Encryption method based on high-dimensional iterative function

Abstract

The invention discloses a data encryption method based on a high-dimensional iterative function, and belongs to the technical field of data encryption. Comprising the following steps: S1, determining calculation parameters, and 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 the plaintext w and the ciphertext c, converting the plaintext to be encrypted into a plaintext space, and performing encryption and decryption. By adopting the method, the IFS different from the previous form is constructed, and an encryption system is constructed based on the two-dimensional IFS. In addition, according to the scheme, the randomness during encryption parameter selection is improved, and due to the fact that the trap door function is designed, nonlinearity exists, and the safety of the system is higher.

Description

Technical Field

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

[0002] Public-key cryptography based on iterated functions is a new type of cryptographic system that incorporates mathematical tools such as fractal theory and matrix operations. Its development is closely related to addressing the limitations of traditional public-key cryptography and the threat of quantum computing. The following is a comprehensive analysis of representative systems, their development history, and their application areas. However, current forensic evidence processing requires addressing several challenges, including the complexity and high cost of key distribution; the explosive growth of key combinations when communicating with multiple parties; the requirement for both parties to share a common key in order to send confidential information; and the security implications of the possibility that a receiver can forge a signature and the sender can deny having sent a message. Therefore, it is necessary to develop a public-key cryptography system based on iterated function systems to address these issues. Summary of the Invention

[0003] The purpose of this invention is to provide an encryption method based on a high-dimensional iterative function, propose an iterative function in two-dimensional space, construct a trapdoor accordingly, and improve the randomness of the secret key and the public key.

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

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

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

[0007] S3: For the plaintext w and ciphertext c, convert the plaintext to be encrypted into the plaintext space and perform encryption and decryption.

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

[0009] For 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 the 0,1 string space of length n;

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

[0013] 0≤a1 <b1≤t1-1;0≤a2<b2≤t2-1;

[0014] Randomly select positive integers δ1 and δ2 to 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 ; Set the following parameters accordingly through 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 Randomly select two integers c1 and c2, set the reversible function u(x,y) = (c1xy,c2y), and set the calculation parameters Get the private key sk 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 the trapdoor function.

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

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

[0025] Set up the binary function as follows:

[0026]

[0027] The formula defining the iterated function system is as follows:

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

[0029]

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

[0031]

[0032] The Φ in the above formula is expressed as Convert the fraction to an integer by dividing it by the remainder of p;

[0033] Setting up a binary function

[0034]

[0035] get:

[0036]

[0037] Get the public key pk 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 S3, the specific process of encryption and decryption is as follows:

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

[0042]

[0043] Where c is 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 based on t1, Write α2 as a decimal based on t2,

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

[0049] Therefore, the present invention adopts the above-mentioned encryption method based on high-dimensional iterative function, which has the following advantages:

[0050] This invention constructs an iterated function system that differs from previous approaches and builds an encryption system based on two-dimensional space. Furthermore, the encryption process of this application improves the randomness of the selected calculation parameters and designs a trapdoor function, making the encryption process nonlinear. This enhances the security of the system and can be applied to existing fields requiring data encryption, such as secure communications, data encryption, digital signatures, identity authentication, and quantum-resistant security infrastructure.

[0051] The technical solution of the present invention is further described in detail below through the accompanying drawings and embodiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0052] Figure 1 The present invention provides an encryption method based on a high-dimensional iterative function; DETAILED DESCRIPTION

[0053] In order to make the purpose, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are part of the embodiments of the present invention, not all of the embodiments. The components of the embodiments of the present invention generally described and shown in the drawings here can be arranged and designed in various different configurations. The specific model specifications need to be selected and determined based on the actual specifications of the device, etc. The specific selection calculation method adopts the existing technology in this field, so it will not be described in detail.

[0054] Example

[0055] like Figure 1 As shown, the present invention provides an encryption method based on a high-dimensional iterative function, which now uses actual data for encryption and decryption calculations, specifically including the following steps:

[0056] S1: Establish calculation parameters, and use the calculation parameters to determine the plaintext space P, ciphertext space C, and private key sk, as follows:

[0057] For a randomly selected positive integer n=1, t1=3, t2=3, the formula for obtaining the plaintext space is as follows:

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

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

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

[0061] 0≤a1 <b1≤t1-1;0≤a2<b2≤t2-1;

[0062] Randomly select positive integers δ1=1, δ2=1 to satisfy the following formula:

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

[0064] Randomly select a prime number p=29, satisfying the formula p>max{t1,t2} n+2 ; Set the following parameters accordingly through 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) from Randomly select two integers c1 = 1 and c2 = 1, then the reversible function is u(x, y) = (xy, y), and set the calculation parameter β = u -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 the trapdoor function.

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

[0072] Set up the binary function as follows:

[0073]

[0074] The formula defining the iterated function system is as follows:

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

[0076]

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

[0078]

[0079] The Φ in the above formula is expressed as Convert the fraction to an integer by dividing it by the remainder of p;

[0080] Setting up a binary function

[0081]

[0082] get:

[0083]

[0084] Get the public key pk 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 the plaintext w and ciphertext c, convert the plaintext to be encrypted into the plaintext space and perform encryption and decryption. The specific process of encryption and decryption 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 ciphertext result is as follows:

[0094]

[0095] Decryption process: For 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] If α1 is written as a decimal in base 3, α1=(1.0)3, and if α2 is written as a decimal in base 3, α2=(1.1)3, then r1=0, s1=1, satisfying a1≤r1≤δ1+a1-1, a2≤s1≤δ2+a2-1, so the plain text w=w1=0.

[0099] Therefore, the present invention adopts an encryption method based on high-dimensional iterative functions, constructs an iterative function system different from the previous form, and builds an encryption system based on two-dimensional space. In addition, for the encryption process of this application, the randomness when selecting calculation parameters is improved, and a trapdoor function is designed to make the encryption process nonlinear, making the system more secure.

[0100] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention rather than to limit the same. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that they can still modify or replace the technical solutions of the present invention with equivalents, and these modifications or equivalent replacements 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: The following steps are involved: S1: Establish calculation parameters, and determine the plaintext space P, ciphertext space C and private key sk through the calculation parameters; S2: Build an iterative function system and obtain the public key pk accordingly; S3: For the plaintext w and ciphertext c, convert the plaintext to be encrypted into the plaintext space and perform encryption and decryption.

2. The encryption method based on a high-dimensional iterative function according to claim 1, characterized in that: The specific process of S1 is as follows: For a randomly selected positive integer n, t1 ≥ 3, t2 ≥ 3, the formula for obtaining the plaintext space is as follows: P＝{0,1} n ； The plaintext space in the above formula is specifically the 0,1 string space of length n; Randomly select integers a1, b1, a2, and b2 that satisfy the following formula: 0≤a1 <b1≤t1-1;0≤a2<b2≤t2-1; Randomly select positive integers δ1 and δ2 to satisfy the following formula: 1≤δ1≤min{t1-b1,b1-a1}; 1≤δ2≤min{t2-b2,b2-a2}; Randomly select a prime number p that satisfies the formula p>max{t1,t2} n+2 ; Set the following parameters accordingly through prime numbers: The formula for obtaining the key space is as follows: from Randomly select a two-dimensional integer vector (x0, y0) from Randomly select two integers c1 and c2, set the reversible function u(x,y) = (c1xy,c2y), and set the calculation parameters Get the private key sk as follows: sk={t=(t1,t2,a1,a2,b1,b2,δ1,δ2),u(x,y)} In the above formula, u(x,y) is set as the trapdoor function.

3. The encryption method based on a high-dimensional iterative function according to claim 1, characterized in that: In S2, the specific process is as follows: The process of obtaining the public key is as follows: Set up the binary function as follows: The formula defining the iterated function system is as follows: For any plaintext, w=w1w2…w n ∈{0,1} n , the definition formula is as follows: In the above formula, Φ w (x,y) is an iterated function system. For positive integers m1, m2, k1, k2, set the function Φ as follows: The Φ in the above formula is expressed as Convert the fraction to an integer by dividing it by the remainder of p; Set up the binary function as follows: get: Get the public key pk as follows: pk={p,g0(x,y),g1(x,y)} In the above formula, p represents a randomly selected prime number.

4. The encryption method based on a high-dimensional iterative function according to claim 1, characterized in that: In S3, the specific process of encryption and decryption is as follows: Encryption process: For any plaintext w∈{0,1} n ,calculate The ciphertext c is obtained as follows: Where c is the ciphertext; Decryption process: For any ciphertext c, calculate m=(m1,m2), The calculation formula is as follows: Write α1 as a decimal based on t1, Write α2 as a decimal based on t2, When a1≤r i ≤δ1+a1-1,a2≤s i ≤δ2+a2-1, then determine w i =0, when b1≤r i ≤δ1+b1-1,b2≤s i ≤δ2+b2-1. When w i =1.

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