DBDH problem under the order group of composite n- 123docz.net

suppose ,p r are two different primes, ,G GT are two multiplication cycle groups with the order of the composite number N= ⋅p r, and e is a bilinear mapping

× →

e G G: GT, which has the following properties:

1 bilinear: ∀g h G a b Z, ∈ , , ∈ N*, then

= ( , ) ( , ) e g ha b e g hab

2 non-degeneracy: ∃g h G, and makes the order , ∈ of (g, h)e is N in the group GT;

3 Computability: mapping can be computed effi- ciently in the polynomial time.

use G Gp, r to represent subgroups in the group G, and the order is ,p r respectively. suppose

= ×

G Gp Gr , and there is such property: if hp∈Gp and hr∈Gr, then e h h( , ) 1p r =

The proof is as follows: suppose ,g g gp, r are the generators of groups ,G G Gp, r respectively, then grcan generate the group Gp, gp can generate the group Gr, then, for certain ∂ ∂ ∈1, 2 ZN* , there are hp=(gr)∂1 and hr =(gp)∂2, then

= ∂ ∂ = ∂ ∂ =

( , ) (( ) ,(1 ) )2 ( 1, 2) 1 e h hp r e gr gp e g g pr

Definition 3.2 (DBDH problem under the order group of composite number)

suppose bilinear mapping e G G: × →GT, ,G GT are the two multiplication cycle groups with the order of composite number N=pr, and ,p r are two different primes, G Gp, r are subgroups of the group of G . The orders are respectively ,p r, marked as

= ×

G Gp Gr. select a generator gp∈G gp, r∈Gr, and the random number a b c Z, , ∈ *N. Judge a six tuple (g g g g g Zp, r, pa, bp, pc, ) as the input, and see whether

= ( , )

Z e g gp p abc is established.

To define an algorithm ℜ, the advantage σcan be used to solve the DbDH problem. If

ℜ = ℜ = −

ℜ = ≥

( ) | Pr[ ( , , , , , ( , ) ) 0]

Pr[ ( , , , , , ( , ) ) 0]|

Adv g g g g g e g g g g g g g e g g

p r pa

p b

p c

p p abc

p r pa

p b

p c

p p Z

Wherein z Z∈ *N . If you can’t find an algorithm of polynomial time to solve the DbDH problem through at least the advantage σ, then the DbDH problem is thorny.

3.2 Basic framework design of the scheme

The entities involved in this scheme can be divided into four parties, respectively the Trusted Authority (TA), Data Provider (DP), Data Demander (DD) and storage Center (sC).

As a trusted third party, TA plays a key role in the scheme. It is responsible for the initialization of the system, the management of system properties and user property, and the issue of corresponding private key for users according to the user’s attribute set.

As the owner of data, DP encrypts the data and then transfers them to the sC. DD is the user who applies to access the data; TA issues the user’s private key; DD decrypts and then obtains the data through using the private key after acquiring the data cipher text from the sC. DD and DP are collectively referred to as the user, and a DP often can also become a DD, and vice versa.

sC is responsible for the storage of mass data information. Due to the weak safety protection of sC, in many cases, it cannot be trusted. Therefore, in this scheme, what sC stores is the encrypted data cipher text. Even if it is attacked, the attacker cannot obtain the data plaintext.

The basic framework of the RPP-AbE is shown in the Figure 1.

The basic process of the RPP-AbE is as follows:

1 setup: It is executed by TA which generates the group and bilinear equivalent algebraic structure that the scheme needs according to the security

parameter. meanwhile, TA generates the public key (PK) and master key (mK) of system, wherein PK is open to all users and mK, as shall be kept secretly by TA.

2 Key Generation: It is executed by TA which uses PK and mK to generate private key for all users according to the attribute uploaded by the user.

3 Encryption: It is locally executed by DP which uses public key PK and the access structure of files to realize the encryption of data. Then DP sends the cipher text of files to sC.

4 Decryption: It is executed by DD which obtains the required data cipher text from the sC. DD uses the decryption cipher text of user private key obtained from TA to get the data plaintext.

The basic process of this scheme is shown as Figure 2.

3.2.1 Setup

setup algorithm is executed by TA, which selects the group that the scheme needs, random number and bilinear equivalent algebraic structure, to generate the public key (PK) and master key (mK) of system. PK is open to all users, while mK, as secret parameters, shall be kept solely of by TA.

The specific algorithm for Setup()→{PK MK is , } as follows,

through using the private key after acquiring the data cipher text from the SC. DD and DP are collectively referred to as the user, and a DP often can also become a DD, and vice versa.

SC is responsible for the storage of mass data information. Due to the weak safety protection of SC, in many cases, it cannot be trusted. Therefore, in this scheme, what SC stores is the encrypted data cipher text. Even if it is attacked, the attacker cannot obtain the data plaintext.

The basic framework of the RPP-ABE is shown in the Figure 1.

Figure 1. The basic framework of the RPP-ABE The basic process of the RPP-ABE is as follows:

1) Setup: It is executed by TA which generates the group and bilinear equivalent algebraic structure that the scheme needs according to the security parameter. Meanwhile, TA generates the public key (PK) and master key (MK) of system, wherein PK is open to all users and MK, as shall be kept secretly by TA.

2) Key Generation: It is executed by TA which uses PK and MK to generate private key for all Figure 1. The basic framework of the RPP-AbE.

Figure 2. Scheme Flow 2.2.1 Setup

Setup algorithm is executed by TA, which selects the group that the scheme needs, random number and bilinear equivalent algebraic structure, to generate the public key (PK) and master key (MK) of system. PK is open to all users, while MK, as secret parameters, shall be kept solely of by TA.

Figure 3. Setup

The specific algorithm forSetup(){PK MK, }

is as follows,

1) Construct a group G with an order of composite number N, wherein

Np r ,GGpGr

andG Gp, r

arerespectively the multiplication cyclic groups of prime order pandr. The generators of subgroups Gp

and Gr are respectively marked as gp

and gr. 2) Randomly selects t, Z*N , and calculate     s t, Z*N,and the result will be marked

1 p

g g . Randomly select

2 p, 3 r

g G g G hZN, mark them asg gph, and

Figure 2. scheme f low.

Figure 2. Scheme Flow 2.2.1 Setup

Figure 3. Setup

The specific algorithm forSetup(){PK MK, } is as follows,

1) Construct a group G with an order of composite number N, wherein Np r ,GGpGr

andG Gp, r

arerespectively the multiplication cyclic groups of prime

Figure 3. setup.

1 Construct a group G with an order of composite number n, wherein N= ⋅p r,G G= p×Gr and G Gp, rare respectively the multiplication cyclic groups of prime order p and r. The generators of subgroups Gpand Gr are respectively marked as gp

and gr.

2 Randomly selects t Z, ∈ N*, and calculate

∂= ⋅ ∂∈s t, Z*N, and the result will be marked as

= ∂

g1 gp. Randomly select g2∈G gp, 3∈Gr h Z∈ N, mark them as g′ =gph, and calculate X e g g . = ( , )1 ′ uniformly select ai∈Z*N(1≤ ≤i n) for each attribute in the system, where n is the number of attributes in the system, and then calculate = 2+

Ai ga ai i2 , wherein 1 i n.≤ ≤

3 Generate system public key

=< , , , , 3, ,{ }1≤ ≤ >

PK g g s t g X Ap r i i n , and master key MK=<g h a2, ,{ }i 1≤ ≤i n>.

3.2.2 Private key generation

Private key generation is executed by TA. After a user sends his attribute list L to TA, TA uses PKand MK to generate private key SK, and sends it to the user.

The following is the specific algorithm

→

( , , )

KeyGen PK MK L SK for private key generation, 1 set users’ attributes as L={ ,v1vn}, and then TA

calculates à =∑ +

∈

(ai ai2)

v Li

for a user’s attributeL.

2 TA randomly selects t Z∈ N*, and calculates D0=gpt and D1=g gph⋅ 2à. The user’s corresponding private key is SK=<D D .0, 1>

3 TA shall send the private key SK to the corresponding user through a secure channel.

3.2.3 Encryption

Encryption of file is executed by DP, and DP uses public key PK and the access structure of file W to realize the encryption of the plaintext of file. Then DP sends the cipher text of file that it gets to sC. The specific algorithm for key encryption algorithm is as follows:

1 The access structure of file is W={ ,w1wn}, in which wi∈Si.

2 The plaintext that needs to be encrypted is M.

Combined with the access structure of plaintext M , calculate C0=g∂p and C′ =M e g g , obtain ⋅ ( , )1 ′ the corresponding part from the public key, and calculate = ∏ ⋅

∈

( )

1 3

C Ai g

v W s i

3 After each plaintext is encrypted, the cipher text obtained is CT=< ′C C C , and then DP will , 0, 1>

send the cipher text CT to sC.

3.2.4 Decryption

When the data demander (DD) needs to access a file, firstly he should obtain cipher text CT of the file from sC, use the private key obtained from TA to decrypt the cipher text, and in the course of decryption, the bilinear operation is used. Whether the user can use his private key to decrypt the cipher text of file successfully, lies in whether the user’s attribute can meet the access structure of file.

The decryption algorithm Decrypt CT SK( , )→M is as follows:

= ′ ⋅ ( , ) ( , )

0 1

1 0

M C e D C

e D C

Any user, who has the decryption ability, namely user’s attribute meets the requirements to visit the access structure of file, can use the decryption algorithm, and use his private key to decrypt the cipher text of file successfully.

4 AnonymITy AnALysIs oF sCHEmE In section 1, it is elaborated that waters’ scheme fails to achieve anonymity, which means that the attacker can obtain valid users’ identity information by DDH test.

In the scheme designed in this paper, due to the fact that an additional multiplication with the ele- ments in group Gr is added in the cipher-text com- ponent ( , )C C , the cipher text is 0 1 = ∂ = ∏ ⋅

∈

, ( )

0 1 3

C g Cp Ai g

v W s

, instead of = ∂ = ∏

∈

, ( )

0 1

C g Cp Ai

v W s i

. If the latter is the cipher text, through selecting a certain access structure W ′ and judging whether ∏ =

∈ ′

( ,0 ) ( ,1 ) e C Ai e C g

v W p

is true, the attacker can verify whether W and W are consistent, ′ and through continuous tests, the attacker can even- tually obtain the identity information of user’s who can visit the file validly; but if the cipher text is

∏

= ∂ = ⋅

∈

, ( )

0 1 3

C g Cp Ai g

v W s i

, then DDH test attacks are inva- lid, because even the right access structure W is used for test, the DDH test e C( ,0 ∏∈ Ai)=e C g( ,1 )

v W p

is false.

Therefore, this scheme can resist the attackers’

DDH test attacks, so as to hide the access structure of file effectively, and ensure that the valid users’ identity privacy will not be obtained by attackers.

5 ExPERImEnT sImuLATIon

Through experiment simulation, the RPP-AbE scheme and the existing AbE scheme are compared in computation overhead and memory overhead

respectively, to verify the performance of the RPP- AbE scheme.

5.1 Experimental environment

The experimental devices are Intel (R) Core (Tm) 2 Dou processor, 2.80GHz, 4.00Gb memory, and 2Tb HP 7200r/s sATA hard disk, with the kernel version of 2.6.31ubuntu10.04 operating system. The experiment is based on the open source code repositories PbC, and uses the toolkit of CPAbE-0.10 in PbC to write the program.

5.2 Experimental results and analysis

In the existing and typical CP-AbE scheme, the memory overheads in literature [4] and [5] are ideal, as its user’s private key and master key are short in length;

the computing overheads in literature [2] and [5] are ideal, as the encryption time and decryption time are short. so compare the RPP-AbE scheme with literature [4] and [5] respectively in the lengths of private key and master key to measure the scheme’s memory overhead; compare the RPP-AbE scheme with literature [2] and [5] respectively in encryption time and decryption time to measure the scheme’s computation overhead. In the experiment, the number of values for each attribute is uniformly set as 4.

1 The comparative analysis on memory overhead Compare the RPP-AbE scheme with schemes designed in literature [4] and [5] separately in the length of private key, and with the increase in the number of system attributes, the comparison in the length of private key for three schemes is shown in Figure 4.

The length of private key in RPP-AbE scheme basically remains unchanged with the increase of the number of system attributes, so it is constant, which is inde- pendent from the number of system attributes; both the lengths of private key in schemes of literature [4] and [5] grow linearly with the increase of system attributes.

Therefore, the scheme in this paper is superior to those in literature [4] and [5] in the length of private key.

Compare the length of master key in RPP-AbE scheme and CP-AbE scheme designed in literature [2]

and [5] respectively, and according to the different num- bers of system attributes, the comparison in the length of master key for the three schemes is shown in Figure 5.

As seen from Figure 5, the length of master key in RPP-AbE scheme basically remains unchanged with the increase of the number of system attributes, so it is constant; both the lengths of master key in schemes of literature [4] and [5] grow linearly with the increase of the number of system attributes. Therefore, the scheme in this paper is superior to those of literature [4] and [5] in the length of master key.

Therefore both the lengths of private key and master key in RPP-AbE scheme are superior to those in schemes of literature [4] and [5].

2 The Comparative Analysis on Computation overhead

Respectively use the encryption algorithm in RPP-AbE scheme and the encryption algorithm in scheme of literature [2] and [5] to encrypt the data file of 128Kb, and the relation between the required encryption time and the number of system attributes is shown in Figure 6.

Figure 6. The comparison in encryption time for different schemes.

Figure 4. The comparison in the length of private key for three schemes.

Figure 5. The comparison in the length of master key for three schemes.

As seen from Figure 6, the encryption time in RPP-AbE scheme and the scheme of literature [2]

has no relation with the number of system attributes, so it is constant; the encryption time in the scheme of literature [5] grows linearly with the increase of the number of system attributes and its encryption time is obviously longer than the above mentioned two.

Although the encryption time in RPP-AbE scheme is a little longer than that in the scheme of literature [2], due to the fact that both of them are in the constant-level, the difference between them in cloud computing environment can be ignored.

Respectively use the decryption algorithm in RPP-AbE scheme and the encryption algorithm in scheme of literature [2] and [5] to decrypt the cipher text of 64Kb, and the relation between the required decryption time and the number of system attributes is shown in Figure 7.

As seen from Figure 7, the required time for decryption of ciphertext in RPP-AbE scheme and the scheme in literature [2] has nothing to do with the number of system attributes, so it is a constant; while the required time for decryption of ciphertext in literature [5] grows linearly with the increase of the number of system attributes; when the number of system attributes is within 6, the decryption time of literature [5] is ideal, but when the number of system attributes is more, the decryption time of literature [5] is much longer than the time in the RPP-AbE scheme and the scheme of literature [2].

Through the experiment, in which the scheme in this paper and the rest CP-AbE schemes are compared in the performance, it is can be seen that on memory overhead, the scheme in this paper is superior to the other schemes; on computation overhead, the schemes in this paper and literature [2] are

superior to the scheme in literature [5]. Although the computation overhead of the scheme in this paper is slightly larger than that of the literature [2], both of them are in constant-level, so the difference is little.

6 ConCLusIons

security issues of cloud storage influence the devel- opment of cloud storage applications, and the reason- able and effective encryption scheme not only can guarantee the security of data stored in the cloud, but also should ensure the anonymity of the data owners and users, thereby improving the cloud storage service users’ trust in the cloud storage service. With the introduction of encryption scheme based on CP - AbE algorithm, this paper ensures the confidentiality of user data, at the same time makes some attempts to realize the protection of user’s private data, and the experimental results show that, the mechanism can ensure the security of data, and at the same time guarantee user’s private data from leakage. Compared with the existing schemes, the various performance of this scheme is more ideal. In the follow-up work, the method will be improved to support more flexible access control policy.

REFEREnCEs

[1] Goyal v, Pandey A, sahai A, Waters b. (2006) Attribute- based encryption for fine-grained access control of encrypted data. In: Juels A, Wright Rn, vimercati sDC, eds. Proc. of the 13th ACM Conf. on Computer and Communications Security, CCS 2006. Alexandria:

ACm Press, pp:89−98.

[2] xiaohui Li, Dawu Gu, yanli Ren, ning Ding, Kan yuan.

Efficient ciphertext-policy attribute based encryption with hidden policy. In: y. xiang et al. (eds.) Proc. of 5th Internet and Distributed Computing Systems (IDCS), 2012, LNCS, (7647):146–159.

[3] sims K. (2007) Ibm introduces ready-to-use cloud computing collaboration services get clients started with cloud computing. Available from: http://www-03.

ibm.com/press/us/en/pressrelease/22613.

[4] Cheung L, newport C. (2007) Provably secure ciphertext policy AbE. Proceedings of the 14th ACM Conference on Computer and Communication Security, pp:456–465.

[5] nishide T, yoneyama K, ohta K. (2008) Attribute- based encryption with partially hidden encryp- tor-specified access structures. In: bellovin sm, Gennaro R, Keromytis A, yung m, (eds). Proc. of the Applied Cryptography and Network Security. berlin, Heidelberg: springer-verlag, pp:111–129.

Figure 7. The comparison in decryption time for different schemes.

Research on the agility of C2 organization

Shuai Chen

Graduate Management Unit No.14, CAPF Engineering University, Xi’an, China

Xuejun Ren

Department of Information Engineering, CAPF Engineering University, Xi’an, China

Wenjing Shao

Graduate Management Unit No.13, CAPF Engineering University, Xi’an, China

AbStRACt: this paper researched on agility of C2 organization, imagined a battle case and assign tasks between platforms by improved MPLDS algorithm, proving the reliability of the result, changing the task demand resources randomly, through the analysis of the case it can be concluded that agility and task finish- ing rate have a certain positive correlation.

KeyWoRDS: Command and control; C2 organization; agility; adaptability; positive correlation.

1 INtRoDUCtIoN

the concept of Agile C2 organization was first brought about by Alberts & Hayes [1] in 2003. Adam Forsyth et al. [2] dug into the issue of information refinery for a C2 organization featured with its agility. Lenard Simpson et al. [3] analyzed the features of an agility-based organization network and issues like dynamic troop distribution, etc. Reiner K. et al.[4] performed qualitative analysis on the improvements in the agility of a C2 organization in terms of the role and influence of individuals in an organization. However, the construction of an agile C2 organization and the process of such construction are still at the early conceptual and experimental stages. this paper is presented in an attempt to provide a new approach of agile organization design on basis of conceptual and case analyses.

2 C2 oRGANIZAtIoN 2.1 Concept

A C2 organization represents a command and control architecture designed for a particular battlefield environment, completed by the chain of command and control between commanding nodes (i.e. platforms, resources and decision-making entities) and the whole operation process to complete the mission. [5]

2.2 Agility of C2 organization

the modern C2 combat is a complex and dynamic process full of uncertainty. Military organizations must improvise new methods and measures, and put

them into work immediately for the achievement of their goals. the Canadian armed forces believe a more agile C2 organizing system should be established in cope with diversified military operations.

Agility represents an organization’s ability to maintain its efficiency to an acceptable level in the changed environment. Agility is defined by NAto SAS-085 in its research works as “the ability to influence, deal with and/or take advance of varied situations”.

Albert and Hayes in their Power to the Edge believed that agility was achieved by the coordination of 6 properties: Robustness, elasticity, Responsiveness, Flexibility, Innovativeness and Adaptability.

Robustness: “An ability to maintain efficiency in diversified missions and under diversified situations [6] ”, described as:

∑ ( )

= O

N N

rob n

i i ri

/ max

(1)

Suppose an organization of is in possess of a resource set {(ri,Nri),ããã(rn,Nrn)}, representing the types and capability of each resource. the task resource demand and resource capability are described as {(ri,Nmaxri),ããã(rn,Nmaxrn)}. So we have the ability of of to successfully deal with changes in missions in progress as orob (i.e. Robustness), which determines the ability of a C2 organization to complete the pro- posed mission and to cope with emergency situations key to the accomplishment of missions in the environment of uncertainty. It should be pointed out that

DBDH problem under the order group of composite number)