Encoding and decoding method for distributed multi-user key sharing problem under strong privacy conditions

Abstract

The present invention provides an encoding and decoding method for the distributed multi - user key sharing problem under strong privacy conditions. The method includes the following steps: Step 1: Initialize an N×(∑ k∈[K] R k +N) generating matrix V according to the access structure and the key information rate array; Step 2: Obtain the corresponding K(K - 1) row index sets according to the access structure and the key information rate array, and extract K sub - matrices from the generating matrix V by these sets. The strong privacy condition is equivalent to the full rank of each sub - matrix; Step 3: Calculate the determinants according to each sub - matrix, multiply them to obtain a polynomial, find a set of feasible solutions that make the polynomial not equal to 0, then set the remaining variables in the generating matrix V to 0, then extract the decoding method from the generating matrix V, and finally perform Gaussian elimination on the generating matrix V to obtain the encoding method. The method of the present invention can be applied to any access structure and can achieve the optimal key information rate.

Description

Technical Field

[0001] The present invention relates to the technical field of information theory, and in particular to an optimal encoding and decoding method for the distributed multi-user key sharing problem for any number of keys K, any number of sub-keys N, and any access structure A. Background Art

[0002] In the multi-user key sharing problem, most of the work focuses on the scenario of weak privacy conditions. However, since a certain user may be able to obtain partial information of the key sets corresponding to the remaining users, such as the linear combination between K-1 keys, it is meaningful to study strong privacy conditions. The strong privacy condition requires that any user cannot obtain any information of the key sets corresponding to all other users, while the weak privacy condition only requires that any user cannot obtain the information of the key corresponding to any other user. The existing research has solved the optimal encoding and decoding problem under weak privacy conditions. Under strong privacy conditions, the current encoding and decoding methods only perform key sharing for each user separately, so the corresponding key information rate is not high. There is no better encoding and decoding method, that is, there is no relevant research report on the optimal encoding and decoding method for the distributed multi-user key sharing problem under strong privacy conditions. Summary of the Invention

[0003] Object of the Invention: To solve the above problems, the present invention proposes an optimal encoding and decoding method for the distributed multi-user key sharing problem under strong privacy conditions, uses the generating matrix to solve polynomials, thereby extracting the decoding method for keys and the encoding method for sub-keys, and obtains the optimal encoding and decoding method for any number of keys K, any number of sub-keys N, and any access structure A.

[0004] Technical Solution: To achieve the object of the present invention, the technical solution adopted by the present invention is:

[0005] An optimal encoding and decoding method for the distributed multi-user key sharing problem under strong privacy conditions, the method comprising the following steps:

[0006] Step 1: First, for the distributed multi-user key sharing problem for any number of keys K, any number of sub-keys N, and any access structure A, in combination with the key information rate array (R1,..., R K ), initialize an N×(∑ k∈[K] R k +N) generating matrix V, where the generating matrix V corresponds to the key W k , k∈[K] and the sub-key Y n , n∈[N] from left to right. For the sub-matrix place different variables in the rows corresponding to the k-th access set A k in the access structure A, and in the remaining rows [N]\A kPlace 0 in it, for the sub - matrix Set it as an N×N identity matrix;

[0007] Step 2: According to the access structure and the key information rate array, obtain the corresponding K(K - 1) sets of row indices. Extract K sub - square matrices from the generating matrix V by these sets. The strong privacy condition is equivalent to the full rank of each sub - square matrix;

[0008] Step 3: Calculate the determinant according to each sub - square matrix, multiply to get a polynomial, find a set of feasible solutions that make the polynomial not equal to 0, then set the remaining variables in the generating matrix V to 0, then extract the decoding method from the generating matrix V, and finally perform Gaussian elimination on the generating matrix V to obtain the encoding method.

[0009] For the optimal encoding and decoding method of the distributed multi - user key sharing problem under the strong privacy condition, the set of row indices described in Step 2 k∈[K], i∈[K]\{k} are required as follows:

[0010]

[0011]

[0012]

[0013] where A k is the k - th access set in the access structure A, refers to the set of row indices contains the number of rows, R i is the information rate of the i - th key, is an empty set. As long as the key information rate array satisfies the following inequality:

[0014]

[0015] Such a set of row indices can always be found.

[0016] After obtaining the K(K - 1) sets of row indices as required, for each element k in the set [K], from the sub - matrix extract a ∑ i∈[K]\{k} according to the set of row indices i∈[K]\{k} R i ×∑ i∈[K]\{k} R i sub - square matrix. The strong privacy condition requires that this sub - square matrix is full - rank, and the set of row indices that meet the requirements can ensure that the determinant of this sub - square matrix is a non - zero polynomial, that is, at least one coefficient in the polynomial is non - zero.

[0017] The optimal encoding and decoding method for the distributed multi - user key sharing problem under strong privacy conditions. The specific steps of step 3 are as follows:

[0018] Step 3.1: The steps for calculating the determinant are as follows:

[0019] Extract K sub - square matrices from the generating matrix V according to the above row index set. Each sub - square matrix only contains variables and 0, so its determinant is a polynomial, and the characteristics of the row index set ensure that at least one coefficient in this polynomial is non - zero;

[0020] Step 3.2: The steps for solving the polynomial are as follows:

[0021] The strong privacy condition is equivalent to the full rank of each sub - square matrix, that is, the K sub - square matrices need to satisfy full rank simultaneously, which is equivalent to the product of the above K determinants not being equal to 0. Since the K sub - square matrices are extracted from the generating matrix V, the degree of the polynomial after multiplication is less than the number of keys K. Therefore, as long as the size of the finite field is greater than or equal to K, a set of feasible solutions that make the polynomial not equal to 0 must exist;

[0022] Step 3.3: The steps for determining the encoding and decoding method are as follows:

[0023] After obtaining a set of feasible solutions that make the polynomial not equal to 0, set the remaining variables in the generating matrix V to 0, and then the decoding method can be obtained from this generating matrix V. Specifically, put the sub - keys Y n , n ∈ [N] into a row vector, and the key W k , k ∈ [K] is equal to the product of this row vector and the matrix The decoding method is thus obtained.

[0024] For the encoding method, Gaussian elimination needs to be performed on the generating matrix V, so that above the new sub - matrix V W[K] is a ∑ i∈[K] R i ×∑ i∈[K] R i identity matrix, and below is all zeros. Then put the key W k , k ∈ [K] and random noises into a row vector, and the sub - key Y n , n ∈ [N] is equal to the product of this row vector and the new matrix The encoding method is thus obtained.

[0025] Beneficial effects:

[0026] The present invention proposes a general encoding and decoding scheme for the distributed multi - user key sharing problem, which can be applied to strong privacy conditions and can achieve the optimal information rate, that is, the size of the key that the user wants to obtain corresponding to the size of the unit sub - key is the largest. Brief Description of the Drawings

[0027] Figure 1 is the flowchart of the method of the present invention. Detailed Embodiments

[0028] The technical solution of the present invention will be further described below with reference to the accompanying drawings.

[0029] As Figure 1 shown, an optimal encoding and decoding method for the distributed multi - user key sharing problem under strong privacy conditions of the present invention includes the following steps:

[0030] Step 1: First, for the distributed multi - user key sharing problem with any number of keys K, any number of sub - keys N, and any access structure A, in combination with the key information rate array (R1,...,R K ), initialize an N×(∑ k∈[K] R k +N) generation matrix V, where the generation matrix V corresponds to the key W k ,k∈[K] and the sub - key Y n ,n∈[N] from left to right. For the sub - matrix place different variables in the rows corresponding to the k - th access set A k in the access structure A, and place 0 in the remaining rows [N]\A k . Set the sub - matrix as an N×N identity matrix;

[0031] Step 2: According to the access structure and the key information rate array, obtain the corresponding K(K - 1) row index sets, and extract K sub - matrices from the generation matrix V. The strong privacy condition is equivalent to the full rank of each sub - matrix;

[0032] Step 3: Calculate the determinants according to each sub - matrix, multiply them to obtain a polynomial, find a set of feasible solutions that make the polynomial not equal to 0, then set the remaining variables in the generation matrix V to 0, then extract the decoding method from the generation matrix V, and finally perform Gaussian elimination on the generation matrix V to obtain the encoding method.

[0033] For the optimal encoding and decoding method for the distributed multi - user key sharing problem under strong privacy conditions, the row index sets k∈[K],i∈[K]\{k} in Step 2 are required as follows:

[0034]

[0035]

[0036]

[0037] Among them, A k is the k-th access set in the access structure A, refers to the set of row indices including the number of rows, R i is the information rate of the i-th key, is an empty set. As long as the key information rate array satisfies the following inequality:

[0038]

[0039] such a set of row indices can always be found.

[0040] After obtaining K(K - 1) sets of row indices as required, for each element k in the set [K], from the submatrix extract a Σ i ∈ [K]\{k} according to the set of row indices i∈[K]\{k} R i × Σ i∈[K]\{k} R i sub-square matrix of. The strong privacy condition requires that this sub-square matrix is full rank, and the set of row indices that meets the requirements can ensure that the determinant of this sub-square matrix is a non-zero polynomial, that is, at least one coefficient in the polynomial is non-zero.

[0041] For the optimal encoding and decoding method of the distributed multi-user key sharing problem under the strong privacy condition described above, the specific steps of step 3 are as follows:

[0042] Step 3.1: The steps for calculating the determinant are as follows:

[0043] Extract K sub-square matrices from the generating matrix V according to the above set of row indices. Each sub-square matrix only contains two forms: variables and 0. Therefore, its determinant is a polynomial, and the characteristics of the set of row indices ensure that at least one coefficient in this polynomial is non-zero;

[0044] Step 3.2: The steps for solving the polynomial are as follows:

[0045] The strong privacy condition is equivalent to the full rank of each sub-square matrix, that is, the K sub-square matrices need to satisfy full rank simultaneously. This is also equivalent to the product of the above K determinants not being equal to 0. Since the K sub-square matrices are extracted from the generating matrix V, the degree of the polynomial after multiplication is less than the number of keys K. Therefore, as long as the size of the finite field is greater than or equal to K, a set of feasible solutions that make the polynomial not equal to 0 must exist;

[0046] Step 3.3: The steps for determining the encoding and decoding method are as follows:

[0047] After obtaining a set of feasible solutions that make the polynomial not equal to 0, then set the remaining variables in the generating matrix V to 0, and then the decoding method can be obtained from this generating matrix V. Specifically, for the sub-key Yn , for \(n\in [N]\), put it into a row vector, the key \(W\) k , for \(k\in [K]\), it is equal to the product of this row vector and the matrix The decoding method is obtained accordingly.

[0048] For the encoding method, Gaussian elimination needs to be performed on the generating matrix \(V\), so that the upper part of the new sub - matrix is a \(\sum\) i∈[K] \(R\) i \(\times\sum\) i∈[K] \(R\) i identity matrix, and at the same time the lower part is all zeros. Then put the key \(W\) k , for \(k\in [K]\) and \(R\) k random noises into a row vector, the sub - key \(Y\) n , for \(n\in [N]\) is equal to the product of this row vector and the new matrix The encoding method is obtained accordingly.

[0049] The following gives an embodiment:

[0050] (1) First, for the distributed multi - user key sharing problem with any number of keys \(K\), any number of sub - keys \(N\) and any access structure \(A\), combined with the key information rate array \((R_1,\cdots,R\) K ), initialize an \(N\times(\sum\) k∈[K] \(R\) k \(+N)\) generating matrix \(V\), where the generating matrix \(V\) corresponds to the key \(W\) k , for \(k\in [K]\) and the sub - key \(Y\) n , for \(n\in [N]\) from left to right. For the sub - matrix place different variables in the rows corresponding to the \(k\) - th access set \(A\) k in the access structure \(A\), and place \(0\) in the remaining rows \([N]\setminus A\) k . For the sub - matrix set it as an \(N\times N\) identity matrix.

[0051] (2) According to the access structure and the key information rate array, obtain the corresponding \(K(K - 1)\) row - index sets, and extract \(K\) sub - square matrices from the generating matrix \(V\) by these sets. The strong privacy condition is equivalent to the full rank of each sub - square matrix.

[0052] (3) Calculate the determinant according to each sub - square matrix, multiply to get a polynomial, find a set of feasible solutions that make the polynomial not equal to \(0\), then set the remaining variables in the generating matrix \(V\) to \(0\), then extract the decoding method from the generating matrix \(V\), and finally perform Gaussian elimination on the generating matrix \(V\) to obtain the encoding method.

[0053]

[0054] Table 1 shows an encoding and decoding method for K keys, N sub-keys, an access structure A, and a key information rate array (R1,...,R K ). During encoding, the coefficients and are obtained from the generating matrix V, and during decoding, the coefficient is obtained from the generating matrix after Gaussian elimination.

[0055] The above is only the preferred embodiment of the present invention, and the protection scope of the present invention is not limited to the above embodiment. Any equivalent modification or change made by those of ordinary skill in the art according to the disclosure of the present invention shall be included in the protection scope recorded in the claims.

Claims

1. An encoding and decoding method for the distributed multi-user key sharing problem under strong privacy conditions, characterized in that It includes the following steps: Step 1: First, for the distributed multi-user key sharing problem with any number of keys K, any number of sub-keys N, and any access structure A, combined with the key information rate array (R1,...,R K ), initialize an N×(Σ k∈[K] R k +N) generation matrix V, where the generation matrix V corresponds to the key W k ,k∈[K] and the sub-key Y n ,n∈[N] from left to right. For the sub-matrix place different variables in the rows corresponding to the k-th access set A k in the access structure A, and place 0 in the remaining rows [N]\A k . For the sub-matrix set it to an N×N identity matrix; Step 2: According to the access structure and the key information rate array, obtain the corresponding K(K - 1) row index sets. From these sets, extract K submatrices from the generating matrix V. The strong privacy condition is equivalent to the full rank of each submatrix; Step 3: Calculate the determinants according to each submatrix, multiply them to get a polynomial, find a set of feasible solutions that make the polynomial not equal to 0, then set the remaining variables in the generating matrix V to 0, and then extract the decoding method from the generating matrix V. Finally, perform Gaussian elimination on the generating matrix V to obtain the encoding method; 2. The encoding and decoding method for the distributed multi-user key sharing problem under strong privacy conditions according to claim 1, characterized in that The set of row indices described in step 2 The requirements are as follows: where A k is the k-th access set in the access structure A, denotes the set of row indices containing the number of rows, R i is the information rate of the i-th key, is the empty set; as long as the key information rate array satisfies the following inequality: Such row index sets can always be found; After obtaining the set of K(K - 1) row indices as required, for each element k in the set [K], a sub - matrix is extracted by the row index set to obtain a sub - square matrix. The strong privacy condition requires that this sub - square matrix is full - rank, and the row index set that meets the requirements can ensure that the determinant of this sub - square matrix is a non - zero polynomial, that is, at least one coefficient in the polynomial is non - zero.

3. The encoding and decoding method for the distributed multi-user key sharing problem under strong privacy conditions according to claim 2, characterized in that, The specific steps of Step 3 are as follows: The steps for calculating the determinant are as follows: Extract K submatrices from the generating matrix V according to the above row index sets. Each submatrix only contains two forms: variables and 0. Therefore, its determinant is a polynomial, and the characteristics of the row index sets ensure that at least one coefficient in this polynomial is non-zero; The steps for solving the polynomial are as follows: The strong privacy condition is equivalent to the full rank of each submatrix, that is, the K submatrices need to satisfy full rank simultaneously, which is equivalent to the product of the above K determinants not being equal to 0; Since the K submatrices are extracted from the generating matrix V, the degree of the polynomial after multiplication is less than the number of keys K. Therefore, as long as the size of the finite field is greater than or equal to K, a set of feasible solutions that make the polynomial not equal to 0 must exist; The steps for determining the encoding and decoding methods are as follows: After obtaining a set of feasible solutions that make the polynomial not equal to 0, then set the remaining variables in the generator matrix V to 0, and then the decoding method can be obtained from the generator matrix V; specifically, the sub-key Y n , n ∈ [N] is put into a row vector, and the key W k , k ∈ [K] is equal to the product of this row vector and the matrix The decoding method is thus obtained; For the encoding method, Gaussian elimination needs to be performed on the generating matrix V, so that above the new submatrix is a ∑ i∈[K] R i ×∑ i∈[K] R i identity matrix, and at the same time, the lower part is all zeros; then the secret key W k , k ∈ [K] and random noises are put into a row vector, and the sub-secret key Y n , n ∈ [N] is equal to the product of this row vector and the new matrix The encoding method is thus obtained.

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