Modeling Costs of Access Control
with Various Key Management Systems
Tomomi Yamasaki, Shunsuke Inenaga, Daisuke Ikeda, and Hiroto Yasuura
Graduate School of Information Science and Electrical Engineering, Kyushu University,
744 Motooka, Fukuoka 819-0395, Japan
Abstract— Access control, a task of managing permission
or denial to the use of particular resources by particular entities, is achieved by credentials such as passwords
and physical keys. Although security of access control has
extensively been studied, researchers have not paid muchattention to comparing costs of the systems with different
credentials. This paper provides a general model of doorkey management systems for access control, where a door
represents a resource to control access and a key does acredential. We show door-key management systems based on
smart cards, biometrics, metal keys, and passwords will well
fit to our model. Then, we introduce management costs tothe proposed model. Finding the best door-key management
system in terms of minimum management cost is a hard task,
since different systems have different restrictions. We present
a general algorithm that computes the minimum cost of doorkey management systems with different types of keys.
Keywords: access control, door-key management system, deterministic finite state machine, cost evaluation

1. Introduction
Access control, a task of managing permission or denial
to the use of a particular resource by a particular entity,
has been critical in human society. Considered resources tocontrol access are in a wide range from physical resources,
such as secure rooms, to digital resources, such as computer
files.
Access control is achieved with credentials, and in fact
there exist many kinds of credentials we can use for access
control. For instance a physical key is a classical, yetpractical, credential for physical access control, and anelectronic key has become popular as a credential. Using
electronic credentials offers us more sophisticated physicalaccess control, as well as it also improves security. We are at
the stage where a variety of possible credentials are available
for access control.
However, this fact poses a new challenging problem inaccess control: How can we select the most suitable typeof credential for a particular measure such as access control
management costs? Here we mean by the costs the burdenof the administrator of the system to manage some changes
in the security policy, e.g., making a particular person able
to enter a secure room. Note that these kinds of costs areheavily dependent on the type of credentials and hence they
are not negligible at all. In fact, it is reported that using some
sorts of electronic credential has increased the costs of theadministrators to maintain the access control policy [1].
However, to the best of our knowledge, there exist no
researches considering to reduce or estimate the costs ofaccess control, although we can find many researches on
security of (digital) access control systems [2], [3] andmany services of identity and access management, such as
Microsoft Identity and Access Management, and that ofphysical access control systems with some fixed type ofcredentials, such as biometrics. In the existing researches,
services, and systems, we do not need to consider thecost of access control management because every cost ofchanging the security policy is the same. For instance, ina digital access control system, the costs for making someperson both able and disable to access some resource are
the same, because they are completed by modifying entriesof an access control database. Therefore, it is important tocompare costs among access control with different typesof credentials. But, we can find no general approaches toevaluate such costs.
In this paper, we first formalize a model of access control
in which a door represents a resource to protect. The modelis based on a deterministic finite state machine in which eachstate represents a binary relation between doors and users.
We define two types of requests in the model, one of whichenables users to open doors, and the other disables users
from opening doors. Then, we propose a model of door-key
management systems in which a key represents a credential.
This model is based on another deterministic finite statemachine, in which each state represents a pair of binary
relations between doors and keys, and between keys andusers. We define four unit operations in the model, to enable
keys to unlock doors, to disable keys from unlocking doors,
to issue keys to users, and to collect keys from users. We
show that electronic door-key systems using smart cards,
biometrics, metal keys, and passwords all well fit in ourmodel.
The access control model we propose describes a policyof access control and a door-key management system modelis an implementation of the policy using keys as credentials.
The separation of the door-key management system from theaccess control policy enables us to uniformly evaluate thecosts of implementations with different types of keys.
We then consider how to evaluate the minimum cost fora given request when some changes occur in the access
control policy. We emphasize this problem of computing theminimum cost is not as easy as it may sound, as the costvaries with the door-key management system. Section 3 will
discuss it in more details. We present a general algorithm tosolve the problem for any type of door-key management
systems, by reduction to the single-source shortest path
problem of a directed graph [4].

2. Modeling Door Access Control andDoor-Key Management
We firstly propose a model of door access control in
Section 2.1, which represents the access control policy todoors. Then in Section 2.2, we introduce a model of doorkey management systems that implement the door access
control using keys as credentials of users to access doors.
Section 2.3 shows that the above model is general enoughto deal with different types of keys such as smart cards,
biometrics, metal keys and passwords.

2.1 Door Access Control
In this section, we consider a model of door access control
where the task is to maintain a binary relation between doors
and users. Here the binary relation implies that the doors
may be opened by the users. We model door access control
based on a deterministic finite state machine where eachstate represents each binary relation.
Definition 1 (Door Access Control) A door access control
is a tuple (D, U , M ) such that D is a finite set of doors, U
is a finite set of users, and M is a deterministic finite statemachine (Q, Σ, δ, q0 ) where
D×U• Q =2is the set of states,
D×U• Σ = {gr, re} × 2
is the input alphabet,• δ : Q × Σ → Q is the transition function, and• q0 ∈ Q is the initial state.
Each state q ∈ Q represents a policy of door access
control. Let d ∈ D and u ∈ U . If (d, u) ∈ q, then we
say that the user u may open the door d in the state q. If(d, u) ∈/ q, then we say that user u must not open the doord in the state q.
Each element of the alphabet Σ = {gr, re} × 2D×U
is called a request to change the policy of door access
control, that is, to transit to another state. Namely, gr
represents a request of enabling users to open doors. For
instance, (gr, {(u, d)}) is a request of enabling the user
u to open the door d. On the other hand, re represents arequest of disabling users from opening doors. For instance,(re, {(u, d)}) is a request of disabling the user u from
opening the door d.
The transition function δ is defined below:

Definition 2 (Transition Function δ) For any states p, q ∈Q,δ(p, (gr, q − p)) = q if p ⊂ q andδ(p, (re, p − q)) = q if p ⊃ q.
The above definition implies that we only consider a gr
request which “newly” enables at least one user to open
at least one door. Similarly, we only consider a re request
which “newly” disables at least one user from opening atleast one door.
Fig. 1 illustrates the finite state machine M of the door
access control ({d}, {u1 , u2 }, M ), where q0 = ∅.
In the sequel we assume D and U are fixed.

2.2 Door-Key Management System
Consider how to implement the door access control ofDefinition 1. A natural, yet practical, solution is to use keys
as mediums of the right of users to open doors. We maintaina binary relation between doors and users using two binary
relations between doors and keys, and between keys andusers.
Definition 3 (Door-Key Management System) A doorkey management system is a quadruple (D, K , U , MK )such that D is a finite set of doors, K is a finite set ofkeys, U is a finite set of users, and MK is a deterministic
finite state machine (QK , ΣK , δK , qK 0 ) where
QK ⊆ 2D×K × 2K ×U is the set of states,
D×K• ΣK ⊆ ({ac, in} × 2) ∪ ({is, co} × 2K ×U ) is the
input alphabet,• δK : QK × ΣK → QK is the transition function, and• qK 0 ∈ QK is the initial state.

•

We call each element qK ∈ QK a k-state, to tell it froma state q ∈ Q of the door access control.
Let d ∈ D, k ∈ K and u ∈ U . Let qK ∈ QK , and denote
qK = (qKa , qKb ). Then qKa ⊆ D × K and qKb ⊆ K × U .
If (d, k) ∈ qKa , then we say that the key k can unlock thedoor d in the k-state qK . If (d, k) ∈/ qKa , then we say thatthe key k cannot unlock the door d in the k-state qK . If(k, u) ∈ qKb , then we say that the user u has the key k in
the k-state qK . If (k, u) ∈/ qKb , then we say that the user u
does not have the key k in the k-state qK .
Each element of the alphabet ΣK ⊆ {ac, in} × 2D×K ∪{is, co}×2K ×U is called an operation. Here, ac represents
an operation of enabling keys to unlock doors, in represents
an operation of disabling keys from unlocking doors, isrepresents an operation of issuing keys to users, and co
represents an operation of collecting keys from users.
We define the transition function δK in a somewhat similar
way to the transition function δ of Definition 2.
Definition 4 (Transition Function δK ) For any k-statesq1(re,{(d,u1)})
q0

u1
d

(gr,{(d,u1)})u1(re,{(d,u2)})u2q3(gr,{(d,u2)})u1(gr,{(d,u1),(d,u 2)})dd(re,{(d,u1),(d,u 2)})u2(gr,{(d,u2)})(gr,{(d,u1)})u1(re,{(d,u2)})

u2

d
q2

(re,{(d,u1)})

u2

Fig. 1: The finite state machine M of the door access control ({d}, {u1 , u2 }, M ), where q0 = ∅. Each state consisting of abinary relation between {d} and {u1 , u2 } is represented by a bipartite graph.
pK = (pKa , pKb ), qK = (qKa , qKb ) ∈ QK ,
δK (pK , (ac, qKa − pKa )) = qK if pKa ⊂ qKa , pKb = qKb ,δK (pK , (in, pKa − qKa )) = qK if pKa ⊃ qKa , pKb = qKb ,δK (pK , (is, qKb − pKb )) = qK if pKb ⊂ qKb , pKa = qKa ,δK (pK , (co, pKb − qKb )) = qK if pKb ⊃ qKb , pKa = qKa .
See Fig. 2 which illustrates an example of the finite state machine M of the door access control({d}, {k}, {u1 , u2 }, MK ), where qK 0 = ∅.
Now we consider the correspondence between a k-stateof a door-key management system and a state of the door
access control.
Definition 5 (Function ζ from K-States to States) Let
qK = (qKa , qKb ) be any k-state in QK and q any state inQ, respectively. Define a binary relation R ⊆ D × U suchthat (d, u) ∈ R iff (d, k) ∈ qKa and (k, u) ∈ qKb for somek ∈ K . If R = q, then we denote ζ(qK ) = q.
If ζ(qK ) = q, we say that the k-state qK in a doorkey management system (D, K , U , MK ) corresponds to thestate q in the door access control (D, U , M ). Note that ζ
may be a many-to-one function, that is, more than one k-state
in QK may correspond to a state in Q. For the door access
control of Fig. 1 and the door-key management system
of Fig. 2, we have that ζ(qK 0 ) = ζ(qK 1 ) = ζ(qK 2 ) =ζ(qK 3 ) = ζ(qK 5 ) = q0 , ζ(qK 4 ) = q1 , ζ(qK 6 ) = q2 , andζ(qK 7 ) = q3 .
Consider to implement the door access control using adoor-key management system. In so doing, the following
extended version of the transition function δK is useful.
Definition 6 (Extended Transition Function δK ) For any
k-state qK ∈ QK ,
δK (qK , ε) = qk for the empty sequence ε,δK (qK , zβ) = δK (δK (qK , z), β) for any z ∈ ΣK , β ∈ Σ∗K .
We say that a door-key management system implements
the door access control if(1) ζ is a surjection, and(2) for any p, q ∈ Q such that δ(p, r) = q with some r ∈Σ, there exists (pK , qK ) ∈ QK such that ζ(pK ) = p,ζ(qK ) = q, and δK (pK , s) = qK with some s ∈ Σ∗K .
Note the door-key management system of Fig. 2 implements the door access control of Fig. 1.
The following proposition gives the lower bound forthe number of keys for a door-key management system to
implement the door access control.
Proposition 1 Anydoor-key
management
system
(D, K , U , MK ) implements the door access control(D, U , M ) only if |K | ≥ min{|D|, |U |}.
Proof. Assume |D| ≤ |U |. Consider state q = {(di , ui ) | 1 ≤i ≤ |D|, di 6= di+1 } ∪ {(d|D| , u` ) | |D| + 1 ≤ ` ≤ |U |, u` 6=
u`+1 }. If |K | < |U |, due to the pigeonhole principle (referto e.g. [5]), no k-states in QK correspond to q. The case
|D| > |U | can be shown similarly.
¤
By Proposition 1 we will only consider door-key management systems with |K | ≥ min{|D|, |U |}.

2.3 Modeling Real-World Door-Key Management Systems
In this section we describe real-world door-key management systems based on our general door-key management
system model of Definition 3.
Definition 7 A smart-card-based door-key management system is a door-key management system of Definition 3 withthe following restrictions:
1) For any qK = (qKa , qKb ) ∈ QK , let (k, u) ∈ qKb . Thenfor any u0 6= u it holds that (k, u0 ) ∈/ qKb .
2) ΣK = ({ac, in} × D × K ) ∪ ({is, co} × K × U ).
Condition 1 implies that at every k-state each key is notshared by two or more users. That is, for each smart card(is,{(k,u 2)})

qK1

qK4

(is,{(k,u1)})u1d(ac,{(d,k)})
qK2
u1

d

k

d

u2

qK7

qK5
u1

(is,{(k,u2)})u2(co,{(k,u2)})k(co,{(k,u1)})d(is,{(k,u1)})(is,{(k,u2)})

d

k
u2

qK3

u1

(ac,{(d,k)})u2(in,{(d,k)})k(in,{(d,k)})

d

k
u2

(co,{(k,u1)})u1(ac,{(d,k)})u1d(co,{(k,u1)})u1(co,{(k,u2)})(co,{(k,u2)})(is,{(k,u 2)})(co,{(k,u1)})(is,{(k,u1)})(is,{(k,u1)})

k
u2

d

u2

(in,{(d,k)})qK0u1(co,{(k,u1)})k(is,{(k,u1)})

k
u2
qK6

(co,{(k,u2)})

Fig. 2: The finite state machine MK of the door access control ({d}, {k}, {u1 , u2 }, MK ), where QK = 2{d}×{k} ∪2{k}×{u1 ,u2 } , ΣK = {ac, in} × ({d} × {k}) ∪ {is, co} × ({k} × {u1 , u2 }) and qK 0 = ∅. Each k-state consisting ofa pair of binary relations between {d} and {k}, and between {k} and {u1 , u2 } is represented by a tripartite graph.

there is a unique user who has it. This describes a smart card
often contains personal information of the user which is notshared by other users. Condition 2 implies that at a single
step of the system we can make an operation to a single pairof a door and a key, or of a key and a user.
Definition 8 A biometrics-based door-key management system is a door-key management system of Definition 3 withthe following restrictions:
1) For any qK = (qKa , qKb ) ∈ QK and any u ∈ U , there
exists k ∈ K such that (k, u) ∈ qKb .
2) For any qK = (qKa , qKb ) ∈ QK and for any(k, u), (k 0 , u0 ) ∈ qKb with u 6= u0 , it holds that k 6= k 0 .
3) For any pK = (pKa , pKb ), qK = (qKa , qKb ) ∈ QK , itholds that pKb = qKb .
4) ΣK = {ac, in} × D × K .
Condition 1 implies that at any k-state, every user has
at least one key. This describes that each user has at least
one biometric key such as fingerprints, iris, and hand vein
patterns. Condition 2 implies that at any k-state, no key isshared by two or more users. This describes the uniquenessof biometric keys. Condition 3 implies that at any k-states,
the binary relation between keys and users is identical. This
describes the permanency of biometric keys. Condition 4
implies that a single operation either enables or disables a
key from unlocking a door. From the other conditions it
is clear that there is no transition with is or co operations.

This assumes that biometric keys cannot be given to or takenfrom a user afterwards.
Definition 9 A metal door-key management system is adoor-key management system of Definition 3 with the following restriction:1) ¡ΣSK = ({ac} × D × K ) ∪ ({is, co}¢× K × U ) ∪qK ∈QK ,d∈D ({in} × {(d, k) ∈ qKa }) .
Condition 1 implies that a single operation enables a key
to unlock a door, issues a key to a user, or collects a key
from a user. A special in operation implies that at any kstate all keys which can unlock the same door are disabled atonce. This describes that if a key hole of a door is changed,
then all metal keys associated with the old key hole become
unable to unlock the door.
Definition 10 A password-based door-key management system is a door-key management system of Definition 3 withthe following restrictions:
1) For any qK = (qKa , qKb ) ∈ QK , let (d, k) ∈ qKa . Thenfor any k 0 6= k, it holds that (d, k 0 ) ∈/ qKa .
2) Let pK = (pKa , pKb ), qK = (qKa , qKb ) ∈ QK be
any k-states such that δK (pK , (in, {(d, k)})) = qK
for some d and k. Then qK has exactly one transition
δK (qK , (co, {(k, u) ∈ qKb })).
3) ΣK = ({ac, in} × D × K ) ∪ ({is} × K × U ) ∪¡S¢
qK ∈QK {co, {(k, u) ∈ qKb }} , where {co, {(k, u) ∈qKb }} is as defined in Condition 2.
Condition 1 implies that for each door there is a unique
key that can unlock the door. This assumes only one password is associated with each door. Condition 2 implies thatonce the key k gets disable from unlocking the door d, then
the key k is immediately collected from all the users who
have the key k. By “a user has a password”, we mean that theuser can open the door at the first trial of typing a password.
Once the password is changed to a new one, then since theuser does not know the new one, the user can open the door
only with a negligible (very small) probability. This suggests
the key has been collected from the user. Condition 3 implies
a single operation enables a key to unlock a door, disables
a key from unlocking a door, or issues a key to a user. A
special co operation takes place at a single step as definedin Condition 2.
It is not difficult to prove the following theorem.
Theorem 1 Every door-key management system of Definition 7, Definition 8, Definition 9 and Definition 10 implements the door access control of Definition 1.

3. Evaluating Costs of Door-Key Management Systems
In this section we consider how to evaluate the minimum
costs of gr and re requests of the door access control. We
remark that the minimum cost may vary with the doorkey management system that implements the door access
control, as real-world systems often have some restrictions
such as those introduced in Section 2.3. Hence generalmeasures of the costs of requests, and an efficient method tocompute them, are significantly important to evaluate doorkey management systems.
Let N be the set of non-negative integers. We define thecost of each single operation of a door-key management
system as follows.
Definition 11 (Cost Function cost) Function cost
:
Σ+
→
N
is
definedasfollows.
KCac ∈ N if σ = (ac, A) ∈ ΣK ,
C ∈ N if σ = (in, A) ∈ Σ ,inKcost(σ) =Cis ∈ N if σ = (is, B) ∈ ΣK ,
Cco ∈ N if σ = (co, B) ∈ ΣK , andcost(s) =|s|X

cost(s[i])if |s| ≥ 2,

i=1

where A ⊆ 2D×K , B ⊆ 2K ×U , and s[i] is the i-th elementof the sequence s ∈ Σ+
K of operations.
The actual values of Cac , Cin , Cis , and Cco may vary withthe door-key management system. Hence, we need a generalframework where we can deal with arbitrary costs to evaluatevarious types of door-key management systems. Computing
the minimum cost of a given request is formalized as thefollowing problem.
Problem 1 The minimum cost problem of a request of thedoor access control (D, U , M ) w.r.t. a door-key management
system (D, K , U , MK ) is as follows.
- Input: A k-state pK ∈ QK and a request r ∈ Σ.
- Output: A positive integer c ∈ N and a k-state qK ∈QK such thatc = min{cost(β) | ζ(δK (pK , β)) = δ(ζ(pK ), r)},qK = δK (pK , s),where s is a sequence of operations such that s ∈argmin{cost(β) | ζ(δK (pK , β)) = δ(ζ(pK ), r)}.
Namely, the problem is: given a k-state pK ∈ QK and arequest r ∈ Σ, find a sequence of operations of minimum
cost which leads to a k-state corresponding to the target stateδ(ζ(pK ), r) ∈ Q.
See Figure 1 and Figure 2. Assume that the inputs of Problem 1 are k-state qK7 and request (re, {(d, u1 ), (d, u2 )}). If
Cin = 3 and Cco = 1, then the solution is minimum costc = 2 (= 2 × Cco ) and k-state qK1 which corresponds tostate q0 = ∅. However, if Cin = 3 and Cco = 2, then thesolution is minimum cost c = 3 (= Cin ) and k-state qK5
which also corresponds to state q0 = ∅.
It is also noteworthy that, even in the simplest unit costmodel where Cac = Cin = Cis = Cco = 1, the minimum
cost varies with the type of door-key management system.
Remark that in the unit cost model, the minimum costequals the minimum number of operations. See Figure 3. Ineach system, the input is a pair of a k-state corresponding
to a state {(d1 , u1 ), (d1 , u2 ), (d2 , u1 ), (d2 , u2 )} anda request (re, {(d1 , u1 ), (d2 , u1 )}). Hence the target
state is {(d1 , u2 ), (d2 , u2 )} in which only the user
u2 can open the doors d1 and d2 . In the smart
card based system of Definition 7, the minimum
cost is cost((co, {(k1 , u1 )})) = 1. In the biometrics
based system of Definition 8, the minimum cost iscost((in, {(d1 , k1 )}), (in, {(d2 , k1 )})) = 2. In the metal
key based system of Definition 9, the minimum cost iscost((in, {(d1 , k1 ), (d1 , k2 )}), (in, {(d2 , k1 ), (d2 , k2 )}), (ac,{(d1 , k3 )}), (ac, {(d2 , k3 )}), (is, {(k3 , u2 )})) = 5. In the
password based system of Definition 10, the minimum cost iscost((in, {(d1 , k1 )}), (co, {(k1 , u1 ), (k1 , u2 )}), (in, {(d2 ,k2 )}), (co, {(k2 , u1 ), (k2 , u2 )}), (ac, {(d1 , k3 )}), (ac, {(d2 ,k4 )}), (is, {(k3 , u2 )}), (is, {(k4 , u2 )})) = 8.
As observed above, the solution to Problem 1 varies withthe cost function and the door-key management system, andtherefore a general algorithm to solve the problem with
arbitrary costs is of significant importance. The following
theorem shows that we indeed have such an algorithm.

d1

k1

u1

d2

k2

u2

1

d1

k1

u1

d1

k1

u1

d2

k2

u2

d2

k2

u2

d1
k1

u1

d1

k1

u1

d2

k2

u2

d1

k1

u1

d2

k2

u2

biometrics

smart card

d1

2

d1

k1

u1

d2

k2

u2

k1

u1
8

5

d2
d2

k2

u2

k3

k2

u2

k3

k3

k4

k4

k3
metal key

password

Fig. 3: The minimum cost for the same request may vary with the door-key management system.
Theorem 2 Problem 1 is solvable in O(|QK |2 ) time.
Proof. Consider an edge-weighted directed graph G =(V, E) s.t. V = QK and E = {(pK , qK , cost(α)) |δK (pK , α) = qK } ⊆ V ×V ×N . It follows from Definition 3
and Definition 11 that a solution to the single source shortest
path problem based on cost(·) from the input vertex pK
to each vertex qK ∈ QK is equal to min{cost(α) |δK (pK , α) = qK }. For each vertex qK ∈ QK , check if
ζ(qK ) = δ(ζ(pK ), r), where r is a given request r ∈ Σ,and keep the solution of the shortest path problem frompK if it is smaller than the previously kept solution forsome vertex already checked. This checking can be done in
O(|D|×|K |×|U |) time for each vertex by e.g. implementing
the binary relations between D and K , and between K
and U , with matrices of size |D| × |K | and |K | × |U |,respectively. It is well know that the single source shortest
path problem can be solved in O(|V |2 + |E|) time, e.g.
see [4]. Note that |V | = |QK | and |E| = O(|V |2 ). Hencethe overall time cost is O(|QK |2 ).
¤4. Conclusions and Future WorkIn this paper we considered door-key management systems for door access control. We model them based ondeterministic finite state machines. Our model is generalenough to describe various door-key management systems
based on smart cards, biometrics, metal keys, and passwords.
We gave an algorithm to compute the minimum cost of adoor-key management systems to realize a given request on
the door access control. The proposed algorithm is based on
Dijkstra’s algorithm [4] that solves the single-source shortest
path problem on a directed graph.
Our future work includes the following.

1) In the proposed model, we did not consider that auser may take some operations. For instance, in thereal world, a user might tell a password to anotheruser who did not know it, independently of the administrator of the system. Yamasaki et al. [6] presented
a mobile phone based door-key management system
in which a user can copy his/her electronic door-key
and can delegate it to another user’s mobile phone
in an “offline” manner (without communicating theadministrator). Our model needs to be extended todealing with such operations by users too.
2) To quickly evaluate large-scaled systems, it is necessary to speed up the algorithm to solve Problem 1.
Our algorithm is inefficient when only a few k-statescorrespond to the target state of the door access control,as Dijkstra’s algorithm preprocesses all vertices (kstates) and the total number of vertices is |QK | =O(2|D|×|K |+|K |×|U | ). Hence, if we can efficiently enumerate every and only k-state corresponding to thetarget state and compute the edit distance between the
input k-state and each enumerated k-state, we can saveconsiderable computational time and space. The edit
operations we consider are insertion and deletion ofedges of the tripartite graphs representing k-states. An
efficient enumeration algorithm of pattern trees matching a given data tree has been studied [7]. It may bepossible to modify the above algorithm to enumerating
all tripartite graphs (k-states) which correspond to agiven bipartite graph (target state).

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