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# The Security of Database Network Model Based on Fractional Differential Equations

###### Recibido: 19 Feb 2022
Detalles de la revista
Formato
Revista
eISSN
2444-8656
Primera edición
01 Jan 2016
Calendario de la edición
2 veces al año
Idiomas
Inglés
Introduction

Database network homomorphic encryption is a subset of secret homomorphism. Fully Homomorphic Encryption has always been a problem that scholars dream of solving. Fully homomorphic encryption can perform arbitrary operations on the ciphertext without encryption. In this way, the corresponding plaintext operation can be implemented. The ciphertext database statistics method is constructed by using the cryptosystem with homomorphic properties [1]. The simulation test adopts the two-tier structure of Client/Server. At the same time, we use MFC to implement it and analyze its efficiency.

Theoretical Framework
Homomorphic encryption system

Homomorphism of addition and multiplication ensures that the calculation result after encryption of two variables is the same as the calculation result before encryption [2]. A brief description of HES is as follows:

Suppose R, S is two rings. R represents the plaintext space. S stands for ciphertext space. We give the following definition: E : RS. If E(x × y) = E(x)y is computed from E(x) and y by mixed multiplication [3]. This allows E(x × y) to be calculated without knowing the value of x. If an encryption algorithm satisfies the following operation conditions, it is called a fully homomorphic encryption algorithm: $E(m1Θm2)←E(m1)ΘE(m2);∀m1,m2 ∈M$ E\left( {{m_1}\Theta {m_2}} \right) \leftarrow E\left( {{m_1}} \right)\Theta E\left( {{m_2}} \right);\forall {m_1},{m_2}\, \in M

M specifies the text space. Θ refers to any kind of operation. ← means only operate on ciphertext, without decrypting the ciphertext information into plaintext for operation.

Ciphertext database statistics

The statistics of ciphertext databases are inseparable from ciphertext retrieval. Domestic and foreign scholars' research focus is to retrieve the plaintext information according to the characteristics of the ciphertext and then realize the statistics of the ciphertext data items. Some scholars have proposed a linear retrieval algorithm and implemented it using an asymmetric encryption algorithm. Each data item corresponds to a string of ciphertext information [4]. The system generates a series of pseudo-random sequences whose length is less than its length and consists of pseudo-random sequences. The user submits the ciphertext sequence corresponding to the plaintext information that meets the conditions. The server performs modulo 2 plus verification and records the number of ciphertext items that meet the conditions. This completes the statistics of plaintext data items. Linear retrieval methods are difficult to apply in the case of large datasets. This retrieval method can only complete the identification of specific ciphertexts but cannot complete the range statistics of ciphertexts.

Some scholars have described a keyword-based public key retrieval scheme. The algorithm encrypts the keyword with the public key and generates the ciphertext for searching. This scheme has not solved the problem of whether the extraction of keywords can represent the quantity calculation between the entire ciphertext and the keyword ciphertext. Some scholars have proposed a secure indexing strategy [5]. The disadvantage of this scheme is that it needs to generate many key sequences. The search cost for each retrieval will increase with retrieval times. At this time, the statistical efficiency will be greatly reduced. It is difficult to be accepted in practical application.

Some scholars have proposed an encrypted database idea. This paper designs an onion encryption scheme that encrypts data layer by layer. The scheme is composed of order-preserving encryption OPE and homomorphic encryption system HOM. Some cryptosystems construct the homomorphic encryption part with single multiplication homomorphism or single addition homomorphism [6]. The text also divides SQL operations into JOIN, RND, SEARCH, etc. They classified the data using four onion-style data encryption schemes. In this way, the homomorphic operation on the ciphertext is realized, and the retrieval efficiency is greatly improved. We use the homomorphic properties to perform different homomorphic operations on the ciphertext. In this way, the analysis and retrieval of the ciphertext data are realized. And the statistics of the ciphertext database are completed.

Fractional Delay Differential Equations

If there is a positive definite matrix P and a semi-positive definite matrix Q, the fractional-order delay differential equation under any state variable x(t) ∈ RN still satisfies $xT(t)PDtαx(t)+xT(t)Qx(t)−xT(t−τ)Qx(t−τ)≤0$ {x^T}\left( t \right)PD_t^\alpha x\left( t \right) + {x^T}\left( t \right)Qx\left( t \right) - {x^T}\left( {t - \tau } \right)Qx\left( {t - \tau } \right) \le 0

Then the fractional differential equation is Lyapunov stable. We design linear feedback controllers: $Dtαx1(t)=a1(x3(t)−x1(t))−k1x1(t)$ D_t^\alpha {x_1}\left( t \right) = {a_1}\left( {{x_3}\left( t \right) - {x_1}\left( t \right)} \right) - {k_1}{x_1}\left( t \right)

We assume that the positive definite matrix P and the semi-positive definite matrix Q are as follows: $P=[ 11111 ],Q=[ 10110 ]$ P = \left[ {\matrix{ 1 & {} & {} & {} & {} \cr {} & 1 & {} & {} & {} \cr {} & {} & 1 & {} & {} \cr {} & {} & {} & 1 & {} \cr {} & {} & {} & {} & 1 \cr } } \right],\,Q = \left[ {\matrix{ 1 & {} & {} & {} & {} \cr {} & 0 & {} & {} & {} \cr {} & {} & 1 & {} & {} \cr {} & {} & {} & 1 & {} \cr {} & {} & {} & {} & 0 \cr } } \right]

Construct a positive definite function: $xT(t)PDtαx(t)+xT(t)Qx(t)−xT(t−τ)Qx(t−τ)=x1T(t)Dtαx1(t)+x2T(t)Dtα x2(t)+x3T(t)Dtα x3(t)+x4T(t)Dtαx4(t)+x5T(t)Dtα x5(t)+xT(t)Qx(t)−xT(t−τ)Qx(t−τ)=(−9−k1)x12(t)+(−10−k2)x22(t)+(1−k3)x32(t)+(1−k4)x42(t)+(−8/3−k5)x52(t)+38x1(t)x3(t)+38x2(t)x4(t)−x3(t)x3(t−τ)−x4(t)x4(t−τ)−x12(t−τ)−x32(t−τ)−x42(t−τ)≤(10−k1)x12(t)+(9−k2)x22(t)+(20−k3)x32(t)+(20−k4)x42(t)+(−8/3−k5)x52(t)−x3(t)x3(t−τ)−x4(t)x4(t−τ)−x12(t−τ)−x32(t−τ)−x42(t−τ)≤(10−k1)x12(t)+(9−k2)x22(t)+(20.5−k3)x32(t)+(20.5−k4)x42(t)+(−8/3−k5)x52(t)−0.5 x32(t−τ)−0.5 x42(t−τ)−x12(t−τ)$ \matrix{ {{x^T}\left( t \right)PD_t^\alpha x\left( t \right) + {x^T}\left( t \right)Qx\left( t \right) - {x^T}\left( {t - \tau } \right)Qx\left( {t - \tau } \right) = } \hfill \cr {x_1^T\left( t \right)D_t^\alpha {x_1}\left( t \right) + x_2^T\left( t \right)D_t^\alpha \,{x_2}\left( t \right) + x_3^T\left( t \right) D_t^\alpha \,{x_3}\left( t \right) + x_4^T\left( t \right)D_t^\alpha {x_4}\left( t \right) + } \hfill \cr {x_5^T\left( t \right)D_t^\alpha \,{x_5}\left( t \right) + {x^T}\left( t \right)Qx\left( t \right) - {x^T}\left( {t - \tau } \right)Qx\left( {t - \tau } \right) = } \hfill \cr {\left( { - 9 - {k_1}} \right)x_1^2\left( t \right) + \left( { - 10 - {k_2}} \right)x_2^2\left( t \right) + \left( {1 - {k_3}} \right)x_3^2\left( t \right) + } \hfill \cr {\left( {1 - {k_4}} \right)x_4^2\left( t \right) + \left( { - 8/3 - {k_5}} \right)x_5^2\left( t \right) + 38{x_1}\left( t \right){x_3}\left( t \right) + } \hfill \cr {38{x_2}\left( t \right){x_4}\left( t \right) - {x_3}\left( t \right){x_3}\left( {t - \tau } \right) - {x_4}\left( t \right){x_4}\left( {t - \tau } \right) - } \hfill \cr {x_1^2\left( {t - \tau } \right) - x_3^2\left( {t - \tau } \right) - x_4^2\left( {t - \tau } \right) \le } \hfill \cr {\left( {10 - {k_1}} \right)x_1^2\left( t \right) + \left( {9 - {k_2}} \right)x_2^2\left( t \right) + \left( {20 - {k_3}} \right)x_3^2\left( t \right) + } \hfill \cr {\left( {20 - {k_4}} \right)x_4^2\left( t \right) + \left( { - 8/3 - {k_5}} \right)x_5^2\left( t \right) - {x_3}\left( t \right){x_3}\left( {t - \tau } \right) - } \hfill \cr {{x_4}\left( t \right){x_4}\left( {t - \tau } \right) - x_1^2\left( {t - \tau } \right) - x_3^2\left( {t - \tau } \right) - x_4^2\left( {t - \tau } \right) \le } \hfill \cr {\left( {10 - {k_1}} \right)x_1^2\left( t \right) + \left( {9 - {k_2}} \right)x_2^2\left( t \right) + \left( {20.5 - {k_3}} \right)x_3^2\left( t \right) + } \hfill \cr {\left( {20.5 - {k_4}} \right)x_4^2\left( t \right) + \left( { - 8/3 - {k_5}} \right)x_5^2\left( t \right) - 0.5\,x_3^2\left( {t - \tau } \right) - } \hfill \cr {0.5\,x_4^2\left( {t - \tau } \right) - x_1^2\left( {t - \tau } \right)} \hfill \cr }

The system is Lyapunov stable if each coefficient of the constructed positive definite function is not greater than 0.

Paillier, RSA homomorphism analysis
Paillier initialization

Randomly select two prime numbers p and q that satisfy gcd(pq, (p − 1) (q − 1)) = 1.

Calculate n = pq and λ = lcm(p − 1, q − 1).

We choose a random number $g(g∈cn2*)$ g\left( {g \in c_{{n^2}}^*} \right) , and we can guarantee that μ = (L(gλ mod n2))−1 mod n exists. where function L is defined as $L(u)=u−1n$ L\left( u \right) = {{u - 1} \over n} .

Paillier encryption and decryption: plaintext m(mcn) and we choose random number $r(r∈cn*)$ r\left( {r \in c_n^*} \right) , then the encryption process is c = gm rn mod n2. The decryption process is m = L(cλ mod n2) μ mod n.

Paillier homomorphism analysis: The plaintext m1, m2 is encrypted to obtain $E(m1)=gm1x1n(modn2)$ E\left( {{m_1}} \right) = {g^{{m_1}}}x_1^n\,\left( {\bmod {n^2}} \right) , $E(m2)=gm2x2n(modn2)$ E\left( {{m_2}} \right) = {g^{{m_2}}}x_2^n\,\left( {\bmod {n^2}} \right)

$E(m1)E(m2)=gm1x1ngm2x2n=gm2+m2(x1,x2)nmodn2=E(m1+m2)$ E\left( {{m_1}} \right)\,E\left( {{m_2}} \right) = {g^{{m_1}}}x_1^n\,{g^{{m_2}}}\,x_2^n = {g^{{m_2} + {m_2}}}{\left( {{x_1},\,{x_2}} \right)^n}\,\bmod {n^2}\, = \,E\left( {{m_1} + {m_2}} \right) at this time. Therefore, the Paillier public-key cryptosystem satisfies the additive homomorphism property.

RSA initialization

RSA encryption and decryption: the plaintext information is m, and the ciphertext information is c.

RSA homomorphism analysis: After plaintext m1, m2 is encrypted, $E(m1)=m1e(modN)$ E\left( {{m_1}} \right) = m_1^e\left( {\bmod N} \right) , $E(m2)=m2e(modN)$ E\left( {{m_2}} \right) = m_2^e\left( {\bmod N} \right) can be obtained. Therefore, the RSA public-key cryptosystem satisfies the multiplicative homomorphism property.

Neither Paillier nor RSA encryption uses any form of padding in this scheme. The purpose is to make better use of the homomorphic properties of Paillier and RSA. Paillier encryption is an unpadded scheme. The scheme can maintain the homomorphic properties of ciphertext addition and subtraction. RSA encryption is an unpadded scheme. This scheme makes the ciphertext keep the same size relationship with the plaintext and is used in range retrieval statistics.

Statistical model design

The user saves the encrypted text to the server of the cloud computing service provider. The purpose is to prevent the leakage of plaintext information. However, service providers are often required to provide search and statistical services for stored content. At this time, the server can only operate on the stored ciphertext [7]. Or the server uses some auxiliary parameters to achieve the user's search and statistical needs. This paper designs a ciphertext statistical scheme based on Paillier and RSA homomorphic properties based on this idea. Table 1 is the ciphertext data stored in the database. Attribute 1 column is encrypted with Paillier. The attribute 1 copy column is still the content-encryption Vi in attribute 1. But it uses RSA. Attribute 1 and the plaintext data in the copy of attribute 1 are taken from $Vi∈cn*$ {V_i} \in c_n^* .

Ciphertext databases based on Paillier and RSA

Physical address User Property 1 Attribute 1 copy …… Attribute n
91 U1 EP(V1) ER(V1) …… Data
92 U2 EP(V2) ER(V2) …… Data
…… …… …… …… ……
00EF Un EP(Vn) ER(Vn) …… Data
Exact statistical models
Initialization Phase

It is assumed that all users User and queryer Q are credible, and only the cloud computing server provider is not credible [8]. Both the user and the queryer hold Paillier's private key (N, g). All users and queryers in the scheme do not use traditional Paillier encryption. The random number is chosen randomly from $r(r∈Zn*)$ r\left( {r \in Z_n^*} \right) . This encryption method redefines $Zn*:Zn*={ R|1≤R≤N,gcd(N,R)=1 }$ Z_n^*:\,\,Z_n^*\, = \,\left\{ {R|1 \le R \le N,\,\gcd \left( {N,R} \right) = 1} \right\} . At this time, the maximum carrying capacity of the unified user is φ(N) − 3. The remaining 3 random numbers are assigned to the queryer Q. And ordinary users specify the random number as (R1, …, Rφ(n) − 3) by the system. The server only holds the public key (λ, μ).

Query Phase

Inquirer Q selects random number RQ from (R1, R2, R3). We use it for query item encryption this time [9]. The random number (Ri, Ri+1,… Ri +k) used by each user in the user group (Ui, Ui+1,… Ui+k) for encryption is handed over to Q. Q calculates the query matching value after getting all random numbers: $MQ,i=(RiRQ)NmodN2,MQ,i+1=(Ri+1RQ)NmodN2,⋯⋯,MQ,i+k=(Ri+kRQ)NmodN2$ {M_{Q,i}} = {\left( {{{{R_i}} \over {{R_Q}}}} \right)^N}\,\bmod {N^2},\,{M_{Q,i + 1}}\, = {\left( {{{{R_{i + 1}}} \over {{R_Q}}}} \right)^N}\,\bmod {N^2},\, \cdots \cdots ,\,{M_{Q,i + k}}\, = \,{\left( {{{{R_{i + k}}} \over {{R_Q}}}} \right)^N}\,\bmod {N^2}

If Q wants to retrieve and count items equal to VQ in database attribute 1, the random number RQ uses Paillier's private key to encrypt VQ to obtain EPRQ (VQ). Then send EPRQ (VQ) and (MQ,i, MQ,i+1,…, MQ,i+k) to the server.

After the server receives the query vector, it matches the data items. First, the server traverses the data items of attribute 1 in table 1. Calculate the quotient of each data item and EPRQ (VQ) to get (M0, M1,…, Mn). We compare (M0, M1,…, Mn) and (MQ,i, MQ,i+1,…, MQ,i+k). The number of equal items is counted as Result. We return it to the client. Result is the statistical result of the number of items equal to VQ.

Correctness Analysis

Express all data items in attribute 1 in the form of the encryption process as: $(gV1R1N,gV2R2N,…,gVnRnN)modN2$ \left( {{g^{{V_1}}}\,R_1^N,\,{g^{{V_2}}}\,R_2^N, \cdots ,\,{g^{{V_n}}}\,R_n^N} \right)\,\bmod {N^2}

Calculate each data item and quotient to match EPRQ (VQ). It can be expressed as: $(gV1R1NEP−RQ(VQ),gV2R2NEP−RQ(VQ),⋯,gVnRnNEP−RQ(VQ))modN2$ \left( {{{{g^{{V_1}}}R_1^N} \over {{E_{P - {R_Q}}}\left( {{V_Q}} \right)}},\,{{{g^{{V_2}}}R_2^N} \over {{E_{P - {R_Q}}}\left( {{V_Q}} \right)}},\, \cdots ,\,{{{g^{{V_n}}}R_n^N} \over {{E_{P - {R_Q}}}\left( {{V_Q}} \right)}}\,} \right)\bmod {N^2} $(gV1R1NgVQRQN,gV2R2NgVQRQN,⋯,gVnRnNgVQRQN)modN2$ \left( {{{{g^{{V_1}}}R_1^N} \over {{g^{{V_Q}}}R_Q^N}},\,{{{g^{{V_2}}}R_2^N} \over {{g^{{V_Q}}}R_Q^N}},\, \cdots ,\,{{{g^{{V_n}}}R_n^N} \over {{g^{{V_Q}}}R_Q^N}}\,} \right)\bmod {N^2}

It can also be expressed as: $gV1−VQ(R1RQ)N,gV2−VQ(R2RQ)N,⋯,gVn−VQ(RnRQ)NmodN2$ {g^{{V_1} - {V_Q}}}{\left( {{{{R_1}} \over {{R_Q}}}} \right)^N},\,\,{g^{{V_2} - {V_Q}}}{\left( {{{{R_2}} \over {{R_Q}}}} \right)^N}, \cdots ,\,{g^{{V_n} - {V_Q}}}{\left( {{{{R_n}} \over {{R_Q}}}} \right)^N}\,\bmod {N^2}

The gV1VQ term in each term can be equal to 1 only when Vi = VQ is present. At this point, the remaining item $(RiRQ)N$ {\left( {{{{R_i}} \over {{R_Q}}}} \right)^N} can be matched with an item in (MQ,i, MQ,i+1,…, MQ,i+k) and the retrieval is completed. The quotient step in the model takes advantage of Paillier's subtractive homomorphism [10]. The algorithm realizes the retrieval and matching with the operation of ciphertext without decryption. We pick an arbitrary random number from (R1, R2, R3). The algorithm can realize that the ciphertext of the same data item is different under different random number selections. This can confuse server-side matching values. At the same time, the algorithm also reduces the retrieval request leakage to a certain extent.

Scope Statistical Model

The Paillier ciphertext has the participation of random numbers. After the plaintext to ciphertext mapping, the size relationship of the ciphertext corresponding to the plaintext no longer maintains the original size order [11]. Therefore, the scheme uses RSA for encryption to obtain a copy of attribute 1.

Initialization Phase

Suppose the queryer is user (U1,⋯, Un) and these users are trusted. Cloud computing server providers are not trustworthy. The user holds the RSA public key (e, d) and the private key d, and the server provider only holds the RSA public key (e, d).

Query Phase

When user Ui wants to query the data within the range of a% above and below Vi, we only need to submit a% to the server. Then the server passes ER(V1) corresponding to user Ui. We compute the ciphertext in the range a%: (ER(V1) (1 − a%)e, ER(V1) (1 + a%)e). We retrieve the column of other rows and determine whether it falls within this field. Count the number of data items that fall within the field Result. This value is the final statistical result.

Correctness Analysis

We express all data items in the copy of attribute 1 in the form of the encryption process as: $(V1e,V2e,…,Vne)modN$ \left( {V_1^e,\,V_2^e, \ldots ,\,V_n^e} \right)\,\bmod N

The retrieved ciphertext value range can be expressed as: $( (VQ(1−a%))e,(VQ(1+a%))e$ \left( {{{\left( {{V_Q}\left( {1 - a\% } \right)} \right)}^e},\,\,{{\left( {{V_Q}\left( {1 + a\% } \right)} \right)}^e}\,} \right.

If Vi satisfies the inequality VQ(1 − a%) < Vi < VQ(1 + a%), then there is $(VQ(1−a%))e {\left( {{V_Q}\left( {1 - a\% } \right)} \right)^e} < V_i^e < \,\,{\left( {{V_Q}\left( {1 + a\% } \right)} \right)^e} .

The ciphertext that meets the conditions can be counted in detecting its ciphertext. This also verifies the correctness of the scheme.

Model Simulation and Analysis

The homomorphic operation in the statistical model of this paper is based on the original RSA and Paillier. We cannot encrypt with plaintext padding. Therefore, the experimental simulation adopts the large integer operation function library in the Open SSL v1.0.1(b) cryptographic algorithm library [12]. We wrote the original RSA and Paillier encryption algorithms and completed the design of some operational functions. The system uses 1024bit RSA and 512bit Paillier. The data storage part of the server adopts Oracle10g Standard Edition. The server environment is AMD Athlon (TM) IIX 2250 Processor dual-core CPU, DDR 31333 MHz 2G memory, Windows XP SP3, Visual C++ 2010.

The experiment was designed with 6 tables. The data volume is 1×104, 2×104, 4×104, 8×104, 16×104, 32×104, respectively. Then we use two schemes to perform some specific statistics on these 6 tables. Scheme 1 uses the method of storage and statistics based on Paillier and RSA homomorphism. Scheme 2 uses plaintext storage and statistics [13]. Figure 1 is a comparison chart of the precise statistics and time-consuming of the two schemes in the matching data items containing 1/100 in the table. Figure 2 is a time-consuming comparison diagram of the two schemes in the range statistics of matching data items containing 1/100 in the table.

The time-consuming of accurate statistics based on Paillier and RSA homomorphism is about 2 times that of statistics in plaintext (Figure 1). The former mainly consumes time in finding matching values and quoting operations. In Figure 2, the two polylines appear almost in one place. This shows that the time-consuming of the homomorphism scheme does not increase significantly when performing a range search. This is related to the number of submitted items for the query. The scheme simply submits a query item. At this point, we can perform an encryption operation. So there is not much extra time consumption as the data items increase.

Conclusion

This paper analyzes the existing ciphertext retrieval schemes. We then perform a homomorphic analysis of the Paillier and RSA cryptosystems. Then we propose a fractional differential equation model based on their homomorphic properties. This paper implements information retrieval and statistics in the case of full ciphertext on the server-side. Although the time-consuming of accurate statistics and range statistics increases with the increase of data volume, the statistical efficiency within a certain number of data items is still acceptable.

#### Ciphertext databases based on Paillier and RSA

Physical address User Property 1 Attribute 1 copy …… Attribute n
91 U1 EP(V1) ER(V1) …… Data
92 U2 EP(V2) ER(V2) …… Data
…… …… …… …… ……
00EF Un EP(Vn) ER(Vn) …… Data

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