A novel access control method using Morton number and prime factorization

A novel access control method using Morton

number and prime factorization

Henry Ker-Chang Chang

, Jing-Jang Hwang

,

Hsing-Hua Liu

a_{Department of Information Management, Chang Gung University, Taoyuan, Taiwan, ROC}

b_{Institute of Information Management, National Chiao Tung University, Hsinchu, Taiwan, ROC} Received 3 March 1999; received in revised form 1 March 2000; accepted 15 July 2000

Abstract

A novel scheme used for controlling access requests in security information system is proposed. In the proposed method, the system administrator chooses distinct prime numbers representing each atomic access right as well as four large prime numbers for encryption. By setting these representative prime numbers as input parameters, the proposed method applies a one-way function combining the Morton number theory transferring into a single value to derive the encrypted compound privilege (ECP). With ECP, veri®cation of right of access can be achieved easily and secretly. Meanwhile, the proposed scheme provides the following advantages: (1) the veri®cation of right of access can be e€ectively implemented using the Morton sequence with coordinate transformation; (2) the problem of dynamic access control also can be e€ectively im-plemented; (3) integrity and con®dentiality while controlling system resources can be ensured; (4) the proposed method can decrease the redundancy of the access matrix in some speci®c circumstances. Ó 2000 Elsevier Science Inc. All rights reserved.

1. Introduction

The Internet and Intranet established a foundation for the global commu-nity and remote access to resources in networks are very popular today. In

www.elsevier.com/locate/ins

*_{Corresponding author. Address: P.O. Box 7-12, Chung-Ho, Taipei, Taiwan, ROC; Tel:} +886-3-3960947; fax: +886-2-22232876.

E-mail addresses: changher@mail.cgu.edu.tw (H.K.-C. Chang), u8434803@cc.nctu.edu.tw (H.-H. Liu).

0020-0255/00/$ - see front matter Ó 2000 Elsevier Science Inc. All rights reserved. PII: S 0 0 2 0 - 0 2 5 5 ( 0 0 ) 0 0 0 7 3 - 6

open environments, security systems must have ability to authorize of re-quested operations; this is the security problem of access control. More spe-ci®cally, access control is a core function for information system security. The access control model o€ers a framework for describing the protection mech-anism. The initial model was introduced by Graham and Denning [1]. In this model, the state of an information protection mechanism is de®ned by a triple (S,O,A), where S is the set of subjects which are active entities of the model, O is the set of protected objects and A is an access control matrix, in which each column consists of subjects representing human or programs, and each row consists of objects representing ®les or records. An entry aij for A‰S; OŠ

de-scribes the right of subject Si to access object Oj. The access right de®nes the

kind of authorized access to the object where r(read), w(write), and e(execute), o(owner), a(append) etc. All these rights are generic and can be combined together for a subject. For example, an object may be applied for right of access in r, w and e separately, or various combinations may be used. A simple access control matrix as shown in Fig. 1 is used to specify rights of subjects to access objects. For example, the subjects S1, and S2 have `read' and `write'

access rights to object O1 while S1 is the owner of object O3.

Based on Graham and Denning's abstraction protection model, Wu and Hwang [2] proposed a single-key-lock (SKL) mechanism in which there is only one key for each subject and one lock for a each object. To derive an access right aij for subject to an object, a function f of a key and lock is used;

mathematically, f …Ki; Lj† ˆ aij.

Several relevant methods have appeared in the literature which are based on SKL work. Chang [3,4] proposed two methods based on the Chinese remainder theorem and Euler's theorem, respectively. Laih et al. [5] used a Newton in-terpolating polynomial to design another method in 1989 while Chang and Jiang [6] presented a binary version of Wu and Hwang's method. Hwang et al. [7] proposed a new SKL scheme using prime factorization. In Hwang's scheme, each subject Si is assigned a distinct prime as the key Ki, and a lock Lj is

produced as LjˆQmiˆ1…Ki†aijfor the object Oj, where aijis the right of subject Si

to access object Oj. Since a lock is the product of some prime powers, it can

easily exceed the limited range of the largest integer allowed in a computer system. Hwang et al. used to decompose each lock value into an X-based representation to solve this problem, where X can be any integer. Chang et al. [8] proposed a scheme based on binary coding and prime factorization. In Chang's method, each access right aij can be represented by a binary form.

Again, each user Uiis assigned a distinct prime as the key Ki. The lock vector Lj

is produced as (Lj…b†; Lj…b ÿ 1†; . . . ; Lj…1†), where Lj…x† ˆQmiˆ1…Ki†a …x†

ij . There is a

problem of over¯ow, which is inevitable a…x†ij 's 2 f1g.

In order to evaluate the e€ectiveness and eciency of an SKL scheme, the following six criteria are considered [7]:

1. the e€ort involved in initializing keys and locks;

2. the e€ort involved in computing an access right from a lock and key; 3. the e€ort involved in revising keys and locks when an access right is

modi-®ed;

4. the e€ort involved in appending and updating keys and locks when a new user or ®le is added;

5. the e€ort involved in removing and updating keys and locks when a user or ®le is deleted;

6. the space needed for storing keys and locks.

These criteria are, therefore, generally applied in performance evaluation of and comparison among various schemes. In this paper, we intend to develop a new method based on the Morton sequence and prime factorization to improve schemes derived by Hwang and Shao [7] and Chang and Lou [8]. According to the six criteria, our method has better performance than the works in [7] and [8]. In particular, the proposed method has a compression e€ect on the matrix in some speci®c circumstances, which decreases redundancy in the access matrix.

The rest of this paper is organized as follows. The Morton sequence is in-troduced ®rst in Section 2; it helps the proposed method work e€ectively. In

Section 3, we describe the proposed method ®rst, and then algorithms are developed for physical application. In Section 4, the performance of the pro-posed method is analyzed and compared with that of other schemes. Conclu-sions and directions for future research are given in Section 5.

2. Morton sequence

Morton [9] proposed the addressing scheme commonly referred to as Morton sequencing. Morton sequencing is created by interleaving the bits of the binary representations of the x and y coordinates (each represented by a ®xed number of digits) of a speci®c position in the matrix. Fig. 2 using 3-bit binary representation gives a simple example. Each sequence of an element is formed by the y-axis and x-axis, such that the y-axis bit is prior to the x-axis bit. The sequence at (3,2) is 13 since 2 ˆ …010†₂ and 3 ˆ …011†₂, so the inter-leaving 001101 is …13†₁₀. On the other hand, the sequence 55 has the binary form 110111, in which the odd-numbered bits 101 are the y-axis and the even-numbered bits 111 are the x-axis, so the coordinates are (7,5). From Fig. 2, it is clear that the Morton sequence in a square matrix which follows a scanning order like the character `Z'.

3. The proposed new method

This section shows our design for access control in a security environment. The server workstation resides a security manager whose job is to monitor and maintain the access control mechanism. A user from a client node has to de-liver a request to ®nd an opportunity to access the resources in the system. Section 3 comprises three parts. Firstly, the design for the access control mechanism will be developed. Secondly, the veri®cation procedure will be proposed. Finally, the design for dynamic access control will be presented. 3.1. The design for the access control mechanism

3.1.1. Basic mechanism

The following will explain how a one-way function can be designed in which the relationship of the Morton sequence and associated access rights within access matrix are embedded.

Step 1: Creating compound access rights. Consider Amn as an access control

matrix, where m is the number of subjects, n is the number of objects, and the access right aij is the …i; j†th element of Amn for the subject Sito the object Oj,

which is composed of atomic access rights, such as read, write, execute, append etc. The system manager chooses distinct prime numbers p for each atomic access right. Thus, the scope of a subject's authorized operations, aij, can be

considered as a compound access right, which is derived through multiplication of a series of prime numbers

aijˆ

Y

p: …1†

Step 2: Morton sequence transformation. In this step, we assign each aij as a

distinguishable Morton number z. According to the Morton sequence, the sequence can be derived as described in Section 2. In other words, the sequence az has an access value aij, which represents the access right for a subject Si to

objects Oj.

Step 3: One-way function transformation. The security manager chooses four prime numbers, q1; q2; q3; and q4, for key identi®cation. Thus, a one-way

function can be used to transform az and qt to derive encrypted compound

privilege (ECP). It can be represented in the following form: ECPsˆ

Yz‡3 zˆ4…sÿ1†

qaz

t ; where t ˆ 1; 2; 3; 4;

s ˆ 1; 2; 3; . . . ; n and az represents the access value of aij

at Morton sequence z: …2†

Using Eq. (2), we can see that ECP applies four keys to encrypt subject± object speci®c access control information. The purpose of ECP is to prevent

from improper authorization Ti access system resources. Once the system administrator computes all the ECP values from the access matrix, the ad-ministrator can store and maintain the ECP values locally on a server work-station.

For example, consider an access control matrix A‰S; OŠ with access rights aij

from eight subjects and objects A88as shown in Fig. 3. The contents of Fig. 3

come from the mapping of Fig. 1. First, the system manager chooses distinct prime numbers for each atomic access right, such as read ˆ 2, write ˆ 3, execute ˆ 5, owner ˆ 7, and append ˆ 11. The compound access rights in the access matrix can be derived as a00ˆ 6…2  3†, a01ˆ 2; a02ˆ 7; . . . ;

a26ˆ 10…2  5†. Second, given each aijin the access matrix as a distinct Morton

number z, such as the Morton sequence a0has value of access right a00; a1has

value a01; a2 has a value a10, etc. Finally, the security manager chooses four

prime numbers, such as q1ˆ 2; q2ˆ 3; q3ˆ 5 and q4 ˆ 7, as keys of

identi-®cation. Using Eq. (2), the corresponding ECP values can be computed as follows:

Once the system manager computes the ECP values, the matrix can be dis-carded. Only the ECP values are stored and maintained.

ECP1ˆ 26 32 56 70ˆ 9 000 000 ECP5ˆ 26 30 50 75ˆ 1 075 648

ECP2ˆ 27 30 50 72ˆ 6272 ECP6ˆ 23 31 53 73ˆ 1 029 000

ECP3ˆ 23 30 52 72ˆ 9800 ECP7ˆ 20 35 52 76ˆ 714 717 675

ECP4ˆ 211 30 50 70ˆ 2048 ECP8ˆ 210 36 53 72ˆ 4 572 288 000

Once the content of the access matrix is determined and computed as ECP values by the security manager, the procedures for verifying access requests is developed. In addition, dynamic access control methods are also provided. 3.2. Veri®cation of access requests

Suppose that the subject Si wants to access the object Oj using the access

request r. He/she issues the request triple …Si; Oj; r†. Two procedures need to be

performed:

1. Find the Morton sequence z, in which the value aij corresponding to Siand

Ojin the access matrix can be found.

2. Perform Algorithm A (Authorization±Validation procedure) below using a one-way function f …qt; ECPs† to compute the compound access rights aij,

where theqt and ECP values can be derived from the Morton number az.

If the derived result r is a factor of the ECP, then the access request is ac-cepted. In other words, if the request is granted, then the request r should be a factor for the ECP values.

Algorithm A (Authorization±Validation procedure …qt; ECPs†).

Input (Si, Oj; r), where

(1). Find az from procedure (1)

(2). Derive ECPs; qt where t ˆ ‰z mod 4Š ‡ 1; s ˆ ‰z=4Š ‡ 1

Set: var T, Q, R, X: integer; X ˆ 1 Step 1: T ECPs;

Step 2: Q T =qt; R T mod qt;

// Set Q be the quotient, and set R to be the remainder.// Step 3: if (Q > 1) and (R ˆ 0) then {X ˆ X ‡ 1; T Q; goto Step 2;} Step 4: output X; //compound access right aij.//

Step 5: if rjX exist, then accept the request r, otherwise reject it.

After the iteration of Step 3 and Step 4, the compound access right of aij is

kept in X. For example, when the subject S0requests a read operation (r ˆ 2) on

the object O0, described as …Si; Oj; r† ˆ …S0; O0; 2†, we derive a Morton sequence

for a00 is 0(z ˆ 0). Then, via Algorithm A, the system administrator gets

qtˆ q1ˆ 2, and ECP value ˆ ECP1 ˆ 9 000 000. Finally, the output X is

computed as f …qt; ECPs† ˆ 6, in which the request r ˆ `2' is a factor of `6';

then, the access request is accepted. 3.3. Dynamic access control

In order for the administrator to be able to maintain the access matrix, dynamic access control, including modi®cation of access rights, insertion of subjects/objects, and deletion of subjects/objects, is required. The following sections describe this control details.

3.3.1. Modi®cation of access rights

Consider the situation in which an access right is changed from aij to a0ij.

Here, only the corresponding ECP values for a0

ijshould be updated; other ECP

values remain to the same ECP0

sˆ ECPs q…a 0 ijÿaij†

t ; where t ˆ …zmod 4† ‡ 1 and s ˆ …z=4† ‡ 1:

For example, suppose the compound access right a00ˆ 6 (read and write) is

changed to 10 (read and execute). First, we derive a Morton sequence, where a00

is 0; then, the system administrator gets the identi®cation key qtˆ q1ˆ 2 and its

corresponding ECP values ˆ ECP1 ˆ 9 000 000, where t ˆ …0 mod 4† ‡ 1 ˆ

1; s ˆ …0=4† ‡ 1 ˆ 1. The new ECP value for the modi®ed access right is computed as follows:

ECP0

1ˆ 9 000 000  2…10ÿ6†ˆ 9 000 000  24ˆ 144 000 000:

3.3.2. Insertion of subjects/objects

There are two possibilities to consider: the ®rst one is to add a subject to the access matrix; the second one is to add an object. To insert a new subject Sm‡1

into the access matrix, where the corresponding access rights are a…m‡1†;j for

j ˆ 1; 2; . . . ; n, the proposed scheme needs to calculate the new ECP values only, without modifying all of the existing ECP values. Due to the character-istic of the Morton sequence within the access matrix, only qtand qt‡1are used

to insert a new subject.

The algorithm for inserting the new subject is shown below. When a new subject is added, the administrator gets qt and qt‡1 ®rst and then sets them as

inputs for Algorithm B.

Algorithm B (//Inserting subjects //). Input (a…m‡1†;j; qt; qt‡1)

Output (ECP0 s)

Begin

for j ˆ 1 to n //* n is the number of objects in the access matrix*// Find the Morton number z for each input a…m‡1†;jand a…m‡1†;j‡1

Derive t ˆ ‰z mod 4Š ‡ 1 s ˆ ‰z=4Š ‡ 1 ECP0

sˆ qatz qat‡1z‡1

j ˆ j ‡ 2 /* each time, two access rights are computed at the same time */ end.

For example, suppose a new subject S8is added to the system for which the

corresponding access rights are a80ˆ 5; a81ˆ 2; a82ˆ 3; a83ˆ 3; a84ˆ 5;

a85ˆ 7; a86ˆ 7; and a87ˆ 7 by performing Algorithm B, such that the new

ECP0 33ˆ 25 32ˆ 288; ECP0 34ˆ 23 33ˆ 216; ECP0 37ˆ 25 37ˆ 69 984; ECP0 38ˆ 27 37ˆ 279 936:

To insert a new object On‡1 into the access matrix, and the access rights are

ai;…n‡1† for i ˆ 1; 2; . . . ; m, and the proposed method needs to calculate the new

ECP values only. The computed Morton sequence z for each added aij is

de-rived as az ®rst, and only qt and qt‡2 are used to insert a new object. The

al-gorithm for inserting a new object is shown below. When a new object is added, the administrator gets qt and qt‡2 ®rst, and then sets them as inputs for

Algorithm C.

Algorithm C (//Inserting objects//). Input (ai;…n‡1†; qt; qt‡2)

Output ( ECP0 s)

Begin

for i ˆ 1 to m //* m is the number of subjects in the access matrix * // Find the Morton number z for each input ai;…n‡1† and ai‡1;…n‡1†

Derive t ˆ ‰z mod 4Š ‡ 1 s‰z=4Š ‡ 1 ECP0

sˆ qatz qat‡2z‡2

i ˆ i ‡ 2 /* each time, two access rights are computed at the same time */ end.

For example, suppose a new object O8 is added to the system for which the

corresponding access rights are a08ˆ 5; a18ˆ 2; a28ˆ 3; a38ˆ 3; a48ˆ 5;

a58ˆ 7; a68ˆ 7, and a78ˆ 7 by performing Algorithm C, such that the new

ECP values are computed as follows: ECP0 17ˆ 25 52ˆ 500; ECP0 19ˆ 23 53ˆ 1000; ECP0 25ˆ 25 57ˆ 2 500 000; ECP0 27ˆ 27 57ˆ 10 000 000:

Fig. 4 gives an example which helps to explain how corresponding ECP values can be calculated when new subjects/objects are inserted into access control matrix.

3.3.3. Deleting subjects/objects

To delete a subject Sr from the access matrix of a system and remove the

method requires that ECP values corresponding to each ar;j be recomputed.

The algorithm for deleting a subject is shown below. When a subject is deleted, the administrator gets qt and qt‡1 ®rst, and then sets them as inputs for

Algorithm D.

Algorithm D (//Deleting Subjects.//). Input (Sr; Oj; qt; qt‡1)

Output (ECP0 s)

for j ˆ 1 to n // n is the number of objects in the access matrix // Find the Morton number z for each input arj with Sr; Oj

Derive t ˆ ‰z mod 4Š ‡ 1 s ˆ ‰z=4Š ‡ 1 Step 1: T ECPs; W qt

Step 2: Q T =W ; R T mod W

Step 3: if (R ˆ 0) then fT Q; Goto Step 2; g else {T Q; W qt‡1; Goto Step 2;}

until {R 6ˆ 0;}//output Q as ECP0 s.//

j ˆ j ‡ 2 /* each time, two access rights are computed at the same time */ end.

For example, assume that the subject S2 is deleted from the system. In Fig. 5,

we see that the ECP values for S2corresponding to Ojare ECP3; ECP4; ECP7, Fig. 4. The results of inserting a subject/object into the access control matrix.

and ECP8. These ECP values will be modi®ed. Using Algorithm D, the

modi®ed values of the ECP values are recomputed as follows: ECP0 3ˆ ……9800=23†=30† ˆ 1225; ECP0 4ˆ ……2048=211†=30† ˆ 1; ECP0 7ˆ ……714 717 675=20†=35† ˆ 2941225; ECP0 8ˆ ……4 572 288 000=210†=36ˆ 6125:

Now, an object Or is to be deleted from the access matrix, so the

corre-sponding access rights, such as ai;rfor i ˆ 1; 2; . . . ; n; have to be deleted. Again,

the proposed method requires that the ECP values for the corresponding access right of ai;r be recomputed. The algorithm for deleting an object is shown

below. When an object is deleted, the administrator gets qtand qt‡2 ®rst, and

then sets them as inputs for the Algorithm E. Algorithm E (// Deleting Objects.//).

Input (Si; Or; qt; qt‡2)

Output (ECP0 s)

for i ˆ 1 to m==m is the number of subjects in the access matrix // Find the Morton number z for each input air with Si; Or

Derive t ˆ ‰z mod 4Š ‡ 1 s ˆ ‰z=4Š ‡ 1 Step 1: T ECPs; W qt

Step 2: Q T =W ; R T mod W

Step 3: if (R ˆ 0) then fT Q; Goto Step2; g else {T Q; W qt‡2; goto Step 2;}

until {R 6ˆ 0;}//output Q as ECP0 s.//

i ˆ i ‡ 2 /* each time, two access rights are computed at the same time */ end.

For example, assume that the object O3is deleted from the system. In Fig. 5,

we see that the ECP values corresponding to O3 and the subjects Si are

ECP2; ECP4; ECP10, and ECP12. These ECP values have to be modi®ed.

Using Algorithm E, the modi®ed values of ECP are recomputed as follows: ECP0 2ˆ ……6272=30†=72† ˆ 128; ECP0 4ˆ ……2048=30†=70† ˆ 2048; ECP0 10ˆ ……44100=32†=72† ˆ 100; ECP0 12ˆ ……44100=32†=72† ˆ 100:

4. Performance analysis and comparison

There is always a serious problem of storing the access matrix for any access control mechanism. The problem appears either as a sparse matrix or occu-pation of a large amount of storage space. Compression of the access matrix is desirable if it is possible. The proposed scheme has the advantage of being able to compress the matrix. The reduction of the amount of required is due to the compression of ECP values. In this section, storage compaction will be ®rst described. The comparative advantages from the six criteria for the proposed method will also be illustrated.

4.1. Storage space compression

Situation may occur in which the identical level of access rights for a group of subjects may be compressed; e.g., all the students in a group have a common access authorization and are assigned to a speci®c directory. Fig. 6 shows this situation for subjects from S4to S7who have a common access right `2' to O2

and O3; as a result, we can compress these data from ECP9 to ECP12 into

ECP9…C†, in which (C) represents a compressed right. The value of the new ECP

is identical to that of the previously computed 44 100. Meanwhile, the vali-dation of access requests from S4to S7follows the same procedure for verifying

authorization described above. In this case, the security manager does not require to manage each access right separately; the new ECP…C† will be used to

4.2. Performance comparisons

In this section, the performance of the proposed method will be analyzed and compared with that of other SKL schemes based on the set of six criteria [7]. Previous methods will be brie¯y discussed. They will be summarized in six tables below.

4.2.1. E€ort involved in initializing keys and locks

Table 1 deals with initialization of m keys and n locks. Solving sets of linear equations is time-consuming in the schemes developed by Wu and Hwang [2] and Chang and Jiang [2,6]. Two methods developed by Chang [3,4] have to solve the same over¯ow problem previously mentioned and may require ap-plication of the decomposition technique. Although the scheme in [8] based on binary coding and prime factorization can avoid the over¯ow problem, the over¯ow problem while the Lock value was calculated will occur in the worst case when the binary value x ˆ 1. However, our method requires only four

keys to generate ECP values. The over¯ow problem can be avoided no matter whether or not the subjects and objects propagate.

4.2.2. E€ort involved in computing an access right from a lock and key

The numbers of operations needed to ®nd access rights for various schemes are listed and compared in Table 2. Hwang et al.'s method [7], Chang's method [8] and our method need only a constant number of operations to compute access rights while others require a number of operations which is proportional to m (i.e., the number of users).

4.2.3. E€ort involved in revising keys and locks when an access right is modi®ed For modi®cation of access rights, Table 3 shows that Hwang and Shao's [7] and Chang's [8] methods can modify the original key or lock value to obtain a new one. Our method only needs to modify the original ECP value; hence, the recomputation e€ort is smaller than that required by the other methods. Modi®cation in two methods by Chang's [3,4] are, however, more complex due to computations of xj; Mjand L.

Table 1

Initialization of the keys and locks

SKL schemes E€ort involved in initializing keys and locks

Wu and Hwang [2] Given m keys, solve n sets of m linear equations for n lock vectors

Chang, 1986 [3] Given n locks, compute KiˆPnjˆ1…L=Lj†  xj aijmod L for m keys Chang, 1987 [4] Given n locks, compute KiˆPnjˆ1daij Lj=ne  n  Mjfor m keys Laih et al. [5] Given m keys, compute Lj…x† ˆPmiˆ1GziQiÿ1sˆ1…x ÿ Ks† for n lock vectors Chang and Jiang [6] Given m keys, solve n sets of bm 0±1 linear equations for n lock matrices Hwang et al. [7] Given m keys, compute LjˆQmiˆ1Kiaijfor n locks in the X-based form

Chang, 1997 [8] Given m keys, compute Lj…x†ˆQmiˆ1K

a…x† ij

i for x ˆ 1; 2; . . . ; b and j ˆ 1; 2; . . . ; n

Our method Given 4 keys, compute ECP values ˆQz‡3

zˆ4…sÿ1†qatz, where t ˆ 1; 2; 3; 4

Table 2

Computation of the access rights

SKL schemes Operations needed to compute the access right aij

Wu and Hwang [2] m multiplications, …m ÿ 1† additions and one division

Chang, 1986 [3] One division

Chang, 1987 [4] Two divisions and one subtraction

Laih et al. [5] …i ÿ 1† multiplications, …i ÿ 1† additions and one division

Chang and Jiang [6] bm ANDs and b…m ÿ 1† XORs

Hwang et al. [7] 6 amax(X-based) divisions (amax: maximal value of access right)

Chang's, 1996 [8] b divisions

4.2.4. E€ort involved in appending and updating keys and locks when a new user or ®le is added

The appendability and removability properties listed in Tables 4 and 5 might be critical issues for dynamic access control in practical applications. For ap-pendability, Laih et al.'s scheme [5] is best because it satis®es both user and ®le appendability. In our method, when a new subject is added, it recomputes only the ECP values of corresponding accessible objects instead of all the ECP values; when a new object is added, it recomputes only the ECP values of the corresponding accessible subjects, so the proposed method is still easy to im-plement.

Table 4 Appendability

SKL schemes User appendability File appendability

Wu and Hwang [2] Recompute all lock vectors Yes

Chang, 1986 [3] Yes Recompute all keys

Chang, 1987 [4] Yes Recompute all keys

Laih et al. [5] Yes (add a coecient to each lock

vector) Yes

Chang and Jiang [6] Recompute all lock matrices Yes

Hwang et al. [7] Recompute the locks of accessible

®les only Yes

Chang, 1997 [8] To add user Um‡1, recompute the

elements of lock vector for a…x†

…m‡1†jˆ 1 only

Yes

Our method To add subject S…m‡1†recompute the

elements of ECP values for s ˆ 1 to n=2

To add object O…n‡1† recompute the elements of ECP values for s ˆ 1 to m=2 Table 3

Modi®cation for access rights

SKL schemes E€orts involved in changing the access right aijto a0ij

Wu and Hwang [2] Solve a new set of m linear equations for new locks L0

j Chang, 1986 [3] Recompute Kiˆ Ki‡ …L=Lj†  xj …aijÿ f …Ki; Lj†† mod L Chang, 1987 [4] Recompute KIˆ Ki‡ … a ij Lj=nÿ f …K i; Lj†  Lj=n†  n  Mj

Laih et al. [5] Recompute the …m ÿ i ‡ 1† coecients Gj

i for Lj

Chang and Jiang [6] Solve a new set of bm 0±1 linear equations for Lj

Hwang et al. [7] Recompute Ljˆ Lj …Ki†…a

0 ijÿaij† Chang, 1997 [8] Recompute Lj…x†ˆ Lj…x† …Ki†a 0…x† ij ÿa…x†ij … _{† for x ˆ 1;2;...;b}

Our method Recompute ECP0

sˆ ECPs q…a

0 ijÿaij†

t for

4.2.5. E€ort involved in removing and updating keys and locks when a user or ®le is deleted

For removability, as a subject Siis deleted, Wu and Hwang's [2], Chang's [3]

and Jiang's [5] methods need to recompute all the locks while our method only needs to recompute the ECP values of the corresponding accessible objects instead of all the ECP values when a new object is deleted. In short, the re-computation e€ort required by our method is relatively small when a subject or object is added to or removed from the system.

4.2.6. Space for storing keys and locks

The required storage space for keys and locks is compared in Table 6. Note that O(m ‡ n) in Chang's [3] method is obtained by ignoring the over¯ow issue.

Table 5 Removability

SKL schemes Subject removability Object

removabili-ty

Wu and Hwang [2] Recompute all lock vectors Yes

Chang's, 1986 [3] Yes Recompute all

keys

Chang's, 1987 [4] Yes Recompute all

keys

Laih et al. [5] To delete Ui, recompute (m ÿ i) coecients

of all for deleting Yes

Chang and Jiang [6] Recompute all lock matrices Yes

Hwang et al. [7] Recompute the locks of accessible ®les only Yes

Chang, 1997 [8] To delete user Um‡1, recompute the

elements of lock vector for a…x†

…m‡1†jˆ 1 only Yes

Our method To delete subject S…m‡1†, recompute the

elements of the ECP values for s ˆ 1 to n=2 To delete objectO…n‡1†, recompute the elements of the ECP values for s ˆ 1 to m=2

Table 6

Storage requirement

SKL schemes The complex of the required

Wu and Hwang [2] O…m2_{‡ mn†}

Chang, 1986 [3] O…m ‡ n†

Chang, 1987 [4] O…m ‡ n†

Laih et al. [5] O…mn†

Chang and Jiang [6] O…m2_{‡ bmn†}

Hwang et al. [7] O…mn†

Chang, 1997 [8] O…mn†

The storage requirement required by Hwang's [7] and Chang's [8] methods is not less than O(mn). In our method, the keys are bounded by four large prime numbers. Let l be the longest among all the elements of the ECP values, and let qmaxbe the maximal key value. ECP values ˆ Qz‡3zˆ4…sÿ1†qatz6 2w, where w is the

bit-length of an integer allowed in a computer system. Since the amount of ECP is bounded by access matrix Amn such as (m/4) or (n/4) a matrix of

max(m,n). Storage for ECP values is hence max …O…m†; O…n††. 5. Conclusions

We have proposed a novel scheme for controlling access requests in a secure information system. Based on prime factorization and the Morton sequence, we have presented an improvement of Hwang et al.'s [7] and Chang et al.'s [8] SKL methods. Di€erent from the conventional SKL scheme, the over¯ow problem while computed ECP does not occur in our scheme. Based on six criteria, the proposed scheme is considerably better for access control than most of the other comparable schemes. The convenient way in which it mod-i®es ECP values while adding or removing objects/subjects is also impressive. Furthermore, with our compression method, the proposed scheme is suitable for implementing a large access control matrix in a distributed computer sys-tem.

Future work may include an attempt to extend the proposed idea to inte-grate both authentication and authorization in a security system. We could choose a distinct prime number to represent each entity-identi®cation and access right, then computed as ECP. An ECP would be assigned to each en-tity's identi®cation and stored in a place where the user who must access to information resources. The proposed method implies entity identi®ers and authorization operations. On receiving the subject requested, the systems manager would verify the subject's identity and its authorization operations by referring to the corresponding ECP value. This would greatly improve integrity and con®dentiality in access control systems.

References

[1] D.E.R. Denning, Cryptography and Data Security, Addison-Wesley, Reading, MA, 1982. [2] M.L. Wu, T.Y. Hwang, Access control with single-key-lock, IEEE Trans. Software Eng. 10 (2)

(1994) 185±191.

[3] C.C. Chang, On the design of a key-lock-pair mechanism in information protection systems, BIT 26 (4) (1986) 410±417.

[4] C.C. Chang, An information protection scheme based upon number theory, The Comput. J. 30 (3) (1987) 249±253.

[5] C.S. Laih, L. Harn, J.Y. Lee, On the design of a single-key-lock mechanism based on Newton's interpolating polynomial, IEEE Trans. Software Eng. 15 (9) (1989) 1135±1137.

[6] C.K. Chang, T.M. Jiang, A binary single-key-lock system for access control, IEEE Trans. Comput. 38 (10) (1989) 1462±1466.

[7] J.J. Hwang, B.M. Shao, P.C. Wang, A new access control method using prime factorization, The Comput. J. 35 (1) (1992) 16±22.

[8] C.C. Chang, D.C. Lou, A binary access control method using prime factorization, Informatics Comput. Sci. 96 (1997) 15±26.

[9] G.M. Morton, A computer oriented geodetic database, and a new technique in ®le sequencing, IBM Canada Ltd, March 1 (1966).