CN110457940B - Differential privacy measurement method based on graph theory and mutual information quantity - Google Patents

Abstract

The invention discloses a differential privacy measurement method based on graph theory and mutual information. The present invention reconstructs the differential privacy protection framework with the information theory communication model, constructs the information communication model of differential privacy, expresses the original data set as the information source, expresses the published data set as the sink, and expresses the query mechanism and the noise mechanism as the communication channel; The proposed differential privacy measurement model is based on the information communication model, and uses the characteristics of graphs combined with information entropy to give a mutual information calculation method for privacy leakage. The bound of privacy leakage only depends on the number of attributes and attribute values of the original data set. And differential privacy budget parameters, for any distributed original data set, any adversary with attack capability is established. The differential implicit metric method proposed by the present invention can provide the upper bound of privacy leakage mutual information for differential privacy protection, has fewer restrictive conditions, is applicable to all channels, and does not depend on the distribution of the original data set.

Description

A differential privacy measurement method based on graph theory and mutual information

Technical Field

The present invention relates to the field of information security technology, and in particular to a differential privacy measurement method based on graph theory and mutual information.

Background Art

The advent of the big data era and the popularization of mobile Internet, while generating huge commercial and social value, have also triggered widespread concern and worry about privacy. More hidden and diverse data collection, storage, and data mining have led to more frequent privacy leaks and thefts, with greater harm and impact. On the one hand, data owners directly publish data containing private information without any protection, which will cause the leakage of personal privacy information; on the other hand, malicious attackers use mature data mining and other technologies to steal sensitive information in the published data. Therefore, it is urgent to solve the problem of privacy leakage.

The issue of data privacy protection has been studied for a long time. It can be traced back to the concept of database privacy information proposed by statistician Dalenuis in 1977. He believed that in the process of accessing data, even if the attacker has background knowledge, he cannot obtain accurate information about any individual. Under this definition, corresponding privacy protection models and methods have been proposed one after another. Early privacy protection technology was mainly based on anonymous models. The basic idea is to anonymize the quasi-identifiers in the records so that all records are divided into several equivalence classes, thereby hiding one record in another set of records. Although traditional anonymous protection models and their derived algorithm models can protect users' personal privacy information to a certain extent, they cannot resist background knowledge attacks, homogeneous attacks and similarity attacks. Until 2006, Dwork of Microsoft Research proposed the concept of differential privacy protection. This model ignores the maximum background knowledge attack and ensures that neighboring data sets that differ by at most one record are indistinguishable in probability output.

Differential privacy protection is a privacy protection technology based on data distortion. It achieves privacy protection by adding noise perturbations to the original data set or statistical results, while keeping certain data attributes or statistical attributes in the data set unchanged. Differential privacy protection technology ensures that changes in a single record in the data set will not affect the query results, and even if the attacker has unlimited background knowledge, it can ensure that queries on adjacent data sets are probabilistically indistinguishable.

Differential privacy protection can be divided into two categories according to different implementation environments: interactive differential privacy and non-interactive differential privacy. The interactive differential privacy protection mechanism refers to the user submitting a query request to the data owner through the query interface. The data owner queries the original data set according to the query request, and then feeds back the query result to the user after adding noise disturbance. The non-interactive differential privacy protection mechanism refers to the data manager directly publishing a published data set that meets the differential privacy protection, and then querying the published data set according to the user's request.

The privacy budget parameter ε of differential privacy represents the strength of privacy protection. The selection of this parameter is highly dependent on experience. There is still a lack of effective information quantification methods to pre-quantify the strength of differential privacy and the amount of privacy leakage. Therefore, how to use information theory methods to quantify the amount of privacy leakage and the quantification method of the upper bound of the differential privacy protection degree for a given data set has become the key to optimizing differential privacy algorithms and designing privacy risk assessment schemes.

Summary of the invention

The technical problem to be solved by the present invention is to provide a differential privacy measurement method based on graph theory and mutual information, which solves the difficulties in quantifying privacy leakage in differential privacy protection mechanisms: (1) At present, the strength and effect of differential privacy protection can only be evaluated a posteriori, and it is highly dependent on the privacy budget parameter ε selected empirically, making it difficult to quantify the privacy protection strength and privacy leakage in advance. (2) In the differential privacy protection mechanism, once the privacy budget ε is exhausted, the differential privacy protection will be destroyed, and the privacy protection algorithm will lose its meaning. The existing privacy measurement method is mainly based on the privacy measurement model of information entropy. How to combine Shannon information theory with differential privacy to quantify the privacy leakage in the differential privacy protection mechanism and prove the upper limit of the privacy leakage in the differential privacy protection mechanism is the difficult problem that the present invention focuses on solving.

The present invention is implemented as follows: A differential privacy measurement method based on graph theory and mutual information includes the following steps:

Step 1: First, reconstruct the differential privacy protection framework based on the information theory communication model, construct the information communication model of differential privacy, represent the original data set in the differential privacy protection mechanism as the information source, represent the published data set as the information destination, and represent the differential privacy protection mechanism as the communication channel;

Step 2: Construct a privacy quantification model and model the communication channel in the differential privacy communication model as a query mechanism and a noise mechanism:

Step 3: The signal source and the signal sink are then regarded as graph structures, so that the channel transfer matrix is regarded as a composite graph of the signal source graph and the signal sink graph;

Step 4: The channel matrix M is converted to the maximum diagonal matrix M′; the maximum value of each column element in the first n columns of the channel matrix M is moved to the diagonal. The matrix M′ still satisfies ε-differential privacy and the conditional entropy H(X|Y) between the original data set and the published data set remains unchanged.

Step 5: Based on the graph distance regularization and point transfer, the maximum diagonal matrix M′ is converted into a Hamming matrix M″, so that the elements on the diagonal are equal and equal to the maximum element in the matrix, that is, the conditional entropy H(X|Y) between the original dataset and the published dataset remains unchanged;

Step 6: Using the automorphism and adjacency relationship of the graph, we prove that there is an upper bound on the privacy leakage of the differential privacy protection mechanism through the scaling formula method, and give a formula for calculating the upper bound of privacy leakage.

In the privacy quantification model based on the communication mechanism in step 2, the original data set in the differential privacy protection mechanism is represented as the source, the published data set is represented as the destination, and the query mechanism and the noise mechanism are represented as the communication channel.

The privacy quantification method based on graph theory in steps 4 and 5 uses the automorphism, point transitivity and regular distance properties of the graph to give an upper bound on privacy leakage, where the permutation σ on the automorphism point set V(G) is called the automorphism of the graph G, that is, for any vertex v, v′∈V, if v～v′ then σ(v)～σ(v′); point transitivity means that for any vertex v, v′∈V in the graph G, there exists an automorphism such that σ(v)＝v′, then the graph G is called a point transitive vertex; distance regularity means that if there exist integers b _d and c _d (d∈0,1,K d _max ) such that for any vertex v, v′ in the graph G, d(v,v′)＝d, vertex v has b _d neighbors belonging to the set V _<d+1> (v), and vertex v' has c _d neighbors belonging to the set V _<d-1> (v), then the graph G is called a distance regular graph.

The present invention reconstructs the differential privacy protection framework based on the information theory communication model, constructs the differential privacy information communication model, represents the original data set as the information source, represents the published data set as the information destination, and represents the query mechanism and the noise mechanism as the communication channel; further regards the information source and the information destination as a graph, and thus regards the channel transfer matrix as a composite graph of the information source graph and the information destination graph, and converts the channel transfer matrix into a Hamming graph based on the distance regularization and point transfer of the graph, and proposes a privacy leakage mutual information quantification method for differential privacy; using the self-isomorphism and adjacency relationship of the graph, the method of scaling formula is used to prove that the privacy leakage amount of the differential privacy protection mechanism has an upper bound, and a formula for calculating the upper bound of privacy leakage is proposed. The proposed differential privacy measurement model is based on the information communication model, and uses the characteristics of the graph combined with information entropy to give a mutual information calculation method for the privacy leakage amount. The bound of the privacy leakage amount only depends on the number of attributes, the number of attribute values and the differential privacy budget parameter of the original data set, and it is applicable to any distributed original data set and any adversary with any attack capability.

The scheme uses a combination of graph theory and mutual information. Compared with the traditional a posteriori evaluation method that relies on empirical selection of privacy budget, this method not only provides a specific privacy concept and quantification method, but also considers the upper bound of privacy leakage in the query and uses the self-isomorphism and adjacency relationship of the graph. It proves that the privacy leakage of the differential privacy protection mechanism has an upper bound by the scaling formula and gives the calculation formula of the upper bound of privacy leakage. It is proved through analysis that the differential implicit measurement method proposed in the present invention can give the upper bound of the privacy leakage mutual information of differential privacy protection, with fewer restrictions, applicable to all channels, and independent of the distribution of the original data set.

BRIEF DESCRIPTION OF THE DRAWINGS

Fig. 1 is a schematic diagram of the process of the present invention;

FIG2 is a diagram of a differential privacy measurement model of the present invention;

FIG. 3 is a diagram showing a channel matrix conversion in the communication model of the present invention.

DETAILED DESCRIPTION

The present invention will be further described below in conjunction with the accompanying drawings and embodiments.

An embodiment of the present invention: A differential privacy measurement method and technical process based on graph theory and mutual information is shown in Figure 1, which includes 6 steps, constructing a differential privacy channel model and a privacy quantification model, and then giving a measurement method based on graph theory and mutual information, proving that the differential privacy protection mechanism has an upper bound on privacy leakage and giving a calculation formula.

The channel transfer process in the communication model is shown in FIG2 , and includes two steps: the channel matrix M is converted into the maximum diagonal matrix M′. The maximum value of each column element in the first n columns of the channel matrix M is moved to the diagonal; based on the distance regularization and point transfer of the graph, the maximum diagonal matrix M′ is converted into a Hamming matrix M″, so that the elements on the diagonal are equal and equal to the maximum element in the matrix.

The differential privacy measurement model based on graph theory and mutual information is shown in Figure 3. The privacy leakage between the original data set and the published data set is quantified based on the differential privacy measurement model in Figure 2. The maximum privacy leakage of the published data set to the original data set in the privacy measurement model is ML. First, the original data set, the distributed data set, and the data set are regarded as undirected graphs, and a channel matrix (channel graph) based on the binary graph structure is constructed. The channel matrix M is transformed by using the distance regularization and point transfer of the graph to obtain the Hamming matrix (Hamming graph); then, the conditional entropy of the original data set and the published data set is proved to have a lower bound by the adjacency relationship of the Hamming matrix. The symmetry and automorphism relationship of the Hamming matrix are further used to obtain the lower bound of the conditional entropy. For any distributed input data set, the upper bound of the privacy leakage is calculated by using the mutual information calculation method.

According to the definition of conditional entropy:

According to the definition of information entropy and the principle of maximum entropy of uniform distribution, we can get

Since M″ _i,j ≤max ^M″ ,

且

故Also because

and

Therefore

After the channel matrix is converted to the Hamming matrix, the conditional entropy between the original data set and the transmitted data set remains unchanged, that is, H ^M (X|Y) = H ^M ″(X|Y), so

H(X|Y)≥-log ₂ max ^M

According to the extended definition of differential privacy, assuming that the channel matrix M satisfies ε-differential privacy, then for any column j and any pair of rows i and h (i～h), we have

When h=j, the elements on the diagonal of the matrix M″ are equal and equal to the maximum element value, so for each element M″ _i,j,

max ^M″ ≤e ^εd(i,j) M″ _i,j

And because any row element in the matrix M" is a probability distribution, ∑ _j M" _i,j = 1, so

And according to the distance grouping of the graphic structure elements, we can get

By transforming the inequality, we get

If the input graph structure of the communication model is a distance regular graph and point transfer, then for each d∈S _G , the value of |X _<d> (i)| is the same and depends only on d. Its value is recorded as N _d , that is, N _d ＝|X _<d> (i)|. Therefore

种可能选择，每一种选择有(v-1)种可能情况，故By changing the value of the individual in the u-tuple representing i, each element j with a distance d from x can be obtained. These individuals have

There are (v-1) possible choices for each choice, so

but

Therefore

Also because

Therefore

When the probability distribution of the original data set is uniform, the information entropy reaches its maximum value, that is, H(X) = log ₂ n = log ₂ v ^u . According to the definition of mutual information,

From the above proof, we can infer that when the probability distribution of the original data set is uniform, the original data set has the maximum information entropy, and the mutual information leakage is the largest at this time. Therefore, when the original data set is an arbitrary probability distribution, the result still holds. Therefore, the upper bound of the mutual information is valid for any distribution on the original data set. In addition, since the proposed model still satisfies the differential privacy mechanism, the upper bound of the mutual information is valid for any background knowledge that the adversary may have.

The present invention has been described in detail above with reference to the specific drawings, which do not constitute a limitation of the invention. Without departing from the principle of the present invention, those skilled in the art may make many variations and improvements, which should also fall within the protection scope of the present invention.

Claims (3)

1. A differential privacy measurement method based on graph theory and mutual information content is characterized by comprising the following steps:

step 1: firstly, reconstructing a differential privacy protection framework by using an information theory communication model, constructing an information communication model of differential privacy, representing an original data set in a differential privacy protection mechanism as an information source, representing a released data set as an information sink, and representing the differential privacy protection mechanism as a communication channel;

step 2: constructing a privacy quantification model, and modeling a communication channel in the differential privacy communication model into a query mechanism and a noise mechanism:

and step 3: then, the information source and the information sink are regarded as graph structures, and the channel transfer matrix is regarded as a composite graph of the information source graph and the information sink graph;

and 4, step 4: converting the channel matrix M into a maximum diagonal matrix M'; moving the maximum value of each row of elements in the first n rows of the channel matrix M to a diagonal line, wherein the matrix M' still meets epsilon-difference privacy and the conditional entropy H (X | Y) between the original data set and the published data set is unchanged;

and 5: transforming the maximum diagonal matrix M 'to a Hamming matrix M' based on graph distance regularization and point transfer such that the elements on the diagonals are all equal and equal to the maximum element in the matrix, i.e., and the conditional entropy H (X | Y) between the original dataset and the published dataset is unchanged; the point transmission refers to that any vertex V in the graph G exists in a self-isomorphism mode, V 'is belonged to V, so that sigma (V) = V', and the graph G is called point transmission;

step 6: the self-isomorphic and adjacency relation of the graph is utilized, the privacy disclosure quantity of the differential privacy protection mechanism is proved to be stored in the upper bound through a scaling formula method, and a formula for calculating the upper bound of the privacy disclosure is provided;

defined by conditional entropy:

then the method is obtained by the definition of information entropy and the principle of uniformly distributing maximum entropy

Also due to M _i,j ≤max ^M″ Therefore, it is

And due to

And->

Therefore->

After the channel matrix is converted into the hamming matrix, the conditional entropy between the original data set and the transmitted data set is unchanged, i.e. H ^M (X|Y)＝H ^M″ (X | Y), so

H(X|Y)≥-log ₂ max ^M

As defined by the extended definition of differential privacy, assuming that the channel matrix M satisfies ε -differential privacy, for any column j, and any pair of rows i and h (i-h), there is

When h = j, the elements on the diagonal of the matrix M "are equal and equal to the maximum element value, so for each element M ″ _i,j Is provided with

max ^M″ ≤e ^εd(i,j) M″ _i,j

Since any row element in the matrix M' is probability distribution, then sigma _j M″ _i,j Is not less than 1, therefore

And is obtained by grouping the distance of the graphic structure elements

Obtained by inequality transformation

If the input graph structure of the communication model is a distance regular graph and point transfer, for each d ∈ S _G ，|X _＜d＞ (i) The values of | are identical and depend only on d, and are denoted as N _d I.e. N _d ＝|X _＜d＞ (i) L, |; therefore, it is

By changing the value of an individual in the u-tuple representing i, each element j at a distance d from x can be obtained; these individuals have

(v-1) possible choices for each choice, therefore

Then

Therefore, it is

And because of

Therefore, it is

When the probability distribution of the original data set is uniform, the entropy of the information has a maximum value, i.e., H (X) = log ₂ n＝log ₂ v ^u (ii) a According to the definition of mutual information quantity

2. The differential privacy measurement method based on graph theory and mutual information quantity according to claim 1, characterized in that: in the step 2, based on the privacy quantization model of the communication mechanism, the original data set in the differential privacy protection mechanism is represented as an information source, the issued data set is represented as an information sink, and the inquiry mechanism and the noise mechanism are represented as communication channels.

3. The differential privacy measurement method based on graph theory and mutual information quantity according to claim 1, characterized in that: the privacy quantification method based on graph theory in the steps 4 and 5 utilizes the self-isomorphism, point transmission and regular distance properties of the graph to give an upper bound of privacy disclosure, wherein the substitution sigma on a self-isomorphism known point set V (G) is called the self-isomorphism of the graph G, namely, sigma (V) -sigma (V ') exists for any vertex V, V ' ∈ V if V-V '; point transfer refers to that any vertex V in the graph G exists in a self-isomorphism mode, V 'is belonged to V, so that sigma (V) = V', and the graph G is called point transfer; distance regular refers to the integer b if present _d And c _d (d∈0,1,...d _max ) Let any vertex v, v 'in the graph G, where d (v, v') = d, vertex v has b _d A neighbor belongs to the set V _＜d+1＞ (v) The vertex v' has c _d A neighbor belongs to the set V _＜d-1＞ (v) Then, the graph G is called a distance regular graph.

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