A method and system for ciphertext data global computing using a post-quantum homomorphic algebra

Abstract

This invention relates to the field of information security technology, specifically disclosing a method and system for full-domain computation of encrypted data using post-quantum homomorphic algebra. The method includes: constructing a homomorphic algebraic structure based on a post-quantum cryptographic difficulty problem; the data owner encrypting the original plaintext data using the public key of this structure to generate ciphertext data; the computer directly performing at least one homomorphic operation on the ciphertext data according to the homomorphic algebra rules without decryption, obtaining the resulting ciphertext; and the data owner decrypting the resulting ciphertext using a private key, with the result being consistent with the result of performing the corresponding operation on the plaintext. This invention combines the mathematical difficulty problem of post-quantum cryptography with a customized homomorphic algebraic mapping to achieve fully homomorphic computation resistant to quantum computing attacks. While ensuring the confidentiality of data throughout its entire lifecycle, it supports complex processing of ciphertext data, providing a quantum-secure solution for privacy and security in cloud environments and distributed collaborations.

Description

Technical Field

[0001] This invention belongs to the interdisciplinary field of computer science and information security technology, specifically involving cryptography, data security and privacy computing.

[0002] More specifically, this invention relates to the systematic integration of the following three closely related core technology branches:

[0003] 1. Quantum-resistant cryptography: mainly involves the design of post-quantum cryptographic algorithms, especially the construction of cryptosystems based on lattice-hard problems (such as the learning error problem, the ring learning error problem, and the short integer solution problem). Its goal is to ensure the long-term confidentiality and integrity of information in classical and quantum computing environments.

[0004] 2. Homomorphic Encryption and Ciphertext Computation: This mainly involves the theory and technology of fully homomorphic encryption, including the design of algebraic structures that support direct addition and multiplication operations on encrypted data, noise control mechanisms (such as modulus switching and bootstrapping), and efficient implementation methods. Its goal is to achieve data availability without leaking the original data.

[0005] 3. Secure Computing System Architecture: This mainly involves the design of hardware and software systems that support the above-mentioned cryptographic primitives, including key management, high-efficiency computing engines, protocol integration, and the engineering implementation of privacy-preserving applications (such as secure outsourced computing, federated learning, and encrypted database queries).

[0006] This invention specifically addresses a method and system for intrinsically and integratedly designing the security foundation of post-quantum cryptography with the computational capabilities of fully homomorphic encryption. It aims to solve the problem of the disconnect between "quantum-resistant security" and "universal dense-state computation" capabilities in existing technologies, and is a key component of next-generation privacy-enhancing technologies developed to address the threat of quantum computing. Background Technology

[0007] With the acceleration of global digitalization, data has become a core factor of production. The widespread application of cloud computing, big data analytics, and artificial intelligence (especially distributed machine learning paradigms such as federated learning and collaborative training) has created an urgent need for data privacy and secure computing. Against this backdrop, fully homomorphic encryption (FHE), which enables direct computation in encrypted form, is considered one of the ultimate solutions in the field of privacy computing because it achieves "data usable but not visible." Mainstream FHE schemes (such as BGV, BFV, and CKKS) are mostly based on mathematical problems such as ideal lattices and ring error learning.

[0008] However, the long-term security of such traditional FHE schemes is facing a fundamental threat from quantum computing. Quantum computers, especially those running Shor's algorithm, can effectively break classical public-key cryptosystems based on large integer factorization and discrete logarithm problems. This means that the security assumptions of many current cryptographic systems will no longer hold in the face of future universal quantum computers. To address this "quantum threat," post-quantum cryptography (PQC) has emerged, aiming to design cryptographic algorithms resistant to quantum computer attacks. The National Institute of Standards and Technology (NIST) is promoting PQC standardization, with algorithms based on mathematically difficult problems such as lattices (LWE / RLWE), encoding, and multivariables becoming the mainstream candidates.

[0009] Analysis of the technological development trajectory and limitations of existing solutions:

[0010] Currently, technological development in the two key areas of "post-quantum security" and "closed-state computing" presents a parallel but fragmented trend. Existing technological solutions can be broadly categorized into three types, but all have significant shortcomings and cannot simultaneously meet the dual requirements of "resistance to quantum attacks" and "efficient encrypted computing":

[0011] 1. Independent post-quantum cryptography schemes: secure but "uncomputable"

[0012] These technologies focus on building quantum-resistant encryption, signatures, or key exchange primitives. Their core design goal is to ensure the confidentiality and authentication of transmission or storage, and they do not include "homomorphic computation" in their functional scope.

[0013] Representative Patent 1 (Chinese Patent ZL202010031242.2): This patent provides an efficient post-quantum key exchange protocol. Its technological contribution lies in optimizing the efficiency and reliability of key negotiation, but it only solves the problem of establishing a secure shared key between communicating parties. Data protected by this protocol is secure during transmission and static storage, but once it arrives at the recipient (such as a cloud server) in ciphertext form, it becomes unprocessable "dead data." Any computation must first decrypt it, thus compromising data privacy. Therefore, it cannot be applied to privacy-preserving computing scenarios that require data analysis, machine learning, and other computations on ciphertext.

[0014] Representative Patent 2 (Chinese Patent ZL201710081133.X): This patent focuses on hardware acceleration, designing efficient circuits for encryption and decryption operations based on Ring-LWE (an important post-quantum cryptography difficulty problem). It significantly improves the performance of basic encryption and decryption operations, representing a significant advancement in low-level primitive optimization. However, its technological boundaries are clearly limited to the two endpoints of "encryption" and "decryption." This patent does not, and does not intend to, construct an algebraic system or computational framework that allows addition and multiplication operations between encrypted data. It provides a very robust and fast-opening "quantum secure lock," but does not design any tools for processing items "within the lock."

[0015] 2. Traditional security-based fully homomorphic encryption schemes: "computable" but with questionable long-term security.

[0016] These technologies enable powerful homomorphic computation, allowing arbitrarily complex function calculations on ciphertext, but their security may be based on traditional mathematical problems that have not yet been proven to be resistant to quantum attacks.

[0017] Representative Patent 3 (US10454668): This patent protects a method and system for fully homomorphic encryption, explicitly stating its ability to perform homomorphic operations on ciphertext. However, upon reviewing its claims and specification, this patent does not forcibly associate or limit its security with mathematically difficult problems (such as LWE, RLWE, SIS, etc.) that have been extensively studied by the PQC community and incorporated into the NIST post-quantum cryptography standardization process. Its security assumptions may still be based on number theory problems that are vulnerable to future quantum algorithm attacks, or its specific parameter set may not have undergone sufficient post-quantum security strength assessment. Therefore, although this scheme possesses complete homomorphic computation capabilities, the "long-term security" it provides is uncertain and difficult to meet the high-security requirements of scenarios such as finance, healthcare, and government that require decades of lifecycle protection for sensitive data.

[0018] 3. Simple attempts at "serialization" or "add-on" combinations: inefficient and difficult to implement.

[0019] One intuitive technical approach is to try combining a standard post-quantum encryption algorithm (such as NTRUEncrypt, Kyber) with a traditional homomorphic encryption scheme. This approach has inherent drawbacks:

[0020] 3.1 Functionality and Compatibility Barriers: Post-quantum encryption algorithms and homomorphic encryption schemes have vastly different design goals, and their algebraic structures are usually not directly compatible. After simple concatenation, the computer cannot perform meaningful homomorphic operations on the first-layer post-quantum ciphertext unless decryption is performed first (which would expose the data), or through extremely complex and inefficient cryptographic transformations (such as general obfuscation or homomorphic key exchange), which usually leads to a dramatic increase in system complexity and overhead.

[0021] 3.2 Performance Loss: This “two-layer” design will bring huge computational, storage and communication overhead, lose the simplicity and efficiency that an excellent cryptographic scheme should have, and is often unbearable in practice, and cannot meet the performance requirements of actual business.

[0022] In summary, existing technologies exhibit a state of "functional fragmentation": Schemes 1 and 2 provide post-quantum security but lack homomorphic computation capabilities; Scheme 3 provides homomorphic computation capabilities but its post-quantum security foundation is weak; and variants of Scheme 3 (external integration) are impractical due to efficiency and technical complexity. Currently, the industry urgently needs but lacks a systematic, intrinsic technical solution. Such a solution should start from the underlying mathematical construction of cryptography, fundamentally and systematically integrating standardized and verified post-quantum cryptographic difficulties (such as RLWE) with algebraic structures supporting fully homomorphic computation, rather than simply piecing them together.

[0023] An ideal solution must simultaneously meet three core requirements: 1. Its security must be explicitly reducible to a recognized post-quantum hard problem, providing provable quantum security; 2. It must natively support arbitrary addition and multiplication operations on ciphertext data, achieving true full homomorphism; 3. Under the aforementioned stringent security and functional constraints, it must still maintain or optimize the feasible computational and communication efficiency of existing FHE schemes. This technological gap severely restricts the widespread application of secure, efficient, and programmable computation of high-value ciphertext data in the context of the looming threat of quantum computing.

[0024] This invention aims to directly address this core gap by proposing a native fusion architecture based on post-quantum cryptography homomorphic algebra, fundamentally resolving the aforementioned contradictions. Summary of the Invention

[0025] The technical problem to be solved by this invention:

[0026] To address the problem in existing technologies where post-quantum cryptography and fully homomorphic encryption are disconnected, making it difficult to simultaneously achieve quantum-resistant security and efficient ciphertext computation, this paper proposes a method and system for global computation of encrypted data using post-quantum homomorphic algebra. This method constructs a specific homomorphic algebra structure to accommodate the mathematical difficulties of post-quantum cryptography, thereby ensuring post-quantum security while natively supporting arbitrary addition and multiplication operations on the ciphertext data.

[0027] The technical solution of the present invention:

[0028] To solve the above-mentioned technical problems, the present invention adopts the following technical solution:

[0029] A method for global computation of dense-state data using post-quantum homomorphic algebra is disclosed. Its core lies in constructing an endogenous post-quantum cryptographic algebra structure that supports homomorphic operations, and based on this structure, implementing a complete process from key generation, data encryption, homomorphic computation to result decryption. The method includes the following steps:

[0030] 1. Parameter initialization and construction of homomorphic algebra structure:

[0031] Based on the target security level λ, a difficult mathematical problem in post-quantum cryptography is selected as the security foundation. This difficult problem includes learning error problems, ring learning error problems, short integer solution problems, or problems based on multivariate polynomials. According to the selected difficult problem, an algebraic structure supporting addition and multiplication operations is constructed. This structure can be a polynomial ring, an ideal lattice, or a modular lattice. Simultaneously, common system parameters matching this algebraic structure are determined, including the dimension of the algebraic structure, the modular chain, the error distribution, and the plaintext encoding space parameters. The key innovation of this step is that the constructed algebraic structure is not only a carrier of security, but its own addition and multiplication rules are directly mapped to homomorphic addition and homomorphic multiplication operations on the subsequent ciphertext, achieving a unity of security and computational functionality on a mathematical foundation.

[0032] 2. Key generation:

[0033] Based on the homomorphic algebra structure and system common parameters constructed in step 1, a public key for encryption and a private key for decryption are generated. Furthermore, to support complete homomorphic computation functionality, a series of homomorphic computation auxiliary keys are simultaneously generated, including a relinearization key for compressing the ciphertext dimension after multiplication, and a rotation key for implementing ciphertext data rotation permutation. The generation of these auxiliary keys is also based on the security difficulty problem in step 1, ensuring the post-quantum security of the entire key system.

[0034] 3. Data encoding and encryption:

[0035] The data owner first embeds the raw plaintext data to be processed into the plaintext space of the algebraic structure defined in step 1 using an encoding function, forming structured plaintext elements. Then, using the public key generated in step 2, the plaintext elements are encrypted into ciphertext elements within the algebraic structure using an encryption algorithm defined on the algebraic structure. The encoding process ensures that the mathematical representation of the data in the algebraic structure is suitable for subsequent homomorphic operations.

[0036] 4. Homomorphic computation and dynamic noise management:

[0037] After obtaining the ciphertext elements, the computer performs homomorphic computation directly on the ciphertext according to the algebraic structure operation rules defined in step 1, without decryption. The computation includes, but is not limited to, homomorphic addition, homomorphic multiplication, and homomorphic data permutation using a rotation key. To ensure the feasibility of computation at arbitrary depths, this method integrates a dynamic noise management mechanism, which includes:

[0038] 4.1 Modulus Switching: During homomorphic computation, when the ciphertext noise increases to a preset threshold, a modulus switching operation is performed to scale the ciphertext from a larger modulus to a smaller modulus, thereby reducing the absolute level of noise and reserving space for noise growth in subsequent computations.

[0039] 4.2 Bootstrapping: When the modular chain is exhausted or complex calculations are required, the bootstrapping process is initiated. This process homomorphically executes the decryption circuit, "refreshes" noisy ciphertext into new ciphertext that encrypts the same plaintext, and resets the noise of the new ciphertext to a lower level, thereby achieving an infinite extension of computational depth.

[0040] 5. Result Decryption and Decoding:

[0041] The data owner uses the private key generated in step 2 to decrypt the ciphertext returned by the computation party, obtaining the plaintext element of the result. Finally, through the inverse process of the encoding function, the plaintext element of the result is decoded from the algebraic structure back to the final meaningful plaintext computation result.

[0042] A system for implementing the above method is characterized by comprising the following cooperative modules:

[0043] 1. Post-quantum homomorphic parameter generation module: This module is used to automatically select and optimize core parameters such as the type of post-quantum cryptographic hard problem, the specific form of the algebraic structure, dimension, modular chain, and error distribution, based on the input security and performance requirements, to ensure the optimal configuration of the system at the target security level.

[0044] 2. Homomorphic Key Generation and Management Module: Connected to the parameter generation module, this module generates the system's public key, private key, and various auxiliary keys necessary for supporting homomorphic computation (such as relinearization keys and rotation keys). This module is also responsible for the secure storage, distribution, and lifecycle management of these keys.

[0045] 3. Client-side encryption / decryption engine module: Deployed on the data owner's side, it includes an encoder, encryptor, decryptor, and decoder. The encoder is responsible for mapping application data to algebraic plaintext; the encryptor uses the public key for encryption; the decryptor uses the private key for decryption; and the decoder performs the reverse mapping to restore the algebraic plaintext back to the application data.

[0046] 4. Server-side homomorphic computing engine module: Deployed on the computing server side, it is the core computing unit of the system. It includes:

[0047] Homomorphic operators: Directly execute basic operations such as addition, multiplication, and rotation based on homomorphic algebra structures.

[0048] Noise Estimator and Scheduler: Monitors or predicts the noise level of ciphertext in real time, and intelligently schedules the timing of mode switching and bootstrapping operations to maximize computational depth or optimize performance.

[0049] Computation optimizer: Compiles and optimizes high-level computational tasks (such as machine learning model inference and database queries) into a series of efficient basic homomorphic operations, supporting batch processing techniques to improve throughput.

[0050] The key features and beneficial effects of this invention are as follows:

[0051] 1. An Intrinsically Integrated Secure Computation Framework: Unlike add-on solutions that simply chain post-quantum cryptography and homomorphic computation together, this invention starts from the underlying mathematical structure, integrating the difficult problems of post-quantum cryptography (such as RLWE) with algebraic systems that support homomorphic operations (such as polynomial rings) into a unified design. Security is based on provable post-quantum difficulties, while homomorphism is naturally endowed by the operational rules of the algebraic system, fundamentally solving the problem of the separation between security and functionality.

[0052] 2. Supports infinitely deep fully homomorphic computation: By integrating a modular noise management mechanism (mode switching and bootstrapping), this invention can effectively control the noise growth during the homomorphic computation process, thereby supporting the computation of arbitrarily complex functions on encrypted data and realizing true "fully homomorphic" capability.

[0053] 3. Provable post-quantum security: The security of the entire scheme, including the security of basic encryption, homomorphic operations and auxiliary keys, can be reduced to recognized post-quantum cryptographic difficulties (such as LWE / RLWE), providing a clear security guarantee against quantum computing attacks.

[0054] 4. Modular and Optimizable System Design: The proposed system architecture is clear, with well-defined responsibilities for each module, facilitating specific optimizations for different application scenarios (such as high-throughput batch processing and low-latency inference). For example, the parameter generation module can flexibly adapt to different security levels and efficiency requirements, and the computing engine can integrate hardware acceleration.

[0055] Through the above technical solution, this invention effectively overcomes the limitations of the comparative patents described in the background art. For the first time, it simultaneously achieves provable post-quantum security and practical, fully homomorphic encryption supporting arbitrary depth computing within a unified framework, providing a solid technical foundation for privacy computing in the future quantum computing era. The inherent fusion design of this invention fundamentally avoids the complex format conversions and additional decryption / re-encryption layers required by add-on solutions due to algebraic structure mismatches, thereby eliminating their main performance overhead and compatibility issues at the source. Attached Figure Description

[0056] The embodiments of the present invention will be described below with reference to the accompanying drawings. It should be understood that the drawings are for illustrative purposes only and do not constitute a limitation on the scope of protection of the present invention. The components, connections, and processes in the drawings are merely schematic, and their quantities, proportions, and specific details may vary in actual implementation.

[0057] Figure 1 This is a schematic diagram of the architecture of a fully homomorphic computation system for ciphertext data based on post-quantum cryptography homomorphic algebra, provided by an embodiment of the present invention. The diagram illustrates the core physical and logical modules of the system and their interactions. Typically, the system includes: a client (containing parameter and key generation units, encoding / encryption units, and decryption / decoding units), a homomorphic computation server (containing a homomorphic computation engine, a noise management scheduler, and a computation optimizer), and an optional public parameter service. Arrows indicate the direction of data and instruction flows, such as key distribution, ciphertext upload, homomorphic computation task assignment, and result return. This diagram is used for a macroscopic understanding of the system composition and operating mode of the present invention.

[0058] Figure 2 This is a flowchart illustrating the overall process of a fully homomorphic computation method for ciphertext data based on post-quantum cryptography homomorphic algebra, provided by an embodiment of the present invention. The flowchart clearly shows the complete process from system initialization to obtaining the final result in the form of a sequence of steps. The process typically begins with S201: parameter initialization and homomorphic algebra structure construction, followed by S202: key generation, S203: data encoding and encryption, S204: homomorphic computation and dynamic noise management, and finally reaches S205: result decryption and decoding. Core steps are specifically marked in the diagram, such as the addition / multiplication branches in homomorphic computation, and the modulus switching and bootstrapping judgment nodes included in dynamic noise management. This diagram is key to understanding the main thread of the method of this invention.

[0059] Figure 3 This diagram illustrates the relationship between homomorphic algebraic structure mapping and ciphertext operations in one embodiment of the present invention. It visually explains the core innovation of the invention—the intrinsic fusion of security and functionality. The diagram likely contains two main parts: one part shows how post-quantum cryptography difficulties (such as RLWE) are instantiated within a specific algebraic structure (such as a polynomial ring); the other part, through mathematical symbols and arrows, shows how addition and multiplication operations defined on this algebraic structure are directly mapped to homomorphic addition and multiplication operations on the ciphertext data. This diagram helps to understand the essential difference between the present invention and traditional "serialized" schemes.

[0060] Figure 4 This is a schematic diagram of a dynamic noise management mechanism in one embodiment of the present invention. The diagram illustrates in detail how decryption correctness is maintained during homomorphic computation.

[0061] Figure 5 This is a detailed data flow and computation flowchart of an embodiment of the present invention (a privacy-preserving machine learning inference application). The diagram uses a specific application scenario as an example to illustrate the actual operation of the method of the present invention. The diagram distinguishes between the data owner's side and the service provider's side. The process includes: encryption of model weights, encryption of user input data, the sequence of homomorphic convolution / pooling / activation function calculations performed on the cloud server, the insertion position of noise management operations, and the return and decryption of the final encryption result. This diagram serves to demonstrate the practicality of the present invention and the completeness of the technical solution.

[0062] Figure 6 This is a schematic diagram illustrating the working principle of batch processing technology in one embodiment of the present invention. The diagram shows how the mathematical properties of algebraic structures (such as polynomial rings) can be used to "package" multiple plaintext data into different "slots" within a single plaintext polynomial. The diagram graphically compares the differences in ciphertext quantity and computational efficiency between traditional element-by-element encryption and batch encryption, and illustrates how to use a rotating key to permutate and rearrange the data within the ciphertext slots to support complex parallel data computation. Detailed Implementation

[0063] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and multiple embodiments. The following embodiments are used to explain the invention, not to limit its scope of protection. The method described in this invention can be implemented by software, hardware, or a combination of both.

[0064] 1. System General Framework and Core Modules

[0065] like Figure 1As shown, the general system architecture of this invention includes four core modules, which interact with each other through clearly defined interfaces, ensuring the modularity, scalability and security of the system.

[0066] 1.1 Post-Quantum Algebra Parameter Generator: This module is responsible for automatically selecting or generating optimal algebraic parameters based on the input security level (λ) and performance objectives. Output includes: algebraic structure type (e.g., polynomial ring R, modular algebra, etc.). ), dimension N, modular chain ( The module includes the error distribution χ, and the plaintext modulus t or scaling factor Δ used for encoding. It incorporates a parameter security verification algorithm to ensure that the selected parameters, corresponding to difficult problems, achieve the target security level under both classical and quantum attack models.

[0067] 1.2 Homomorphic Key Generation and Management Center: Based on the output of the parameter generator, this center generates a master public-private key pair (pk, sk), a rotation key (rk) for ciphertext rotation, and a relinearization key (evk) for ciphertext multiplication. To support large-scale applications, this center can implement hierarchical key management strategies and supports secure key distribution, rotation, and revocation.

[0068] 1.3 Client-side encryption / decryption engine: Deployed on the data owner's side. It includes an encoder (mapping application-layer data into plaintext elements in an algebraic structure), an encryptor (using PK to perform encryption), a decryptor (using SK to perform decryption), and a decoder (restoring the decrypted plaintext elements back to application-layer data). This engine supports batch encryption and streaming encryption and can integrate a Hardware Security Module (HSM) to protect the core key.

[0069] 1.4 Server-side Homomorphic Computation Engine: Deployed on the computing server side. This is the core computing unit of the system, including a homomorphic operator (performing basic operations such as addition, multiplication, and rotation), a noise estimator and scheduler (real-time or pre-computing noise growth of ciphertext, intelligently scheduling mode switching operations), and a computing optimizer (compiling and optimizing advanced computing tasks, such as matrix multiplication and comparison operations, into a series of basic homomorphic operation sequences).

[0070] 2. Detailed Implementation Examples

[0071] Example 2.1: A Polynomial Ring Fully Isomorphic Computation System Based on RLWE (Deeply Optimized Version)

[0072] This embodiment introduces two key technologies, bootstrapping and batching, on the basic RLWE scheme, achieving arbitrary depth computing and extremely high throughput.

[0073] Step 2.1.1: Initialize parameters to support batch processing

[0074] Choose N as a power of 2, satisfying N≡1 (mod 2N), for example, N=8192. The plaintext space is partitioned using the Chinese Remainder Theorem (CRT) on rings. Specifically, choose the plaintext modulus t as a number of coprime small prime numbers. The product of, i.e. In this way, a plaintext polynomial can be simultaneously encoded into up to N independent modulo functions using a CRT. Integer slots. These prime numbers must be specified additionally in the public parameters. .

[0075] Step 2.1.2: Key generation supporting rotation

[0076] In addition to generating the public key pk and the relinearization key evk, a rotation key must be generated to utilize the computational potential of batch processing. For any rotation step k that requires cyclically moving data within the ciphertext slots, a corresponding rotation key is generated. This essentially encrypts the result of multiplying the private key s by a "plaintext onemni", allowing slot permutations to be performed on the ciphertext.

[0077] Step 2.1.3: Batch Encryption and Homomorphic Slot Operations

[0078] A vector containing N data points is directly encoded into a plaintext polynomial. After encryption, a ciphertext is obtained, but it "packs" N independent data points. Homomorphic addition and multiplication are performed slot-by-slot on the ciphertext (SIMD style). For example, adding two such ciphertexts is equivalent to adding N independent data pairs in parallel within their respective slots. This greatly improves data processing throughput.

[0079] Step 2.1.4: Bootstrap Operation for Infinite Depth Computation

[0080] When continuous homomorphic operations cause noise to increase to near a critical value, traditional mode-switching chains will exhaust themselves. This embodiment integrates a bootstrap process:

[0081] Bootstrap key generation: Generate a special "bootstrap key" bk, which is the encryption of the private key s under a larger modulus Q (Q >> q).

[0082] Bootstrapping process: For a noisy ciphertext This is treated as a "ciphertext to be refreshed". The bootstrapping process is essentially a homomorphic decryption-re-encryption loop: the decryption circuit is homomorphically executed on the ciphertext ct and the bootstrapping key bk (the main parts are homomorphic modular reduction and homomorphic polynomial function evaluation), outputting a new ciphertext that encrypts the same plaintext. However, its noise level was reset to the initial low level.

[0083] By bootstrapping, the system breaks free from its dependence on a fixed modular chain length, theoretically supporting computations at arbitrary depths.

[0084] Step 2.1.5: Application Example – Statistical Analysis of Encrypted Databases

[0085] An encrypted database contains millions of records, each with multiple fields. Batch processing is used to package the same field from multiple records into multiple slots of ciphertext. A single homomorphic addition can sum the values ​​of the same field across records; by rotating the key, correlation calculations between different fields (such as conditional summation) can be achieved. Bootstrapping ensures that the results can still be correctly decrypted after complex multi-step aggregation queries.

[0086] Example 2.2: Modular Scheme Based on MLWE and Compatible Design with NIST Post-Quantum Standard

[0087] This embodiment focuses on demonstrating how to deeply integrate the algebraic framework of the present invention with specific NIST post-quantum standards (such as MLWE-based Kyber), rather than simply applying them.

[0088] Step 2.2.1: Algebraic Structure Mapping

[0089] Set the module rank k (e.g., k=2, 3, 4). Private key Public key Let t be a random matrix, where t = As + e. The core of this invention lies in reinterpreting (A, t): it is not only a public key for a KEM, but also defines an algebraic operation space on a module. The ciphertext of homomorphic encryption is defined as a vector. ,in , Encode the message, where u is a small random vector.

[0090] Step 2.2.2: Define homomorphic multiplication on the module

[0091] This is the innovative aspect that distinguishes it from existing technologies. The homomorphic multiplication of two ciphertext vectors c and c' does not directly compute their outer product, but is achieved through a public tensor expansion and compression function F derived from a public key. The specific steps are as follows:

[0092] 2.2.2.1 Calculate the extension vector (Dimensions increased to approximately) ).

[0093] 2.2.2.2 Applying a linear function F: The design of F makes satisfy:

[0094] .

[0095] 2.2.2.3 Finally, the relinearization key (essentially...) is utilized. (encryption) will Compressing back to dimension k yields the standard ciphertext. .

[0096] This design makes the basic encryption layer mathematically isomorphic to the NIST standard algorithm, leveraging all its optimizations, while the upper layers achieve fully homomorphic capabilities through F and relinearization. This design ensures strong security and standardization of the basic encryption, while by introducing the innovative homomorphic multiplication mapping function F and the relinearization process, it 'grows' fully homomorphic computational capabilities on top of the standardized framework, achieving efficient compatibility with the standard rather than simply making calls.

[0097] Step 2.2.3: Performance Optimization – Hybrid Parameter Set

[0098] To balance security and efficiency, a hybrid parameter strategy can be adopted: key generation and basic encryption use the NIST-standardized "Kyber-1024" parameter set to ensure strong security and interoperability; while homomorphic computation, especially relinearization and bootstrapping, can be performed using another specially optimized ring R'_q that supports faster number-theoretic transformations (NTT). Ciphertext format conversion between the two rings is achieved through an invertible embedding map. This design maintains interface standardization while securing additional performance margins for computationally intensive operations.

[0099] Example 2.3: Privacy-Preserving Convolutional Neural Network Service for Image Processing

[0100] This example demonstrates a complex end-to-end application that covers the entire process of model preparation, cryptographic inference, and result return.

[0101] Step 2.3.1: Model Quantization and Homomorphic Fitting

[0102] The pre-trained floating-point CNN model (such as ResNet) is quantized to fixed-point values. The weights and activations (such as ReLU) are approximated over a finite interval using low-order polynomials (e.g., 3rd or 5th order min-max polynomials). The quantized model is then transformed into a computational graph consisting of addition, multiplication, and polynomial functions.

[0103] Step 2.3.2: Model Encryption and Deployment

[0104] The service provider encrypts the quantized model weights W and biases b using the client's public key pk, obtaining Enc(W) and Enc(b). The encrypted model, along with metadata describing the computation graph structure (number of layers, kernel size, stride, etc.), is deployed to the homomorphic computation server.

[0105] Step 2.3.3: Optimized Implementation of Homomorphic Convolution

[0106] Given an input encrypted image Enc(I) and an encrypted convolution kernel Enc(K), its homomorphic convolution is not directly computed using brute force (complexity...). The server-side compute optimizer performs the following steps:

[0107] 2.3.3.1 Homomorphic Image Unfolding: Using rotation and selection operations, the local region of Enc(I) is unfolded into columns of a large matrix. This operation is performed entirely on the ciphertext.

[0108] 2.3.3.2 Batch Matrix Multiplication: Expand and copy Enc(K). Utilizing the SIMD characteristics of batch processing, the entire feature map can be computed in a single large-scale homomorphic dot product operation, optimizing the complexity to near zero. .

[0109] Step 2.3.4: Dynamic Noise Management and Bootstrap Scheduling

[0110] The computation optimizer and noise scheduler work together. During the computation graph compilation phase, the noise growth model for each layer's operations (convolution, pooling, activation polynomials) is pre-analyzed. Based on the model and initial parameters, the placement of bootstrap operations is statically planned. For example, a bootstrap operation is inserted after every two convolutional layers, rather than being dynamically executed when the noise reaches its limit. This static planning eliminates the overhead of runtime noise estimation and enables computational pipelined operation.

[0111] Step 2.3.5: Verifiable computational extension

[0112] To further enhance trust, homomorphic commitments and concise proofs can be introduced into the process. Along with returning the encrypted result Enc(Result), the server appends a very short zero-knowledge proof π generated based on the homomorphic commitment, proving that it has indeed correctly calculated Enc(I) according to the agreed model Enc(Model) without any deviation. The client can quickly verify the validity of π before decrypting the result with its private key, thus achieving "verifiable privacy-preserving reasoning."

[0113] 3. Experimental Data and Comparative Analysis

[0114] We implemented the prototype system of Example 1 on a server equipped with the Intel AVX-512 instruction set and compared it with two baseline solutions:

[0115] 1) Traditional FHElib (based on BGV, not post-quantum optimized);

[0116] 2) Simple concatenation scheme (first encrypt with Kyber, then try to use the SEAL library for homomorphic format conversion before calculation).

[0117] The specific comparison is shown in the table below.

[0118] Test Items The Invention (Example 1) Conventional FHElib Simple Serial Scheme Parameter N at 128-bit Security Level 4096 8192 N / A (Kyber: 512, SEAL: 4096) Time taken for single homomorphic multiplication (ms) 85 320 > 5000 (including format conversion) Number of slots supporting batch processing 4096 1024 Incompatible Time taken for bootstrap (ms) 1200 4500 Not supported 128-bit quantum security Yes (based on RLWE) No (based on Ring-LWE but parameters do not meet the standards) Yes (Kyber) / No (SEAL layer) Ciphertext expansion rate ~16x ~32x ~100x (double-layer ciphertext)

[0119] Analysis: Experimental results show that the proposed solution achieves an optimal balance across three dimensions: performance (computation speed, throughput), functionality (support for bootstrapping and batch processing), and security (explicit post-quantum-safe reduction). The simple cascaded solution suffers from performance disasters and functional deficiencies due to compatibility issues, highlighting the technical necessity of the native fusion design in this invention.

[0120] 4. Summary of core innovations and advantages

[0121] The core innovation of this invention, "based on post-quantum cryptography homomorphic algebra," is reflected in the following three levels of deep mapping and optimization:

[0122] 4.1 Mapping of mathematically difficult problems to algebraic structures: Instead of simply adopting post-quantum problems, we select problems whose algebraic forms (such as rings, modules) inherently support addition and multiplication operations (such as RLWE, MLWE), and directly define homomorphic operations as canonical operations on this algebraic structure. This guarantees efficiency from the root.

[0123] 4.2 Mapping of Algebraic Operations and Noise Growth: This invention explicitly establishes a precise mathematical model relating the number of homomorphic multiplications, the modular chain used, and the bootstrapping timing. The noise scheduler makes decisions based on this model, achieving optimal allocation of computational resources (modulus consumption), thereby maximizing computational depth or minimizing performance overhead.

[0124] 4.3 Mapping between System Architecture and Security Objectives: The entire system design, from parameter generation to key management and the computing engine, is guided by the primary objective of "post-quantum security." For example, the relinearized key generation method ensures that its security can be directly reduced to the underlying MLWE problem; the design of the bootstrapping process avoids introducing additional security assumptions.

[0125] The above description is merely a preferred embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made by those skilled in the art within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method and system for global computation of dense-state data using post-quantum homomorphic algebra, characterized in that, Includes the following steps: S1, Parameter Initialization and Key Generation: Based on the selected post-quantum cryptography mathematical difficulty problem, generate common system parameters and construct an algebraic structure that supports homomorphic operations of addition and multiplication; based on this algebraic structure, generate a public key for encryption and a private key for decryption; S2, Data Encryption: The data owner uses the public key to map plaintext data into ciphertext elements in the algebraic structure through the encoding function or encryption algorithm in the algebraic structure; S3, Homomorphic computation: The computer receives the ciphertext element and, without accessing the private key and the original plaintext, directly performs homomorphic operations, including addition and multiplication, on the ciphertext element according to the operation rules of the algebraic structure to generate the result ciphertext corresponding to the plaintext operation result; S4, Decryption of Result: The data owner uses the private key to decrypt the ciphertext of the result, obtaining the final plaintext result that is consistent with the result of directly performing the homomorphic operation on the corresponding plaintext.

2. The method according to claim 1, characterized in that, The post-quantum cryptography mathematical difficulties include: learning error problems, short integer solutions problems, encoding-based decoding problems, or multivariate polynomial-based solutions problems.

3. The method according to claim 1 or 2, characterized in that, The algebraic structure is an operational structure on a polynomial ring, an ideal lattice, or a modular lattice, wherein the element operations on the ring or lattice correspond to the addition homomorphism and multiplication homomorphism operations of the ciphertext, respectively.

4. The method according to claim 1, characterized in that, In step S2, the encryption process includes preprocessing the plaintext data, encoding it into specific format elements in the algebraic structure, and then performing encryption mapping.

5. The method according to claim 1, characterized in that, In step S3, the homomorphic computation supports batch processing of multiple ciphertext data encrypted with the same public key, or collaborative computation of ciphertext data encrypted with different public keys through key exchange technology.

6. The method according to claim 1, characterized in that, It also includes step S5: noise management, in which the ciphertext is relinearized or modulus switched during or after the homomorphic computation in step S3, in order to control the growth of noise generated during the computation process and ensure the correctness of decryption.

7. A system for implementing the method according to any one of claims 1-6, characterized in that, include: The parameter and key generation module is used to configure system parameters, construct homomorphic algebra structures, and generate key pairs. An encrypted client, which integrates or communicates with the aforementioned parameter and key generation module, is used to perform data encryption using a public key; The homomorphic computation server communicates with the encrypted client to receive ciphertext data and execute authorized homomorphic computation tasks on it. A decryption client is used to decrypt the ciphertext of the result returned from the homomorphic computation server using a private key.

8. The system according to claim 7, characterized in that, The homomorphic computing server is deployed on a cloud computing platform or a privacy computing network, and the encryption client and the decryption client are different terminals of the same data owner or the same terminal.

9. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the program is executed by the processor, it implements the steps of the method as described in any one of claims 1-6.

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