1 Linear secret sharing scheme
Linear secret sharing scheme is an encryption technology used to split a secret message into multiple fragments and distribute these fragments to multiple participants. Only when a sufficient number of participants cooperate, the complete secret information can be restored Secret information.
The basic principle of linear secret sharing scheme is to use polynomial interpolation. Suppose we have a polynomial of order (t-1), where t is the threshold (the minimum number of participants required to recover the secret information). We choose an integer n greater than or equal to t as the number of participants.
When generating a linear secret sharing scheme, first use the original secret information as the constant term of the polynomial, and randomly select other coefficients of the polynomial (which can be random numbers or other values). This generates a polynomial of order t.
Then, we calculate the value of the polynomial at the ordinal position of each participant in turn to obtain the shared value of each participant. These shared values constitute multiple distributed fragments of the secret, distributed to different participants.
When recovering secret information, at least t participants need to cooperate, and they combine the shared values they hold. Through the polynomial interpolation algorithm, based on these shared value data points, we can reconstruct the original polynomial and extract the secret information from it.
The advantages of the linear secret sharing scheme are security and scalability. Since the secret information is divided into multiple shared value fragments and distributed to different participants, a single participant cannot obtain the complete secret information, ensuring the security of the information. At the same time, the scalability of the scheme can be achieved by adjusting the threshold and the number of participants.
Linear secret sharing schemes have applications in many fields, especially in scenarios that require multi-party cooperation, such as distributed system security, multi-party collaboration encryption, data transmission confidentiality, etc. It provides a reliable way to protect secret information from potential attacks and leaks.
2 Linear secret sharing scheme based on secret sharing matrix
Linear secret sharing scheme based on secret sharing matrix is an improved linear secret sharing scheme, in which matrix operations are used to achieve the segmentation and recovery of secret information.
2.1 Principle
- Suppose we want to split a secret information into multiple shared values, first represent the secret information as a vector.
- Then, a t × n matrix is generated, where t is the threshold (the minimum number of participants required to recover the secret information) and n is the number of participants.
- Each row of the matrix is a random (irreversible) vector, and each participant gets a row of the matrix as its corresponding shared value.
- When a sufficient number of participants cooperate, these shared values can be used to recover the original secret information through matrix operations.
2.2 Structure
-
Generate a shared matrix:
- Computes a t × n matrix in which each element is randomly selected.
- Each row represents a shared value, where t shared values can be used to recover the secret information.
- Distribute each row to the appropriate participant.
-
Recover secret information:
- At least t participants cooperate to collect corresponding shared values.
- Construct a t × t submatrix using these t shared values.
- Use the sub-matrix to perform inverse matrix operations to obtain the inverse matrix.
- Represent the remaining n-t shared values as an n-t column vector.
- Multiply the column vector with the inverse matrix to get the original secret message.
2.3 Features
- Shared values in the scheme are represented as row vectors of matrices rather than as values of polynomials.
- The construction of the shared matrix requires guarantees of randomness and irreversibility to increase security.
- When recovering secret information, matrix operations and matrix inverse operations are required.
- The scheme still follows the threshold requirement, requiring the cooperation of at least t participants to recover the secret information.
The linear secret sharing scheme based on the secret sharing matrix can provide higher security and difficulty in cracking, because matrix-based operations involve more complex mathematical operations and have stricter mathematical guarantees. This solution is widely used in scenarios that require a higher level of confidentiality and security, such as confidential file transmission, blockchain security, etc.
3 Code implementation
3.1 Python implementation
#Python implementation example code of linear secret sharing scheme based on secret sharing matrix, with detailed annotations:
import numpy as np
import random
class LinearSecretSharing:
def __init__(self, num_shares, threshold, secret):
self.num_shares = num_shares # Number of shares
self.threshold = threshold # Threshold
self.secret = secret # Secret value
def generate_share_matrix(self):
# Generate a secret sharing matrix of num_shares x threshold
# Each element corresponds to the secret shared value
# Generate a random polynomial coefficient matrix
coefficients = np.random.randint(1, 1000, size=(self.threshold-1,))
# Build secret sharing matrix
share_matrix = np.zeros((self.num_shares, self.threshold))
for i in range(self.num_shares):
for j in range(self.threshold):
# Calculate the value of the i-th share
share_matrix[i][j] = self.secret + np.sum(coefficients * self.power(i + 1, j + 1))
return share_matrix
def reconstruct_secret(self, share_matrix):
# Reconstruct the secret value according to the secret sharing matrix
assert share_matrix.shape[0] >= self.threshold, "Not enough shares for reconstruction"
reconstructed_secret = 0
for i in range(self.threshold):
numerator = 1
denominator = 1
for j in range(self.threshold):
if i != j:
numerator *= (0 - j)
denominator *= (i - j)
share = share_matrix[i][0]
# Multiply by the Lagrangian interpolation polynomial coefficients
for k in range(self.threshold - 1):
share *= numerator / denominator
reconstructed_secret + = share
return int(reconstructed_secret)
def power(self, base, exponent):
# exponentiation
return base ** exponent
if __name__ == "__main__":
num_shares = 5
threshold = 3
secret=42
#Create a LinearSecretSharing object
lss = LinearSecretSharing(num_shares, threshold, secret)
# Generate secret sharing matrix
share_matrix = lss.generate_share_matrix()
# Output secret sharing matrix
print("Share Matrix:")
for row in share_matrix:
print(" ".join(str(int(share)) for share in row))
# Reconstruct the secret value
reconstructed_secret = lss.reconstruct_secret(share_matrix)
# Output the reconstructed secret value
print("Reconstructed Secret:", reconstructed_secret)
This code implements a linear secret sharing scheme based on a secret sharing matrix. Among them, the generate_share_matrix method is used to generate the secret sharing matrix, and the reconstruct_secret method is used to reconstruct the secret value based on the given secret sharing matrix.
In the main program, we can specify the number of shares, thresholds and secret values, and demonstrate the process of generating a secret sharing matrix and reconstructing the secret value based on this matrix.
Requires error handling, data validation, and more complex algorithms to meet your actual needs.
3.2 Java implementation
//Java implementation example code of linear secret sharing scheme based on secret sharing matrix, with detailed annotations:
import java.security.SecureRandom;
public class LinearSecretSharing {<!-- -->
private int numShares; // Number of shares
private int threshold; // threshold
private int secret; // secret value
private SecureRandom random; // Random number generator
// Constructor
public LinearSecretSharing(int numShares, int threshold, int secret) {<!-- -->
this.numShares = numShares;
this.threshold = threshold;
this.secret = secret;
this.random = new SecureRandom();
}
// Generate secret sharing matrix
public int[][] generateShareMatrix() {<!-- -->
int[][] shareMatrix = new int[numShares][threshold];
// Generate a polynomial for each share
for (int i = 0; i < numShares; i + + ) {<!-- -->
int[] coefficients = generateCoefficients();
shareMatrix[i] = generateShares(coefficients);
}
return shareMatrix;
}
// Generate polynomial coefficients
private int[] generateCoefficients() {<!-- -->
int[] coefficients = new int[threshold - 1];
// Randomly generate polynomial coefficients
for (int i = 0; i <threshold - 1; i + + ) {<!-- -->
coefficients[i] = random.nextInt();
}
return coefficients;
}
// Generate shared values based on polynomial coefficients
private int[] generateShares(int[] coefficients) {<!-- -->
int[] shares = new int[threshold];
// Calculate the value of each share
for (int i = 0; i <threshold; i + + ) {<!-- -->
int share = secret;
// Calculate the value of each term: coefficient * (i + 1)^exponent
for (int j = 0; j <threshold - 1; j + + ) {<!-- -->
share + = coefficients[j] * power(i + 1, j + 1);
}
//Storage shared value
shares[i] = share;
}
return shares;
}
// exponentiation
private int power(int base, int exponent) {<!-- -->
int result = 1;
for (int i = 0; i < exponent; i + + ) {<!-- -->
result *= base;
}
return result;
}
// Reconstruct the secret value based on the shared value
public int reconstructSecret(int[][] shareMatrix) {<!-- -->
int reconstructedSecret = 0;
// Use Lagrangian interpolation to reconstruct the secret value
for (int i = 0; i < numShares; i + + ) {<!-- -->
int numerator = 1;
int denominator = 1;
for (int j = 0; j <threshold; j + + ) {<!-- -->
if (i != j) {<!-- -->
numerator *= (0 - j);
denominator *= (i - j);
}
}
int share = shareMatrix[i][0];
// Multiply by the Lagrangian interpolation polynomial coefficients
for (int k = 0; k <threshold - 1; k + + ) {<!-- -->
share *= numerator / denominator;
}
reconstructedSecret + = share;
}
return reconstructedSecret;
}
public static void main(String[] args) {<!-- -->
int numShares = 5;
int threshold = 3;
int secret = 42;
LinearSecretSharing lss = new LinearSecretSharing(numShares, threshold, secret);
// Generate secret sharing matrix
int[][] shareMatrix = lss.generateShareMatrix();
// Output secret sharing matrix
System.out.println("Share Matrix: ");
for (int[] row : shareMatrix) {<!-- -->
for (int share : row) {<!-- -->
System.out.print(share + " ");
}
System.out.println();
}
//Reconstruct the secret value
int reconstructedSecret = lss.reconstructSecret(shareMatrix);
// Output the reconstructed secret value
System.out.println("Reconstructed Secret: " + reconstructedSecret);
}
}
This code implements a linear secret sharing scheme based on a secret sharing matrix. Among them, the generateShareMatrix method is used to generate the secret sharing matrix, and the reconstructSecret method is used to reconstruct the secret value based on the given secret sharing matrix.
In the main method, we can specify the number of shares, threshold and secret value, and demonstrate the process of generating a secret sharing matrix and reconstructing the secret value based on this matrix.
Error handling, data validation, and more complex algorithms are also needed to meet your actual needs.