A public key encryption and decryption algorithm based on high-dimensional extended chebyshev polynomials

Abstract

The application provides a public key encryption and decryption algorithm based on high-dimensional extended Chebyshev polynomials, which comprises generating high-dimensional extended Chebyshev polynomials, calculating binary matrix multiplication of the high-dimensional extended Chebyshev polynomials, key generation, information encryption and information decryption. The application designs a public key encryption algorithm based on the recursive polynomials, enhances the resistance of the public key encryption algorithm based on Chebyshev polynomials to ciphertext-only attacks, and allows multiple plaintexts to be encrypted at one time, thereby improving the efficiency of the public key encryption algorithm.

Description

Technical Field

[0001] This invention belongs to the field of information security technology and relates to a public key encryption and decryption algorithm based on a high-dimensional extended Chebyshev polynomial. Background Technology

[0002] Since Diffie and Hellman proposed the idea of ​​public-key cryptography in 1976, cryptography has broken through the limitations of symmetric cryptosystems and entered a new stage of development. On the other hand, with the rapid development of chaos theory and its applications, introducing chaotic systems into public-key cryptosystems has become an important research direction in the field of public-key cryptography.

[0003] In 2003, Kocarev et al. first proposed a public-key encryption scheme based on Chebyshev chaotic maps (also known as Chebyshev polynomials). However, since the trigonometric and inverse trigonometric functions in the definition of the Chebyshev map both have obvious inverse processes, Bergamo et al. quickly proposed an attack method against this public-key encryption algorithm. Subsequently, Kocarev et al. extended the Chebyshev polynomials over the real number field to the finite field Z. N They also designed a public-key encryption algorithm.

[0004] Although Chebyshev polynomials over finite fields do not exhibit an obvious inverse process, the period of the numerical sequence generated by their iteration is a factor of N+1 or N-1 (where N is the modulus). Due to this periodicity, if the initial value of the Chebyshev polynomial or the finite field in the encryption scheme is poorly chosen, the period of the sequence generated by Chebyshev iteration can be too small. Attackers can then deduce the cryptographic key by observing the messages exchanged between legitimate communicating parties, rendering the algorithm vulnerable to ciphertext-only attacks and leading to serious security problems. Summary of the Invention

[0005] The purpose of this invention is to address the problems existing in the prior art by providing a public-key encryption and decryption algorithm based on a high-dimensional extended Chebyshev polynomial, which solves the problem that the sequence period generated by the iteration of existing Chebyshev polynomials is too short, resulting in low security of public-key encryption algorithms based on it.

[0006] Therefore, the present invention adopts the following technical solution:

[0007] A public-key encryption / decryption algorithm based on a high-dimensional extended Chebyshev polynomial includes the following steps:

[0008] Step 1: Generate a high-dimensional extended Chebyshev-like polynomial;

[0009] Step 2: Calculate the binary matrix multiplication of the Chebyshev polynomial;

[0010] Step 3: Key generation;

[0011] Step 4: Encrypt the information;

[0012] Step 5: Decrypt the information.

[0013] Furthermore, the recursive formula for the high-dimensional extended Chebyshev-like polynomial in step 1 is as follows:

[0014] T n+1 (X)=aXT n (X)-T n-1 (X)(mod N)

[0015] The recursive formula for the block matrix form of the high-dimensional extended Chebyshev polynomial is as follows:

[0016]

[0017] Where N is a large prime number, a is an integer such that 0 < a < N, and X is a finite field Z. N A k-order square matrix on the t-axis, T0 = 2a -1 E k (E k For the finite field Z N (a k-order identity matrix on the x-axis), T1 = X.

[0018] Further, step 2 includes the following steps:

[0019] Input: Finite field Z N Let X be a k-order square matrix, a be the iteration parameter, and n be the number of iterations (a and n are integers and 0 < a < N);

[0020] Output: Finite field Z N A k-order square matrix T;

[0021] Step 2a: Represent the integer n in binary form, that is:

[0022] n = b r 2 r +b r-1 2 r-1 +…+b12+b0

[0023] Among them, b i =1 or 0, i = 0, 1, ..., r;

[0024] Step 2b: Construct the block matrix D, which has the following form:

[0025]

[0026] Where X is a finite field Z N E is a k-order square matrix.k For the finite field Z N Let a be a k-order identity matrix on Z, a be the iteration parameter of a high-dimensional extended Chebyshev-like polynomial, and D be the finite field Z. N A 2k-order square matrix on top;

[0027] Step 2c: Use binary matrix multiplication to find D n Its form is:

[0028]

[0029] Where D is the finite field Z N A 2k-order square matrix on the top, b i The value is 0 or 1, and 0 ≤ i ≤ r;

[0030] Step 2d: D n Decompose it into a block matrix, which has the following form:

[0031]

[0032] Among them, G ij For the finite field Z N A k-order square matrix on the , i = 1, 2, j = 1, 2;

[0033] Step 2e: Calculate T, which is in the form of:

[0034] T=(2a -1 G 11 +G 12 X)(mod N)

[0035] Wherein, T, X, G 11 G 12 G 21 G 22 All are finite fields Z N Let N be a k-order square matrix, where N is a large prime number and a is the iteration parameter.

[0036] Furthermore, step 3 includes the following steps:

[0037] Step 3a: Choose a large prime number N, a k-order square matrix X over a finite field, and satisfy X N ≠X(mod N);

[0038] Step 3b: Select a random integer s, where s < 4N, and use the binary matrix multiplication of the high-dimensional extended Chebyshev polynomial from Step 1 to calculate A = Ts(X) (mod N);

[0039] Step 3c: Publish the matrix (A, X, N, a) as the public key and save the random integer s as the private key.

[0040] Furthermore, step 4 includes the following steps:

[0041] Input: Finite field Z N A plaintext square matrix M of order k on the key, with public key (A, X, N, a);

[0042] Output: Encrypted ciphertext matrices C1 and C2;

[0043] Step 4a: Select a random integer r, where r < 4N;

[0044] Step 4b: Calculate B = T using binary matrix multiplication of Chebyshev polynomials with high-dimensional extension. r (A)(modN), C1=T r (X)(modN);

[0045] Where B and C1 are both finite fields Z N k-order square matrix on;

[0046] Step 4c: Determine whether B is invertible. If B is invertible, calculate C2 = MB (mod N); otherwise, calculate C2 = M⊙B (mod N).

[0047] Where B, M, and C2 are all finite fields Z N MB(modN) represents a k-order square matrix in the finite field Z. N The ordinary product of k-order matrices M and B, M⊙B(mod N) represents the product of the k-order matrices M and B in the finite field Z. N The Hadamard product of k-order matrices M and B.

[0048] Furthermore, step 5 includes the following steps:

[0049] Input: Finite field Z N The k-order ciphertext matrix C1, C2, private key s, and public key (A, X, N, a) are given.

[0050] Output: A decrypted k-order plaintext matrix M′;

[0051] Step 5a: Calculate B′=T using binary matrix multiplication of Chebyshev-like polynomials with high-dimensional extension. s (C1)(modN);

[0052] Step 2: Determine if B′ is invertible. If B′ is invertible, calculate M′ = C²B′. -1 Otherwise, calculate M′=C2⊙B1(mod N), where the element b1 in B1 is... ij Let b' be an element in B'. ij The inverse element, i.e., b1 ij =b′ ij-1 (modN).

[0053] The beneficial effects of this invention are as follows:

[0054] 1. This invention extends the original Chebyshev polynomial while retaining its semigroup properties, thus enriching the range of commutative polynomials to be selected in public-key encryption algorithms.

[0055] 2. In this invention, the iteration parameter 'a' in the high-dimensional extended Chebyshev-like polynomial can be flexibly selected, which increases the key space of the algorithm to a certain extent.

[0056] 3. This invention can reduce the short-period characteristics of the iterative sequence to a negligible level by imposing simple constraints on the initial matrix in the high-dimensional extended Chebyshev-like polynomial, thereby improving the ability of public-key encryption algorithms to resist ciphertext-only attacks.

[0057] 4. This invention increases computational complexity by introducing matrix operations, thereby improving the algorithm's ability to resist exhaustive attacks. Attached Figure Description

[0058] Figure 1 This is a schematic diagram of binary matrix multiplication in this invention;

[0059] Figure 2 This is a statistical chart showing the encryption and decryption time consumption of Embodiment 3 of the present invention;

[0060] Figure 3 This is a statistical chart showing the time consumption of an exhaustive attack in Embodiment 4 of the present invention.

[0061] Specific implementation

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

[0063] A total of four embodiments are provided here, in which:

[0064] Example 1 is an example used to intuitively understand the algorithm flow of the present invention;

[0065] Example 2 observes the characteristics of a high-dimensional extended Chebyshev-like polynomial through experiments. It mainly observes the short-period probability of the high-dimensional extended Chebyshev-like polynomial under certain conditions of initial matrix, and compares it with the conventional Chebyshev polynomial to highlight the technical advantages of the present invention.

[0066] Examples 3 and 4 are observations of the public-key encryption algorithm based on high-dimensional extended Chebyshev polynomials through experiments. The algorithm execution time and the execution time of the simulated exhaustive attack are observed respectively. By comparing with the public-key encryption algorithm based on Chebyshev polynomials, the advantages of the algorithm in terms of efficiency and security are highlighted.

[0067] Example 1

[0068] A public-key encryption / decryption algorithm based on a high-dimensional extended Chebyshev polynomial is characterized by the following steps:

[0069] Step 1: Generate a high-dimensional extended Chebyshev-like polynomial:

[0070] The recurrence relation for the high-dimensional extended Chebyshev polynomial in step 1 is:

[0071] T n+1 (X)=aXT n (X)-T n-1 (X)(modN)

[0072] The recurrence relation for the block matrix form of the high-dimensional extended Chebyshev polynomial is as follows:

[0073]

[0074] Where N is a large prime number, a is an integer such that 0 < a < N, and X is a finite field Z. N A k-order square matrix on the t-axis, T0 = 2a -1 E k (E k For the finite field Z N (a k-order identity matrix on the x-axis), T1 = X.

[0075] Step 2: Calculate the binary matrix multiplication of the high-dimensional extended Chebyshev polynomials:

[0076] Input: Finite field Z N Let X be a k-order square matrix, with recursion parameter a and recursion number n (a and n are integers and 0 < a < N);

[0077] Output: Finite field Z N A k-order square matrix T;

[0078] Here, we set N = 251, n = 100, a = 120, and the second-order square matrix X is as follows:

[0079]

[0080] Step 2a: Convert the integer n into its binary representation, which is as follows:

[0081] n = 1 * 2 6 +1*2 5 +0*2 4 +0*2 3 +1*2 2 +0*2 1 +0*2 0

[0082] Step 2b: Construct the block matrix D as follows:

[0083]

[0084] like Figure 1 The diagram illustrates the binary matrix multiplication method for quickly calculating high-dimensional extended Chebyshev-like polynomials. It can quickly calculate the nth iteration result of high-dimensional extended Chebyshev-like polynomials with minimal memory usage.

[0085] Step 2c: Use binary matrix multiplication to find D n Its form is as follows:

[0086]

[0087] Step 2d: D n Represented as a block matrix, its form is as follows:

[0088]

[0089]

[0090]

[0091]

[0092]

[0093] Step 2e: Calculate T n (X), its formula is as follows:

[0094]

[0095] Step 3, Key Generation:

[0096] Step 3a: Choose a large prime number N, a k-order square matrix X over a finite field, and satisfy X N ≠X(modN), here N = 251, the second-order square matrix X is as follows:

[0097]

[0098] Step 3b: Select random integers a and s, where s < 4N and 0 < a < N. Here, we set a = 13 and s = 134.

[0099] Step 3c: Input X, a, and s to quickly compute the binary matrix multiplication of the high-dimensional extended Chebyshev polynomial to obtain A, which has the following form:

[0100]

[0101] Step 3d: Publish the matrix (A, X, N, a) as the public key and save s as the private key.

[0102] Step 4: Encrypt the information.

[0103] Input: Finite field Z N A plaintext square matrix M of order k on the key, with public key (A, X, N, a);

[0104] Output: The encrypted ciphertext matrices C1 and C2;

[0105] Step 4a: Select a random integer r, where r < 4N. Here, we set r = 191. The plaintext matrix M takes the following form:

[0106]

[0107] Step 4b: Input X, a, and r to quickly compute the binary matrix multiplication of the high-dimensional extended Chebyshev polynomial to obtain C1, which has the following form:

[0108]

[0109] Step 4c: Input A, a, r to quickly compute the binary matrix multiplication of the high-dimensional extended Chebyshev polynomial to obtain B, which has the following form:

[0110]

[0111] Step 4d: Determine whether B is an invertible matrix. Here, B is an invertible matrix.

[0112] Step 4e: Based on the judgment result in Step 4, B is an invertible matrix. Calculate C2 using B and M. The specific calculation method is as follows:

[0113]

[0114] Step 5, Information Decryption:

[0115] Input: Finite field Z N The k-order ciphertext matrix C1, C2, private key s, and public key (A, X, N, a) are given.

[0116] Output: A decrypted k-order plaintext matrix M′;

[0117] Step 5a: Input X, a, and s to quickly compute the binary matrix multiplication of the high-dimensional extended Chebyshev polynomial to obtain B′, which has the following form:

[0118]

[0119] Step 5b: Determine if B′ is invertible. Here, B′ is an invertible matrix, and its inverse matrix B′ -1 The format is as follows:

[0120]

[0121] Step 5c: Based on the judgment result of step 5b, calculate M′. The specific calculation method is as follows:

[0122]

[0123] To more intuitively demonstrate the advantages of high-dimensional extended Chebyshev polynomials over finite-field Chebyshev polynomials, experiments were conducted to measure the short-period probabilities of the iterative sequences of both, providing Examples 2 and 3 respectively.

[0124] In Example 2, when a = 2 and N is 31, 71, 101, 251, 1009, and 1997 respectively, the probability that the period of the high-dimensional extended Chebyshev-like iterative sequence under the 2nd to 5th order square matrix X is less than N-1 is selected, and the probability that the period of the finite field Chebyshev polynomial iterative sequence with the initial value in real form is less than N-1 is statistically analyzed.

[0125] As can be seen from Table 1, the short-period probability of the high-dimensional extended Chebyshev polynomial is significantly lower than that of the short-period probability of the finite field Chebyshev polynomial, making it more suitable for public-key encryption algorithms.

[0126] Table 1 Short-period probability statistics for Example 1

[0127]

[0128] In Example 3, a = 2 and N = 2. 1279 -1, the average time consumption statistics of the algorithm of this invention and the public key encryption algorithm based on Chebyshev polynomials for selecting 2nd to 5th order square matrices that meet the conditions respectively.

[0129] Figure 2 This is a statistical comparison of the time taken by using the public key encryption algorithm of the present invention and the time taken by using the public key encryption algorithm based on Chebyshev polynomials in Example 3.

[0130] The second column of Table 2 compares the average execution time of the encryption algorithm of this invention with the average execution time of the public key encryption algorithm based on Chebyshev polynomials. As can be seen from the table, the algorithm of this invention has a certain increase in execution time compared with the public key encryption algorithm based on Chebyshev polynomials.

[0131] Table 2. Average execution time statistics of the algorithm in Example 3.

[0132] real numbers 5.142059652805329 5.142059652805329 2nd order square matrix 15.75331128835678 3.938327822089195 3rd order square matrix 16.638419733047485 1.8487133036719428 4th order square matrix 34.20149348497391 2.1375933428108693 5th order square matrix 37.665357356071475 1.506614294242859

[0133] Considering that the encryption algorithm of this invention uses a k-order matrix, the number of ciphertexts transmitted is k. 2 The Chebyshev polynomial-based public-key encryption algorithm can only transmit one ciphertext at a time. To better compare algorithm efficiency, the third column of Table 2 normalizes the time consumption in the second column to each ciphertext, i.e., the average time consumption per ciphertext transmission. As can be seen from the table, the encryption algorithm of this invention is more efficient than the Chebyshev polynomial-based public-key encryption algorithm.

[0134] Explanation of the complexity of exhaustive attack:

[0135] For Chebyshev sequences in finite fields, the period is N-1 or N+1, and the probability of small periods occurring is very high. This can lead to serious security problems in Chebyshev public-key encryption algorithms, namely, the complexity of its brute-force attack is O(T). min ), T min ≤N+1, and T min The probability of <N-1 is relatively high. Meanwhile, the high-dimensional extended Chebyshev-like polynomial has an initial value matrix X ≠ X. N The probability of a small cycle occurring when (mod N) is very small, although the complexity of its exhaustive attack is also O(T). min However, usually T min The value of is usually greater than N, which greatly resists exhaustive attacks and makes the security performance better.

[0136] Exhaustive attack time calculation:

[0137] Because the public-key encryption algorithm proposed in this invention operates in matrix form, and matrix operations are more complex than real-number operations, even if the attack complexity of the two algorithms is the same, the attack time of the scheme proposed in this invention will be longer.

[0138] To simulate a brute-force attack, Example 4 was set up with a small amount of data.

[0139] In Example 4, a = 2 and N = 2. 17 When -1, the time taken to directly recursively derive N-1 times the initial value matrix of the high-dimensional extended Chebyshev polynomial using 2- to 5-stage matrices that meet the conditions is compared with the time taken to directly recursively derive N-1 times the initial value matrix of the finite field Chebyshev polynomial using random real numbers. Table 3 shows the average time taken in multiple experiments. Figure 3 As can be seen from Table 3, when the high-dimensional extended Chebyshev-like polynomial and the Chebyshev polynomial are directly recursively applied to the same number of times, the high-dimensional extended Chebyshev-like polynomial takes significantly longer than the Chebyshev polynomial. This demonstrates that the public-key encryption algorithm based on the Chebyshev polynomial of the finite field in this invention has a higher resistance to exhaustive attacks.

[0140] Table 3. Statistics on the average execution time of exhaustive attacks.

[0141] real numbers 0.044979915618896485 2nd order square matrix 1.2834997367858887 3rd order square matrix 1.3276434564590454 4th order square matrix 1.398881368637085 5th order square matrix 1.5174826169013977

Claims

1. A public-key encryption / decryption algorithm based on a high-dimensional extended Chebyshev polynomial, characterized in that, Includes the following steps: Step 1: Generate a high-dimensional extended Chebyshev-like polynomial; Step 1 includes the following steps: Generate high-dimensional extended Chebyshev-like polynomials: ； The recursive formula for generating the block matrix form of the high-dimensional extended Chebyshev polynomial is as follows: ； in, It is a large prime number. It is an integer and For a finite field On Square array , For a finite field On An identity matrix of order 1. ; Step 2: Calculate the binary matrix multiplication of the high-dimensional extended Chebyshev polynomial; Step 2 includes the following steps: Input: Finite field On Square Array Iteration parameters recursion count ,in All are integers, and ; Output: Finite field On Square Array ; Step 2a, convert the integer Represented in binary form, that is: ； in , ; Step 2b: Construct the block matrix Its form is: ； in, For a finite field On Square array For a finite field On An identity matrix of order 1. For the iterative parameters of the high-dimensional extended Chebyshev polynomial, For a finite field On Square matrix; Step 2c: Use binary matrix multiplication to find Its form is: ； in, For a finite field On Square array for or , ; Step 2d: Decompose it into a block matrix, which has the following form: ； in, For a finite field On Square Array ; Step 2e: Calculation Its form is: ； in, , , , , , All are finite fields On Square array It is a large prime number. For the iteration parameters of the high-dimensional extended Chebyshev polynomial; Step 3: Key generation; Step 3 includes the following steps: Step 3a: Choose a large prime number On a finite field Square Array And must meet ; Step 3b: Select a random integer And it must satisfy one of them. The binary matrix multiplication of the high-dimensional extended Chebyshev polynomial in step 1 is calculated using fast computation. ; Step 3c: Convert the matrix Published as a public key, random integer Stored as a private key; Step 4: Encrypt the information; Step 4 includes the following steps: Input: Finite field On Rank plaintext square array Public key ; Output: The encrypted ciphertext matrix ; Step 4a: Select a random integer And it must satisfy one of them. ; Step 4b: Calculate binary matrix multiplication using high-dimensional extended Chebyshev polynomials. = , = ; in, , All are finite fields On Square matrix; Step 4c: Determine Is it reversible? Reversible, computation Otherwise calculate ; in , All are finite fields On Square array Represents in a finite field superior 1-th order matrix and ordinary product, Represents in a finite field superior 1-th order matrix and Hadamaji; Step 5: Information decryption; Step 5 includes the following steps: Input: Finite field On Secret Array private key Public key ; Output: Decrypted Rank plaintext square array ; Step 5a: Calculate binary matrix multiplication using high-dimensional extended Chebyshev polynomials. ; Step 5b: Determine Is it reversible? Reversible, computation Otherwise calculate ,in, elements in for Middle elements The inverse, that is .

Images