Computer method and system for determining cryptographic keys

Abstract

To provide a computer method and system for determining an encryption key. [Solution] The method includes the steps of: constructing a tensor network using parameters representing candidate cryptographic keys; adjusting the parameters of the tensor network; generating a new candidate key sample and obtaining a candidate ciphertext obtained using the candidate cryptographic key; calculating a cost function for a target ciphertext; measuring the overlap between the target ciphertext and the candidate ciphertext; and determining whether the overlap has reached a threshold. If the threshold is not reached, the method is repeated by further adjusting the parameters of the tensor network; and if the threshold is reached, the candidate cryptographic key is determined to be a cryptographic key.

Description

Technical Field

[0001] (Cross - reference to related applications) This application claims priority to European Patent Application EP24383492.6 filed on December 31, 2024, and US Patent Application US19 / 284,077 filed on July 29, 2025, the entire contents of which are incorporated herein by reference.

[0002] The field of the present invention relates to computer - implemented methods and systems for determining cryptographic keys.

Background Art

[0003] Maintaining the security of information is required in various fields, including but not limited to defense, finance, and personal privacy. Currently, cryptographic protocols are commonly used to secure confidential data by using cryptographic keys to convert plaintext into encrypted ciphertext. Conventionally, attackers have employed brute - force attacks or sophisticated algorithms to recover these cryptographic keys. However, with the advent of quantum computing, the threat to current cryptographic protocols has increased due to quantum algorithms such as Shor's algorithm and Grover's algorithm, which can break current encryption protocols faster than using classical methods.

[0004] Currently, due to noisy intermediate - scale quantum (NISQ) hardware, there are limitations regarding the use of quantum computers for their current adaptability. NISQ hardware is characterized by a limited number of qubits and other performance bottlenecks. Quantum - inspired approaches, such as variational quantum algorithms (VQAs), have been developed to address these constraints. These approaches still rely on quantum hardware and pose significant barriers due to high costs and operational complexities.

[0005] One example of a cryptographic protocol is the Simplified Data Encryption Standard (S-DES). S-DES, or Simplified Data Encryption Standard, is a cryptographic protocol designed to provide a fundamental understanding of symmetric-key cryptography. The S-DES protocol uses a 10-bit key, operates on 8-bit blocks of data, and uses a series of rearrangement and substitution techniques to convert plaintext to ciphertext and vice versa. A brute-force attack attempting to decrypt the ciphertext requires an average of 512 iterations. In a brute-force attack, all possible keys are tried until the correct key is found. Assuming the S-DES protocol uses a 10-bit key, there are 2 to the power of 10 = 1024 possible keys, and on average, half of these (512) need to be tried to find the correct key.

[0006] Another example of an encryption standard is the Simplified Advanced Encryption Standard (S-AES), which is an educational adaptation of the widely used Advanced Encryption Standard (AES) algorithm. S-AES simplifies the essential components of the encryption process. The S-AES algorithm typically operates on small 16-bit data blocks, uses shorter key lengths compared to the standard AES algorithm, and employs reorder-substitution to convert plaintext to ciphertext and ciphertext to plaintext.

[0007] Blowfish is another symmetric-key block cipher encryption algorithm known for its simplicity and efficiency. The Blowfish encryption algorithm operates on fixed-size blocks of data and supports key lengths ranging from 32 to 448 bits, making it adaptable to various security requirements. Its key setup phase is particularly fast, enabling rapid encryption and decryption processes. Despite its age, Blowfish is widely used and valued due to its robust security features and speed. Its open design and the absence of any license restrictions have contributed to its popularity in both commercial and open-source applications. The algorithm's resistance to various cryptanalysis attacks has solidified its reputation as a reliable choice for secure data encryption. To date, no effective cryptanalysis of Blowfish has been discovered, and brute-force attacks remain the standard method. While the Blowfish encryption algorithm is considered vulnerable to birthday attacks, these are also brute-force collision attacks based on the so-called birthday paradox.

[0008] (Conventional technology) Numerous publications and patent documents address the challenges of breaking cryptographic protocols. For example, the preprint "Hacking Cryptographic Protocols with Advanced Variational Quantum Attacks" by Borja Aizpurua et al., arXiv:2311, published April 15, 2023, available at https: / / arxiv.org / abs / 2311.02986 (downloadable October 21, 2024). This paper introduces a Variational Quantum Attack Algorithm (VQAA) for hacking cryptographic protocols using quantum circuits.

[0009] U.S. Patent Application US18 / 088,042, filed on May 16, 2024, entitled "Variational Quantum Attack for Cryptographic Protocol," focuses on using quantum circuits for cryptographic key recovery via VQAA. The method proposed in this draft patent uses a tensor network instead of quantum circuits.

[0010] U.S. Patent Application US18 / 106,555, “Method and System for Modifying Document Without Changing Hash Value,” filed on August 8, 2024, instructs a method for recovering plaintext from a hashed message using quantum circuits for the recovery process.

[0011] U.S. Patent Application US18 / 375,714, filed on August 8, 2024, entitled "Method and System for Modifying Document Without Changing Hash Value," teaches the use of a tensor network approach that can be implemented on classical hardware to recover plaintext from hashed messages. This approach provides a more efficient and cost-effective alternative to quantum solutions for cryptographic attacks against encrypted messages. [Prior art documents] [Patent Documents]

[0012] [Patent Document 1] U.S. Patent Application US18 / 088,042 [Patent Document 2] U.S. Patent Application US18 / 106,555 [Patent Document 3] U.S. Patent Application US18 / 375,714 [Non-patent literature]

[0013] [Non-Patent Document 1] “Hacking Cryptographic Protocols with Advanced Variational Quantum Attacks”, Borja Aizpurua et al., arXiv:2311 [Overview of the Initiative] [Problems that the invention aims to solve]

[0014] There is a need to develop classically feasible and efficient methods for attacking the aforementioned cryptographic protocols without relying on quantum hardware. The method described in this application leverages matrix product states (MPS) and tensor networks (TN), such as the Flexible-PEPS Quantum Circuit Simulator (FQCS), to provide a novel approach to improve the efficiency of cryptographic attacks, particularly for targeted key cryptography, while avoiding the need for quantum processors. The goal is to simulate quantum-like attacks on classical hardware and significantly enhance the performance of cryptographic analysis in a practical and scalable manner. [Means for solving the problem]

[0015] A method for determining an encryption key in a key space for encrypting plaintext into a corresponding encrypted ciphertext is disclosed in this application. The method includes the steps of: constructing a tensor network based on the encrypted ciphertext; encoding the key space into matrix product states (MPS) or a flexible PEPS quantum circuit simulator (FQCS); simulating the encryption process to obtain a simulated superposition of the ciphertexts; and determining the overlap between the simulated superposition of the ciphertexts and the encrypted ciphertexts. When a predetermined overlap value is reached for the overlap, the key space is folded to determine the encryption key, or, if not, the parameters of the tensor network are adjusted.

[0016] Projected entangled pair states (PEPS) are a class of tensor network states used in quantum many-body physics and quantum information theory to describe the quantum states of systems on a lattice. Each lattice site is associated with a tensor, and the entanglement structure of the quantum state is encoded in the connections (or indices) between these tensors.

[0017] Flexible PEPS is an extension or generalization of the standard PEPS framework, designed to enhance its adaptability and efficiency, particularly in describing quantum states in systems with irregular geometry, heterogeneity, or specific physical constraints.

[0018] This specification proposes a method for determining an encryption key. This method includes the steps of constructing a tensor network using parameters representing a candidate encryption key, adjusting the parameters of the tensor network, generating a new candidate key sample and obtaining a candidate ciphertext obtained using the candidate encryption key, calculating a cost function for the target ciphertext, measuring the overlap between the target ciphertext and the candidate ciphertext, determining whether the overlap reaches a threshold, and repeating the method by further adjusting the parameters of the tensor network when the threshold is not reached, and determining that the candidate encryption key is the encryption key when the threshold is reached.

[0019] A system for determining an encryption key is also proposed. The system includes at least one input / output device for inputting a ciphertext and outputting an encryption key, at least one tensor network having parameters representing a candidate encryption key, an optimizer for adjusting the parameters of the at least one tensor network by classical optimization, and a comparator for determining the overlap between the target ciphertext and the candidate ciphertext.

[0020] This specification proposes a computer system for determining an encryption key. The computer system is configured to construct a tensor network using parameters representing a candidate encryption key, adjust the parameters of the tensor network, generate a new candidate key sample and obtain a candidate ciphertext obtained using the candidate encryption key, calculate a cost function for the target ciphertext, measure the overlap between the target ciphertext and the candidate ciphertext, determine whether the overlap reaches a threshold, and repeat the method by further adjusting the parameters of the tensor network when the threshold is not reached, and determine that the candidate encryption key is the encryption key when the threshold is reached, and includes a classical processor.

[0021] A computer program product including logic for executing, by a classical processor, a method for determining an encryption key, the method comprising: constructing a tensor network using parameters representing candidate encryption keys; adjusting the parameters of the tensor network; generating a new candidate key sample and obtaining a candidate ciphertext obtained using the candidate encryption key; calculating a cost function for a target ciphertext; measuring the overlap between the target ciphertext and the candidate ciphertext; determining whether the overlap has reached a threshold; repeating the method by further adjusting the parameters of the tensor network when the threshold is not reached; and determining that the candidate encryption key is the encryption key when the threshold is reached.

Brief Description of the Drawings

[0022] [Figure 1] A diagram showing an overview of a computing system for implementing the method of this specification. [Figure 2] A diagram showing an overview of the method of this specification. [Figure 3] A diagram showing an overview of the system of this specification. [Figure 4] A flowchart of the method of this specification.

Modes for Carrying Out the Invention

[0023] The present invention will be described below based on the drawings. It is understood that the embodiments and aspects of the present invention described in this specification are merely examples and do not limit the scope of protection of the claims in any way. The present invention is defined by the claims and their equivalents. It is understood that the features of one aspect or embodiment of the present invention can be combined with the features of one or more different aspects and / or embodiments of the present invention.

[0024] This specification describes a method for cryptographic key recovery using tensor network (TN) technology. This method utilizes matrix product states (MPS) and a flexible PEPS quantum circuit simulator (FQCS) to recover cryptographic keys more efficiently on classical hardware, without requiring the use of quantum hardware. While this method has been found to be applicable to any cryptographic protocol, it has shown particularly high efficiency in symmetric-key cryptography.

[0025] Figure 1 shows an overview of a typical system that may be used to carry out the method described herein. Figure 1 shows an overview of a computing system 10 for carrying out the method herein. In this example, the computing system 10 comprises a classic central processing unit 20 connected to a data storage unit 25 (i.e., one or more memory devices), and a number of input / output devices 30. The input / output devices 30 allow for the input of one or more values ​​for tensor parameters, such as cryptographic keys obtained from tensors, and the output of results, as will be described later.

[0026] A graphics processing unit 35 for processing vector computations and a field-programmable gate array (FGPA) 40 for control logic can also be connected to the central processing unit 20. The central processing unit 20 and / or graphics processing unit 35 have an optimizer 320 for optimizing tensor parameters and a comparator 330 for determining cryptographic keys, as will be described later.

[0027] The computing system 10 may be connected to a computer network 60, such as the Internet. The computing system 10 in Figure 1 is merely illustrative, and it should be understood that other units or elements may be present in the computing system 10. It should also be understood that there may be many input / output (I / O) devices 30 located in multiple locations, and similarly, multiple data storage units 25 located in multiple locations. These multiple I / O devices 30 and data storage units 25 are connected by the computer network 60.

[0028] The method described herein is implemented in the computing system 10 shown in Figure 1. This method leverages the compression and optimization capabilities of tensor networks (TNs) to reduce the exponential computational requirements of computing system 10 when performing a brute-force attack to recover cryptographic keys. These classical implementations of TN methods can outperform quantum-inspired approaches by avoiding the need for hardware-dependent quantum simulators. This makes TN-based methods practical and powerful for attacking cryptographic protocols of varying complexity.

[0029] This method can use two types of tensor networks (TNs): specifically, matrix product states (MPS) for smaller cryptographic keys and flexible PEPS (Projected Entangled Pair States) for larger and more complex key spaces. The TN method enables handling large-scale data by compressing the data and minimizing the computational resources required to explore large key spaces. Unlike brute-force methods, which scale exponentially with key size, the TN method described herein reduces the dimensionality of the problem through efficient representation and optimization of the key space.

[0030] This method can be applied to both symmetric and asymmetric cryptographic protocols. It has been tested on symmetric-key protocols such as S-DES, S-AES, and Blowfish. These symmetric-key protocols are used, for example, for data transmission.

[0031] Tensor network-based methods for cryptographic key recovery offer a more efficient alternative by constructing a tensor network that represents the potential key combinations of encrypted ciphertext and cryptographic keys. This method has been shown to outperform brute-force methods, particularly in large and complex key spaces. Even as key size and complexity increase, TN-based methods remain computationally viable, reducing the search space and accelerating key recovery even for larger cryptographic protocols like Blowfish, where brute-force methods become impractically costly.

[0032] In one embodiment, this method uses the concept of a matrix multiplication state (MPS) for the recovery of small cryptographic keys. Such small cryptographic keys are used in protocols such as the Simplified Data Encryption Standard (S-DES). The MPS is a one-dimensional tensor network that can efficiently represent quantum states. In this embodiment, the MPS directly represents the cryptographic key space, and by iteratively tuning the parameters of the tensor network, the algorithm can rapidly converge to the correct cryptographic key. The MPS optimization process involves tuning the join dimension and internal parameters to maximize the overlap with the correct ciphertext, thereby enabling the recovery of cryptographic keys using fewer computational resources than conventional methods. This algorithm will be described in more detail later in this specification.

[0033] The Flexible PEPS Quantum Circuit Simulator (FQCS) is used for larger cryptographic keys, such as those used in the Blowfish protocol. PEPS is a two-dimensional tensor network, which makes it suitable for representing more complex systems with higher-dimensional data. The FQCS method simulates the behavior of quantum circuits without requiring any quantum hardware, thus enabling efficient key recovery even in large key spaces. By optimizing the tensor network to maximize the overlap between simulated and actual ciphertexts, FQCS enables key recovery in larger and more complex protocols, demonstrating higher efficiency than brute-force attacks.

[0034] A description of the method for determining the encryption key is given below, along with Figure 2. Assume that an attack against the encryption standard is performed to decrypt the encrypted ciphertext and recover the plaintext. Candidate encryption keys are encoded in MPS.

[0035] The encryption standard may be the S-DES encryption standard, and the candidate encryption key may be a 10-bit encryption key for the S-DES standard.

[0036] Plaintext is a statement encrypted using a candidate cryptographic key. Plaintext is understood to be any set or combination of data presented in its unencrypted form. A possible superposition of plaintexts yields encrypted ciphertext.

[0037] This method begins with constructing a tensor network 500 that represents the candidate ciphertext obtained by the candidate encryption key 310. The tensor network 500 contains a set of tensors. One non-restrictive example is to create a tensor for each character of the encryption key.

[0038] The goal is to optimize the internal parameters of the tensor within the MPS to tune the tensor by maximizing the overlap 390 between the simulated ciphertext 370, i.e., the candidate ciphertext obtained using the candidate encryption key 310, and the base ciphertext 380. The simulated ciphertext is data obtained by encrypting known plaintext using a candidate encryption key sampled from a superposition of possible keys.

[0039] Once the tensor network 500 is constructed, the cryptographic key determination process begins by optimizing the internal parameters of the tensor network 500 (step S2). This involves adjusting the join dimension, efficiently modifying the tensor network 500 by applying techniques such as singular value decomposition (SVD), and updating the tensors of the tensor network 500 using classical optimization methods such as Adam optimization.

[0040] In MPS, the optimization process involves sweeping the Tensor Network 500 from left to right and right to left, sequentially adjusting the tensors. In Flexible PEPs, the optimization follows a similar iterative process, although it is more complex due to the higher number of dimensions.

[0041] During each iteration, the candidate cryptographic key 310 is modified, and a candidate ciphertext 370 is generated. The candidate ciphertext 370 can be compared with the reference ciphertext 380 in the comparator 360. The comparator 360 measures the overlap between the reference ciphertext 380 and the candidate ciphertext 370 using the Hamming distance.

[0042] The duplicate 390 is used as a criterion to determine whether the correct cryptographic key has been found. If the duplicate 390 reaches a predetermined threshold (e.g., minimum Hamming distance), the algorithm concludes that the correct cryptographic key has been recovered (or at least nearly recovered). If the duplicate 390 is below the predetermined threshold, the parameters of the tensor network 500 are further adjusted in the classical central processing unit 20 via the classical optimization algorithm 390.

[0043] The optimization algorithm is used to adjust the input parameters of the tensor network 500 so that the candidate ciphertext 370 has overlap 390 with the reference ciphertext in step S4. Overlap occurs when the Hamming distance is zero (or very close to zero). The iterative process continues until the correct plaintext is revealed.

[0044] The construction and optimization of tensor networks are outlined with respect to MPS with reference to Figure 4, which shows a flowchart of the method described herein.

[0045] The first step S200 in the method involves constructing a tensor network representation of the encrypted ciphertext and key space. In the case of a matrix multiplication state (MPS), the tensor network S200 contains a set of tensors, each tensor representing a possible bit in the cryptographic key. The MPS is:

number

[0046] The MPS network is initialized in step S210 with random values ​​for the tensor parameters. The algorithm is then unfolded, iteratively updating the tensors in the MPS network to converge to the correct cryptographic key.

[0047] In the next step, S215, the hyperparameters of the MPS are set. Hyperparameters are parameters that control the structure of the MPS. Examples of hyperparameters include BondDim (the virtual bond dimension of the MPS), step length (the learning rate for parameter updates), step (the number of sequential updates for the same tensor in each iteration), cutoff value (a threshold for discarding smaller eigenvalues ​​in singular value decomposition (SVD)), reset value (a threshold for detecting local minima and resetting parameters), and temperature (an acceptance criterion for undesirable changes).

[0048] In the next step S220, the MPS is left-normalized, and key samples (i.e., candidate cryptographic keys) for the cryptographic key are generated in step S225 by sampling the MPS. The cost function is then computed in step S230 by using the Hamming distance. The generation of the cost function will be described in more detail later. A sweep is then performed in step S232, in which, for each tensor from the rightmost to the leftmost tensor, the current tensor is merged with its neighboring tensors to form a combined matrix called the merged matrix. The merged matrix is ​​updated in step S235 by applying random modifications, normalized in step S240, and then decomposed back into the original tensors via SVD in step S245.

[0049] The change is a small, random change applied to the elements of the merged matrix, with different elements receiving different perturbations. The energy difference (ΔE) is then calculated to evaluate the effect of this change on the cost function. Based on the Metropolis-Hastings criterion, the change is accepted if ΔE < 0, or if ΔE ≥ 0, with a probability proportional to exp(-ΔE / T), where T is the temperature parameter. If the change is accepted, the updated matrix is ​​normalized (step S240) and decomposed back into the original tensor via singular value decomposition (SVD) in step S245; otherwise, the matrix returns to its previous state and the algorithm proceeds.

[0050] In the next step S250, the MPS is left-normalized to ensure numerical stability and proper normalization. A new key sample is generated by sampling through the MPS in step S255. In step S260, the cost function, specifically the Hamming distance, is recalculated using the new sample key. If the ciphertext generated with the new key is closer to the target ciphertext, the modification is considered favorable and accepted in step S265. If the modification is unfavorable, it is accepted with a probability determined by the Metropolis-Hastings criterion, thereby allowing the algorithm to avoid exploring the solution space and getting trapped in a local minimum. The probability is given as follows:

number

[0051] It should be noted that simply ignoring random changes can lead to getting trapped in a local minimum. Avoiding local minima is preferable, especially in complex, high-dimensional optimization landscapes.

[0052] The Adam optimizer is applied in step S270 to update the merged matrix by calculating the gradient and updating the parameters of the MPS to which the changes were applied, only if the changes were accepted. The updated merged matrix is ​​normalized and decomposed back into the original tensor via SVD in step S275.

[0053] In the next step S280, all parameters of the MPS tensor, in this case the gradient norm of the gradient from the changes to the two tensors for the two-site MPS update, are monitored to detect a local minimum. If a local minimum is detected, the parameters are reset in step S285.

[0054] The first and second moments of the gradient can be used. These refer to statistical summaries used in optimization. The first moment is the mean (average) of the gradient, which captures the direction of the optimization path. The second moment is the uncentered variance (mean of the squared gradient), which captures the scale of the gradient change and helps adapt the learning rate. The absolute value of the gradient provides a simple measurement of the magnitude of the gradient at each step.

[0055] The gradient norm typically quantifies the overall size of the gradient vector via the L2 norm (i.e., the square root of the sum of squared gradients). This is used to monitor convergence and detect local minima. These metrics are used, for example, in the Adam optimizer to adaptively update parameters during iterative tuning of tensor networks.

[0056] The MPS is then right-normalized in step S290, and the sweep is updated in step S295, sweeping from left to right. The left-to-right and right-to-left sweeps are repeated until the target cryptographic key is found. In each iteration, the tensor network simulates the encryption process and generates a sample cryptographic key by comparing the simulated ciphertext resulting from the simulated encryption process with the actual (target) ciphertext. A cost function, such as the Hamming distance between the generated ciphertext and the actual / target ciphertext, guides the optimization process, allowing the algorithm to focus on key samples that are closer to the correct solution.

[0057] A crucial step in this method, S297, is to measure the overlap between the simulated ciphertext generated by the tensor network and the actual / target ciphertext. The algorithm calculates this overlap after each iteration and uses the value of the overlap as a criterion to determine in step S298 whether the correct encryption key has been found. If the overlap reaches a predetermined threshold, the algorithm concludes that the correct encryption key has been recovered. If the value of the overlap is lower than the predetermined threshold, the parameters of the tensor network are further adjusted until the target encryption key is found.

[0058] For larger cryptographic keys, Flexible PEPs are used as described above. Flexible PEPs extends the tensor network to two dimensions, adjusting the network geometry based on the complexity of the key and ciphertext, and removing less relevant connections based on bond entanglement entropy (BEE). This adaptability reduces the computational burden and allows for the simulation of larger systems. The FQCS method simulates quantum circuits with high accuracy even in the absence of quantum hardware, making it ideal for larger cryptographic key recovery problems.

[0059] Tensor networks can be extended by adding new edges (connections) between tensors, allowing them to reflect potential correlations in unknown cryptographic keys. Each added edge represents a possible interaction or dependency between parts of a variable string. If a tensor exceeds a set number of connections (cutoff κ), the least relevant edges are removed. The least relevant edges are determined based on a measure of the importance of the connections. This allows for efficiency to be maintained while adapting to the complexity of the message.

[0060] PEPS networks can be dynamically tuned, retaining important connections while removing less relevant connections between tensors in a tensor network, making the process computationally efficient even for large hashed messages. More precisely, PEPS networks can be dynamically tuned by monitoring the number of connections (joins) per tensor. If a tensor exceeds a predetermined limit (cutoff κ), the network removes the least relevant connections to maintain efficiency. A connection is considered less relevant if it carries low entanglement entropy (meaning it contributes only a small amount to the overall correlation in the data). The term "dynamic" means that the network's geometry is not fixed; i.e., connections are added or removed during execution time based on ongoing important evaluations, allowing the network to adapt to the structure of the hashed message.

[0061] The methods described herein primarily focus on symmetric-key cryptography. However, the described methods can be extended to all cryptographic protocols. For asymmetric protocols, tensor networks can be adapted so that the encryption and decryption keys search different key spaces, providing a novel way to attack public-key systems without requiring quantum resources.

[0062] This invention introduces a powerful cryptographic attack method that leverages tensor networks to recover encryption keys with improved efficiency and scalability. By enabling classical simulations of quantum-like attacks, many of the limitations of existing quantum cryptography methods are overcome. The use of MPS and flexible PEP enables performance improvements in both small and large cryptographic systems, making it a versatile and valuable tool for modern cryptography. [Explanation of Symbols]

[0063] 10 Computing Systems 20. Central Processing Unit (CPU) 25 Data Storage Unit 30 Input / Output Devices 35. Graphics Processing Unit (GPU) 40 Field-Programmable Gate Arrays 60 Computer Networks 310 Tensor Networks 320 Optimizer 330 Comparator

Claims

1. A computer method for determining an encryption key, wherein the method is The steps include constructing a tensor network using parameters representing candidate cryptographic keys, The steps include adjusting the parameters of the tensor network, The steps include generating a new candidate key sample and obtaining a candidate ciphertext obtained using the candidate cryptographic key, The steps include: calculating a cost function for the target ciphertext, A step of measuring the overlap between the target ciphertext and the candidate ciphertext, The steps include determining whether the aforementioned overlap has reached a threshold, If the threshold is not reached, the method is repeated by further adjusting the parameters of the tensor network, and if the threshold is reached, it is determined that the candidate cryptographic key is the cryptographic key. Methods that include...

2. The method according to claim 1, wherein the cost function is based on the Hamming distance between the simulated ciphertext and the target ciphertext.

3. The method according to claim 1, wherein the step of adjusting the parameters of the tensor network includes applying a random change to the parameters of the tensor network, wherein the random change has a magnitude of 1 to 1e-4.

4. The method according to claim 1, wherein the step of adjusting the parameters of the tensor network includes either left-normalizing or right-normalizing the tensor network.

5. The method according to claim 1, wherein the step of adjusting the parameters of the tensor network includes sweeping the tensor network to create a merged matrix.

6. The method according to claim 5, wherein the random changes are applied to the merged matrix.

7. The method according to claim 5, further comprising the step of optimizing the merged matrix.

8. The method according to claim 5, further comprising the step of decomposing the merged matrix.

9. The method according to claim 1, further comprising the step of determining a local minimum.

10. The method according to claim 1, wherein the tensor network is implemented as either a matrix product state or a flexible projection entanglement state.

11. A system for determining an encryption key, wherein the system is At least one input / output device for inputting ciphertext and outputting the cipher key, A tensor network having parameters representing candidate cryptographic keys, An optimizer for adjusting the parameters of the at least one tensor network by classical optimization, A comparator for determining duplication between the target ciphertext and the candidate ciphertext, A system equipped with these features.

12. The system according to claim 11, wherein the at least one tensor network is implemented as either a matrix product state or a flexible projection entanglement state.

13. A computer system for determining an encryption key, wherein the computer system is A tensor network is constructed using parameters representing candidate cryptographic keys. The parameters of the tensor network are adjusted, A new candidate key sample is generated, and a candidate ciphertext obtained using the said candidate cryptographic key is retrieved. Calculate the cost function for the target ciphertext, The overlap between the target ciphertext and the candidate ciphertext is measured, Determine whether the aforementioned overlap has reached a threshold, If the threshold is not reached, the above method is repeated by further adjusting the parameters of the tensor network, and if the threshold is reached, it is determined that the candidate cryptographic key is the cryptographic key. A computer system equipped with a classic processor configured in such a way.

14. A computer program product comprising logic for executing a method for determining an encryption key using a classical processor, wherein the method is The steps include constructing a tensor network using parameters representing candidate cryptographic keys, The steps include adjusting the parameters of the tensor network, The steps include generating a new candidate key sample and obtaining a candidate ciphertext obtained using the candidate cryptographic key, The steps include: calculating a cost function for the target ciphertext, A step of measuring the overlap between the target ciphertext and the candidate ciphertext, The steps include determining whether the aforementioned overlap has reached a threshold, The method is repeated by further adjusting the parameters of the tensor network if the threshold is not reached, and if the threshold is reached, it is determined that the candidate cryptographic key is the cryptographic key. Computer program products, including [the following].

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