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A secret sharing scheme is one of cryptographies. A threshold scheme, which is introduced by Shamir in 1979, is very famous as a secret sharing scheme. We can consider that this scheme is based on Lagrange's interpolation formula. A secret sharing scheme has one key. On the other hand, a multi-secret sharing scheme has more than one keys; that is, a multi-secret sharing scheme has p ( ≥ 2) keys. Dealers distribute shares of keys among n participants. Gathering t ( ≤ n ) participants, keys can be reconstructed. In this paper, we give a scheme of a (t,n) multi-secret sharing based on Hermite interpolation, in the case of p ≤ t .

A secret sharing scheme is one of cryptographies. A secret sharing scheme was introduced by Shamir in 1979 [

The secret sharing scheme is a method to distribute shares of a secret value―we call it a key, too―

A secret sharing scheme has one key

among

Various schemes are proposed about a multi-secret sharing scheme. In 1994, Jackson et al. [

In this paper, we give a scheme of a

In this section, we describe two famous interpolation formula, that is, Lagrange’s interpolation and Hermite interpolation. In numerical analysis, Lagrange’s interpolation and Hermite interpolation is a method of interpolating data points as a polynomial function.

Suppose that a function

Here, we can get an

where an

Let

Hermite interpolation is an extension of Lagrange’s interpolation. When using divided differences to calculate the Hermite polynomial of a function

Suppose that a function

Here, it is known that we can get an unique

where two

and

This is called Hermite interpolation.

In this section, we describe a multi-secret sharing scheme based on Lagrange’s interpolation, which is proposed by Yang et al. in 2004. We refer to [

In Yang et al.’s scheme, for secret distribution, the secret are distributed in two separate cases,

1) System parameters. Let

Here, we use

2) Secret distribution. In the case of

2a) Construct a

2b) Randomly choose an integer

2c) Publish

3) Secret reconstruction. In the case of

In this section, we describe our new scheme, that is, a multi-secret sharing scheme based on Hermite interpolation in the case

In the case of

Theorem 1. [

We expand this theorem. In our scheme, at first, we prepare system parameters which we need. Secondly, we describe secret distribution. Finally, we describe secret reconstruction.

1) System parameters. Let

Here, we use

2) Secret distribution. In the case of

2a) He constructs a

2b) He computes

2c) He publishes

3) Secret reconstruction. In the case of

3a) They pool their pseudo shadows

3b) They compute

3c) They compute

3d) By Hermite interpolation polynomial, with the knowledge of

From the obtained polynomial

As stated above, we obtain the following theorem.

Theorem 2. Suppose that

Corollary 1. Suppose that

Proof. In the case

(Q.E.D.)

Corollary 2. Theorem 2 contains Theorem 1.

Proof. In the case

(Q.E.D.)

In this section, we compare computational complexity of our scheme which we describe in Section 4, and that of Yang et al.’s scheme which we describe in Section 3.

As regards phase 1) system parameters, the both schemes have the same amount of parameters.

As regards phase 2) secret distribution, computational complexity of our scheme is twice of that of Yang et al.’s scheme. Since, in our scheme, there are

As regards phase 3) secret reconstruction, computational complexity of our scheme is twice of that of Yang et al.’s scheme. Since, in 3b) of our scheme, there are

are not only

Hence, computational complexity of our scheme is twice of that of Yang et al.’s scheme. This is suitable, since computational complexity of Hermite interpolation is twice of that of Lagrange’s interpolation.

We can propose a new scheme of a multi-secret sharing scheme with many keys based on Hermite interpolation. Hermite interpolation is a higher precision analysis and needs more complex computation than Lagrange’s interpolation. The merit on our scheme is that we can use many keys with fine distinctions. On the other hand, the demerit on our scheme is that its computation is complex for participants.

We thank the editor and the referee for their comments.