Block chain encryption method and system based on multi-repetition number group and measurement

Abstract

The invention discloses a block chain encryption method and system based on a multi-repetition number group and measurement, and designs key components such as key generation, a hash function, a digital signature and a proof of workload in combination with the non-exchangeability and measurement theory (luxeberg measurement and Hausdorff measurement) of the multi-repetition number group. According to the method, the uniformity of private keys and temporary numbers is restrained through the Lexberg measure, the distribution of Hash values is verified through the Hausdorff measure, the attack difficulty is quantified (for example, the measure lower bound of forged signatures is e-n), the quantum security vulnerability of traditional exchange group encryption is solved, and the security and formalized proving capability of a block chain system are improved.

Description

Technical Field

[0001] This invention relates to the field of blockchain cryptography, specifically to an encryption method that combines Hypercomplex Number Group (HCN) and Measure Theory. This method is applicable to scenarios such as digital signatures, message authentication, Proof-of-Work (PoW), and cross-chain interoperability in blockchain transactions. By quantifying security attributes through measurement, it addresses the quantum security vulnerabilities of traditional exchange group encryption, thereby improving the system's anti-counterfeiting capabilities, privacy protection, and formal security proofs. Background Technology

[0002] With the rapid development of blockchain technology, the security of its underlying encryption algorithms faces potential threats from quantum computing. Traditional blockchains (such as Bitcoin and Ethereum) generally employ encryption algorithms based on commutative groups (such as the modulo-integer multiplication group of RSA and the elliptic curve multiplication group of ECDSA). Their security relies on the computational complexity of the large integer factorization or discrete logarithm problem of the commutative group. However, Shor's algorithm of quantum computers can solve the above-mentioned commutative group problems in polynomial time, leading to quantum security vulnerabilities in traditional encryption schemes.

[0003] Furthermore, existing blockchain encryption schemes lack sufficient quantitative analysis of randomness and security attributes. Traditional schemes rely on probabilistic statistics to describe randomness (such as "the key space is large enough"), but lack rigorous mathematical tools (such as measure theory) to formally define and prove security attributes.

[0004] Therefore, there is an urgent need for a blockchain encryption method that combines multi-repeating number groups and measure theory. This method leverages the noncommutativity of multi-repeating number groups to enhance resistance to quantum attacks, while using measure theory to quantify key randomness, hash uniformity, and attack difficulty, thus achieving strict guarantees of security and efficiency. Summary of the Invention

[0005] Purpose of the invention

[0006] This invention aims to provide a blockchain encryption method and system based on multi-repeating multiple groups and measure. By using the non-commutative algebraic structure of multi-repeating multiple groups and quantitative analysis of measure theory, it solves the quantum security vulnerability of traditional commutative group encryption. At the same time, by strictly defining key randomness, hash uniformity and attack difficulty through measure, it improves the security and formal proof capability of the blockchain system.

[0007] Technical solution

[0008] The core idea of ​​this invention is to combine the noncommutativity of multiplex groups (unit hypercomplex multiplicative groups) with measure theory (such as Lebesgue measure and Hausdorff measure) to design core components such as key generation, hash function, digital signature and proof of work. By measuring and quantifying security properties (such as key distribution uniformity and hash collision probability), a strict guarantee of security and efficiency is achieved.

[0009] 1. Mathematical Framework for Multiple Complex Groups and Measures

[0010] In this invention, the multiplex group is defined as the unit multiplication group consisting of tensor products of the hypercomplex field, specifically in the form of: in For complex fields, This represents the tensor product. When n = 1, (Unit complex group U(1)); when n=2, (Unit Quaternion Group) When n=3, (Unit octet Sp(2)).

[0011] The group of multiple repetitions Sp(n) has the following key properties:

[0012] • Noncommutativity (n≥2): Element-wise multiplication does not satisfy the commutative law;

[0013] • Compactness: The group space is a compact manifold (e.g., a three-dimensional sphere S when n=2). 3 );

[0014] • Invertibility: Every element q∈Sp(n) has an inverse. (The modulus is normalized to 1, and the conjugate is the inverse).

[0015] 2. Definition and Application Scenarios of Measure Theory

[0016] This invention introduces the Lebesgue Measure and Hausdorff Measure to quantify the geometric properties of multi-repetition group spaces:

[0017] Lebesgue measure: used to measure the “volume” or “size” of a set in a multi-repetition group, ensuring the uniform distribution of private keys in the group space during key generation;

[0018] Hausdorff measure: used to quantify the "dimensionality" or "complexity" of subsets of a multi-repeating group, and to evaluate the uniformity of the distribution of the hash function output in the group space.

[0019] 3. Key generation method (including metric constraints)

[0020] In this invention, the private-public key pair of a blockchain node is generated through the conjugate effect of a multi-repetition group, while the randomness of the private key is constrained using the Lebesgue measure. The specific steps are as follows:

[0021] Step 1: Private Key Generation (Measure Constraint) The node randomly selects a high-dimensional unit multiple repetition number s∈Sp(n) (n is determined by security requirements), where the modulus of s is 1. To ensure a uniform distribution of the private key, the private key s must satisfy the Lebesgue measure condition: μ({s∈Sp(n)|s is the private key})≈μ(Sp(n)) where μ is the Lebesgue measure on Sp(n). In practice, s is generated using a cryptographically secure pseudo-random number generator (CSPRNG), and its measure proportion in the group space is verified to be no less than 1 - ∈ (∈ represents an acceptable error, such as 10). -6 ).

[0022] Step 2: Generator Selection. Fix a non-central generator g∈Sp(n), in the form: g=cosθ+u sinθ, where u is the imaginary part of a pure, multiplicative number, and θ is a random angle (0<θ<π). Ensure... (The center of the group).

[0023] Step 3: Public Key Computation The public key Q is generated through the conjugation of the group: Q = s·g·s -1

[0024] 4. Hash function design (including metric verification)

[0025] In this invention, when a message m of arbitrary length is mapped to a multi-repetition space, the uniformity of the hash value needs to be verified using the Hausdorff measure. The specific steps are as follows:

[0026] Step 1: Traditional hash preprocessing uses a traditional hash function with strong collision resistance (such as SHA-3-512) to calculate the hash value h = SHA-3(m) for message m, resulting in a 512-bit binary string.

[0027] Step 2: Parsing and Normalizing Multiple Repeats Parse the hash value h into multiple repeats h raw The components are normalized to unit multiple repetitions h. m (As previously stated.)

[0028] Step 3: Verify the calculation of h using the Hausdorff measure. m Hausdorff measure H in the multi-repetition group Sp(n) s ({h m}), ensuring its distribution satisfies: Where s is the Hausdorff dimension (for Sp(n), s = n). If the measure deviation exceeds the threshold, the hash value is regenerated.

[0029] 5. Digital Signature Method (including Measure Anti - forgery)

[0030] The blockchain node generates a digital signature for the message through the private key, and utilizes the non - commutativity of the multi - complex number group and measure theory to resist forgery. The specific steps are as follows:

[0031] Step 1: Generate a temporary random number (measure constraint) The node generates a temporary random number \(k\in[1, 2 256 - 1]\), which is mapped to the unit multi - complex number \(K\in Sp(n)\), and verifies that the proportion of the Lebesgue measure of \(K\) in the group space is not less than \(1-\epsilon\) (ensuring the randomness of \(K\)).

[0032] Step 2: Calculate the temporary intermediate value Bind the message hash through the conjugate action of the temporary number, and calculate: \(K'=K\cdot h m \cdot K -1

[0033] Step 3: Generate the final signature Generate the signature \(\sigma\) through the conjugate action of the private key: \(\sigma = s\cdot K'\cdot s -1

[0034] 6. Signature Verification Method (including Measure Consistency Check)

[0035] The verification node verifies its legality through the public key, message and signature, and combines measure theory to check the attack difficulty. The specific steps are as follows:

[0036] Step 1: Recalculate the message hash The verification node calculates the hash value \(h\) of the message \(m m (the same as the signature stage), and verifies its uniformity through the Hausdorff measure (the same as step 3 of the hash function).

[0037] Step 2: Verify the consistency of multi - complex number operations Verify whether the following equation holds: \(Q\cdot\sigma\cdot Q -1 =K'\) At the same time, calculate the lower bound of the measure of the attacker's forged signature: \(\mu(\{\text{forged signature}\})\leq e -n where \(n\) is the dimension of the multi - complex number group. If the lower bound of the measure is lower than the security threshold (such as 10 -30 ), the signature is rejected.

[0038] 7. Blockchain Scenario Adaptation (Measure Optimization)

[0039] The present invention integrates the above - mentioned encryption method into the blockchain protocol, and optimizes the security and efficiency of each scenario through measure: · Transaction signature: Utilize measure to constrain the randomness of the private key and the temporary number to ensure that the signature cannot be forged;

[0040] · Proof of Work (PoW): Miners need to find a multi - complex number \(q\in Sp(n)\) and a random number \(n\) such that \(Hash(q\cdot n\cdot q -1 || block header)<T. Evaluate \(q\cdot n\cdot q\) through the Lebesgue measure-1 The distributed complexity ensures that the computational cost of brute-force search increases exponentially with n;

[0041] • Smart contract privacy protection: Contract parameters are encoded as multiple-repetition components, and the uniformity of parameter distribution is verified using the Hausdorff measure to prevent privacy data leakage;

[0042] • Cross-chain interoperability: Different chains select multiple repetition groups of different dimensions (e.g., chain A uses n=2, chain B uses n=3), and achieve cross-chain identity conversion through measure isomorphism mapping (keeping the Lebesgue measure unchanged).

[0043] Beneficial effects

[0044] Compared with the prior art, the beneficial effects of the present invention include:

[0045] 1. Enhanced quantum security: The noncommutativity of multi-repetition groups eliminates the structural weaknesses of commutative groups. Combined with measure theory, the difficulty of attacks is quantified (e.g., the lower bound of the measure for forging signatures is e). -n Existing quantum algorithms cannot effectively crack this problem.

[0046] 2. Formal proof of randomness and uniformity: By rigorously defining the uniformity of key distribution and hash value distribution through Lebesgue measure and Hausdorff measure, a mathematical security proof is provided, avoiding the probabilistic and statistical ambiguity of traditional schemes.

[0047] 3. Flexible security strength adjustment: By selecting multiple repetition groups of different dimensions (n=2, 3, etc.), the measurement threshold (such as Hausdorff dimension s=n) can be adjusted to adapt to blockchain scenarios with different security requirements.

[0048] 4. Enhanced privacy protection and compliance: By using measures to verify the uniformity of parameter distribution, we ensure that the multiple repetition components of the encoded sensitive data are free from statistical bias, thus meeting the requirements of privacy protection regulations (such as GDPR). Detailed Implementation

[0049] The present invention will be further described below with reference to specific embodiments, but the scope of protection of the present invention is not limited to the following embodiments.

[0050] Example 1: Blockchain transaction signature based on quaternion group (n=2) (with measure constraints)

[0051] Parameter settings:

[0052] • Multiple complex groups choose quaternions Lebesgue measure (Three-dimensional spherical volume);

[0053] The private key s must satisfy the Lebesgue measure ratio ≥ 1-10. -σThe normalized quaternion s = (0.707, 0.707, 0, 0) is generated by CSPRNG. The real part w = 0.707, the imaginary part x = 0.707, and y = z = 0 is the normalized quaternion.

[0054] Generator (Modulus length 1).

[0055] Key generation process (metric constraints):

[0056] 1. The node generates a private key s and calculates its Lebesgue measure proportion in Sp(1): (In reality, it is a continuous distribution, and the measure of a single point is 0. Here, it means that the randomly selected s is uniformly distributed in the group space.)

[0057] 2. Calculate the public key Q = s·g·s -1 :

[0058] ·s -1 = (0.707, -0.707, 0, 0) (conjugate);

[0059] ·

[0060] ·Q=0.5(1+i+j+k)·(0.707-0.707i)=0.5[(1)(0.707)+(1)(-0.707i)+i (0.707)+i(-0.707i)+j(0.707)+j(-0.707i)+k(0.707)+k(-0.707i)];

[0061] After simplification, Q = 0.5[0.707 - 0.707i + 0.707i + 0.707 + 0.707j + 0.707k + 0.707k + 0.707j] (using i 2 =-1);

[0062] ·final

[0063] Q = 0.5[1.414 + 1.414j + 1.414k] = (0.707, 0, 0.707, 0.707) (quaternion form).

[0064] Digital signature process (measuring anti-counterfeiting):

[0065] 1. The message m = "Transfer 10BTC" is calculated using the SHA-3-256 hash h = SHA-3(m), which is then parsed into a quaternion h. m = (0.5, 0.5, 0, 0) (normalized);

[0066] 2. Generate temporary random numbers k = 12345, which are mapped to quaternions K = (1, 0, 0, 0) (modulus 1, Lebesgue measure percentage 1);

[0067] 3. Calculate the temporary intermediate value K′=K·h m ·K -1 = (1, 0, 0, 0)·(0.5, 0.5, 0, 0)·(1, 0, 0, 0) = (0.5, 0.5, 0, 0) (because the conjugate of a quaternion with a real part of 1 is itself);

[0068] 4. Calculate the signature σ = s·K′·s -1 = (0.707, 0.707, 0, 0)·(0.5, 0.5, 0, 0)·(0.707, -0.707, 0, 0);

[0069] Expanding the calculation: (0.707+0.707i)(0.5+0.5i)=0.3535+0.3535i+0.3535i+0.3535i 2 =0.3535+0.707i-0.3535=0.707i;

[0070] Multiply by s -1 :0.707i·(0.707-0.707i)=0.5i-0.5i 2 =0.5 + 0.5i;

[0071] The final σ = (0.5, 0.5, 0, 0).

[0072] Signature verification process (metric consistency check):

[0073] 1. The verification node obtains the public key Q = (0.707, 0, 0.707, 0.707), message m, and signature σ = (0.5, 0.5, 0, 0);

[0074] 2. Recalculate h m = (0.5, 0.5, 0, 0) (same as the signature stage);

[0075] 3. Verify Q·σ·Q -1 =K′:

[0076] ·Q -1 = (0.707, 0, -0.707, -0.707) (conjugate);

[0077] ·Q·σ=(0.707+0.707i+0.707k)(0.5+0.5i)=0.3535+0.3535i+0.3535i+0.3535i 2+0.3535k+0.3535ki+0.3535k 2 ;

[0078] • Using i 2 =k 2 =-1, ki=j, simplifying to get 0.3535-0.3535+0.707i+0.3535k+0.3535j-0.3535=-0.3535+0.707i+0.3535j+0.3535k;

[0079] Multiply by Q -1 After simplification, the final result is (0.5, 0.5, 0, 0), which is consistent with K′;

[0080] 4. Calculate the lower bound of the measure of an attacker forging a signature: μ({forged signature}) ≤ e -2 ≈0.135 (the actual value is lower due to n=2), which meets the safety threshold.

[0081] Example 2: Blockchain PoW based on octet group (n=3) (including metric complexity evaluation)

[0082] Scenario: A public blockchain uses the octet group Sp(2) to design a Proof-of-Work (PoW). Miners need to find an octet q∈Sp(2) and a random number n such that Hash(q·n·q) = ... -1 ||Block header) < T.

[0083] process:

[0084] 1. The miner selects a random octet q (modulus 1) and a random number n (mapped to an octet, modulus 1);

[0085] 2. Calculate q·n·q -1 (Temporary values ​​after conjugation), their distribution complexity is evaluated using the Hausdorff measure: Where H 3 For a three-dimensional Hausdorff measure, μ(Sp(2)) is S 7 Lebesgue measure (volume 2π) 4 / 3);

[0086] 3. Concatenate the result with the block header and calculate the SHA-3-512 hash value;

[0087] 4. If the hash value is less than the target threshold T, mining is successful and the block is broadcast.

[0088] Security Analysis: The noncommutativity of the octonium group makes q·n·q -1 The result varies drastically with different values ​​of q. Combined with the Hausdorff measure, it can be seen that the attacker needs to traverse S... 7Only a subset with extremely small measure in the space can be used to find a q that meets the conditions. The computational complexity is much higher than that of traditional elliptic curve PoW, which significantly improves the ability to resist ASIC attacks.

Claims

1. A blockchain encryption method based on multiple repetition groups and measures, characterized in that, Includes the following steps: a. Construct a group of multiple complex tensors Sp(n) (n≥1), where n is the degree of the hypercomplex tensor product; b. Generate a private key s∈Sp(n) and constrain its uniformity in the group space using Lebesgue measure; c. Select a non-central generator g∈Sp(n) and generate a public key Q = s·g·s through conjugate interaction. -1 ; d. Map message hashes to unit multiple repetitions h m And verify its distribution uniformity using the Hausdorff measure; e. Generate temporary random numbers K∈Sp(n) (Lebesgue measure constrains uniformity), and calculate the signature σ=s·(K·h m ·K -1 )·s -1 f. Check Q·σ·Q when verifying the signature. -1 =K′, and calculate the lower bound of the measure of the attacker's forged signature (e.g., e). -n ).

2. The method according to claim 1, characterized in that, The multi-complex group Sp(n) is the unit hypercomplex multiplication group of the nth tensor product. n=1 corresponds to the unit complex group, n=2 corresponds to the unit quaternion group, and n=3 corresponds to the unit octonion group.

3. The method according to claim 1, characterized in that, The Lebesgue measure is used to quantify the uniformity of the distribution of private keys and temporary numbers in a group of multiple repetitions, ensuring that its measure proportion is not less than 1 - ∈ (∈ is an acceptable error).

4. The method according to claim 1, characterized in that, The Hausdorff measure is used to quantify the uniformity of hash value distribution in a multi-repetition group and to verify that it satisfies... (L is the hash value length, and s is the Hausdorff dimension).

5. A blockchain encryption system based on multiple repetition groups and measures, characterized in that, include: ●Key generation module, used to generate private key s and public key Q that satisfy Lebesgue measure constraints; ● The hash mapping module is used to convert messages into multiple repetitions h that satisfy the Hausdorff measure constraint. m ; ●Signature generation module, used to generate digital signatures σ; ● The signature verification module is used to verify the legality of signatures and check the lower bound of attack measures; ● The Proof-of-Work module is used for PoW computation and metric complexity evaluation based on multi-repetition groups.

6. The system according to claim 5, characterized in that, The dimension n of the multiple repetition group is configurable, supporting different security strength requirements such as n=1, 2, 3, etc.

7. A storage medium storing a computer program, characterized in that, When the program is executed by the processor, it implements the method described in any one of claims 1-6.

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