resource distribution in multiple attacks against a single target
DESCRIPTION
Resource Distribution in Multiple Attacks Against a Single Target. Author: Gregory Levitin ,Kjell Hausken Risk Analysis, Vol. 30, No. 8, 2010. Agenda. Introduction & Background Problem Description (Goal) The model Assumption Target vulnerability(V) Expenditure(E) Resource Distribution - PowerPoint PPT PresentationTRANSCRIPT
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Resource Distribution in Multiple Attacks Against a Single Target
Author: Gregory Levitin ,Kjell HauskenRisk Analysis, Vol. 30, No. 8, 2010
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Agenda
• Introduction & Background• Problem Description (Goal)• The model
• Assumption• Target vulnerability(V)• Expenditure(E)
• Resource Distribution• Even Resource Distribution(V,E)• Geometric Resource Distribution(V,E)
• Numerical simulations• Conclusion
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Introduction & Background
• It has been common to consider a nonstrategic attacker , either by assuming a fixed attack or a fixed attack probability.
• Bier et al. (1) assume that a defender allocatesdefense to a collection of locations while an attacker chooses a location to attack.
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Introduction & Background
• In this article, we consider a target (object) that a defender seeks to protect and an attacker seeks to destroy through multiple sequential attacks.
• The defender tries to keep the object undestroyed in each attack launched by the attacker.The phenomenon is modeled as a contest between a defender and an attacker.
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• Introduction & Background• Problem Description (Goal)• The model
• Assumption• Target vulnerability(V)• Expenditure(E)
• Resource Distribution• Even Resource Distribution(V,E)• Geometric Resource Distribution(V,E)
• Numerical simulations• Conclusion
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Problem Description
• Basic definitions:–Vulnerability:
Probability of target destruction by the attacker.
– Effort:Amount of intentional force aimed at destruction or protection of a system element (in this article, it is measured as the amount of attacker’s resource allocated to each attack and amount of defender’s resource allocated to defense)
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Problem Description
• 1. Whether the attacker should allocate its entire resource into one large attack or distribute it among several attacks.
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Problem Description
• 1. Whether the attacker should allocate its entire resource into one large attack or distribute it among several attacks.
Attack strategy
One large attack
Several attacks
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Problem Description
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Problem Description
Attack strategy
One large attack
Several attacks
Even Resource Distribution
Geometric Resource
Distribution
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Problem Description
• 2.Whether geometrically increasing or decreasing resource distribution into a fixed number of sequential attacks is more beneficial than equal resource distribution
Attack strategy
One large attack
Several attacks
Even Resource Distribution
Geometric Resource
Distribution
Geometrically increasing
Geometrically decreasing
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Problem Description
• 3.How the optimal attack strategy depends on the contest intensity(m).
• Two objectives:– 1.To maximize the target vulnerability(V).– 2.To minimize the expected attacker resource expenditure(E).
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Optimal attack straregy
Problem Description
• 3.How the optimal attack strategy depends on the contest intensity(m).
Attack strategy
One large attack
Several attacks
Even Resource Distribution
Geometric Resource
Distribution
Geometrically increasing
Geometrically decreasing
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• Introduction• Problem Description (Goal)• The model
• Assumption• Target vulnerability(V)• Expenditure(E)
• Resource Distribution• Even Resource Distribution(V,E)• Geometric Resource Distribution(V,E)
• Numerical simulations• Conclusion
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The model- Assumption• Assumption:– (1) We consider a target (single target) that a defender
seeks to protect and an attacker seeks to destroy through multiple sequential attacks.
– (2)Both the defender and the attacker have limited resources.
– (3)The attacker can observe the outcome of each attack and stop the sequence of attacks when the target is destroyed.
– (4)The attacker distributes its resource over time.
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The model- Assumption
• Assumption:– (5)We model the common case that the protection is
static and cannot be changed over time.• Target is destroyed->The protection is destroyed.• Target is not destroyed->the protection remains in place also
for the subsequent attack.
– (6) We assume that the defender uses the same protection during the series of K attacks and allocatesits entire resource into this protection.
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• Introduction & Background• Problem Description (Goal)• The model
• Assumption• Target vulnerability(V)• Expenditure(E)
• Resource Distribution• Even Resource Distribution(V,E)• Geometric Resource Distribution(V,E)
• Numerical simulations• Conclusion
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Target vulnerability(V)
• For any single attack, the vulnerability of atarget is determined by a contest between the defender exerting effort t and the attacker exerting effort T in this attack.
->Contest success function
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Target vulnerability(V)
• Contest success function
:The attack success probability.
T :Attacker’s effort to attack a target.
t :Defender’s effort to protect a target.
m: Attacker-defender contest intensity.
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Target vulnerability(V)
• Contest success function
• Two factors influence the :
– 1.The relation between the resources(t/T) in each attack.
– 2.Contest intensity m .
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Target vulnerability(V)
• 1.The relation between the resources(t/T) in each attack.
– If the attacker exerts high effort(T>t), it is likely to win the contest that gives high vulnerability.
– If the defender exerts high effort(T<t), it is likely to win the contest that gives low vulnerability.
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Target vulnerability(V)
• 2.Contest intensity m :Measures whether the agents’ efforts have low or high impact on the target vulnerability
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Target vulnerability(V)
• According to assumption (6), We assume that the defender uses the same protection during the series of K attacks and allocates its entire resource into this protection:
t=r• NOMENCLATURE
t :Defender’s effort to protect a targetr :Defender’s resource
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Target vulnerability(V)
• On the contrary, the attacker distributes its entire resource R among K attacks such that the resource allocated to attack
• NOMENCLATURE R :Attacker’s resource
:Attacker’s effort (resource used) in the th attack K :Number of consecutive attacks
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Target vulnerability(V)
• The success probability of the th attack according to Contest success function is:
• The probability that the target survives in th attacks is:
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Target vulnerability(V)
• The probability that the target survives all K attacks is:
• Thus, the target vulnerability in K attacks is:
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Expenditure(E)
• According to assumption (3), The attacker can observe the outcome of each attack and stop the sequence of attacks when the target is destroyed.
• If the target is destroyed in the attack, the attacker spends the resource:
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Expenditure(E)
• NOMENCLATURET : attacker’s effort (resource used) in the attack(for even resource distribution T ≡ T)
• If the probability that the target is destroyed in the th attack is ,the expected attacker’s resource expenditure can be obtained as:
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Expenditure(E)
If the target is destroyed in the th attack , the resource attacker spends .
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Expenditure(E)
The expected attacker’s resource expenditure when target is destroyed .
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Expenditure(E)
The probability that the target survives all K attacks.
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Expenditure(E)
The expected attacker’s resource expenditure when after K attacks the attacker fails to destroy the target.
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Expenditure(E)
• The expected attacker’s resource expenditure can be obtained as:
• We will present the expected resource expenditure as a fraction of the total of attacker’s resource(R):
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• Introduction & Background• Problem Description (Goal)• The model
• Assumption• Target vulnerability(V)• Expenditure(E)
• Resource Distribution• Even Resource Distribution(V,E)• Geometric Resource Distribution(V,E)
• Numerical simulations• Conclusion
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Even Resource Distribution-V
• The attacker can choose the number of attacks K and distribute its resource evenly among the attacks such that T = R/K and the probability of target destruction in any attack is:
=K/R *r = 1/T *r=r/T
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Even Resource Distribution-V
• The target vulnerability is:
Even resource distribution ->1- are equal in all K attacks,so
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Even Resource Distribution-V
• Parameter values exist where the derivative in Equation (6) is negative, but it is often positive.
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Even Resource Distribution-V
• Example:– Negative:
(m = 2, R = r)
1 2 3 4 5 6 7 8 9 10 110
0.1
0.2
0.3
0.4
0.5
0.6
m=2,R=r
K
v
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Even Resource Distribution-V
• Example:– Positive: (m = 0 ; m=1,R=r )
1 2 3 4 5 6 7 8 9 100
0.2
0.4
0.6
0.8
1
1.2
m=0
K
V
1 2 3 4 5 6 7 8 9 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
m=1 ,R=r
K
V
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Even Resource Distribution-V
• m = 0 ,V increases concavely from 0 to 1 as a function of K and the attacker benefits from unlimitedly increasing the number of attacks.
• In realistic situations, the number of attacks is limited by time constraints by limited minimal cost of a single attack, etc. Therefore, the upper limit of K always exists.
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Even Resource Distribution-V
• Fig. 1 presents the target vulnerability as a function of the contest intensity m for different K and r/R.
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Even Resource Distribution-V
• It can be seen that the smaller the contest intensity(m), the more beneficial it is to increase the number of attacks.
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Even Resource Distribution-E
• If the target is destroyed in the th attack, the probability of this event is:
The probability that the target survives in all -1 attacks.
The probability that the target is destroyed.
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Even Resource Distribution-E
• The attacker spends the resource T =R/K* .
* T
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Even Resource Distribution-E
R/K* R*
R/T* *
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Even Resource Distribution-E
• Fig. 2 presents the expected attacker’s resource expenditure as a function of the contest intensity m for different r/R.
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• Introduction & Background• Problem Description (Goal)• The model
• Assumption• Target vulnerability(V)• Expenditure(E)
• Resource Distribution• Even Resource Distribution(V,E)• Geometric Resource Distribution(V,E)
• Numerical simulations• Conclusion
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Geometric Resource Distribution-V
• Now we assume that the attacker can change the amount of resources allocated to each of the K attacks.
• To model the resource distribution, we use the geometric progression since it is simple and flexible.
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Geometric Resource Distribution-V
• Assume that the attacker allocates effort to the first attack and changes the effort according to the geometric progression.
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Geometric Resource Distribution-V
• The parameter q(Attack effort variation factor) determines the strategy of effort variation through the K sequential attacks:
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Geometric Resource Distribution-V
Attack strategy
One large attack
q=0
Several attacks
Even Resource
Distributionq=1
Geometric Resource
Distribution
Geometrically increasing
q>1
Geometrically decreasing
q<1
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Geometric Resource Distribution-V
• For the given resource R and effort variation parameter q, we obtain:
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Geometric Resource Distribution-V
• We obtain the probability of success in the th attack as:
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Geometric Resource Distribution-V
• The total system vulnerability in K sequential attacks with effort variation parameter q is:
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Geometric Resource Distribution-E
• The probability that the target is destroyed in the th attack is:
• Thus, the expected resource expenditure is:
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Geometric Resource Distribution-E
• Fig. 3 illustrates how maximum target vulnerability and minimum resource expenditure can be competing attacker objectives.
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• Introduction & Background• Problem Description (Goal)• The model
• Assumption• Target vulnerability(V)• Expenditure(E)
• Resource Distribution• Even Resource Distribution(V,E)• Geometric Resource Distribution(V,E)
• Numerical simulations• Conclusion
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Numerical simulations
• For low contest intensity m = 0.5– Maximal V:
q=1 (Even distribution)V=0.842E=0.545
– Minimal E:q=1.79E=0.47
V=0.8175
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Numerical simulations
• For high contest intensity m = 2– Maximal V:
q=0 (Single attack)V=0.5E=1
– Minimal E:q=0.47E=0.8855
V=0.2875
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Numerical simulations• Using numerical simulations to demonstrate the
methodology of model analysis.
• Fig. 4 presents the values of q and K that maximize V and the values of q and K that minimize E and the corresponding values of V
and E as functions of m for different values of the ratio r/R and K is allowed to vary from 1 to 10.
• There is a reasonable upper limit of K, which we here set to 10.
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Numerical simulations-Maximal V
r/R=0.5 r/R=1 r/R=2
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Numerical simulations-Maximal V
r/R=0.5m 1 = 1.08
The optimal number of attacks ismaximal for m < m 1
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Numerical simulations-Maximal V
For m> m2, the single attack with K = 1 and q = 0 becomes preferable for the attacker.
m 2 = 1.6
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Numerical simulations-Maximal V
r/R=0.5
r/R=1 r/R=2
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Numerical simulations-Maximal V• The optimal number of attacks is maximal for m < m 1
(where for r/R = 0.5 m 1 = 1.08, for r/R = 1 m 1 = 1.04, and for r/R = 2 m 1 = 1.02).
• Then it drops gradually and for m < m 2 (where for r/R = 0.5 m 2 = 1.6, for r/R = 1 m 2 = 1.28, and for r/R = 2 m 2 = 1.16) the single attack with K = 1 and q = 0 becomes preferable for the attacker.
• The values of q between 1 and 0 are never optimal when the system vulnerability is maximized.
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Numerical simulations-Minimal E
r/R=0.5 r/R=1 r/R=2
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Numerical simulations-Minimal E
• For low contest intensities the attack effort should increase through the K attacks (q > 1), whereas for more intensive contests it should decrease (q < 1).
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Numerical simulations-Optimal
r/R=0.5 r/R=1 r/R=2
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Numerical simulations-Optimal
• The attacker needs to strike a V versus E balance (tradeoff) for the attack strategy.
• Fig. 5 presents the differences in V and E obtained for strategies maximizing V and minimizing E.
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Numerical simulations-Optimal
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Numerical simulations-Optimal
• For m = 0.3(Low intensity), E can be reduced by 51.4% by the price of a 1.6% decrease of V.
Move from 0.3->1
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Numerical simulations-Optimal
• On the contrary, when m = 3(High intensity) ,E can be reduced by only by 7.74%, which causes 49% decrease of V.
Move from3->1
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Numerical simulations-Optimal• When the contest intensity is low, a great
reduction of E without significant sacrifice of V is possible.
• Highly intensive contests even a small reduction of E causes a drastic reduction of V.
• The benefit of choosing the minE strategy forsmall m increases with the growth of the attacker’s resource superiority (decrease of r/R).
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Numerical simulations-Optimal
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• Introduction & Background• Problem Description (Goal)• The model
• Assumption• Target vulnerability(V)• Expenditure(E)
• Resource Distribution• Even Resource Distribution(V,E)• Geometric Resource Distribution(V,E)
• Numerical simulations• Conclusion
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Conclusion
• When the contest intensity is low, the attacker benefits from distributing its resource among several attacks and attacking the target with an effort that increases for each subsequent attack.
• For highly intensive contests, concentrating the entire attacker’s effort in one single attack ispreferable.