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 formed by the internal P-set and the outer P-set . In P-sets, the attribute of element 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 which has attribute set and dynamic characteristics. The dynamic characteristics of are as follows:
If some attributes are supplemented within , some data elements are deleted within , and generates internal embedded data , ;
If some attributes are deleted within , is supplemented with some data elements , and generates outer embedded data , ;
Under the condition that 1 and 2 coexist, generates embedded data . The logical characteristics of attribute of data element here are the same as those of the P-sets.
If no attribute is supplemented in , then the internal embedded data is hidden in . If no attribute is deleted in , then the outer embedded data is hidden outside . In this case, , and 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
is given, and
is the attribute set of
;
is called an internal P-set generated by
; moreover,
is the
-element deleted set of
,
If attribute set
of
meets
in (1),
; in (3),
changes
into
.
is called an outer P-set generated by
; moreover,
is the
-element supplemented set of
,
If the attribute set
of
meets,
in (4), ; in (6), , .
The set pair which is composed of internal packet set
and outer packet set
is called P-sets generated by
; moreover,
(8) is the family of P-sets generated by , and and are indicator sets.
Proposition 1. If , then and meet
Proposition 2. If , then and meet
Remark 1. is the finite element domain and the finite attribute, respectively;
and are element (attribute) transfer; and are the family of element (attribute) transfer;
characteristics are as follows: changes into and changes into ;
characteristics are as follows: changes into and changes into ;
The dynamic characteristics of the internal P-set are the same as those of down-counter;
The dynamic characteristics of outer P-set are the same as those of accumulator . Such as, , let , , so on.
Conclusion 1. When we add attributes continuously to , an internal P-set chain is generated by ;
Conclusion 2. When we delete attributes continuously in , an outer P-set chain is generated by ;
Conclusion 3. When we add attributes and deleted attributes at the same time in , the P-sets chain is dynamically generated by .
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: is the internal embedding generated by , ; is the outer embedding generated by , ; is the embedding generated by , .
Assumption 1. and are data, (or or ) 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
are respectively referred to as the internally embedded data and externally embedded data generated by
, if
are satisfied, respectively,
The pair of embedded data formed by
and
is called embedded data
, which is denoted as:
is the embedded data family generated by .
are called the internal embedded measure and outer embedded measure of
, respectively, if
satisfies
is the embedded measure of
Where, in (11), (12) and (14), (15), .
Propositions 3–5 are obtained from Equations (11)–(15).
Proposition 3. The necessary and sufficient condition for to be internal embedded data of is that the internal embedded measure of is the inner point of the unit discrete interval (0,1], that is Proposition 4. The necessary and sufficient condition for to be outer embedded data of is that the outer embedded measure of is the outer point of the unit discrete interval (0,1], that is Proposition 5. The necessary and sufficient condition for to be embedded data of is that the discrete interval formed by the internal embedded measure of and the outer embedded measure of and the unit discrete interval (0,1] satisfy:
where (0,1] in Equations (16)–(18) are discrete intervals composed of numerical values 0 and , and is the embedded measure of 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 , such that the attribute set of and the attribute set of satisfy: The proof of Theorem 1 is directly obtained from Equations (1)–(3); moreover, .
Theorem 2. (Attribute theorem for outer embedded data) There exists an attribute set , such that the attribute set of and the attribute set of satisfy: The proof of Theorem 2 is directly obtained from Equations (4)–(6); moreover, .
Inference 1. There exists an attribute set , , such that the attribute set of and the attribute set of satisfy: where Formula (21) represents .
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 of data element is the explicit attribute of data element .
Where ,; , , and are the set of attributes for and , respectively.
- (2)
Generation criteria for outer embedded data:
The implicit attribute of data element is the explicit attribute of data element .
Where , ; , , and are the set of attributes for and , respectively.
4. Embedded Reasoning and Intelligent Retrieval of Embedded Data
If
and
satisfy
Equation (22) is called internal embedded reasoning generated by embedded data, is called the internal embedded reasoning condition, is called the internal embedded reasoning conclusion. Where in (22), “” is equal to “”, and are the set of attributes for and , respectively.
If
and
satisfy
Equation (23) is called outer embedded reasoning generated by embedded data, is called the outer embedded reasoning condition, is called the outer embedded reasoning conclusion; and are the set of attributes for and , respectively.
If
and
satisfy
Equation (24) is referred to as embedded reasoning generated by embedded data, is referred to as the embedded reasoning condition, is referred to as the embedded reasoning conclusion.
Where Equation (24) indicates that both “” and “” are satisfied. The recursive form of Equation (24) is “”, where is the set of attributes of .
From Equations (22)–(24), we can get Theorems 3–5 and Inferences 1–2.
Theorem 3. If and satisfy Equation (22), then
- (1)
The internal embedded data is intelligently retrieved and obtained within ; and satisfy:
is redundant data composed of data elements within .
- (2)
The attribute sets of and of satisfy: - (3)
The implicit attribute of data element is exposed.
Proof: (1) Using Equations (1)–(3) and (22), under the condition of , some elements in are deleted from , which form , and generates , moreover , and satisfy , or , Equation (25) is obtained.
(2) Using Equations (1)–(3) in
Section 2, it can be obtained that, assuming
is the set of attributes of
, and
has the attribute
, or in other words,
is the explicit attribute of
, then the attribute of
satisfies the normal form of attribute conjunction
;
and
have the same set of attributes: that is,
, and
has an attribute set
, so it is obvious that
and
satisfy
, thus we get Equation (26).
(3) Under the condition of , the attribute of changes from implicit to explicit. □
Theorem 4. If and satisfy Equation (23), then
- (1)
The outer embedded data is intelligently retrieved and obtained outside ; and satisfy:
Here, is composed of missing data element within , which is a supplement within .
- (2)
The attribute sets of and of satisfy: - (3)
The explicit attribute of data element is hidden.
The proof of Theorem 4 is similar to the proof of Theorem 3, which is omitted here.
Theorem 5. If
and satisfy Equation (24), then
- (1)
The embedded data is intelligently retrieved, moreover - (2)
The attribute set of and the attribute set of satisfy (26) and (28), respectively.
Inference 2. The internal embedded data satisfying the recursive form internal embedded inference “” are intelligently retrieved in in order of , and they form the internal embedded data chain
Inference 3. The outer embedded data satisfying the recursive form outer embedded inference “” are intelligently retrieved outside in order of , and they form the outer embedded data chain
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]:
the
determined by Equation (32) is called an ellipse curve as follows
where discriminant
,
is a number field,
;
is the infinite point, and
is a prime number. Let
be the additive operation of a point on
. For any two points
, there is a point
on
generated by
.
If
, then
If
, then
Assumption 2. The internal embedded data in definition 4 is plaintext, denoted by , namely ; is the ciphertext of ; is the encryptor of and the sender of ; is the decryptor of and the recipient of . Both and jointly choose the same elliptical curve , where , and the same base point .
5.2. Encryption of Plaintext and Decryption of Ciphertext
I.
chooses
, and
is the private key to
; moreover,
is the public key of
, then
gives the ciphertext
of
as follows:
Then sends to .
II.
receives
, and chooses
;
is the private key to
; moreover,
is the public key of
;
takes
and calculates
thus
obtains plaintext
.
If is the encryptor of and the sender of , and is the receiver, decryptor and receiver of , then the process is similar to Equations (34)–(41).
Theorem 6. (Abel Group Generation Theorem). If is a set of points on an ellipse curve, and is a point addition operation on , then is an Abel group.
Theorem 7. (Closed loop theorem for generating base point ). If the base point is taken on , and is the point addition operation on , then there exists a positive integer , and the base point generates a closed loop .
Theorem 8. (Start-end theorem) If is a closed loop generated by the base point over , then the base point is taken at , ; and the end point satisfies Theorem 6–8 is a direct result of ellipse curve; the proof is omitted.
From Equation (39), it can be seen that obtains and sends it to , while must send it for signature to . After accepts signature, it provides authentication of the signature. At this point, can make two choices: accept or reject .
5.3. Sending and Authentication of Signatures
Assumption 3. is the provider of signature, is the recipient and authenticator of the signature. , jointly choose elliptical curve , a prime number , and basic point . chooses the secret key , , and computes . is public to , . is the signature plaintext chosen by ; is the signature ciphertext provided by .
chooses: ; is the conversation key.
is the signature of m, is sent to .
Where is a hash function, .
- 2.
B accepts the signature of m and makes an authentication;
makes the authentication: if , then is a legal signature sent by , not from others; accepts ; from (41), it can be seen that obtains .
Theorem 9. (Signature authentication theorem). Suppose is any point on , ; if is a prime number, then there is a unique , and it satisfies
here,
, and
are two fixed points,
.
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.
secure retrieval algorithm for outer embedded data in intelligent retrieval is omitted.
Input: ; is a large prime number and is a basic point; output: internal embedded data and secure authentication results obtained.
Based on and , generate data database and attribute database;
Generate recursive inference database with internal embedded data;
Generate internal embedded data , ;
Given the standard , compare with ; if , then return 2; if , then proceed to the next step;
Suppose plaintext ; encrypts to obtain ciphertext , and sends and signature to ;
B certification signature ; If , then proceed to the next step, otherwise end, cannot obtain ;
If , then accepts and obtains .
Algorithm structure and process
In
Figure 2, the internal embedded data
is subjected to recursive inference and compared with the standard internal embedded data
to obtain
that satisfies
. The internal embedded data
completes the secure acquisition under the condition that the ellipse curve
is satisfied. Among them,
is plaintext
, that is
, and
is the ciphertext of
. 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
selected by the investor is
, and
is the set of attributes that
can obtain profits from. Moreover,
Here, represents ability to generate profits. The names of and are omitted.
At time
, if the investor acquires new attributes
, then
are embedded into
, and
generates
, moreover
Under the condition of
, using the inference (22) of internal embedded data, we get:
are deleted from
, then the unknown embedded data
is intelligently retrieved in
, and
At time
, if the investor acquires new attributes
, then
generates
, and
Under the condition of
,
is obtained by the recursive inference form “
, and
Here, Equation (57) indicates that under the condition of , is deleted from and is intelligently obtained within .
In Equations (52)–(57), the attribute of , the attribute of , and the attribute c of satisfy the normal form of attribute conjunction , , and , respectively.
The
in (57) represents the profitable company determined by the investor, and the estimated profits obtained by
are shown in
Table 1.
In
Table 1,
represents the profit value of
.
6.2. Secure Acquisition and Confirmation of True Profit Value
Assume
is the investor determined by investment company
, and
is the profit determiner determined by investment company
. In
Table 1,
is defined as plaintext
;
is the ciphertext of
;
is an elliptical curve jointly determined by
and
. According to Equations (34)–(37) in
Section 4, we get:
; is the base point chosen by and ; , .
Using Equation (39), A takes plaintext
,
,
and obtains ciphertext
; and, using Equations (43)–(45),
provides the signature
:
sends and to .
After receiving the ciphertext
and signature
, B determines
, if so, then the authentication signature
comes from
;
decrypts
using (40) and obtains:
Thus, obtains the true profit values of as .The encryption–decryption process of is similar to , and is omitted.
Special note: Equation (57) and
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
, and then sends them to the manager using the secure retrieval algorithm in
Figure 2. The manager provides authentication for Equation (57) and
through the algorithm.
7. Conclusions
In application research fields such as data retrieval, data recognition, data hiding, and data camouflage, data
always exists simultaneously with its attribute set
(the feature set of data elements). The dynamic changes in data (decreases or increases in data elements within
) are synchronized with the dynamic changes of attributes (increases or decreases of attributes within
). 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
of data element
meet the attribute disjunctive characteristics. Function inverse P-sets [
19,
20] are obtained by replacing the internal data element
of (
x) with a function
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.