Intelligent Retrieval and Secure Acquisition of Embedded Data

Abstract

Data has attributes, and synchronous changes between data and attributes are the basic characteristics of a type of data: under the condition of supplementing (deleting) some attributes, some data elements are deleted (supplemented) from data. The attributes of data elements follow the conjunctive normal form. Based on the dynamic mathematical model P-sets, firstly, the conception and generation criteria of embedded data are proposed, and the inference theory of embedded data are studied. Secondly, intelligent retrieval and retrieval theorems for embedded data are proposed. Then, intelligent retrieval and secure acquisition algorithms for embedded data are designed. Finally, an application example is presented.

1. Introduction

Reference [1] adds a “dynamic characteristic” and proposes the concept of P-sets (packet sets) in the finite ordinary set (classical Cantor set), which makes up for the deficiency of the classical set only having a “static characteristic”; it creates P-sets, widely used in dynamic information problems such as data intelligent mining and obtaining, health big data, financial risk identification and so on. Several researchers have investigated the theory and application of p-sets and have achieved some outstanding results: The function P-sets is a functional form of the P-sets [2]. The concept of a P-augmented matrix based on P-sets is proposed and its application is given [3]. P-sets have an algebraic model [4]. A new method of intelligent information retrieval in dynamic information systems is present with the use of P-sets [5,6]. References [7,8,9] put forward the concept of P-augmented matrix based on P-sets, and gives the application of the P-augmented matrix in the dynamic findings of unknown information. More dynamic characteristics and applications of P-sets are discussed [10,11,12].

Formally, P-sets are the set pair $(X^{\overline{F}},X^F)$ formed by the internal P-set $X^{\overline{F}}$ and the outer P-set $X^F$. In P-sets, the attribute ${\alpha }_i$ of element $x_i$ satisfies the “conjunction normal form” in mathematical logic. Under certain conditions, p-sets can be reduced to ordinary sets of elements. Using P-sets, this paper presents the theoretical research of embedded data, and applies it to the problem of intelligent retrieval and secure acquisition of data.

In the study of unknown data acquisition and its application, there exists a type of data $(x)$ which has attribute set $\alpha$ and dynamic characteristics. The dynamic characteristics of $(x)$ are as follows:

• If some attributes are supplemented within $\alpha$, some data elements $x_i$ are deleted within $(x)$, and $(x)$ generates internal embedded data ${(x)}^{\overline{F}}$, ${(x)}^{\overline{F}}\subseteq (x)$;
• If some attributes are deleted within $\alpha$, $(x)$ is supplemented with some data elements $x_j$, and $(x)$ generates outer embedded data ${(x)}^F$, $(x)\subseteq {(x)}^F$;
• Under the condition that 1 and 2 coexist, $(x)$ generates embedded data $({(x)}^{\overline{F}},{(x)}^F)$. The logical characteristics of attribute ${\alpha }_i$ of data element $x_i$ here are the same as those of the P-sets.

If no attribute is supplemented in $\alpha$, then the internal embedded data ${(x)}^{\overline{F}}$ is hidden in $(x)$. If no attribute is deleted in $\alpha$, then the outer embedded data ${(x)}^F$ is hidden outside $(x)$. In this case, ${(x)}^{\overline{F}}$, ${(x)}^F$ and $({(x)}^{\overline{F}},{(x)}^F)$ become unknown data.

This article uses the P-sets dynamic mathematical model to introduce dynamic characteristics into ordinary sets, replacing the “statics” of ordinary sets with “dynamism”, then the embedded data inference is discussed, and the intelligent retrieval and acquisition theorem of embedded data is given; based on the above theories and the encryption–decryption algorithm of the ellipse curve model, the intelligent retrieval and secure acquisition algorithm of embedded data are designed, and the application is given. The application examples come from venture capital identification.

The purpose of this study is to characterize the essential characteristics of complex data systems and extract important information from them, as well as to use the dynamic changes of complex data systems to achieve data security authentication, provided new theories and methods for data security transmission and authentication.

2. Dynamic Model with Attribute Conjunction

Cantor set $X=\{x_1,x_2,\cdots ,x_q\}\subset U$ is given, and $\alpha =\{{\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_k\}\subset V$ is the attribute set of $X$; $X^{\overline{F}}$ is called an internal P-set generated by $X$; moreover,

$$X^{\overline{F}}=X-X^{-}$$

(1)

$X^{-}$ is the $\overline{F}$-element deleted set of $X$,

$$X^{-}=\left\{x_i\right|x_i\in X,\stackrel{\_}{f}(x_i)=u_i\overline{\in }X,\stackrel{\_}{f}\in \stackrel{\_}{F}\}$$

(2)

If attribute set ${\alpha }^F$ of $X^{\overline{F}}$ meets

$${\alpha }^F=\alpha \cup \left\{{{\alpha }_i}^{\prime }\right|f({\beta }_i)={{\alpha }_i}^{\prime }\in \alpha ,f\in F\}$$

(3)

in (1), $X^{\overline{F}}\ne \varnothing ,$ $X^{\overline{F}}=\{x_1,x_2,\cdots ,x_p\},$$p<q;p,q\in N^{+}$; in (3), ${\beta }_i\in V,{\beta }_i\overline{\in }\alpha ,f\in F$ changes ${\beta }_i$ into $f({\beta }_i)={{\alpha }_i}^{\prime }\in \alpha$.

$X^F$ is called an outer P-set generated by $X$; moreover,

$$X^F=X\cup X^{+}$$

(4)

$X^{+}$ is the $F$-element supplemented set of $X$,

$$X^{+}=\left\{u_i\right|u_i\in U,u_i\overline{\in }X,f(u_i)={x_i}^{\prime }\in X,f\in F\}$$

(5)

If the attribute set ${\alpha }^{\overline{F}}$ of $X^F$ meets,

$${\alpha }^{\overline{F}}=\alpha -\left\{{\beta }_i\right|\stackrel{\_}{f}({\alpha }_i)={\beta }_i\overline{\in }\alpha ,\stackrel{\_}{f}\in \stackrel{\_\_}{F}\}$$

(6)

in (4), $X^F=\{x_1,x_2,\cdots ,x_r\},$$q<r,q,r\in N^{+}$; in (6), ${\alpha }_i\in \alpha ,\stackrel{\_}{f}\in \stackrel{\_\_}{F}$, ${\alpha }^{\overline{F}}\ne \varnothing$.

The set pair which is composed of internal packet set $X^{\overline{F}}$ and outer packet set $X^F$ is called P-sets generated by $X$; moreover,

$$X=(X^{\overline{F}},X^F)$$

(7)

$$\{(X_i^{\overline{F}},X_j^F)|i\in I,j\in J\}$$

(8)

(8) is the family of P-sets generated by $X$, and $I$ and $J$ are indicator sets.

Proposition 1.

If $F=\overline{F}=\varnothing$, then $(X^{\overline{F}},X^F)$ and $X$ meet

$${(X^{\overline{F}},X^F)}_{F=\overline{F}=\varnothing }=X$$

(9)

Proposition 2.

If $F=\overline{F}=\varnothing$, then $\{(X_i^{\overline{F}},X_j^F)|i\in I,j\in J\}$ and $X$ meet

$${\{(X_i^{\overline{F}},X_j^F)|i\in I,j\in J\}}_{F=\overline{F}=\varnothing }=X$$

(10)

Remark 1.  

• $U,V$ is the finite element domain and the finite attribute, respectively;
• $f\in F$ and $\stackrel{\_}{f}\in \stackrel{\_\_}{F}$ are element (attribute) transfer; $F=\{f_1,f_2\cdots f_n\}$ and $\stackrel{\_\_}{F}=\{\stackrel{\_}{f_1},\stackrel{\_}{f_2},\cdots ,\stackrel{\_}{f_n}\}$ are the family of element (attribute) transfer;
• $f^{\prime }\mathrm{s}$ characteristics are as follows: $f$ changes $u_i$ into $f(u_i)={x_i}^{\prime }\in X$ and changes ${\beta }_i$ into $f({\beta }_i)={{\alpha }_i}^{\prime }\in \alpha$ ;
• ${\overline{f}}^{\prime }s$ characteristics are as follows: $\overline{f}$ changes $x_i$ into $\overline{f}(x_i)=u_i\overline{\in }X$ and changes ${\alpha }_i$ into $\overline{f}({\alpha }_i)={\beta }_i\overline{\in }\alpha$; ${\alpha }_i$
• The dynamic characteristics of the internal P-set are the same as those of down-counter$T=T-1$;
• The dynamic characteristics of outer P-set are the same as those of accumulator $T=T+1$ . Such as, ${X_1}^F=X\cup {X_1}^{+}$, let $X={X_1}^F$, ${X_2}^F=X\cup {X_2}^{+}=(X_1^F\cup {X_1}^{+})\cup {X_2}^{+},\cdot \cdot \cdot$, so on.

Conclusion 1.

When we add attributes continuously to $\alpha$, an internal P-set chain $X_n^{\overline{F}}\subseteq X_{n-1}^{\overline{F}}\subseteq \cdots \subseteq X_2^{\overline{F}}\subseteq X_1^{\overline{F}}$ is generated by $X$;

Conclusion 2.

When we delete attributes continuously in $\alpha$, an outer P-set chain $X_1^F\subseteq X_2^F\subseteq \cdots \subseteq X_{n-1}^F\subseteq X_n^F$ is generated by $X$;

Conclusion 3.

When we add attributes and deleted attributes at the same time in $\alpha$, the P-sets chain $(X_n^{\overline{F}},X_1^F)\subseteq (X_{n-1}^{\overline{F}},X_2^F)\subseteq \cdots \subseteq (X_2^{\overline{F}},X_{n-1}^F)\subseteq (X_1^{\overline{F}},X_n^F)$ is dynamically generated by $X$.

Figure 1 shows the two-dimensional distribution of the relationship between P-sets and the finite set of ordinary elements.

New concepts for the embedding are derived from Conclusions 1–3: $X_i^{\overline{F}}$ is the internal embedding generated by $X$, $X_i^{\overline{F}}\subseteq X$; $X_j^F$ is the outer embedding generated by $X$, $X\subseteq X_j^F$; $(X_i^{\overline{F}},X_j^F)$ is the embedding generated by $X$, $X_i^{\overline{F}}\subseteq X\subseteq X_j^F$.

Assumption 1.

$(x)=X,{(x)}^{\overline{F}}=X^{\overline{F}}$${(x)}^F=X^F,({(x)}^{\overline{F}},{(x)}^F)=(X^{\overline{F}},X^F)$ $(x),{(x)}^{\overline{F}},{(x)}^F$ and $({(x)}^{\overline{F}},{(x)}^F)$ are data, $\forall x_i\in (x)$ (or $\forall x_j\in {(x)}^{\overline{F}}$ or $\forall x_k\in {(x)}^F$) is called a data element, and these symbols and names will be used directly without special explanation.

3. Embedded Generation of Data and Its Measure

${(x)}^{\overline{F}},{(x)}^F$ are respectively referred to as the internally embedded data and externally embedded data generated by $(x)$, if ${(x)}^{\overline{F}},{(x)}^F$ are satisfied, respectively,

$$\mathrm{car}d({(x)}^{\overline{F}})-\mathrm{car}d((x))\le 0$$

(11)

$$\mathrm{car}d({(x)}^F)-\mathrm{car}d((x))\ge 0$$

(12)

The pair of embedded data formed by ${(x)}^{\overline{F}}$ and ${(x)}^F$ is called embedded data $(x)$, which is denoted as:

$$({(x)}^{\overline{F}},{(x)}^F)$$

(13)

$\{({(x)}_i^{\overline{F}},{(x)}_j^F)|i\in I,j\in J\}$ is the embedded data family generated by $(x)$.

${\gamma }^{\overline{F}},{\gamma }^F$ are called the internal embedded measure and outer embedded measure of ${(x)}^{\overline{F}},{(x)}^F$, respectively, if ${\gamma }^{\overline{F}},{\gamma }^F$ satisfies

$${\gamma }^{\overline{F}}=\mathrm{car}d({(x)}^{\overline{F}})/\mathrm{car}d((x))$$

(14)

$${\gamma }^F=\mathrm{car}d({(x)}^F)/\mathrm{car}d((x))$$

(15)

$({\gamma }^{\overline{F}},{\gamma }^F)$ is the embedded measure of $({(x)}^{\overline{F}},{(x)}^F)$

Where, in (11), (12) and (14), (15), $card=cardinal number$.

Propositions 3–5 are obtained from Equations (11)–(15).

Proposition 3.

The necessary and sufficient condition for ${(x)}_k^{\overline{F}}$ to be internal embedded data of $(x)$ is that the internal embedded measure ${\gamma }_k^{\overline{F}}$ of ${(x)}_k^{\overline{F}}$ is the inner point of the unit discrete interval (0,1], that is

$${\gamma }_k^{\overline{F}}\in (0,1]$$

(16)

Proposition 4.

The necessary and sufficient condition for ${(x)}_k^{\overline{F}}$ to be outer embedded data of $(x)$ is that the outer embedded measure ${\gamma }_k^{\overline{F}}$ of ${(x)}_k^{\overline{F}}$ is the outer point of the unit discrete interval (0,1], that is

$${\gamma }_k^F\overline{\in }(0,1]$$

(17)

Proposition 5.

The necessary and sufficient condition for $({(x)}_k^{\overline{F}},{(x)}_k^F)$ to be embedded data of $(x)$ is that the discrete interval $[{\gamma }_k^{\overline{F}},{\gamma }_k^F]$ formed by the internal embedded measure ${\gamma }_k^{\overline{F}}$ of ${(x)}_k^{\overline{F}}$ and the outer embedded measure ${\gamma }_k^{\overline{F}}$ of ${(x)}_k^F$ and the unit discrete interval (0,1] satisfy:

$$[{\gamma }_k^{\overline{F}},{\gamma }_k^F]\cap (0,1]\ne \varnothing$$

(18)

where (0,1] in Equations (16)–(18) are discrete intervals composed of numerical values 0 and $1=\gamma =\mathrm{car}d((x))/\mathrm{car}d((x))$, and $\gamma =1$ is the embedded measure of $(x)$ itself.

The proof of Proposition 4 and 5 is obtained directly from Equations (11)–(15), and the proof is omitted.

Theorem 1.

(Attribute theorem for internal embedded data) There exists an attribute set $\Delta \alpha \ne \varnothing$ , such that the attribute set${\alpha }^F$ of ${(x)}^{\overline{F}}$ and the attribute set $\alpha$ of $(x)$ satisfy:

$$\alpha -({\alpha }^F-\Delta \alpha )=\varnothing$$

(19)

The proof of Theorem 1 is directly obtained from Equations (1)–(3); moreover, $\Delta \alpha =\left\{{\alpha }_i^{\prime }\right|f({\beta }_i)={\alpha }_i^{\prime }\in \alpha ,f\in F\}$.

Theorem 2.

(Attribute theorem for outer embedded data) There exists an attribute set $\nabla \alpha \ne \varnothing$ , such that the attribute set${\alpha }^{\overline{F}}$ of ${(x)}^F$ and the attribute set $\alpha$ of $(x)$ satisfy:

$$({\alpha }^{\overline{F}}\cup \nabla \alpha )-\alpha =\varnothing$$

(20)

The proof of Theorem 2 is directly obtained from Equations (4)–(6); moreover, $\nabla \alpha =\left\{{\beta }_i\right|\overline{f}({\alpha }_i)={\beta }_i\overline{\in }\alpha ,\overline{f}\in \overline{F}\}$.

Inference 1.

There exists an attribute set $\Delta \alpha \ne \varnothing$,  $\nabla \alpha \ne \varnothing$, such that the attribute set $({\alpha }^F,{\alpha }^{\overline{F}})$ of $({(x)}^{\overline{F}},{(x)}^F)$ and the attribute set $\alpha$ of $(x)$ satisfy:

$$({\alpha }^F-\Delta \alpha ,{\alpha }^{\overline{F}}\cup \nabla \alpha )-\alpha =\varnothing$$

(21)

where Formula (21) represents $({\alpha }^F-\Delta \alpha )-\alpha =\varnothing ,\hspace{1em}({\alpha }^{\overline{F}}\cup \nabla \alpha )-\alpha =\varnothing$.

The generation criteria for embedded data obtained from Propositions 3–5, Theorem 1–2, and Inference 1 are as follows.

(1)

Generation criteria for internal embedded data:

The implicit attribute ${\alpha }_t\in \alpha$ of data element $x_i\in (x)$ is the explicit attribute ${\alpha }_t\in {\alpha }_k^F$ of data element $x_i\in {(x)}_k^{\overline{F}}$.

Where $(x)=\{x_1,x_2,\cdots ,x_m\}$,${(x)}_k^{\overline{F}}=\{x_1,x_2,\cdots ,x_n\},n<m$; $\alpha =\{{\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_{t-1}\}$, ${\alpha }_k^F=\{{\alpha }_1,{\alpha }_2,\cdots ,$ ${\alpha }_{t-1},{\alpha }_t\}$, $\alpha$ and ${\alpha }_k^F$ are the set of attributes for $(x)$ and ${(x)}_k^{\overline{F}}$, respectively.

(2)

Generation criteria for outer embedded data:

The implicit attribute ${\alpha }_t\in \alpha$ of data element $x_i\in {(x)}_k^F$ is the explicit attribute ${\alpha }_t\overline{\in }{\alpha }_k^{\overline{F}}$ of data element $x_i\in {(x)}_k^F$.

Where $(x)=\{x_1,x_2,\cdots ,x_m\}$, ${(x)}_k^F=\{x_1,x_2,\cdots ,x_n\},m<n$; $\alpha =\{{\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_t\}$, ${\alpha }_k^{\overline{F}}=\{{\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_{t-1}\}$, $\alpha$ and ${\alpha }_k^{\overline{F}}$ are the set of attributes for $(x)$ and ${(x)}_k^F$, respectively.

4. Embedded Reasoning and Intelligent Retrieval of Embedded Data

If $(x),{(x)}_k^{\overline{F}}$ and $\alpha ,{\alpha }_k^F$ satisfy

$$\mathrm{If}\hspace{0.33em}\hspace{0.33em}\alpha \Rightarrow {\alpha }_k^F,\hspace{0.33em}\hspace{0.33em}\mathrm{then}\hspace{0.33em}\hspace{0.33em}{(x)}_k^{\overline{F}}\Rightarrow (x).$$

(22)

Equation (22) is called internal embedded reasoning generated by embedded data, $\alpha \Rightarrow {\alpha }_k^F$ is called the internal embedded reasoning condition, ${(x)}_k^{\overline{F}}\Rightarrow (x)$ is called the internal embedded reasoning conclusion. Where in (22), “$\Rightarrow$” is equal to “$\subseteq$”, $\alpha$ and ${\alpha }_k^F$ are the set of attributes for $(x)$ and ${(x)}_k^{\overline{F}}$, respectively.

If $(x),{(x)}_k^F$ and $\alpha ,{\alpha }_k^{\overline{F}}$ satisfy

$$\mathrm{If}\hspace{0.33em}\hspace{0.33em}{\alpha }_k^{\overline{F}}\Rightarrow \alpha ,\hspace{0.33em}\hspace{0.33em}\mathrm{then}\hspace{0.33em}\hspace{0.33em}(x)\Rightarrow {(x)}_k^F.$$

(23)

Equation (23) is called outer embedded reasoning generated by embedded data, ${\alpha }_k^{\overline{F}}\Rightarrow \alpha$ is called the outer embedded reasoning condition, $(x)\Rightarrow {(x)}_k^F$ is called the outer embedded reasoning conclusion; $\alpha$ and ${\alpha }_k^F$ are the set of attributes for $(x)$ and ${(x)}_k^{\overline{F}}$, respectively.

If $({(x)}_k^{\overline{F}},(x)),((x),{(x)}_k^F)$ and $(\alpha ,{\alpha }_k^{\overline{F}}),({\alpha }_k^F,\alpha )$ satisfy

$$\mathrm{If}\hspace{0.33em}\hspace{0.33em}(\alpha ,{\alpha }_k^{\overline{F}})\Rightarrow ({\alpha }_k^F,\alpha ),\hspace{0.33em}\hspace{0.33em}\mathrm{then}\hspace{0.33em}\hspace{0.33em}({(x)}_k^{\overline{F}},(x))\Rightarrow ((x),{(x)}_k^F)$$

(24)

Equation (24) is referred to as embedded reasoning generated by embedded data, $(\alpha ,{\alpha }_k^{\overline{F}})\Rightarrow ({\alpha }_k^F,\alpha )$ is referred to as the embedded reasoning condition, $({(x)}_k^{\overline{F}},(x))\Rightarrow ((x),{(x)}_k^F)$ is referred to as the embedded reasoning conclusion.

Where Equation (24) indicates that both “$\mathrm{If}\hspace{0.33em}\hspace{0.33em}\alpha \Rightarrow {\alpha }_k^F,\hspace{0.33em}\hspace{0.33em}\mathrm{then}\hspace{0.33em}\hspace{0.33em}{(x)}_k^{\overline{F}}\Rightarrow (x).$” and “$\mathrm{If}\hspace{0.33em}\hspace{0.33em}{\alpha }_k^{\overline{F}}\Rightarrow \alpha ,\hspace{0.33em}\hspace{0.33em}\mathrm{then}\hspace{0.33em}\hspace{0.33em}(x)\Rightarrow {(x)}_k^F.$” are satisfied. The recursive form of Equation (24) is “$\mathrm{If}\hspace{0.33em}\hspace{0.33em}({\alpha }_k^F,{\alpha }_{k+1}^{\overline{F}})\Rightarrow ({\alpha }_{k+1}^F,{\alpha }_k^{\overline{F}}),\hspace{0.33em}\hspace{0.33em}\mathrm{then}\hspace{0.33em}\hspace{0.33em}({(x)}_{k+1}^{\overline{F}},{(x)}_k^F)\Rightarrow ({(x)}_k^{\overline{F}},{(x)}_{k+1}^F)$”, where $({\alpha }_k^F,{\alpha }_{k+1}^{\overline{F}})$ is the set of attributes of $({(x)}_k^{\overline{F}},{(x)}_{k+1}^F)$.

From Equations (22)–(24), we can get Theorems 3–5 and Inferences 1–2.

Theorem 3.

If $(x),{(x)}_k^{\overline{F}}$ and $\alpha ,{\alpha }_k^F$ satisfy Equation (22), then

(1)

The internal embedded data${(x)}_k^{\overline{F}}$ is intelligently retrieved and obtained within $(x)$; $(x),{(x)}_k^{\overline{F}}$ and $\nabla (x)$ satisfy:

$${(x)}_k^{\overline{F}}=(x)-\nabla (x)$$

(25)

$\nabla (x)\ne \varnothing$ is redundant data composed of data elements $x_i$ within $(x)$.

(2)

The attribute sets $\nabla \alpha$ of $\nabla (x)$ and ${\alpha }_k^F$ of ${(x)}_k^{\overline{F}}$ satisfy:

$${\alpha }_k^F-\nabla \alpha \ne \varnothing$$

(26)

(3)

The implicit attribute${\alpha }_i$ of data element $x_i\in {(x)}_k^{\overline{F}}$ is exposed.

Proof: 

(1) Using Equations (1)–(3) and (22), under the condition of $\alpha \subseteq {\alpha }_k^F$, some elements $x_i$ in $(x)=\{x_1,x_2,\cdots ,x_t,x_{t+1},\cdots ,x_n\}$ are deleted from $(x)$, which form $\nabla (x)$, and $(x)$ generates ${(x)}_k^{\overline{F}}$, moreover ${(x)}_k^{\overline{F}}$, $(x)$ and $\nabla (x)$ satisfy ${(x)}_k^{\overline{F}}=(x)-\nabla (x)$, or ${(x)}_k^{\overline{F}}\subseteq (x)$, Equation (25) is obtained.

(2) Using Equations (1)–(3) in Section 2, it can be obtained that, assuming $\alpha =\{{\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_{\lambda }\}$ is the set of attributes of $(x)$, and $\forall x_i\in (x)$ has the attribute ${\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_{\lambda }$, or in other words, ${\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_{\lambda }$ is the explicit attribute of $x_i$, then the attribute of $x_i$ satisfies the normal form of attribute conjunction ${\alpha }_1\wedge {\alpha }_2\wedge \cdots \wedge {\alpha }_{\lambda }={{\displaystyle \wedge }}_{t=1}^{\lambda }\hspace{0.17em}{\alpha }_t$; $\nabla (x)$ and $(x)$ have the same set of attributes: that is, $\nabla \alpha =\alpha$, and ${(x)}_k^{\overline{F}}$ has an attribute set ${\alpha }_k^F=\{{\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_{\lambda },$ ${\alpha }_{\lambda +1},\cdots ,{\alpha }_n\}$, so it is obvious that ${\alpha }_k^F,\alpha$ and $\nabla \alpha$ satisfy ${\alpha }_k^F-\alpha ={\alpha }_k^F-\nabla \alpha \ne \varnothing$, thus we get Equation (26).

(3) Under the condition of $\alpha \Rightarrow {\alpha }_k^F$, the attribute ${\alpha }_{\lambda +1},{\alpha }_{\lambda +2},\cdots ,{\alpha }_n$ of $\forall x_i\in {(x)}_k^{\overline{F}}$ changes from implicit to explicit. □

Theorem 4.

If$(x),{(x)}_k^F$ and $\alpha ,{\alpha }_k^{\overline{F}}$ satisfy Equation (23), then

(1)

The outer embedded data ${(x)}_k^F$ is intelligently retrieved and obtained outside $(x)$ ; $(x),{(x)}_k^F$ and $\Delta (x)$ satisfy:

$${(x)}_k^F=(x)\cup \Delta (x)$$

(27)

Here, $\Delta (x)\ne \varnothing$ is composed of missing data element $x_i$ within $(x)$, which is a supplement within $(x)$.

(2)

The attribute sets $\Delta \alpha$ of $\Delta (x)$ and ${\alpha }_k^{\overline{F}}$ of ${(x)}_k^F$ satisfy:

$${\alpha }_k^{\overline{F}}\cap \Delta \alpha =\varnothing$$

(28)

(3)

The explicit attribute of data element $x_i\in {(x)}_k^F$ is hidden.

The proof of Theorem 4 is similar to the proof of Theorem 3, which is omitted here.

Theorem 5.

If  $({(x)}_k^{\overline{F}},(x)),\hspace{0.33em}((x),{(x)}_k^F)$ and $(\alpha ,{\alpha }_k^{\overline{F}}),\hspace{0.17em}({\alpha }_k^F,\alpha )$ satisfy Equation (24), then

(1)

The embedded data $({(x)}_k^{\overline{F}},{(x)}_k^F)$ is intelligently retrieved, moreover

$$({(x)}_k^{\overline{F}},{(x)}_k^F)=((x)-\nabla (x),(x)\cup \Delta (x))$$

(29)

(2)

The attribute set $\nabla \alpha$ of $\nabla (x)$ and the attribute set $\Delta \alpha$ of $\Delta (x)$ satisfy (26) and (28), respectively.

Inference 2.

The internal embedded data ${(x)}_k^{\overline{F}}$ satisfying the recursive form internal embedded inference “$\mathrm{If}\hspace{0.33em}\hspace{0.33em}{\alpha }_k^F\Rightarrow {\alpha }_{k+1}^F,\hspace{0.33em}\hspace{0.33em}\mathrm{then}\hspace{0.33em}\hspace{0.33em}{(x)}_{k+1}^{\overline{F}}\Rightarrow {(x)}_k^{\overline{F}}.$” are intelligently retrieved in $(x)$ in order of $k=1,2,\cdots ,n$ , and they form the internal embedded data chain

$${(x)}_n^{\overline{F}}\subseteq {(x)}_{n-1}^{\overline{F}}\subseteq \cdots \subseteq {(x)}_2^{\overline{F}}\subseteq {(x)}_1^{\overline{F}}$$

(30)

Inference 3.

The outer embedded data ${(x)}_k^F$ satisfying the recursive form outer embedded inference “$\mathrm{If}\hspace{0.33em}\hspace{0.33em}{\alpha }_{k+1}^{\overline{F}}\Rightarrow {\alpha }_k^{\overline{F}},\hspace{0.33em}\hspace{0.33em}\mathrm{then}\hspace{0.33em}\hspace{0.33em}{(x)}_k^F\Rightarrow {(x)}_{k+1}^F.$” are intelligently retrieved outside $(x)$ in order of $k=1,2,\cdots ,n$ , and they form the outer embedded data chain

$${(x)}_1^F\subseteq {(x)}_2^F\subseteq \cdots \subseteq {(x)}_{n-1}^F\subseteq {(x)}_n^F$$

(31)

Special note: Theorem 3–5 and Inference 2–3 are hidden in the application research of data retrieval, data filtering and dynamic data analysis. Equation (22)–(31) provide theoretical preparation for intelligent retrieval and discovery of unknown data.

5. Secure Acquisition Algorithm for Embedded Data

5.1. Ellipse Curve and Its Structure

The following equation is proposed by Koblitz N [13,14]:

$$y^2=x^3+ax+b$$

(32)

the $E(k)$ determined by Equation (32) is called an ellipse curve as follows

$$E(k)=\left\{(x,y)\left|x,y\in k,p(x,y)=0\right.\right\}\cup \left\{\mathcal{O}\right\}$$

(33)

where discriminant $D=(4a^3+27b^2)\mathrm{mod}m\ne 0$, $k$ is a number field, $a,b\in k$; $\mathcal{O}$ is the infinite point, and $m$ is a prime number. Let $\oplus$ be the additive operation of a point on $E(k)$. For any two points $P(x_1,y_1),Q(x_1,y_1)\in E(k)$, there is a point $R(x_3,y_3)$ on $E(k)$ generated by $P(x_1,y_1)\oplus Q(x_1,y_1)$.

• If $P(x_1,y_1)\ne Q(x_2,y_2)$, then

  $$\lambda ={\displaystyle \frac{y_2-y_1}{x_2-x_1}}$$

  $$x_3=({\lambda }^2-x_1-x_2)\mathrm{mod}m$$

  (34)

  $$y_3=(\lambda (x_1-x_3)-y_1)\mathrm{mod}m$$

  (35)
• If $P(x_1,y_1)=Q(x_2,y_2)$, then

  $$\lambda ={\displaystyle \frac{3x_1^2+a}{2y_1}}$$

  $$x_3=({\lambda }^2-x_1-x_2)\mathrm{mod}m$$

  (36)

  $$y_3=(\lambda (x_1-x_3)-y_1)\mathrm{mod}m$$

  (37)

Assumption 2.

The internal embedded data ${(x)}^{\overline{F}}$ in definition 4 is plaintext, denoted by $P_m$, namely $P_m={(x)}^{\overline{F}}$; $C_m$ is the ciphertext of $P_m$; $A$ is the encryptor of $P_m$ and the sender of $C_m$; $B$ is the decryptor of $C_m$ and the recipient of $P_m$. Both $A$ and $B$ jointly choose the same elliptical curve $E(K)$, where $D\ne 0$, and the same base point $G\in E(k)$.

5.2. Encryption of Plaintext $P_m$ and Decryption of Ciphertext $C_m$

I. $A$ chooses $n_A\in N^{+}$, and $n_A$ is the private key to $A$; moreover,

$$P_A=n_{A}G$$

(38)

$P_A$ is the public key of $A$, then $A$ gives the ciphertext $C_m$ of $P_m$ as follows:

$$C_m=\{kG,P_m+k{P}_B\}$$

(39)

Then $A$ sends $C_m$ to $B$.

II. $B$ receives $C_m$, and chooses $n_B\in N^{+}$;$n_B$ is the private key to $B$; moreover,

$$P_B=n_{B}G$$

(40)

$P_B$ is the public key of $B$; $B$ takes $C_m$ and calculates

$$\begin{array}{cc}& P_m+k{P}_B-n_B(kG)\hfill \\ \hfill =& P_m+k{P}_B-k(n_{B}G)\hfill \\ \hfill =& P_m+k{P}_B-k{P}_B\hfill \\ \hfill =& P_m\hfill \end{array}$$

(41)

thus $B$ obtains plaintext $P_m$.

If $B$ is the encryptor of $P_m$ and the sender of $C_m$, and $A$ is the receiver, decryptor and receiver of $C_m$, then the process is similar to Equations (34)–(41).

Theorem 6.

(Abel Group Generation Theorem). If $E(K)$ is a set of points on an ellipse curve, and $\oplus$ is a point addition operation on $E(K)$ , then $<E(K),\oplus >$ is an Abel group.

Theorem 7.

(Closed loop theorem for generating base point $G$). If the base point $G$ is taken on $E(K)$, and $\oplus$ is the point addition operation on $E(K)$, then there exists a positive integer $n$, and the base point $G$ generates a closed loop $D$.

Theorem 8.

(Start-end theorem) If $D$ is a closed loop generated by the base point $G$ over $E(K)$ , then the base point $G^{\prime }$ is taken at $D$ , $G^{\prime }\ne G$ ; $G^{\prime }$ and the end point $G^{″}$ satisfies

$$G^{\prime }=G^{″}$$

(42)

Theorem 6–8 is a direct result of ellipse curve; the proof is omitted.

From Equation (39), it can be seen that $A$ obtains $C_m$ and sends it to $B$, while $A$ must send it for signature to $B$. After $B$ accepts $A^{\prime }s$ signature, it provides authentication of the signature. At this point, $B$ can make two choices: accept $C_m$ or reject $C_m$.

5.3. Sending and Authentication of Signatures

Assumption 3.

$A$ is the provider of signature, $B$ is the recipient and authenticator of the signature. $A$, $B$ jointly choose elliptical curve $E(K),D\ne 0$, a prime number $n$, and basic point $P\in E(K)$. $A$ chooses the secret key $d$, $1<d<n-1$, and computes $Q=dP$. $(E,P,Q,n)$ is public to $A$, $B$. $m$is the signature plaintext chosen by $A$; $(r,s)$ is the signature ciphertext provided by $A$.

• A gives the signature $(r,s)$ of $m$ and sends it to $B$.

$A$ chooses: $1<k<n-1$; $k$ is the conversation key.

$A$ calculates:

$$R=(x_R,y_R)=K_P$$

(43)

$$r=x_R\hspace{0.33em}\mathrm{mod}\hspace{0.33em}n$$

(44)

$$S=k^{-1}(h(m)+dr)\hspace{0.33em}\mathrm{mod}\hspace{0.33em}n$$

(45)

$(r,s)$ is the signature of m, $(r,s)$ is sent to $B$.

Where $h(m)$ is a hash function, $r,s\ne 0$.

2.

B accepts the signature $(r,s)$ of m and makes an authentication;

$A$ gives:

$$W=r^{-1}\mathrm{mod}\hspace{0.33em}n\hspace{0.33em}$$

(46)

$$u_1=h(m)w\hspace{0.33em}\mathrm{mod}\hspace{0.33em}n$$

(47)

$$u_2=rw\hspace{0.33em}\mathrm{mod}\hspace{0.33em}n$$

(48)

$B$ calculates:

$$R^{\prime }=(x_R^{\prime },y_R^{\prime })=u_{1}P+u_{2}Q$$

(49)

$$r^{\prime }=x_R^{\prime }\mathrm{mod}\hspace{0.33em}n$$

(50)

$B$ makes the authentication: if $r=r^{\prime }$, then $(r,s)$ is a legal signature sent by $A$, not from others; $B$ accepts $C_m$; from (41), it can be seen that $B$ obtains $P_m$.

Theorem 9.

(Signature authentication theorem). Suppose $R$ is any point on $E(K)$, $\alpha ,\beta \in N^{+}$; if$n$ is a prime number, then there is a unique $R^{\prime }=\alpha P+\beta Q$, and it satisfies

$$x_R^{\prime }\hspace{0.33em}\mathrm{mod}\hspace{0.33em}n=x_R^{\prime }\hspace{0.33em}\mathrm{mod}\hspace{0.33em}n$$

(51)

here, $R=(x_R,y_R),\hspace{0.33em}{R}^{\prime }=(x_R^{\prime },y_R^{\prime })$, and $P,Q$ are two fixed points, $P,Q\in E(K)$.

Theorem 9 is obtained from Equations (43)–(50); its proof is omitted.

5.4. Security Acquisition Algorithm for Intelligent Retrieval of Embedded Data

This article only provides a secure acquisition algorithm for internal embedded data in intelligent retrieval, and its flowchart is shown in Figure 2. $A$ secure retrieval algorithm for outer embedded data in intelligent retrieval is omitted.

Input: $(x),\alpha ,E(K)$; $n$ is a large prime number and $P\in E(K)$ is a basic point; output: internal embedded data ${(x)}_k^{\overline{F}}$ and secure authentication results obtained.

• Based on $(x)$ and $\alpha$, generate data ${(x)}^{\overline{F}}$ database and attribute ${\alpha }^F$ database;
• Generate recursive inference database with internal embedded data;
• Generate internal embedded data ${(x)}_k^{\overline{F}}$, $k=1,2,\cdots n$;
• Given the standard ${(x)}^{\overline{F},\ast }$, compare ${(x)}_k^{\overline{F}}$ with ${(x)}^{\overline{F},\ast }$; if ${(x)}_k^{\overline{F}}\ne {(x)}^{\overline{F},\ast }$, then return 2; if ${(x)}_k^{\overline{F}}={(x)}^{\overline{F},\ast }$, then proceed to the next step;
• Suppose plaintext $P_m={(x)}_k^{\overline{F}}$; $A$ encrypts $P_m$ to obtain ciphertext $C_m$, and sends $C_m$ and signature $(r,s)$ to $B$;
• B certification signature $(r,s)$; If $r=r^{\prime }$, then proceed to the next step, otherwise end, $B$ cannot obtain $P_m$;
• If $r=r^{\prime }$, then $B$ accepts $C_m$ and obtains $P_m$.

Algorithm structure and process

In Figure 2, the internal embedded data ${(x)}^{\overline{F}}$ is subjected to recursive inference and compared with the standard internal embedded data ${(x)}^{\overline{F},\ast }$ to obtain ${(x)}_k^{\overline{F}}$ that satisfies ${(x)}_k^{\overline{F}}={(x)}^{\overline{F},\ast }$. The internal embedded data ${(x)}^{\overline{F}}$ completes the secure acquisition under the condition that the ellipse curve $E(K)$ is satisfied. Among them, ${(x)}_k^{\overline{F}}$ is plaintext $P_m$, that is $P_m={(x)}_k^{\overline{F}}$, and $C_m$ is the ciphertext of $P_m$. This algorithm is a combination of dynamic intelligent data retrieval algorithms and data security acquisition algorithms.

6. Application of Embedded Intelligent Data Retrieval and Secure Acquisition

For the sake of simplicity and generality, only the application of intelligent retrieval and secure acquisition of internal embedded data is given here. The example comes from the investment risk estimation in the financial system. The funds are invested in multiple companies, and the investor wants to know which companies will return satisfactory profits.

6.1. Given Basic Data and Attribute Sets and Selection of Investment Companies

Suppose that the basic data of the investment company $x_i$ selected by the investor is $(x)$, and $\alpha$ is the set of attributes that $x_i$ can obtain profits from. Moreover,

$$(x)=\{x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8\}$$

(52)

$$\alpha =\{{\alpha }_1,{\alpha }_2,{\alpha }_3\}$$

(53)

Here, ${\alpha }_j\in \alpha$ represents ${x_i}^{\prime }s$ ability to generate profits. The names of $x_i\in (x)$ and ${\alpha }_i\in \alpha$ are omitted.

At time $t_k\in T$, if the investor acquires new attributes ${\alpha }_4,{\alpha }_5$, then ${\alpha }_4,{\alpha }_5$ are embedded into $\alpha$, and $\alpha$ generates ${\alpha }_k^F$, moreover

$${\alpha }_k^F\hspace{0.17em}=\alpha \cup \{{\alpha }_4,{\alpha }_5\}=\{{\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4,{\alpha }_5\}$$

(54)

Under the condition of $\alpha \Rightarrow {\alpha }_k^F$, using the inference (22) of internal embedded data, we get: $x_4,x_6,x_7$ are deleted from $(x)$, then the unknown embedded data ${(x)}_k^{\overline{F}}$ is intelligently retrieved in $(x)$, and

$${(x)}_k^{\overline{F}}\hspace{0.17em}=(x)-\{x_4,x_6,x_7\}=\{x_1,x_2,x_3,x_5,x_8\}$$

(55)

At time $t_{k+\lambda }\in T$, if the investor acquires new attributes ${\alpha }_6$, then $\alpha$ generates ${\alpha }_k^F$, and

$${\alpha }_{k+\lambda }^F\hspace{0.17em}={\alpha }_k^F\cup \left\{{\alpha }_6\right\}=\{{\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4,{\alpha }_5,{\alpha }_6\}$$

(56)

Under the condition of ${\alpha }_k^F\Rightarrow {\alpha }_{k+\lambda }^F$, ${(x)}_{k+\lambda }^{\overline{F}}$ is obtained by the recursive inference form “$\mathrm{If}\hspace{0.33em}\hspace{0.33em}{\alpha }_k^F\Rightarrow {\alpha }_{k+\lambda }^F,\hspace{0.33em}\hspace{0.33em}\mathrm{then}\hspace{0.33em}\hspace{0.33em}{(x)}_{k+\lambda }^{\overline{F}}\Rightarrow {(x)}_k^{\overline{F}}$, and

$${(x)}_{k+\lambda }^{\overline{F}}\hspace{0.17em}={(x)}_k^{\overline{F}}-\left\{x_2\right\}=\{x_1,x_3,x_5,x_8\}$$

(57)

Here, Equation (57) indicates that under the condition of ${\alpha }_k^F\Rightarrow {\alpha }_{k+\lambda }^F$, $x_2$ is deleted from ${(x)}_k^{\overline{F}}$ and ${(x)}_{k+\lambda }^{\overline{F}}$ is intelligently obtained within ${(x)}_k^{\overline{F}}$.

In Equations (52)–(57), the attribute ${\alpha }_i$ of $\forall x_i\in (x)$, the attribute ${\alpha }_j$ of $\forall x_j\in {(x)}_k^{\overline{F}}$, and the attribute ${\alpha }_k$c of $\forall x_k\in {(x)}_{k+\lambda }^{\overline{F}}$ satisfy the normal form of attribute conjunction ${\alpha }_i={{\displaystyle \wedge }}_{t=1}^3\hspace{0.17em}{\alpha }_t$, ${\alpha }_j=\left({{\displaystyle \wedge }}_{t=1}^3\hspace{0.17em}{\alpha }_t\right)\wedge {\alpha }_4\wedge {\alpha }_5={{\displaystyle \wedge }}_{t=1}^5\hspace{0.17em}{\alpha }_t$, and ${\alpha }_k=\left({{\displaystyle \wedge }}_{t=1}^5\hspace{0.17em}{\alpha }_t\right)\wedge {\alpha }_6=$${{\displaystyle \wedge }}_{t=1}^6\hspace{0.17em}{\alpha }_t$, respectively.

The ${(x)}_{k+\lambda }^{\overline{F}}$ in (57) represents the profitable company determined by the investor, and the estimated profits obtained by $x_1,x_3,x_5,x_8$ are shown in

Table 1. Estimates of the value of $x_1,x_3,x_5,x_8$ acquired profits.

${(\mathit{x})}_{\mathit{k}+\mathit{\lambda }}^{\overline{\mathit{F}}}$	${\mathit{x}}_1$	${\mathit{x}}_3$	${\mathit{x}}_5$	${\mathit{x}}_8$
${(y)}_{k+\lambda }^{\overline{F}}$	9	7	13	16

.

In Table 1, $y_i\in {(y)}_{k+\lambda }^{\overline{F}}$ represents the profit value of $x_i\in {(x)}_{k+\lambda }^{\overline{F}}$.

6.2. Secure Acquisition and Confirmation of True Profit Value

Assume $A$ is the investor determined by investment company $D$, and $B$ is the profit determiner determined by investment company $D$. In Table 1, ${(y)}_{k+\lambda }^{\overline{F}}=\{y_1,y_3,y_5,y_8\}=\{(9,7),(13,16)\}$ is defined as plaintext $P_m=\{P_{m,1},P_{m,2}\}=\{(9,7),(13,16)\}$; $C_m=\{C_{m,1},C_{m,2}\}$ is the ciphertext of $P_m$; $y^2=x^3+ax+b\hspace{0.33em}\mathrm{mod}\hspace{0.33em}m=x^3+x+1\hspace{0.33em}\mathrm{mod}\hspace{0.33em}23$ is an elliptical curve jointly determined by $A$ and $B$. According to Equations (34)–(37) in Section 4, we get:

$$\begin{array}{c}E(K)\hspace{0.17em}=\{(0,1),(0,22),(1,7),(1,16),(3,10),(3,13),(4,0),(5,4),(5,19),(6,4),(6,19),(7,11),(7,12),(9,7),(9,16),\\ \hspace{1em}\hspace{0.17em}(11,3),(11,20),(12,4),(12,19),(13,17),(13,16),(17,3),(17,20),(18,3),(18,20),(19,5),(19,18)\};\end{array}$$

$27G+G=\mathcal{O}$; $G=(5,4)\in E(K)$ is the base point chosen by $A$ and $B$; $n_A=5,n_B=17$, $P_A=n_{A}G=(7,11),\hspace{0.33em}$ $P_B=n_{B}G=(18,3)$.

Using Equation (39), A takes plaintext $P_{m,1}=(9,7)$, $k=3$, $P_B=(18,3)$ and obtains ciphertext $C_{m,1}$; and, using Equations (43)–(45), $A$ provides the signature $(r,s)$:

$$C_{m,1}=\{kG,P_{m,1}+k{P}_B\}=\{3G,14G+72G\}=\{3G,86G\}=\{3G,2G\}=\{(18,3),(0,22)\}$$

$A$ sends $C_{m,1}$ and $(r,s)$ to $B$.

After receiving the ciphertext $C_{m,1}$ and signature $(r,s)$, B determines $r=r^{\prime }$, if so, then the authentication signature $(r,s)$ comes from $A$; $B$ decrypts $C_{m,1}$ using (40) and obtains:

$$P_{m,1}+k{P}_B-n_B(kG)=2G-17(24G)=2G-408G=2G-16G=-14G=14G=(9,7)=P_{m,1}$$

Thus, $B$ obtains the true profit values of $x_1,x_3$ as $y_1,y_3$.The encryption–decryption process of $P_{m,2}$ is similar to $P_{m,1}$, and is omitted.

Special note: Equation (57) and ${(y)}_{k+\lambda }^{\overline{F}}$ are trade secrets and should be confirmed by the investment company manager. A uses the intelligent retrieval algorithm in Figure 2 to obtain Equation (57) and ${(y)}_{k+\lambda }^{\overline{F}}$, and then sends them to the manager using the secure retrieval algorithm in Figure 2. The manager provides authentication for Equation (57) and ${(y)}_{k+\lambda }^{\overline{F}}$ through the algorithm.

7. Conclusions

In application research fields such as data retrieval, data recognition, data hiding, and data camouflage, data $(x)$ always exists simultaneously with its attribute set $\alpha$ (the feature set of data elements). The dynamic changes in data (decreases or increases in data elements within $(x)$) are synchronized with the dynamic changes of attributes (increases or decreases of attributes within $\alpha$). Dynamic data often appears in the research of artificial intelligence systems, while static data is only a special case for dynamic data. References [15,16] only provide application research on static data using classical methods. Considering the dynamic characteristics of data in practical applications, this paper presents an application study of a class of dynamic data using the dynamic mathematical model P-sets as the research tool. This article proposes the concept of embedded data, provides a reasoning theory and intelligent retrieval of embedded data, designs intelligent retrieval and secure acquisition algorithms for embedded data, and provides their applications.

This paper is just the beginning of the research on dynamic data problems. As the research on the application of inverse P-sets, function P-sets and function inverse P-sets in data retrieval, data analysis, data image segmentation-hiding, data image camouflage-reduction and other problems, the research is still in progress. Inverse P-sets [17,18] are obtained by improving P-sets, which are the dual model of P-sets. In inverse P-sets, the attributes ${\alpha }_i$ of data element $x_i$ meet the attribute disjunctive characteristics. Function inverse P-sets [19,20] are obtained by replacing the internal data element ${\overline{x}}_i$ of (x) with a function $\overline{s}{(x)}_i$ and improving inverse P-sets. The logical characteristics of function inverse P-sets are the same as those of inverse P-sets. These dynamic mathematical models provide theoretical and methodological support for intelligent data retrieval, discovery of unknown data, and intelligent mining of big data applications.