An encryption method based on high-dimensional iterative functions

Abstract

The application discloses a data encryption method based on high-dimensional iterative functions and belongs to the technical field of data encryption; the method comprises the following steps: S1, establishing calculation parameters, determining a plaintext space P, a ciphertext space C and a private key sk through the calculation parameters; S2, constructing an iterative function system and correspondingly obtaining a public key pk; and S3, for plaintext w and ciphertext c, converting the plaintext to be encrypted into the plaintext space, and performing encryption and decryption. The method is used to construct an IFS different from the previous forms, and an encryption system is constructed based on a two-dimensional IFS. In addition, the scheme improves the randomness when encryption parameters are selected, and the security of the system is stronger due to the designed trapdoor function and the existence of nonlinearity.

Description

Technical Field

[0001] This invention relates to the technical field of data encryption, and in particular to an encryption method based on a high-dimensional iterative function. Background Technology

[0002] Public-key cryptosystems based on mathematical iterative functions are a new type of cryptosystem that combines fractal theory, matrix operations, and other mathematical tools. Their development is closely related to addressing the limitations of traditional public-key cryptography and the threats posed by quantum computing. The following is a comprehensive analysis of representative systems, their development history, and application areas: However, current forensic processes present several problems that need to be addressed, including the complexity and high cost of key distribution; the explosive growth in the number of key combinations during multi-person communication; the requirement for both parties to unify their keys to send confidential information; and security vulnerabilities such as the receiver being able to forge signatures and the sender being able to deny having sent certain information. Therefore, it is necessary to provide a public-key cryptosystem based on iterative function systems to solve these problems. Summary of the Invention

[0003] The purpose of this invention is to provide an encryption method based on a high-dimensional iterative function. A two-dimensional iterative function is proposed, a trapdoor is constructed accordingly, and the randomness of the key and public key is improved.

[0004] To achieve the above objectives, this invention provides an encryption method based on a high-dimensional iterative function, comprising the following steps:

[0005] S1: Establish computation parameters, and determine the plaintext space P, ciphertext space C, and private key sk based on the computation parameters.

[0006] S2: Construct an iterative function system and obtain the public key pk accordingly;

[0007] S3: For plaintext w and ciphertext c, convert the plaintext to be encrypted into plaintext space, and then encrypt and decrypt it.

[0008] Preferably, the specific process of S1 is as follows:

[0009] Given a randomly selected positive integer n, t1≥3, t2≥3, the formula for obtaining the plaintext space is as follows:

[0010] P = {0, 1} n ;

[0011] The plaintext space in the above formula is specifically a 0,1 string space of length n;

[0012] Randomly selected integers a1, b1, a2, and b2 satisfy the following formula:

[0013] 0≤a1 <b1≤t1-1;0≤a2<b2≤t2-1;

[0014] Randomly selected positive integers δ1 and δ2 satisfy the following formula:

[0015] 1≤δ1≤min{t1-b1,b1-a1}; 1≤δ2≤min{t2-b2,b2-a2};

[0016] Randomly select a prime number p that satisfies the formula p > max{t1,t2} n+2 The parameters are set accordingly using prime numbers:

[0017]

[0018] The formula for obtaining the key space is as follows:

[0019]

[0020] from Randomly select a two-dimensional integer vector (x0, y0) from the given data. Two integers c1 and c2 are randomly selected. An invertible function u(x,y) = (c1xy,c2y) is set, and the calculation parameters are set. The private key sk is obtained as follows:

[0021] sk={t=(t1,t2,a1,a2,b1,b2,δ1,δ2),u(x,y)}

[0022] In the above formula, u(x,y) is set as a trapdoor function.

[0023] Preferably, in step S2, the specific process is as follows:

[0024] The process of obtaining the public key is as follows:

[0025] The binary function is set as follows:

[0026]

[0027] The formula for defining an iterative function system is as follows:

[0028] For any plaintext, w = w1w2…w n ∈{0,1} n The formula is defined as follows:

[0029]

[0030] In the above formula, φ w (x, y) is an iterative function system. For positive integers m1, m2, k1, k2, the function Φ is set as follows:

[0031]

[0032] In the above formula, Φ represents: Convert the fraction to an integer by taking the remainder when divided by p.

[0033] Set a binary function

[0034]

[0035] get:

[0036]

[0037] The public key PK is obtained as follows:

[0038] pk = {p, g0(x, y), g1(x, y)}

[0039] In the above formula, p represents a randomly selected prime number.

[0040] Preferably, in step S3, the specific encryption and decryption processes are as follows:

[0041] Encryption process: For any plaintext w∈{0,1} n ,calculate The ciphertext c is obtained as follows

[0042]

[0043] Where c represents the ciphertext;

[0044] Decryption process: For any ciphertext c, calculate m = (m1, m2),

[0045] The calculation formula is as follows:

[0046]

[0047] Write α1 as a decimal with base t1. Write α2 as a decimal with base t2.

[0048] When a1≤r i ≤δ1+a1-1,a2≤s i If w ≤ δ² + a² - 1, then determine w. i =0, when b1≤r i ≤δ1+b1-1,b2≤s i ≤δ²+b²-1. When, then determine w i =1.

[0049] Therefore, the encryption method based on a high-dimensional iterative function described above, as used in this invention, has the following advantages:

[0050] This invention constructs an iterative function system that differs from previous forms and builds an encryption system based on two-dimensional space. Furthermore, the encryption process of this application improves the randomness of selecting computational parameters and designs a trapdoor function, making the encryption process nonlinear and enhancing the system's security. This makes it applicable to existing fields requiring data encryption, such as secure communication, data encryption, digital signatures, identity authentication, and quantum-resistant secure infrastructure.

[0051] 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

[0052] Figure 1 This invention provides an encryption method based on a high-dimensional iterative function; Detailed Implementation

[0053] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. The components of the embodiments of the present invention described and shown in the accompanying drawings can generally be arranged and designed in various different configurations. Specific model specifications need to be selected and determined according to the actual specifications of the device, etc. The specific selection calculation method adopts existing technology in the art, and therefore will not be described in detail.

[0054] Example

[0055] like Figure 1 As shown, this invention provides an encryption method based on a high-dimensional iterative function. The encryption and decryption calculations are performed using actual data, specifically including the following steps:

[0056] S1: Establish the computation parameters, and determine the plaintext space P, ciphertext space C, and private key sk based on the computation parameters, as follows:

[0057] The formula for obtaining the plaintext space by randomly selecting positive integers n=1, t1=3, t2=3 is as follows:

[0058] P = {0, 1} n ;

[0059] The plaintext space in the above formula is specifically a 0,1 string space of length n;

[0060] Randomly selected integers a1 = 0, b1 = 1, a2 = 1, b2 = 2 satisfy the following formula:

[0061] 0≤a1 <b1≤t1-1;0≤a2<b2≤t2-1;

[0062] Randomly selected positive integers δ1 = 1 and δ2 = 1 satisfy the following formula:

[0063] 1≤δ1≤min{t1-b1,b1-a1}; 1≤δ2≤min{t2-b2,b2-a2};

[0064] A prime number p = 29 is randomly selected, satisfying the formula p > max{t1,t2}. n+2 The parameters are set accordingly using prime numbers:

[0065]

[0066] The formula for obtaining the key space is as follows:

[0067]

[0068] from Randomly select a two-dimensional integer vector (x0, y0) = (3, 3), and from... If two integers c1 = 1 and c2 = 1 are randomly selected, then the invertible function is u(x,y) = (xy,y), and the calculation parameter β = u is set. -1 (x0,y0)=(1,3), the private key sk is obtained as follows:

[0069] sk={t=(t1,t2,a1,a2,b1,b2,δ1,δ2),u(x,y)}

[0070] In the above formula, u(x,y) is set as a trapdoor function.

[0071] S2: Construct an iterative function system and obtain the public key pk accordingly; the process of obtaining the public key is as follows:

[0072] The binary function is set as follows:

[0073]

[0074] The formula for defining an iterative function system is as follows:

[0075] For any plaintext, w = w1w2…w n ∈{0,1} n The formula is defined as follows:

[0076]

[0077] In the above formula, φ w (x, y) is an iterative function system. For positive integers m1, m2, k1, k2, the function Φ is set as follows:

[0078]

[0079] In the above formula, Φ represents: Convert the fraction to an integer by taking the remainder when divided by p.

[0080] Set a binary function

[0081]

[0082] get:

[0083]

[0084] The public key PK is obtained as follows:

[0085] pk = {p0, g0(x, y), g1(x, y)}

[0086] In the above formula, p represents a randomly selected prime number.

[0087] S3: For plaintext w and ciphertext c, convert the plaintext to be encrypted into plaintext space, and then perform encryption and decryption. The specific encryption and decryption process is as follows:

[0088] Encryption process: For any plaintext w = 0 ∈ {0, 1} 1 ,calculate The ciphertext c is obtained as follows

[0089]

[0090] The calculation results are as follows:

[0091] 3 -1 =10mod29

[0092]

[0093] The final encrypted result is as follows:

[0094]

[0095] Decryption process: For the ciphertext c = (8, 11), calculate m = (m1, m2) = u(c) = (1, 11), m = (m1, m2), then m1 = 1, m2 = 11.

[0096] The calculation results are as follows:

[0097]

[0098] Write α1 as a base-3 decimal, α1 = (1.0)3, and write α2 as a base-3 decimal, α2 = (1.1)3. Therefore, r1 = 0 and s1 = 1, satisfying a1 ≤ r1 ≤ δ1 + a1 - 1 and a2 ≤ s1 ≤ δ2 + a2 - 1. Hence, the plaintext w = w1 = 0.

[0099] Therefore, this invention employs an encryption method based on a high-dimensional iterative function, constructing an iterative function system different from previous forms, and building an encryption system based on two-dimensional space. Furthermore, the encryption process of this application improves the randomness of selecting computational parameters and designs a trapdoor function, making the encryption process nonlinear and thus enhancing the system's security.

[0100] 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. An encryption method based on a high-dimensional iterative function, characterized in that: Includes the following steps: S1: Establish calculation parameters and determine the plaintext space based on these parameters. ciphertext space and private key The specific process is as follows: Randomly selected positive integers , , The formula for obtaining the plaintext space is as follows: ； The plaintext space in the above formula is specifically the length. The 0,1 string space; Randomly select integers 、 、 Satisfy the following formula: ； ； Randomly select positive integers , The formula is as follows: ； ； Randomly select prime numbers Satisfying the formula The parameters are set accordingly using prime numbers: ， The formula for obtaining the key space is as follows: ； from Randomly select a two-dimensional integer vector ,from Two integers are randomly selected from the data. and Set an invertible function Set calculation parameters Obtain the private key as follows: In the above formula, Set as a trapdoor function S2: Construct an iterative function system and obtain the corresponding public key. The specific process is as follows: The process of obtaining the public key is as follows: The binary function is set as follows: ， ； The formula for defining an iterative function system is as follows: For any plaintext, The formula is defined as follows: ； In the above formula, For an iterative function system, for positive integers Set function as follows: ； In the above formula Represented as Divide by The remainder converts the fraction to an integer; The binary function is set as follows: ， ， get: ， Obtain the public key as follows: In the above formula, Represents a randomly selected prime number; S3: For plaintext and ciphertext The plaintext to be encrypted is converted into plaintext space for encryption and decryption.

2. The encryption method based on a high-dimensional iterative function according to claim 1, characterized in that: In S3, the specific encryption and decryption processes are as follows: Encryption process: For any plaintext ,calculate Received the ciphertext as follows: ； in It is encrypted; Decryption process: For any ciphertext ,calculate , , The calculation formula is as follows: 、 , Will Writing decimals with base 1 ,Will Writing decimals with base 1 ; when Then determine ,when When, then determine .

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