,

we assume there are N nodes in a given network,

then, the scale of vulnerabilities is O(N), and the

scale of possible plans is 2

O(N)

, it is NP hard to find

the effective minimum-cost hardening plan. It will

become unfulfillable to solve this problem

accurately, when facing large-scale network

environment.

We proposed Self-ad-MCNHA-SLOS to find

minimum-cost plans (Min-Plan) iteratively from an

array of stochastic sparse sub-spaces (Sparse-Space)

of the whole possible hardening plan space (Plan-

Space), and updates the approximate optimal

hardening plan (Approx-Opt-Plan) according to the

validity of those Min-Plans judged by the validity

judgment function (Valid()). It also uses the history

validity information of those Min-Plans to adjust the

parameter (density) of the hardening plan generator

(GeneratePlan()) to make sure those Sparse-Spaces

converge quickly to the optimal hardening plan

372

Sun Y., Wang Y. and Zhang Y..

Self-ad-MCNHA-SLOS - A Self-adaptive Minimum-Cost Network Hardening Algorithm based on Stochastic Loose Optimize Strategy.

DOI: 10.5220/0004022803720378

In Proceedings of the International Conference on Security and Cryptography (SECRYPT-2012), pages 372-378

ISBN: 978-989-8565-24-2

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

(Opt-Plan). In this way, it can make sure the

Approx-Opt-Plan obtained by iterations of limited

steps closes enough to the Opt-Plan. Testified both

by theoretical analysis and experiments evaluation,

Self-ad-MCNHA-SLOS has the merits of high-

efficiency, controllable and asymptotically optimal,

and is very suitable for large-scale network

environment.

2 RELATED WORK

MCNH is based on attack graph, (Swiler, 2001) first

proposed the attack graph model. (Sheyner, 2002)

studied the automation of attack graph generation

and analysis. (Ammann, 2002) first proposed

attacker’s ability monotonic assumption to improve

the efficiency of attack graph construction. (R. P.

Lippmann, 2005) proposed network security

assessment and hardening method based on attack

graph. (Ou, 2005) proposed a method with

computational complexity between O(N

) and O(N

)

to construct single-objective attack graph, and found

the phenomenon of circle-included attack paths.

(Ingols, 2006) implement a prototype system to

generate multiple-prerequisite attack graph for large-

scale enterprise network using raw data collected

from real network environment. (Chen, 2009)

proposed an algorithm to generate multi-objective

attack graph. Besides, researchers have also

proposed various methods to enhance the

intelligibility and usability of attack graph for its

complicated (Mehta, 2006); (Lippmann, 2007);

(Homer, 2008).

As for MCNH, (Jha, 2002) proposed the original

concept of MCNH based on state attack graph, their

work is to looking for the minimum safety measure

set to guarantee the safety of the key information

assets of a given network. Since then, (Noel, 2003)

proposed MCNH measure set based on attribute

attack graph, however, it can’t be applied to large

attack graph with circles. Though (Wang, 2006)

used one-pass search strategy to avoid logic loops in

attack graph, it still can’t be applied to large-scale

attack graph. (Homer, 2008) proposed an automatic

network configuration management method to help

for making network configuration decision in an

iterative approach, but it still not efficiency enough

when attack graph is moderate large. (Chen, 2008)

proposed an accurate method to calculate optimal

hardening plan based on Binary Decision Diagrams

(BDD) and an approximate one based on greedy

strategy, the former is better than Lingyu Wang’s

method, but still a method for small-scale network,

while the latter one is theoretically appropriate for

large-scale network, but their experiment results are

not enough to testify its good performance.

MCNH is actually a minimum-cost satisﬁability

problem (MinCostSAT). MinCostSAT, as a Boolean

Satisfiability (SAT) problem, is to minimize the cost

of the satisfying assignment, and is suitable for

Automatic Test Pattern Generation (ATPG), FPGA

Routing, AI Planning, etc. (Fu, 2006). (Li, 2004)

have done in-depth research on optimization

algorithms for the MinCostSAT in his PhD thesis.

However, these algorithms for the MinCostSAT is

not suitable for the minimum-cost network

hardening problem.

3 SLOS

SLOS is come from Stochastic Loose Optimize

Principle (SLOP). If a set named Universe is

partitioned to 2 parts: Low and High, and the ratio of

element numbers of Low and Universe is P

, then,

the probability of an element selected stochastically

from the Universe belongs to Low is P

, and to High

is 1-P

. If repeat the selection N times, then, the

probability of selected an element from the Low is

1-(1-P

)N. No matter how many elements in the

Universe, and how tiny the P

is, we can ensure the

probability of the above selection success is close

enough to 1, as long as the N is moderate big.

As Figure 1 shown, the proof of SLOP is very

simple.

Premise:

|Low|/|Universe| = P

;

, r

, … , r

∈ Universe;

, r

, … , r

is an array of random numbers.

Conclusion:

P({ r

, r

, … , r

}∩Low≠∅) = 1-(1-P

)

where P(event) means the probability of the event happens.

Proof:

∵ r

, r

, … , r

∈ Universe, and they are random numbers.

∴ P(r

∈ Low) = P(r

∈ Low) = … = P(r

∈ Low) = P

∴ P(r

∉ Low) = P(r

∉ Low) = … = P(r

∉ Low) = 1-P

and, ∵ the set of events r

∉ Low, r

∉ Low, … , r

∉ Low are

stochastic independence.

∴ P({ r

, r

, … , r

}∩Low =∅)

= P((r

∉ Low)∩(r

∉ Low)∩…∩(r

∉ Low))

= P(r

∉ Low)×P(r

∉ Low) ×…×P(r

∉ Low)

= (1-P

)

∴ P({ r

, r

, … , r

}∩Low≠∅)

= 1- P({ r

, r

, … , r

}∩Low =∅)

= 1- (1-P

)

Figure 1: Proof of the stochastic loose optimize principal.

SLOS is a strategy for state space searching

which could be used to find an approximate solution

Self-ad-MCNHA-SLOS - A Self-adaptive Minimum-Cost Network Hardening Algorithm based on Stochastic Loose

Optimize Strategy

373

of NP-hard problem quickly. The basic idea is:

select an array of states from the whole state space

to form a stochastic state sub-space, named Sparse-

Space, and the distance between the optimal state of

the Sparse-Space and the optimal state of the whole

space will become smaller gradually with the scale

of the Sparse-Space increasing, according to

concerned partial order. If the scale of the whole

state space is too large to be traveled in limited time,

SLOS could be used to find an approximate optimal

state in a Sparse-Space, and the approximate optimal

state is better and better with the scale of Sparse-

Space increasing. That is, we could find an

approximate solution of a NP-hard problem in a

Sparse-Space with moderate scale, when computing

resource is limited.

MCNH comes down to a state space searching

problem with double constraint conditions, that is,

the Opt-Plan must be both validity and with

minimum cost. The validity depends on the plan

itself, while, the judgment of the minimality need to

search the hardening plan space. We used the SLOS

iteratively both in the MNCHA-SLOS and the Self-

ad-MCNHA-SLOS to ensure the Approx-Opt-Plan’s

validity and approximate minimality.

Table 1: Symbols and their meanings.

Plan

Network Hardening Plan

Approx-Opt-Plan

Approximate Optimal Plan

Opt-Plan

Optimal Plan

Plan-Space

Network Hardening Plan Space

Sparse-Space

Subspace of the Plan-Space contains Some

Stochastic Plans

Valid-Space

Subspace of the Plan-Space contains all of

the Valid Plans

Superior-Space

Subspace of the Plan-Space contains those

Superior Plans

Goal-Space

Subspace of the Plan-Space contains those

Plans both Valid and Superior

Rand()

Random number generator

Valid()

A Function to judge the Validity of Plan

Cost()

A Function to compute the cost of Plan

GeneratePlan()

Plan Generator

density

Mathematical Expectation of the 1-density of

Plan

UpdateDensity()

The Updater of the density

NSparse

The Scale of Sparse-Space

Niterate

Iterate times

PValid

The Ratio of Valid plans

PSuperior

The Ratio of Superior plans

PGoal

The Probability of the Approx-Opt-Plan in

Goal-Space

4 MCNHA-SLOS

4.1 Basic Appointments and Concepts

Appointment 1. The costs of network hardening

plans are given.

Appointment 2. The hardening plan space (Plan-

Space) and hardening plan’s validity judgment

function (Valid()) are given.

Appointment 3. Symbols used in this article with

their meanings in Table. 1.

4.2 Algorithm Description

Basic Thought. Use the SLOS iteratively to meet the

two measures: minim-cost and validity, select the

minimum-cost Plan, noted as Min-Plan, in a N

Sparse

scale Sparse-Space in every iteration, and update the

Approx-Opt-Plan according to the Min-Plan’s

validity, ensure the probability of the Approx-Opt-

Plan belongs to the Goal-Space is close to 1 by

iterate

times iterations. It could be easily proved that

the probability of the Approx-Opt-Plan belongs to

the Goal-Space is P

Goal

, as formula (1) shown, by

iterate

times of iterations on N

Sparse

scale Sparse-

Spaces.

(1 (1 ) ) (1 (1 ) )

Sparse

iterate

Goal Superior Valid

P P P     

(1)

The pseudo-code of the MCNHA-SLOS is shown in

Figure 2.

Figure 2: Pseudo-code of the MCNHA-SLOS.

4.3 Algorithm Analysis

It is obvious that accurate solving the MCNHP need

to search the whole Plan-Space to find the Opt-Plan,

and the theoretical complexity is 2

, while, the

MCNHA-SLOS only need N

iterate

×N

Sparse

times

checking to ensure the Approx-Opt-Plan meet the

user’s expectation, as formula (1) shown. If the scale

of the Plan-Space n is moderate big, those accurate

solving method will inevitable failure, under the Von

Begin:

Approx-Opt-Plan ← 2

-1;

for round = 1 to N

iterate

;

Min-Plan ← Approx-Opt-Plan;

for count = 1 to N

Sparse

;

Plan = Rand(t)%2

;

if( Cost(Plan) < Cost(Min-Plan) )

Min-Plan ← Plan;

if( Valid(Min-Plan) )

Approx-Opt-Plan ← Min-Plan;

Output: Approx-Opt-Plan;

End.

SECRYPT 2012 - International Conference on Security and Cryptography

374

Neumann architecture. While, according to formula

(1), P

Valid

is fixed for a given goal of a given network,

Superior

is given by user, therefore, no matter how

tiny the P

Superior

and the P

Valid

, there are moderate

iterate

and N

Sparse

to ensure the P

Goal

is close to 1,

unless the Goal-Space is empty. We can also find

from the formula (1) that the N

Sparse

is mainly to

control the distance between the Approx-Opt-Plan

and Opt-Plan, while, the N

iterate

is mainly to ensure

the validity of the Approx-Opt-Plan.

Generally speaking, the MCNHA-SLOS

transforms the accurate problem solving of 2

scale

to approximate problem solving of N

iterate

×N

Sparse

scale, and transforms an couldn’t accomplishable

problem of finding the optimal solution in limited

time to an accomplishable problem of finding a

satisfying approximate optimal solution in limited

time, and is able to control the precision of the

solving process according to the available

computing resource and the user’s expectation.

5 Self-ad-MCNHA-SLOS

According to a mass of experiments, we find some

interesting phenomenon. Given a network and a goal,

if fix the N

iterate

, the probability of the Approx-Opt-

Plan is nontrivial valid reducing with the N

Sparse

increasing; if give N

Sparse

a relative big or tiny value,

the changing of N

iterate

has not significant impact on

the probability of the Approx-Opt-Plan is nontrivial

valid, however, if give N

Sparse

a suitable value, the

impact will be significant, and the suitable values of

the N

Sparse

for different goals are various.

For a given network and a given goal, the

probability of a stochastically selected Plan has a

positive correlation relationship with the 1-density

of its binary representation, that is, the more

vulnerabilities of a Plan contains, the higher of the

probability of the Plan is valid, besides, those valid

Plans definitely have a 1-density lower bound, since

the Plan (0)

is invalid for any goal of any network.

The increasing of N

Sparse

, actually, is reducing the

probability of those Min-Plans selected from each

Sparse-Space, because the cost of a Plan also has a

positive correlation relationship with the 1-density

of the Plan’s binary representation. When the N

Sparse

becomes big enough, those Min-Plans will be

invalid with high probability, therefore, the impact

of the increasing of N

iterate

to the probability of the

Approx-Opt-Plan is valid will be insignificant.

If we could find the suitable N

Sparse

quickly for

different goals in different networks, it will

significantly improve the efficiency of the MCNHA-

SLOS, however, the N

Sparse

is a relative stable

number mainly used to control the computation

complexity. Fortunately, we find a substitutable way

to adjust the 1-density of Sparse-Spaces according to

the history validity statistic of those Min-Plans to

make sure the sequence of Sparse-Spaces are

convergent to the Opt-Plan. There are still two

important problems to address, one is how to

generate those Sparse-Spaces according to different

required 1-density, and the other is how to update

the 1-density according to the history validity

information of those Min-Plans. Intuition tell us, if

Min-Plans are valid continuously, we should lower

the 1-density, while, if Min-Plans are invalid

continuously, we should increase the 1-density, and

the lower’s preconditions should be weaker than the

increase’s, since we want to got the minimum-cost

valid Plan.

Based on the above considerations, we proposed

Self-ad-MCNHA-SLOS shown in Figure 3,

Compared with the MCNHA-SLOS, it add a special

function GeneratePlan(density) to generate

stochastic Plans and a function UpdateDensity() to

adjust the density dynamically in each iteration,

where the density is the parameter to control the 1-

density of Plans of GeneratePlan(density) generated,

and its range is (0,1] and is assigned 0.9 to be its

initial value.

Figure 3: Pseudo-code of the Self-ad-MCNHA-SLOS.

GeneratePlan(density) is defined as formula (2). If

fixed the density to 0.5, it will become Rand(t)%2

as in MCNHA-SLOS.

1 2 2

( , ) ( ... ) ,

1 ( )%10 10,

0 ( )%10 10.

, ( 1) .

Sparse

GeneratePlan t density x x x

if Rand t density

where x

if Rand t density

where t round N i



















   

(2)

And we define the UpdateDensity() as formula (3),

where the upFlag is to record the count of those

Min-Plans continuously invalid and the downFlag is

to record the count of continuously valid, and both

Begin:

Approx-Opt-Plan ← 2

-1;

density = 0.9;

for round = 1 to N

iterate

;

Min-Plan ← Approx-Opt-Plan;

for count = 1 to N

Sparse

;

Plan = GeneratePlan(density)

if( Cost(Plan) < Cost(Min-Plan) )

Min-Plan ← Plan;

if( Valid(Min-Plan) )

Approx-Opt-Plan ← Min-Plan;

density = UpdateDensity();

Output Approx-Opt-Plan;

End.

Self-ad-MCNHA-SLOS - A Self-adaptive Minimum-Cost Network Hardening Algorithm based on Stochastic Loose

Optimize Strategy

375

of them are assigned 0 to be their initial value.

Where the UpTraction and DownDrang are the

upper limit of the upFlag and the downFlag

respectively, if any one of the two variables reached

its upper limit, the density will be updated according

to the formula (3), and both of the two variables will

be assigned 0 again. Generally speaking, the upFlag

like a traction to increase the density, while the

downFlag like a drang to lower the density, and the

direction of the state space searching could be

dynamically adjusted by suitable UpTraction and

DownDrang.

, , ,()

0.1 ,

0.1 .

upFlag UpTraction downFlag DownDrangUpdateDensity

density if upFlag UpTraction

density if downFlag DownDrang

















(3)

6 EXPERIMENT

To testify and evaluate algorithms proposed above,

we generate simulative networks and build their

attack graphs using our Network Modeling and

Demonstrating System (Net-MDs) and Network

Vulnerability Analyzing System (Net-VAs), as

Figure 4 and Figure 5 shown. To be concise, we

assigned every vulnerability’s remedy cost 1,

therefore, the cost of hardening plan only depend on

the total vulnerabilities it contains. Besides, as

shown in Figure 4, the single target refers to the

server circled by the bold red loop, and the multiple

targets refers to 4 servers circled by red loops.

Figure 4: Simulative network generated by Net-MDs.

Figure 5: Attack Graph Generated by Net-Vas.

We designed a series of experiments to analyze

the parameters of the Self-ad-MCNHA-SLOS for

finding the best parameter setting principles, and

compared and evaluated the traits and performance

of the MCNHA-SLOS and the Self-ad-MCNHA-

SLOS (see Appendix). In the end, We find: 1.

UpTraction(N

iterate

/e), DownDrang(1) and N

Sparse

(7)

are good parameters for Self-ad-MCNHA-SLOS; 2.

Self-ad-MCNHA-SLOS can significantly improve

MCNHA-SLOS’s performance.

We also compared the Self-ad-MCNHA-SLOS

with (Feng Chen, 2009)’s approximate method

weighted-Greedy in same environment, since those

accurate methods aren’t able to deal with large-scale

network, radically. Due to the complicity of the

weighted-Greedy is firmed as |C|×|L| for a given

goal and given n-valid attack path length n, where C

denotes all of the initial attributes and L denotes all

of the n-valid attack path, we assign appropriate

values to N

iterate

to ensure that N

iterate

×N

Sparse

approximate to |C|×|L| for the purpose of comparing

the accuracy of the two approximate hardening plans:

Greedy-Plan and Approx-Opt-Plan. We observe the

costs of the two approximate plans, respectively, in

four simulated networks: Net

with 200 nodes and

10 vulnerabilities, Net

with 200 nodes and 20

vulnerabilities, Net

with 200 nodes and 30

vulnerabilities and Net

with 200 nodes and 40

vulnerabilities.

Figure 6: Comparison of the Self-ad-MCNHA-SLOS and

the weighted-greedy.

As Figure 6 shown, the cost of Greedy-Plan is

lower than the average cost of Approx-Opt-Plans

when network scale is small but significant bigger

when network scale is moderate big. Besides, the

average cost of Approx-Opt-Plans generated by Self-

ad-MCNHA-SLOS relative stable while the cost of

Greedy-Plan generated by weighted-Greedy increase

obviously with the network scale, on condition that

the Self-ad-MCNHA-SLOS spends equivalent

computing resource with the weighted-Greedy. The

above results illustrate that the Self-ad-MCNHA-

SLOS is more efficient than the weighted-Greedy,

especially when facing large-scale network.

According to the above experiments, the Self-ad-

MCNHA-SLOS could make those Sparse-Spaces

converge quickly to the Opt-Plan by adjusting the

SECRYPT 2012 - International Conference on Security and Cryptography

376

parameter density, therefore, it could get an Approx-

Opt-Plan very close to the Opt-Plan by limited

amount of searches in the Plan-Space. The merits of

high-efficiency, controllable, and asymptotically

optimal, ensure that it could make full use of

available computing resource to find possible better

result, therefore very suitable for large-scale network.

7 CONCLUSIONS

In this paper, we proposed the Self-ad-MCNHA-

SLOS to address the MCNH problem using SLOS

and self-adaptive parameter adjust strategy. It could

find an approximate optimal hardening plan close

enough to the optimal hardening plan by limited

amount of searches, and has the merits of high-

efficiency, controllable and asymptotically optimal,

therefore, can make full use of available computing

resource to find possible better result, and is very

suitable for large-scale network environment.

Considering the Self-ad-MCNHA-SLOS’ ability of

transforming NP-hard problem to P-hard iterations,

we will study the generalization of the algorithm to

solve more hard problems in future.

ACKNOWLEDGEMENTS

This paper is supported by the National High

Technology Research and Development Program of

China (863 Program) under Grant

No.2009AA01Z432, the National Natural Science

Foundation of China under Grant No.60873215 and

the Hunan Provincial Natural Science Foundation of

China under Grant No.s2010J5050.

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Self-ad-MCNHA-SLOS - A Self-adaptive Minimum-Cost Network Hardening Algorithm based on Stochastic Loose

Optimize Strategy

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APPENDIX

Figure A1: Comparison of DownDrang.

Figure A2: Comparison of UpTraction.

Figure A3: Comparison of ratios.

Figure A5: Comparison of ratios, meticulously.

Figure A4: Comparison of ratios.

Figure A7: Comparison of density.

Figure A6: Comparison of NSparse.

Figure A8: Algorithms comparison.

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