A public-key cryptographic method based on iterative function systems

Abstract

This invention discloses a public-key cryptographic encryption method based on an iterative function system (IFS), relating to the field of cryptographic algorithms, including affine cryptosystems and projective cryptosystems. Both of these cryptosystems are constructed using iterative functions based on Cantor sets. The construction of an affine cryptosystem includes the following steps: S1: Constructing an iterative function (IFS); S2: Constructing a public-key cryptosystem; S3: Identity and message integrity verification. The construction of a projective cryptosystem includes the following steps: S11: Defining a projective space of dimension *n* over a field, constructing an IFS, where all elements are in the IFS; S22: Introducing matrix elements; S33: Generating a key, obtaining a public key and a private key, and then encrypting and decrypting to obtain the complete plaintext. This invention employs the above-mentioned cryptographic system, resulting in high efficiency and stronger security in the encryption and decryption processes.

Description

Technical Field

[0001] This invention relates to the field of cryptographic algorithm technology, and in particular to a public-key cryptographic encryption method based on an iterative function system. Background Technology

[0002] Before public-key cryptography, all cryptographic algorithms were based on the two fundamental tools of substitution and permutation. In 1976, Whitefield Difie and Martin Helman pioneered the concept of public-key cryptography, also known as asymmetric cryptography, in their book *New Directions in Cryptography*. Public-key cryptography provided a new theoretical and technological foundation for the development of cryptography: 1. The basic attacks on public-key cryptographic algorithms are no longer substitution and permutation, but mathematical functions; 2. Public-key cryptography uses two keys in an asymmetric manner, and the use of these two keys has profound implications for confidentiality, key distribution, and authentication.

[0003] The concept of public-key cryptography arose from the problems of digital signatures and public key distribution. Solving the public key distribution problem requires that the communicating parties either already have a shared key or a central key distribution point. Solving the digital signature problem considers how to provide a method for signing digitized messages or documents, similar to signing a written document, primarily to ensure that transmitted data cannot be modified. The most significant characteristic of public-key cryptography algorithms is the use of two related keys to separate encryption and decryption capabilities. One key, publicly known and used for encryption, is used in conjunction with a private key, which is kept secret and used for decryption.

[0004] Currently, there are three main public-key cryptosystems: RSA, based on the large integer factorization problem; ElGamal, based on the discrete logarithm problem over the multiplicative group of finite fields; and elliptic curve cryptography, based on the discrete logarithm problem over elliptic curves. They all suffer from the following problems:

[0005] ① The key distribution process is complex and costly; ② When multiple people communicate, the number of key combinations can explode; ③ Both parties must use the same key to send confidential information; ④ The receiver can forge a signature, and the sender can deny sending certain information.

[0006] Therefore, it is necessary to provide a public-key cryptographic method based on an iterative function system to solve the above problems. Summary of the Invention

[0007] The purpose of this invention is to provide a public-key cryptographic encryption method based on an iterative function system. The key generation is relatively simple. According to the construction of the IFS, each bit of the ciphertext is distributed in different terms, which increases security. No matter how long the plaintext is, the transmitted ciphertext is actually a number or a matrix. This greatly reduces the amount of communication during transmission, improves the efficiency of the cryptographic system, is conducive to multi-person communication, has corresponding identity and message integrity verification procedures, and avoids forgery attacks.

[0008] To achieve the above objectives, this invention provides a public-key cryptographic encryption method based on an iterative function system, including an affine cryptosystem and a projective cryptosystem. Both the affine cryptosystem and the projective cryptosystem are constructed using iterative functions based on Cantor sets. The construction of the affine cryptosystem includes the following steps:

[0009] S1: Construct the iterative function IFS;

[0010] S2: Construct a public-key cryptosystem;

[0011] S3: Identity and message integrity verification;

[0012] The construction of the projection cryptosystem includes the following steps:

[0013] S11: Definition For domain The upper dimension is Constructing the IFS in the projection space The elements therein are all in superior;

[0014] S22: Introducing matrix elements;

[0015] S33: Generate the private key and obtain the public key. and private key Then, encryption and decryption are performed to obtain the complete plaintext.

[0016] Preferably, step S1 includes the following steps:

[0017] S1A: Construction For sets with separation properties Mapped to The set of mappings;

[0018] S1B: Separation characteristics are for All Define a from arrive The double shot, and ;Calculate the Hausdorff dimension of the limit set of the above IFS;

[0019] S1C: Plaintext Define an iterative function system , ;

[0020] S1D: Plaintext length is and points At that time, find the Solution: ;

[0021] S1E: Obtain the complete plaintext ,

[0022] The formula has no solution.

[0023] If the formula has a solution, then , For plain text First place;

[0024]

[0025]

[0026] Based on the above formula, we obtain the new... and By analogy, the complete plaintext can be obtained.

[0027] Preferably, in step S2, the construction of the public-key cryptosystem involves the following specific steps:

[0028] S2A: For prime numbers, define A sub-ring: , defined from Mapped to mapping Extending to polynomial rings: yes The least common multiple of the denominators of the middle coefficients. choose Each value range and Intervals, .

[0029] S2B: Key generation, prime number selection Set function and , , defined in IFS in: and ,definition , To define the left endpoint of the interval, obtain the public key. and private key :

[0030]

[0031] S2C: Encryption process:

[0032]

[0033] ;

[0034] S2D: Decryption Process: Calculation

[0035]

[0036]

[0037]

[0038] in, Using the results obtained in step S1F To obtain the complete plaintext .

[0039] Preferably, in step S11, there is A reversible matrix For plaintext , preset .

[0040] Preferably, in step S22, matrix elements are introduced, using known... get Specifically, this includes: selecting a point , ,exist Make ,get , making the new , Continue until only one identity matrix remains, at which point the complete plaintext is obtained. .

[0041] Preferably, in step S33, the key generation selects a prime number. Greater than Take twice the largest coefficient. The actual length and the maximum length of the plaintext. Using integer coefficients to construct a determinant is not multiples Dimensional Matrix ,make , Obtain the public key and private key :

[0042] ；

[0043] Encryption process,

[0044]

[0045] ;

[0046] Decryption process: ;get , and complete plaintext .

[0047] The preferred method is the Cantor tripartite set. To continuously trisect an interval and remove the intermediate interval after each trisection, the steps are as follows:

[0048] initial interval The reserved interval after the first division , Divide the two remaining intervals into three equal parts and remove the middle interval. This second division results in four remaining intervals. , , , Continue according to the three-part division plan The division, in the... After the second partitioning, we get each interval , This completes the division of the interval.

[0049] Therefore, the present invention employs the above-mentioned public-key cryptography encryption method based on an iterative function system, which has the following beneficial effects:

[0050] (1) The generation of the key of the present invention is relatively simple, and the encryption and decryption process is relatively simple without losing security.

[0051] (2) Based on the construction of IFS, the present invention distributes each bit of the ciphertext into different terms, which increases the security to a certain extent.

[0052] (3) Due to the design of IFS, the ciphertext transmitted is actually a number or a matrix, regardless of the length of the plaintext. This greatly reduces the amount of communication during transmission and improves the efficiency of the cryptographic system to a certain extent.

[0053] (4) After each participant in this invention constructs its own IFS, the ciphertext transmitted is only a number or a matrix, which is beneficial for multiple participants to transmit information and facilitates multi-person communication.

[0054] (5) The present invention has a corresponding identity and message integrity verification procedure, which avoids forgery attacks to a certain extent and increases security.

[0055] (6) The encryption and decryption processes of this invention are highly efficient, and the final plaintext can be obtained more quickly by combining the properties of Cantor set and IFS.

[0056] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Attached Figure Description

[0057] Figure 1 This is a flowchart of an affine cryptosystem based on a public-key cryptographic encryption method using an iterative function system, as described in this invention.

[0058] Figure 2 This is a flowchart of a projection cryptosystem based on a public-key cryptographic encryption method using an iterative function system, according to the present invention.

[0059] Figure 3 This is a flowchart illustrating the identity and message integrity verification process according to an embodiment of the present invention;

[0060] Figure 4 This is the Cantor ternary set graph of the public-key cryptographic encryption method based on iterative function systems of the present invention;

[0061] Figure 5 This is the five-part encrypted ternary plaintext diagram in Embodiment 2 of the present invention. Detailed Implementation

[0062] The technical solution of the present invention will be further described below with reference to the accompanying drawings and embodiments.

[0063] Unless otherwise defined, the technical or scientific terms used in this invention shall have the ordinary meaning as understood by one of ordinary skill in the art to which this invention pertains.

[0064] The terms "comprising" or "including" as used in this invention mean that the element preceding the term encompasses the element listed after the term, and do not exclude the possibility of encompassing other elements. Terms such as "inner," "outer," "upper," and "lower" indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings, and are only for the convenience of describing the invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, they should not be construed as limitations on the invention. When the absolute position of the described object changes, the relative positional relationship may also change accordingly. In this invention, unless otherwise explicitly specified and limited, the term "attached" and similar terms should be interpreted broadly. For example, it can refer to a fixed connection, a detachable connection, or an integral part; it can refer to a direct connection or an indirect connection through an intermediate medium; it can refer to the internal communication of two elements or the interaction relationship between two elements. Those skilled in the art can understand the specific meaning of the above terms in this invention according to the specific circumstances.

[0065] like Figures 1-2 As shown, this invention provides a public-key cryptographic encryption method based on an iterative function system, including an affine cryptosystem and a projective cryptosystem. Both the affine cryptosystem and the projective cryptosystem are constructed from iterative functions based on Cantor sets. Cantor has three sets, as shown in the figure. Figure 4 As shown, take To continuously trisect an interval and remove the intermediate intervals after each trisection, the steps include: Initial interval The reserved interval after the first division , Divide the two remaining intervals into three equal parts and remove the middle interval. This second division results in four remaining intervals. , , , Continue according to the three-part division plan The division, in the... After the second partitioning, we get each interval , This completes the division of the interval.

[0066] The construction of an affine cipher system includes the following steps:

[0067] S1: Construct the iterative function IFS; Step S1 includes the following steps:

[0068] S1A: Construction For sets with separation properties Mapped to The set of mappings;

[0069] S1B: Separation characteristics are for All Define a from arrive The double shot, and ;Calculate the Hausdorff dimension of the limit set of the above IFS;

[0070] S1C: Plaintext Define an iterative function system , ;

[0071] S1D: Plaintext length is and points At that time, find the Solution: ;

[0072] S1E: Obtain the complete plaintext ,

[0073] The formula has no solution.

[0074] If the formula has a solution, then , For plain text First place;

[0075]

[0076]

[0077] Based on the above formula, we obtain the new... and By analogy, the complete plaintext can be obtained.

[0078] S2: Constructing a public-key cryptosystem; In step S2, the construction of the public-key cryptosystem involves the following specific steps:

[0079] S2A: For prime numbers, define A sub-ring: , defined from Mapped to mapping Extending to polynomial rings: yes The least common multiple of the denominators of the middle coefficients. choose Each value range and Intervals, .

[0080] S2B: Key generation, prime number selection Set function and , , defined in IFS in: and ,definition , To define the left endpoint of the interval, obtain the public key. and private key :

[0081]

[0082] S2C: Encryption process:

[0083]

[0084] ;

[0085] S2D: Decryption Process: Calculation

[0086]

[0087]

[0088]

[0089] in, Using the results obtained in step S1F To obtain the complete plaintext .

[0090] S3: Identity and message integrity verification;

[0091] The construction of a projection cryptosystem includes the following steps:

[0092] S11: Definition For domain The upper dimension is Constructing the IFS in the projection space The elements therein are all in It has reversible homology; in step S11, there is A reversible matrix For plaintext , preset , and They have homophony.

[0093] S22: Introducing matrix elements; In step S22, matrix elements are introduced, through known... get Specifically, this includes: selecting a point , ,exist Make ,get , making the new , Continue until only one identity matrix remains, at which point the complete plaintext is obtained. .

[0094] S33: Generate the private key and obtain the public key. and private key Then, encryption and decryption are performed to obtain the complete plaintext. In step S33, a prime number is selected for key generation. Greater than Take twice the largest coefficient. The actual length and the maximum length of the plaintext. Using integer coefficients to construct a determinant is not multiples Dimensional Matrix ,make , Obtain the public key and private key :

[0095] ；

[0096] Encryption process,

[0097]

[0098] ;

[0099] Decryption process: ;get , and complete plaintext .

[0100] Example 1

[0101] Identity and message integrity verification, such as Figure 3 As shown, Alice selected a projection with separation properties. And it has a key , Bob based on the selected Generate ciphertext from plaintext After that, another projection at the same position was selected. There is a key , Alice arbitrarily chooses two lengths. of , And according to private key in And Bob's public key in get and and send For Bob, here refer to and The concatenation of the data. Assume some of the bits encode the plaintext length. Bob got After that, Bob can use private key Obtained based on separation characteristics Bob according to Length in Bob then uses the information to verify the integrity of the message. Alice receives the ciphertext that was originally meant to be transmitted. Bob is the only one who can get it easily. This step verifies the identity and integrity of the message by verifying the participants.

[0102] Example 2

[0103] The construction of iterative function systems is based on Cantor sets, and the key to constructing IFS is the construction of... Make In the Cantor trilogy, the choice was made... With the above conditions met, the construction of the IFS is basically complete. The next step is to combine the IFS with a cryptographic system.

[0104] plaintext Through IFS, the final result was:

[0105]

[0106] Pick ,get This is through get Abstract representation of, i.e., judgment The specific method for determining which interval it falls into is to obtain it bit by bit as described in Part Three.

[0107] It can be used for encryption The binary plaintext. Based on the different numbers of equal parts divided into intervals and The difference in the number of mappings is ultimately used for encryption.

[0108] The plaintext. For example... Figure 5 As shown, the second and fourth intervals are discarded to encrypt the ternary plaintext.

[0109] Example 3

[0110] Cantor's three parts and For example, The choice can be directly obtained. Write it in ternary form: , The value is the number after the decimal point. Bit.

[0111]

[0112] like ,like .

[0113] Similarly, different choices of equal division can be represented in different number systems to quickly obtain the plaintext. The value of .

[0114] Therefore, the present invention adopts the above-mentioned public-key cryptography encryption method based on iterative function system. The key generation is relatively simple. According to the construction of IFS, each bit of the ciphertext is distributed in different terms, which increases security. No matter how long the plaintext is, the transmitted ciphertext is actually a number or a matrix. This greatly reduces the amount of communication during transmission, improves the efficiency of the cryptographic system, is conducive to multi-person communication, has corresponding identity and message integrity verification procedures, and avoids forgery attacks.

[0115] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the technical solutions of the present invention, and these modifications or equivalent substitutions cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A public-key cryptographic encryption method based on an iterative function system, characterized in that: This includes affine cryptosystems and projective cryptosystems, both constructed using iterative functions based on Cantor sets. The construction of the affine cryptosystem includes the following steps: S1: Construct the iterative function IFS; S2: Construct a public-key cryptosystem; S3: Identity and message integrity verification; The construction of the projection cryptosystem includes the following steps: S11: Definition For domain The upper dimension is Constructing IFS in the projection space The elements therein are all in superior; S22: Introducing matrix elements; S33: Generate the private key and obtain the public key. and private key Then, encryption and decryption are performed to obtain the complete plaintext.

2. The public-key cryptographic encryption method based on an iterative function system according to claim 1, characterized in that: Step S1 includes the following steps: S1A: Construction For sets with separation properties Mapped to The set of mappings; S1B: Separation characteristics are for All Define a from arrive The double shot, and ;Calculate the Hausdorff dimension of the limit set of the above IFS; S1C: Plaintext Define an iterative function system , ; S1D: Plaintext length is and points At that time, find the Solution: ; S1E: Obtain the complete plaintext , The formula has no solution. If the formula has a solution, then , For plain text First place; Based on the above formula, we obtain the new... and By analogy, the complete plaintext can be obtained.

3. The public-key cryptographic encryption method based on an iterative function system according to claim 2, characterized in that: In step S2, the construction of the public-key cryptosystem involves the following steps: S2A: For prime numbers, define A sub-ring: , defined from Mapped to mapping Extending to polynomial rings: yes The least common multiple of the denominators of the middle coefficients. choose Each value range and Intervals, ; S2B: Key generation, prime number selection Set function and , , defined in IFS in: and ,definition , To define the left endpoint of the interval, obtain the public key. and private key : S2C: Encryption process: ； S2D: Decryption Process: Calculation in, Using the results obtained in step S1F To obtain the complete plaintext .

4. The public-key cryptographic encryption method based on an iterative function system according to claim 3, characterized in that: In step S11, there are A reversible matrix For plaintext , preset .

5. The public-key cryptographic encryption method based on an iterative function system according to claim 4, characterized in that: In step S22, matrix elements are introduced, using known... get Specifically, this includes: selecting a point , ,exist Make ,get , making the new , Continue until only one identity matrix remains, at which point the complete plaintext is obtained. .

6. The public-key cryptographic encryption method based on an iterative function system according to claim 5, characterized in that: In step S33, the key generation selects prime numbers. Greater than Take twice the largest coefficient. The actual length and the maximum length of the plaintext. Using integer coefficients to construct a determinant is not multiples Dimensional Matrix ,make , Obtain the public key and private key : ； Encryption process, ； Decryption process: ; get , and complete plaintext .

7. The public-key cryptographic encryption method based on an iterative function system according to claim 1, characterized in that: Cantor in three parts, take To continuously trisect an interval and remove the intermediate interval after each trisection, the steps are as follows: initial interval The reserved interval after the first division , Divide the two remaining intervals into three equal parts and remove the middle interval. This second division results in four remaining intervals. , , , Continue according to the three-part division plan The division, in the... After the second partitioning, we get each interval , This completes the division of the interval.

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