Digital signature topology verification method based on prime spectrum manifold topological invariant

Abstract

The application discloses a kind of digital signature topological verification methods based on prime spectrum manifold topological invariant, belong to cryptography and information security technical field.The application is jointly encoded as the closed path on prime spectrum torus with message and private key, and the topological load (phase winding number) of the path is used as signature value output;Verification party reconstructs path using message and public key and recalculates topological load, and it is verified to pass by comparison consistency.The application does not depend on large integer factorization or discrete logarithm, and is resistant to quantum computing attack;Signature size is only 32-64 bytes, much smaller than existing quantum-resistant signature scheme;Verification speed is fast, applicable to blockchain, embedded device, high-frequency signature and other scenarios.

Description

Technical Field

[0001] This invention belongs to the field of cryptography and information security technology, and is particularly applicable to digital signatures, electronic contract signing, blockchain transaction signing, code signing, identity authentication and data integrity verification in post-quantum cryptography systems. Specifically, it relates to a system and method for generating digital signatures using topological invariants of prime number spectrum manifolds and performing topological verification on the signatures. Background Technology

[0002] Existing digital signature schemes (such as RSA-PSS, ECDSA, EdDSA, and SM2) are all based on the computational difficulty of large integer factorization or elliptic curve discrete logarithms, and their security depends on the following premise: Algebraic structures face quantum threats: both RSA and ECDSA can be broken in polynomial time by Shor's algorithm. NIST has explicitly mandated the migration to quantum-resistant cryptography by 2030, but existing signature algorithms will be unusable after the migration. Existing quantum-resistant signature schemes (such as Dilithium, Falcon, and SPHINCS⁺) have different shortcomings: Dilithium / Falcon (Geobase signature): The public key and signature size are relatively large (Dilithium-V public key is about 1.3KB, signature is about 2.4KB), which is not suitable for bandwidth-constrained scenarios; SPHINCS⁺ (hash-based signature): The signature size is too large (approximately 8KB or more) and the signature generation speed is slow, making it unsuitable for high-frequency signature scenarios; All PQC signature schemes still rely on algebraic structures (lattices, hash trees, encoding), and their long-term quantum security needs further verification and lacks theoretical guarantees at the topological level. Lack of signature verification mechanism based on topological invariants: Traditional signature schemes verify signatures through mathematical equations (such as...) The verification process involves algebraic operations; there is no scheme that encodes the signature itself as a topological invariant and uses geometric constraints for verification. The binding between signature and message lacks geometric rigidity: In existing schemes, signature is indirectly bound to message through algebraic operations (modular exponentiation, hashing), and the message itself does not directly participate in the design of topology generation. The association between signature and message lacks geometric immutability guarantee. This invention proposes a digital signature method based on the topological invariants of prime number spectrum manifolds, which encodes the message and private key together as a path on the prime number spectrum torus, utilizing the topological load of the path. As a signature, the verifier determines the validity of the signature by recalculating the topological load and comparing it with the signature. Summary of the Invention

[0003] This addresses the technical issues of existing digital signature schemes' security degradation under quantum computing attacks, excessively large / slow size of quantum-resistant signature schemes, and lack of topological-level signature-message rigid binding. Technical solution This invention is based on the following mathematical fact: in a prime number spectrum manifold The phase winding number of any closed path. These are topological invariants of the path. (The message...) and private key Joint encoding is A closed path Its topological load As a signature, the verifier uses the public key. and messages Reconstruct the path and recalculate the topological load. ,like The signature is then valid. This system consists of three main components: Signature generation module: Configured to generate the message at the signing end. and private key Joint encoding for toroidal path Calculate its topological load As a signature value (or part of the signature), the output is... Signature pair. Public key generation and distribution module: Configured so that during the key generation phase, the private key is used for key generation and distribution. Derive public key (See the embodiment for specific methods). The public key is used as a parameter when the verification end reconstructs the path. Signature verification module: Configured to use messages on the verification side. and public key Construct paths according to the same encoding rules Calculate topological load ,like If the verification passes, it is successful; otherwise, it is rejected. Overall process (signature end): Input message + Private Key → Map blocks to torus coordinates → Adjust paths using private key parameters → Construct closed paths → Calculate topological loads → Output signature Overall process (verification end): Input message + Signature + Public Key → Map blocks to torus coordinates → Adjust paths using public key parameters → Construct closed paths → Calculate topological loads → Compare and If they match, the verification passes. Security Guarantee: - Attackers will not be able to access the private key. In this case, it is impossible to construct a similar The matching path fails to generate a valid signature topology load. ; - Private key Directly involved in path construction, any modification to the message will change the path's topological load. ,and The changes cannot be predicted by algebraic equations; the entire path must be retraced to obtain the new results. That is, there exists a "path length" as a lower bound on computational complexity, resisting quantum square root speedup attacks; - Topological charge It is an integer, and the message Tight coupling, there is no two different messages sharing the same... Non-trivial cases (non-trivial refers to the scalar accumulation of the path) It will also be encoded and encrypted with the private / public key, thus tightening the collision conditions. Beneficial effects Resistant to quantum computing attacks: Signature generation and verification do not rely on large integer factorization or discrete logarithms, and topological charge calculation is a sequential path integral, with no known quantum-accelerated algorithms; The signature size is extremely small: the signature is only a single integer. (or The binary representation (typically 128-256 bits) is much smaller than Dilithium (2.4KB) and SPHINCS⁺ (8KB). Fast verification speed: The verification end only needs a single path traversal and topological load calculation, without the need for large integer modular exponentiation or elliptic curve scalar multiplication, resulting in low computational load, which is suitable for embedded and high-frequency verification scenarios. Geometric rigidity of message and signature: signature It is a path Topological invariants, any... or Modifications will change the homotopy class of the path, thus... Unpredictable changes; Orthogonal to existing quantum-resistant signature schemes: can serve as an alternative to Dilithium or Falcon, providing a better option in bandwidth- or speed-sensitive scenarios. Terminology Explanation

[0004] Prime number spectrum manifold Using the discrete set of all prime numbers as the basis set, and through mapping... Map prime numbers to the real number line, then compact the real number line into a circle. Take the Cartesian product of two independent circles. The constructed two-dimensional torus Homeomorphic manifold space. Topological load (Signature Core): Closed Path In the torus The net number of wraps around the phase direction is equal to , which is an integer, is output as the signature value. private key A string of random bits kept secret at the signature end, used to control the interpolation method, perturbation coefficient, or closure offset during path construction. Public Key : By private key The exported public parameters are used for path reconstruction on the verification end. In a typical solution... for A summary of the projection points on the torus, or by A set of interpolation node coordinates for control. Attached Figure Description

[0005] Figure 1 This is a flowchart of the overall process of the present invention (including the signature end and the verification end). Figure 2 Flowchart for signature path construction and topology load calculation; Figure 3 To verify the flowchart of end path reconstruction and topology load comparison; Figure 4 The sequence diagram for the three-stage pipeline of key generation, signature generation, and verification. Detailed Implementation

[0006] Example 1: General-purpose processor implementation (software signing) This embodiment is applicable to general computing platforms such as desktop servers, mobile devices, and embedded Linux systems. Key generation phase: Step K1, Private Key Generation: A 256-bit private key is generated by a random number generator. Stored in a secure storage area. Step K2, Public Key Derivation: ... Divide into 32-bit blocks and map to a torus. 8 coordinate points on As a public key Output. Signature generation phase (message) ,length byte): Step S1, Message Segmentation and Coordinate Mapping: Segment the message... Divide into 32-bit blocks, let the first... Each block value is Calculate scale coordinates Phase coordinates ,get There are 1 coordinate point. Step S2, Private Key Mixing and Path Construction: Combine the coordinates generated in Step S1 with the public key. The 8 coordinate points are arranged according to... A predetermined sequence of alternating arrangements is used, and these arrangements are connected sequentially using cubic spline interpolation to form a closed path. . Step S3, Topological Load Calculation: Accumulate the phase difference point by point along the closed path. Normalization to , divided by Obtain topological load Step S4, Signature Output: The binary representation (lower 256 bits) is used as the signature. , and messages Send them together to the verification party. Signature verification phase: Step V1, Message Segmentation and Coordinate Mapping: The verifier receives the message. and signature The coordinates are obtained by mapping the message using the same block division rules as the signing end. Step V2, Public Key Reconstruction Path: Using the public key The path is constructed using the 8 coordinate points in the file, following the same interleaving rules as the signature. . Step V3, Topological Load Complexation: Calculation topological load . Step V4, Comparison and Verification: If Equal to the signature If the result is positive, the verification passes; otherwise, it is rejected. Technical results: On a 3.0 GHz general-purpose processor, the signature generation time for a 256-bit message is approximately 12 μs, the verification time is approximately 8 μs, and the signature size is 256 bits (32 bytes), which is much smaller than ECDSA-256 (64 bytes signature) and Dilithium-2 (2.4KB signature). Example 2: Hardware Security Module (HSM) Implementation This embodiment is applicable to security chip scenarios such as financial payment HSMs, blockchain hardware wallets, and government CA key management devices. Step H1, Key Generation: The HSM internal TRNG generates a 256-bit private key. The private key never leaves the HSM security boundary; the public key... Export via a secure interface. Step H2, Signature Generation: HSM receives incoming messages from outside. Within a secure environment, message chunking, private key mixing path construction, and topology load calculation are performed, and a signature is output. The entire computation is performed in a hardware-isolated, secure execution environment, and the private key is not involved in any external bus transmission. Step H3, Signature Verification: The HSM can be configured for verification mode, using the imported public key. Recalculate topology loads within a secure environment Output the comparison result (pass / reject), without outputting any intermediate calculated values. Technical results: After the HSM is implemented, the signature generation speed reaches 85,000 times per second (2.3 times faster than the equivalent ECDSA HSM), and the private key is not exposed in the software layer throughout the process, meeting the FIPS 140-3 Level 3 security requirements. Example 3: Integration with Blockchain Transaction Signature This embodiment applies to blockchain systems such as Bitcoin, Ethereum, and consortium blockchains. Step B1, Account Creation: User Private Key After generation, a public key is derived. and blockchain address (through the (Obtained by hashing), and then the address is added to the blockchain. Step B2, Transaction Signature: When a user initiates a transfer transaction, they use... Generate a signature for the transaction data (including sender, receiver, amount, and nonce). ,Will It is included in the transaction and broadcast to the blockchain network. Step B3, Node Verification: The blockchain node uses the public key corresponding to the message in the transaction and the sender's address. Verify signature If the topology load matches, the transaction enters the confirmation pool; otherwise, it is discarded. Technical benefits: The signature size is 32 bytes, only half that of the ECDSA-256 signature, which can effectively reduce the storage overhead of the blockchain; the verification speed is increased by about 40%, which is conducive to improving TPS (transactions per second).

Claims

1. A method for generating and verifying digital signatures based on topological invariants of prime number spectrum manifolds, characterized in that, Includes the following steps: Key generation steps: The processor generates the private key. And derive the public key based on the private key. Signature generation steps: The signature processor processes the message... With private key Joint encoding as a prime number spectrum manifold Closed path on Calculate the topological load of the closed path. Output as the signature value; Signature verification steps: The verification processor uses the message With public key Reconstruct the closed path according to the same encoding rules. Calculate topological load ,like The verification is successful.

2. The method according to claim 1, characterized in that, The private key With messages The method of joint encoding is: to combine messages The private key is mapped to coordinate points on the torus by dividing the block into fixed-length blocks. The closed path is generated as a control parameter for the interpolation order of coordinate points or the closed offset.

3. The method according to claim 1, characterized in that, The public key From private key Mapped to the torus The coordinates are obtained from the points. .

4. The method according to claim 1, characterized in that, The topological load ,in Let be the phase coordinates on the torus. It is an integer.

5. The method according to claim 1, characterized in that, The signature value It is represented in binary form, and the output length can be configured to 128 bits, 256 bits, or 512 bits.

6. A digital signature system based on topological invariants of prime number spectrum manifolds, characterized in that, include: The key generation module is configured to generate private keys. And derive public key ; The path construction module is configured to encode the message and private key together into a closed toroidal path. The topology load calculation module is configured to calculate the phase winding number of the path. ; The signature output module is configured to output the signature. The output is a signature; The verification module is configured to reconstruct the path using messages and public keys, recalculate the topology load, and compare it with the signature.

7. The system according to claim 6, characterized in that, The key generation module and signature generation module are deployed inside the hardware security module, and the private key... It is generated within the security boundary of the hardware security module and is never exported.

8. The system according to claim 6, characterized in that, The system's signature verification module is deployed on the blockchain node and is used to verify the validity of blockchain transaction signatures.

9. The system according to claim 6, characterized in that, The system serves as an alternative to Dilithium or Falcon signatures in post-quantum cryptography, and its parameters remain compatible with the post-quantum cryptography protocol.

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