A Study on Multisecret-Sharing Schemes Based on Linear Codes

Secret sharing has been a subject of study since 1979. In the secret sharing schemes there are some participants and a dealer. The dealer chooses a secret. The main principle is to distribute a secret amongst a group of participants. Each of whom is called a share of the secret. The secret can be retrieved by participants. Clearly the participants combine their shares to reach the secret. One of the secret sharing schemes is a(t, n) − threshold secret sharing scheme. A a(t, n) − threshold secret sharing scheme is a method of distribution of information among n participants such that t ≥ 1 can recover the secret but (t − 1) cannot. The coding theory has been an important role in the constructing of the secret sharing schemes. Since the code of a symmetric (v, k, λ) − design is a linear code, this study is about the multisecret-sharing schemes based on the dual code C of F2 − code C of a symmetric (v, k, λ) − design. We construct a multisecret-sharing scheme Blakley’s construction of secret sharing schemes using the binary codes of the symmetric design. Our scheme is a threshold secret sharing scheme. The access structure of the scheme has been described and shows its connection to the dual code. Furthermore, the number of minimal access elements has been formulated under certain conditions. We explain the security of this scheme.

Page | 264 schemes based on field extensions. However, we constructed an image secret sharing method based on Shamir secret sharing in Calkavur (2018) study [17].
There are several known constructions of linear codes as row spaces of incidence matrices of designs. This paper deals with constructions of multisecret-sharing schemes based on binary linear codes of symmetric designs.
In this work we consider a multisecret-sharing scheme of the construction Blakley's method. The next section gives the basic preliminaries used in the paper. The construction is presented in Section 3. In this section we explain the access structure and the number of minimal access elements of the scheme. Section 4 collects concluding remarks.

2-Background and Preliminaries
In this section we give the basic preliminaries and some necessary mathematical information used in this work.

2-1-Linear Codes
Let q be a prime power and denote the finite field of order q by F q . An [ , ] −code C over F q is a subspace in ( ) , where n is length of the code C and k is dimension of C. The dual code of C is defined to be the set of those vectors ( ) which are orthogonal to every codeword of C. It is denoted by ⊥ . The code ⊥ is a [ , − ] − code. A generator matrix G for a linear code C is a × matrix for which the rows are a basis of C.
Let C be an  ] , [ k n code over q F with generator matrix G. C contains codewords and can be used to communicate any one of distinct messages. We encode the message vector = 1 , 2 , … , as the codeword .
If G is a generator matrix for C, then = { | ∈ ( ) } . → maps the vector space onto a − dimensional subspace of ( ) .

2-2-Secret Sharing
Secret sharing refers to methods for distributing a secret amongst a group of participants, each of whom is allocated a share of the secret. The secret can be reconstructed only when a sufficient number, of possibly different types, of shares are combined together; individual shares are of no use their own.
In one type of secret sharing scheme there is one dealer and players. The dealer gives a share of the secret to the players, but only when specific conditions are fulfilled will the players be able to reconstruct the secret from their shares. The dealer accomplishes this by giving each player a share in such a way that any group of t (for threshold) or more players can together reconstruct the secret but no group of fewer than players can such a system is called a  ) , ( n t threshold scheme.
Shamir's secret sharing method is an old cryptography algorithm. This scheme is a  ) , ( n t threshold scheme. Shamir's scheme was based on polynomial interpolation. Blakley's scheme is also a  ) , ( n t threshold scheme. Blakley uses hyperplane geometry to solve the secret sharing problem.

2-3-Symmetric Designs
A symmetric ( , , ) −design consists of a set of P of points and a set of subsets of P called blocks such that In a symmetric ( , , ) − design the value ( − ) is called n the order of a symmetric ( , , ) − design, where = − . The incidence matrix = [ ] of a symmetric ( , , ) −design is the × matrix whose rows are indexed by blocks and whose columns are indexed by points. The entries of matrix are defined as follows.

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The − code of a symmetric ( , , ) −design is a subspace of ( ) generated by the incidence matrix A of the symmetric design. The extended − code . of a symmetric ( , , ) −design is a code generated by the rows of the extended matrix is:

3-Multisecret-Sharing Schemes Based on the Dual Code of the Binary Code of a Symmetric
( , , ) −Design

3-1-Scheme Description
The code of a symmetric ( , , ) −design is also linear code. It is known that every linear code can be used to construct the secret sharing schemes. In this section we examine a multisecret-sharing scheme based on the dual code of the binary code of a symmetric ( , , ) −design. Let: subspace of ( 2 ) v generated by the rows of the incidence matrix A of the symmetric design. 12 11 is a × matrix. Now we construct a multisecret-sharing scheme based on ⊥ , where ⊥ is the dual code of the binary code C of a symmetric ( , , ) −design.
Let ( 2 ) be the secret space and a vector in subspace of ( 2 ) v be the secret. Let C be a [ , ] − code over 2 generated by a symmetric ( , , ) −design and ⊥ be an [ , − ] − code over 2 .

3-2-Proposed Method
Consider the matrix: Where ⊥ is a generator matrix of ⊥ . Let any element of ⊥ be the secret = ( 1 , 2 , … , ). All of rows of generator matrix ⊥ are minimal access elements and all of codewords of  C are participants in this scheme. We consider the row vectors of ⊥ to calculate the shares i y , = 1, 2, … , .
We write the following linear equation system for each participant.
It can reach by solving this equation system.

Theorem 1.
In this multisecret-sharing scheme we have the following.
1) The access structure consists of the ( − ) elements.
2) No element of number less than ( − ) can be used in recovering the secret.

Proof.
1) The secret is recovered thanks to the rows of ⊥ and their number is ( − ).
2) The number of rows of ⊥ cannot be less than ( − ) by definition. Otherwise the secret cannot be reached. So only ( − ) elements can be used to recover the secret but ( − − 1) cannot.

Corollary 1.
The multisecret-sharing scheme satisfying the hypothesis of the above theorem is also a ( − , 2 − ) − threshold secret sharing scheme.

Proof.
It is clear that the participants are all of elements of ⊥ and their number is 2 − . The ( − ) out of 2 − participants can be reached the secret by combining their shares.

Theorem 2.
Let C be an [ , ] − code over F 2 generated by a symmetric ( , , ) −design, where is the length of C and r is dimension of C. In a multisecret-sharing scheme based on ⊥ the number of minimal coalitions is ( 2 − − ), where ⊥ is the dual code of C.

Proof.
Recall that our scheme is a ( − , 2 − ) −threshold scheme. This means ( − ) out of 2 − participants can recover the secret. These ( − ) participants consist of minimal access sets. So the number of minimal coalitions is

Theorem 3.
Suppose that C is 2 − code of a symmetric ( , , ) −design D. If , then there are altogether m minimal access elements in the multisecret-sharing scheme based on ⊥ of C and m satisfy the inequality If C is the 2 − code of a symmetric ( , , ) −design D and 2|( − ), then: [20] (1) First we have to give the proof of this statement. Suppose that 2|( − ). If 2| , then we find that 2| . So the scalar product of any two blocks is equal to zero since the 2 −code C of D is generated by the blocks of D and C is self-orthogonal ( ⊆ ⊥ ) with respect to Euclidean scalar product [18]. Therefore we obtain dim ≤ − dim , dim ≤ 2 .

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If 2|( − ), but 2 ∤ . Then consider ) (  v v y y y y y be any two rows vectors of B, where B is the extended incidence matrix of a symmetric ( , , ) −design.
There are four options for x and y : For the case 1) The scalar products of two blocks are equal to 0 (mod 2). Therefore dim . ≤ +1 2 . Now we have to prove that dim = dim . .

If 2 ∤ The sum of first v columns of B is equal to
The last column of B is equal to − times of the sum of first v columns of B with respect to modulo 2. On the other hand the sum of first v rows of B is equal to Hence the last row of B is equal to −1 times the sum of first v rows of B. Therefore we obtain the last row of B is a linear combination of first v rows of B and also the last column of B is the linear combination of first v columns of B. We conclude that It is also clear that dim ≥ 2 [18].
By Theorem 1, there are altogether ( − ) = dim ⊥ minimal access elements in the multisecret-sharing scheme based on ⊥ . Now we consider the inequality (1).

Theorem 4.
Let C be 2 −code of a symmetric ( , , ) −design D. If 2 ∤ ( − ) and 2| , then in the multisecret-sharing scheme based on ⊥ of 2 − code C of D there are altogether 1 minimal access elements.
Suppose that 2 ∤ ( − ) and 2| . Every row of A is orthogonal to (1, 1,…, 1) with respect to the scalar product: The sum all of rows containing 0 in the ℎ column is the vector By Theorem 1, there are altogether ( − ) = dim ⊥ minimal access elements in the multisecret-sharing scheme based on ⊥ . If we combine these results, then we obtain there are altogether dim ⊥ = − dim = − ( − 1) = 1 minimal access elements.

Theorem 5.
Let C be 2 − code of a symmetric ( , , ) −design D. If 2 ∤ ( − ) and 2 ∤ , then in the multisecret-sharing scheme based on ⊥ of 2 − code C of D there is no minimal access element.

Proof.
This is similar to the proof of Theorem 4. If C is 2 − code of a symmetric ( , , ) −design D and 2 ∤ ( − ) and 2 ∤ , then v C  dim [18]. Now we will prove it.
Suppose that 2 ∤ ( − ) and 2 ∤ . By Proposition 1, Thus the matrix A is invertible over 2 . Hence v C  dim . We combine this result with Theorem 1. So we obtain: This means there is no minimal access element. In this case, the secret cannot be reached.

Example 1.
Consider the symmetric (7, 3, 1) − symmetric design, where = 7, = 3, = 1. The set of points is 4 , 5 } 6 and the blocks are , 0 The incidence matrix of this design is: The 2 − code C of the symmetric (7, 3, 1) −design is a subspace of ( 2 ) 7 generated by the rows of the incidence matrix A. We examine a multisecret-sharing scheme based on the dual code of the binary code of the symmetric (7, 3, 1) − design. C is the binary [7,4] − code. The dual code of ⊥ of C is a linear [7,3] − code. So | ⊥ | = 2 3 = 8.   It is seen that the secret = (1101000) by solving the above linear system. This scheme is also a (3, 8) − threshold secret sharing scheme.

3-4-Security Analysis
Our scheme has been constructed based on the dual code of 2 − code of a symmetric ( , , ) −design. We use Blakley's method. The secret can be reached by the rows of generator matrix of the dual code. It is needed the following linear equation system. algorithm. We use Blakley's method to recover the secret in the new scheme. Since the system has a unique solution, it is very difficult to find the secret by attackers. So our new system is too safe.

4-Conclusion
In the present article, we have introduced a new multisecret-sharing scheme based on the dual code of the binary code of a symmetric ( , , ) −design. The reconstruction algorithm is based on Blakley's method. We determine the access structure and calculate the information rate of this scheme. We give the number of minimal access elements under certain conditions. We compare our scheme with the other schemes in the literature. The new system stands well, in terms of security.

5-Acknowledgments
I would like to thank to Dr. Patrick Solé for his valuable ideas.

6-Conflict of Interest
The author declares that there is no conflict of interests regarding the publication of this manuscript. In addition, the ethical issues, including plagiarism, informed consent, misconduct, data fabrication and/or falsification, double publication and/or submission, and redundancies have been completely observed by the authors.