Gamesec23 - Scalable Learning of Intrusion Response through Recursive Decomposition

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Scalable Learning of Intrusion Response
through Recursive Decomposition
GameSec 2023, Avignon, France
Conference on Decision and Game Theory for Security
Kim Hammar & Rolf Stadler
kimham@kth.se
Division of Network and Systems Engineering
KTH Royal Institute of Technology
Oct 18, 2023

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Use Case: Intrusion Response
I A defender owns an infrastructure
I Consists of connected components
I Components run network services
I Defender defends the infrastructure
by monitoring and active defense
I Has partial observability
I An attacker seeks to intrude on the
infrastructure
I Has a partial view of the
infrastructure
I Wants to compromise specific
components
I Attacks by reconnaissance,
exploitation and pivoting
Attacker Clients
. . .
Defender
1 IPS
1
alerts
Gateway
7 8 9 10 11
6
5
4
3
2
12
13 14 15 16
17
18
19
21
23
20
22
24
25 26
27 28 29 30 31

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System Model
I G = h{gw} ∪ V, Ei: directed tree
representing the virtual infrastructure
I V: finite set of virtual components.
I E: finite set of component
dependencies.
I Z: finite set of zones.
r&d zone
App servers Honeynet
dmz
admin
zone
workflow
Gateway idps
quarantine
zone
alerts
Defender
. . .
Attacker Clients
2
1
3 12
4
5
6
7
8
9
10
11
13
14
15
16
17
18
19
20
21
22 23
24
25
26
27
28
29
30 31
32
33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48
49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

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State Space
I Each i ∈ V has a state
vt,i = (v
(Z)
t,i
|{z}
D
, v
(I)
t,i , v
(R)
t,i
| {z }
A
)
I System state st = (vt,i )i∈V ∼ St.
I Markovian time-homogeneous
dynamics:
st+1 ∼ f (· | St, At)
At = (A
(A)
t , A
(D)
t ) are the actions.
s1 s2 s3
s4 s5 s4
.
.
.
.
.
.
.
.
.

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State Model
I Each i ∈ V has a state
vt,i = (v
(Z)
t,i
|{z}
D
, v
(I)
t,i , v
(R)
t,i
| {z }
A
)
I System state st = (vt,i )i∈V ∼ St.
I Markovian time-homogeneous
dynamics:
st+1 ∼ f (· | St, At)
At = (A
(A)
t , A
(D)
t ) are the actions.
s1 s2 s3
s4 s5 s4
.
.
.
.
.
.
.
.
.

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Workflows
I Services are connected into workflows W = {w1, . . . , w|W|}.

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Workflows
I Services are connected into workflows W = {w1, . . . , w|W|}.
gw fw idps lb
http
servers
auth
server
search
engine
db
cache
Dependency graph of an example workflow representing a web
application; gw, fw, idps, lb, and db are acronyms for gateway,
firewall, intrusion detection and prevention system, load balancer, and
database, respectively.

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Workflows
I Services are connected into
workflows
W = {w1, . . . , w|W|}.
I Each w ∈ W is realized as a
subtree Gw = h{gw} ∪ Vw, Ewi
of G
I W = {w1, . . . , w|W|} induces a
partitioning
V =
[
wi ∈W
Vwi such that i 6= j =⇒ Vwi ∩ Vwj = ∅
Zone a
Zone b Zone c
gw
1 2 3
4 5 6
7
A workflow tree

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Workflow
I Services are connected into
workflows
W = {w1, . . . , w|W|}.
I Each w ∈ W is realized as a
subtree Gw = h{gw} ∪ Vw, Ewi
of G
I W = {w1, . . . , w|W|} induces a
partitioning
V =
[
wi ∈W
Vwi such that i 6= j =⇒ Vwi ∩ Vwj = ∅
Zone a
Zone b Zone c
gw
1 2 3
4 5 6
7
A workflow tree

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Clients
Client population
. . .
Arrival rate λ Departure
Service time µ
. . .
.
.
.
.
.
.
.
.
.
w1 w2 w|W|
Workflows (Markov processes)
I Homogeneous client population
I Clients arrive according to Po(λ), Service times Exp(1
µ)
I Workflow selection: uniform
I Workflow interaction: Markov process

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Observations
I idpss inspect network traffic and
generate alert vectors:
ot ,

ot,1, . . . , ot,|V|

∈ N
|V|
0
ot,i is the number of alerts related to
node i ∈ V at time-step t.
I ot = (ot,1, . . . , ot,|V|) is a realization
of the random vector Ot with joint
distribution Z
idps
idps
idps
idps
alerts
Defender
. . .
Attacker Clients
2
1
3 12
4
5
6
7
8
9
10
11
13
14
15
16
17
18
19
20
21
22 23
24
25
26
27
28
29
30 31
32
33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48
49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

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probability
ZO1
ZO2
ZO3
ZO4
ZO5
ZO6
ZO7
ZO8
probability
ZO9
ZO10
ZO11
ZO12
ZO13
ZO14
ZO15
ZO16
probability
ZO17
ZO18
ZO19
ZO20
ZO21
ZO22
ZO23
ZO24
probability
ZO25
ZO26
ZO27
ZO28
ZO29
ZO30
ZO31
ZO32
probability
ZO33
ZO34
ZO35
ZO36
ZO37
ZO38
ZO39
ZO40
probability
ZO41
ZO42
ZO43
ZO44
ZO45
ZO46
ZO47
ZO48
probability
ZO49
ZO50
ZO51
ZO52
ZO53
ZO54
ZO55
ZO56
250 500 750
O
probability
ZO57
250 500 750
O
ZO58
250 500 750
O
ZO59
250 500 750
O
ZO60
250 500 750
O
ZO61
250 500 750
O
ZO62
250 500 750
O
ZO63
250 500 750
O
ZO64
Distributions of # alerts weighted by priority ZOi
(Oi | S
(D)
i , A
(A)
i ) per node i ∈ V
no intrusion intrusion

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Defender
I Defender action:
a
(D)
t ∈ {0, 1, 2, 3, 4}|V|
I 0 means do nothing. 1 − 4 correspond
to defensive actions (see fig)
I A defender strategy is a function
πD ∈ ΠD : HD → ∆(AD), where
h
(D)
t = (s
(D)
1 , a
(D)
1 , o1, . . . , a
(D)
t−1, s
(D)
t , ot) ∈ HD
I Objective: (i) maintain workflows; and
(ii) stop a possible intrusion:
J ,
T
X
t=1
γt−1
η
|W|
X
i=1
uW(wi , st)
| {z }
workflows utility
− (1 − η)
|V|
X
j=1
cI(st,j, at,j)
| {z }
intrusion and defense costs
!
dmz
rd
zone
admin
zone
Old path
New path
Honeypot App server
Defender
Revoke
certificates
Blacklist
IP
1) Server migration 2) Flow migration and blocking
3) Shut down server 4) Access control

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Attacker
I Attacker action: a
(A)
t ∈ {0, 1, 2, 3}|V|
I 0 means do nothing. 1 − 3 correspond
to attacks (see fig)
I An attacker strategy is a function
πA ∈ ΠA : HA → ∆(AA), where HA
is the space of all possible attacker
histories
h
(A)
t = (s
(A)
1 , a
(A)
1 , o1, . . . , a
(A)
t−1, s
(A)
t , ot) ∈ HA
I Objective: (i) disrupt workflows; and
(ii) compromise nodes:
− J
.
.
.
Attacker
login attempts
configure
Automated
system
Server
2) Brute-force
1) Reconnaissance
3) Code execution
Attacker Server
TCP SYN
TCP SYN ACK
port open
Attacker Service Server
malicious
request inject code
execution

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The Intrusion Response Problem
maximize
πD∈ΠD
minimize
πA∈ΠA
E(πD,πA) [J] (1a)
subject to s
(D)
t+1 ∼ fD · | A
(D)
t , A
(D)
t

∀t (1b)
s
(A)
t+1 ∼ fA · | S
(A)
t , At

∀t (1c)
ot+1 ∼ Z · | S
(D)
t+1, A
(A)
t ) ∀t (1d)
a
(A)
t ∼ πA · | H
(A)
t

, a
(A)
t ∈ AA(st) ∀t (1e)
a
(D)
t ∼ πD · | H
(D)
t

, a
(D)
t ∈ AD ∀t (1f)
E(πD,πA) denotes the expectation of the random vectors
(St, Ot, At)t∈{1,...,T} when following the strategy profile (πD, πA).
(1) can be formulated as a zero-sum Partially Observed Stochastic
Game with Public Observations (a PO-POSG):
Γ = hN, (Si )i∈N , (Ai )i∈N , (fi )i∈N , u, γ, (b
(i)
1 )i∈N , O, Zi

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Existence of a Solution
Theorem
Given the po-posg Γ (2), the following holds:
(A) Γ has a mixed Nash equilibrium and a value function
V ∗ : BD × BA → R that maps each possible initial pair of
belief states (b
(D)
1 , b
(A)
1 ) to the expected utility of the
defender in the equilibrium.
(B) For each strategy pair (πA, πD) ∈ ΠA × ΠD, the best response
sets BD(πA) and BA(πD) are non-empty and correspond to
optimal strategies in two Partially Observed Markov Decision
Processes (pomdps): M (D) and M (A). Further, a pair of
pure best response strategies (π̃D, π̃A) ∈ BD(πA) × BA(πD)
and a pair of value functions (V ∗
D,πA
, V ∗
A,πD
) exist.

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The Curse of Dimensionality
I While Γ has a value, computing it is intractable. The state,
action, and observation spaces of the game grow
exponentially with |V|.
1 2 3 4 5
104
105
2
105
|S|
|O|
|Ai |
|V|
Growth of |S|, |O|, and |Ai | in function of the number of nodes |V|

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The Curse of Dimensionality
I While (1) has a solution (i.e the game Γ has a value (Thm
1)), computing it is intractable since the state, action, and
observation spaces of the game grow exponentially with |V|.
1 2 3 4 5
104
105
2
105
|S|
|O|
|Ai |
|V|
Growth of |S|, |O|, and |Ai | in function of the number of nodes |V|
We tackle the scability challenge with decomposition

15/33
Intuitively..
rd zone
dmz
admin
zone
workflow
Gateway idps
quarantine
zone
alerts
Defender
. . .
Attacker Clients
2
1
3 12
4
5
6
7
8
9
10
11
13
14
15
16
17
18
19
20
21
22 23
24
25
26
27
28
29
30 31
32
33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48
49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 65
The optimal
action here...
Does not directly
depend on the state or
action of a node
down here

15/33
Intuitively..
rd zone
dmz
admin
zone
workflow
Gateway idps
quarantine
zone
alerts
Defender
. . .
Attacker Clients
2
1
3 12
4
5
6
7
8
9
10
11
13
14
15
16
17
18
19
20
21
22 23
24
25
26
27
28
29
30 31
32
33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48
49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 65
The optimal
action here...
But they are
not completely
independent either.
How can we
exploit this
structure?
Does not directly
depend on the state
or action of a node
down here

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Our Approach: System Decomposition
To avoid explicitly enumerating the very large state, observation,
and action spaces of Γ, we exploit three structural properties.
1. Additive structure across workflows.
I The game decomposes into additive subgames on the
workflow-level, which means that the strategy for each
subgame can be optimized independently
2. Optimal substructure within a workflow.
I The subgame for each workflow decomposes into subgames on
the node-level that satisfy the optimal substructure property
3. Threshold properties of local defender strategies.
I The optimal node-level strategies for the defender exhibit
threshold structures, which means that they can be estimated
efficiently

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Additive Structure Across Workflows (Intuition)
“=”
I If there is no path between i and j in G, then i and j are
independent in the following sense:
I Compromising i has no affect on the state of j.
I Compromising i does not make it harder or easier to
compromise j.
I Compromising i does not affect the service provided by j.
I Defending i does not affect the state of j.
I Defending i does not affect the service provided by j.

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Additive Structure Across Workflows
Definition (Transition independence)
A set of nodes Q are transition independent iff the transition
probabilities factorize as
f (St+1 | St, At) =
Y
i∈Q
f (St+1,i | St,i , At,i )
Definition (Utility independence)
A set of nodes Q are utility independent iff there exists functions
u1, . . . , u|Q| such that the utility function u decomposes as
u(St, At) = f (u1(St,1, At,1), . . . , u1(St,|Q|, At,Q))
and
ui ≤ u0
i ⇐⇒ f (u1, . . . , ui , . . . , u|Q|) ≤ f (u1, . . . , u0
i , . . . , u|Q|)

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Additive Structure Across Workflows
Theorem (Additive structure across workflows)
(A) All nodes V in the game Γ are transition independent.
(B) If there is no path between i and j in the topology graph G,
then i and j are utility independent.
Corollary
Γ decomposes into |W| additive subproblems that can be solved
independently and in parallel.
π
(w1)
k
π
(w2)
k
π
(w|W|)
k
ot,w1
ot,w2
ot,w|W|
.
.
.
⊕
a
(k)
w1
a
(k)
w2
a
(k)
w|W|
a
(k)
t

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Additive Structure Across Workflows: Example
Zone 1 Zone 2
1
2
3
gw
w1
w2
V = {1, 2, 3},
E = {(1, 2)},
W = {w1, w2},
Z = {1, 2}
a) IT infrastructure b) Transition dependencies
St, At St+1, Ot+1
S
(D)
t+1,1
S
(D)
t,1
A
(D)
t,1 S
(A)
t+1,1
S
(A)
t,1
Ot,1
A
(A)
t,1
S
(D)
t+1,2
S
(D)
t,2
A
(D)
t,2 S
(A)
t+1,2
S
(A)
t,2
Ot,2
A
(A)
t,2
S
(D)
t+1,3
S
(D)
t,3
A
(D)
t,3 S
(A)
t+1,3
S
(A)
t,3
Ot,1
A
(A)
t,3
c) Utility dependencies
St, At Ut
S
(D)
t,1
A
(D)
t,1
Ut,1
S
(A)
t,1
S
(D)
t,2
A
(D)
t,2
Ut,2
S
(A)
t,2
S
(D)
t,3
A
(D)
t,3
Ut,3
S
(A)
t,3

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efficiently

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Optimal Substructure Within a Workflow
I Nodes in the same workflow are utility
dependent.
I =⇒ Locally-optimal strategies for
each node can not simply be added
together to obtain an optimal strategy
for the workflow.
I However, the locally-optimal strategies
satisfy the optimal substructure
property.
I =⇒ there exists an algorithm for
constructing an optimal workflow
strategy from locally-optimal
strategies for each node.
Zone 1 Zone 2
1
2
3
gw
w1
w2
V = {1, 2, 3},
E = {(1, 2)},
W = {w1, w2},
Z = {1, 2}
IT infrastructure
Utility dependencies
St, At Ut
S
(D)
t,1
A
(D)
t,1
Ut,1
S
(A)
t,1
S
(D)
t,2
A
(D)
t,2
Ut,2
S
(A)
t,2
S
(D)
t,3
A
(D)
t,3
Ut,3
S
(A)
t,3

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Algorithm for Combining Locally-Optimal Node Strategies
into Optimal Workflow Strategies
π
(2)
D
π
(1)
D
π
(3)
D
π
(4)
D
π
(5)
D
π
(6)
D
(π
(i)
D )i∈Vw : local strategies in the same workflow w ∈ W

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π
(2)
D
π
(1)
D
π
(3)
D
π
(4)
D
π
(5)
D
π
(6)
D
(π
(i)
D )i∈Vw : local strategies in the same workflow w ∈ W

24/33
π
(2)
D
π
(1)
D
π
(3)
D
π
(4)
D
π
(5)
D
π
(6)
D
Can redefine the utility function for each node i
to take into account the utility impact on its ancestors.
e.g. utility of node 6 need to include utility impact for 1, 3, 5.

24/33
π
(2)
D
π
(1)
D
π
(3)
D
π
(4)
D
π
(5)
D
π
(6)
D
Can prove that this utility transformation makes the nodes utility independent.
=⇒ Optimal substructure.

25/33
efficiently

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Threshold Properties of Local Defender Strategies.
I The local problem of the defender can be decomposed in the
temporal domain as
max
πD
T
X
t=1
J = max
πD
τ1
X
t=1
J1 +
τ2
X
t=1
J2 + . . . (2)
where τ1, τ2, . . . are stopping times.
I =⇒ (1) selection of defensive actions is simplified; and (2)
the optimal stopping times are given by a threshold strategy
that can be estimated efficiently:
Belief space B
(j)
D
Switching curve
Υ
Continuation set
C
Stopping set
S
(1, 0, 0)
j healthy
(0, 1, 0)
j discovered
(0, 0, 1)
j compromised

27/33
Threshold Properties of Local Defender Strategies.
Belief space B
(j)
D
Switching curve
Υ
Continuation set
C
Stopping set
S
(1, 0, 0)
j healthy
(0, 1, 0)
j discovered
(0, 0, 1)
j compromised
I A node can be in three attack states s
(A)
t : Healthy,
Discovered, Compromised.
I The defender has a belief state b
(D)
t

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Proof Sketch (Threshold Properties)
I Let L(e1, b̂) denote the line segment
that starts at the belief state
e1 = (1, 0, 0) and ends at b̂, where b̂ is
in the sub-simplex that joins e2 and e3.
I All beliefs on L(e1, b̂) are totally
ordered according to the Monotone
Likelihood Ratio (MLR) order. =⇒ a
threshold belief state αb̂ ∈ L(e1, b̂)
exists where the optimal strategy
switches from C to S.
I Since the entire belief space can be
covered by the union of lines L(e1, b̂),
the threshold belief states αb̂1
, αb̂2
, . . .
yield a switching curve Υ.
Belief space B
(j)
D
(the 2-dimensional unit simplex)
sub-simplex B
(j)
D,e1
joining e2 and e3
b̂5
b̂4
b̂3
b̂2
b̂1
b̂6
b̂7
b̂8
b̂9
L(e1, b̂5)
Switching curve
Υ
Threshold
belief state αb̂9
e1
(1, 0, 0)
e2
(0, 1, 0)
e3
(0, 0, 1)

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Scalable Learning through Decomposition
1 2 3 4 5 6 7 8 9 10
2
4
6
8
10
linear
measured
# parallel processes n
|V| = 10
Speedup
S
n
Speedup of completion time when computing best response strategies for
the decomposed game with |V| = 10 nodes and different number of
parallel processes; the subproblems in the decomposition are split evenly
across the processes; let Tn denote the completion time when using n
processes, the speedup is then calculated as Sn = T1
Tn
; the error bars
indicate standard deviations from 3 measurements.

30/33
Decompositional Fictitious Play (DFSP)
π̃2 ∈ B2(π1)
π2
π1
π̃1 ∈ B1(π2)
π̃0
2 ∈ B2(π0
1)
π0
2
π0
1
π̃0
1 ∈ B1(π0
2)
. . .
π∗
2 ∈ B2(π∗
1)
π∗
1 ∈ B1(π∗
2)
Fictitious play: iterative averaging of best responses.
I Learn best response strategies iteratively through the parallel
solving of subgames in the decomposition
I Average best responses to approximate the equilibrium

31/33
Learning Equilibrium Strategies
0 20 40 60 80 100
running time (h)
0
5
b
δ = 0.4
Approximate exploitability b
δ
0 20 40 60 80 100
running time (h)
0.0
0.5
1.0
Defender utility per episode
dfsp simulation dfsp digital twin upper bound oi,t 0 random defense
Learning curves obtained during training of dfsp to find optimal
(equilibrium) strategies in the intrusion response game; red and blue
curves relate to dfsp; black, orange and green curves relate to baselines.

32/33
Comparison with NFSP
0 10 20 30 40 50 60 70 80
running time (h)
0.0
2.5
5.0
7.5
Approximate exploitability
dfsp nfsp
Learning curves obtained during training of dfsp and nfsp to find
optimal (equilibrium) strategies in the intrusion response game; the red
curve relate to dfsp and the purple curve relate to nfsp; all curves show
simulation results.

33/33
Conclusions
I We study an intrusion response use
case.
I We formulate the use case as a POSG
I We design a novel decompositional
approach to approximate equilibria
I We show that the decomposition
allows scalable approximation of
equilibria.
s1,1 s1,2 s1,3 . . . s1,n
s2,1 s2,2 s2,3 . . . s2,n
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Emulation
Target
System
Model Creation
System Identification
Strategy Mapping
π
Selective
Replication
Strategy
Implementation π
Simulation
Learning

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