Quantum Optimal Multiple Assignment Scheme for Realizing General Access Structure of Secret Sharing

TL;DR

This paper introduces a quantum version of the multiple assignment scheme for secret sharing, enabling realization of general access structures, and proposes an optimization method to minimize share size.

Contribution

It presents a novel quantum multiple assignment scheme for secret sharing and an integer optimization approach to reduce share sizes.

Findings

Quantum scheme successfully realizes general access structures.

Optimization reduces average share size in secret sharing.

Framework extends classical schemes to quantum context.

Abstract

The multiple assignment scheme is to assign one or more shares to single participant so that any kind of access structure can be realized by classical secret sharing schemes. We propose its quantum version including ramp secret sharing schemes. Then we propose an integer optimization approach to minimize the average share size.

Equations

$$C_1/C_2=\{\vec{a}+C_2\mid \vec{a}\in C_1\}.$$

$$\dim P_I(C_1)-\dim P_I(C_2)=\dim C_1-\dim C_2,$$

$$\dim P_I(C_1)-\dim P_I(C_2)=0.$$

$$\frac{1}{\sqrt{q^L}}\sum _{\vec{x}\in f(\vec{s})}|\vec{x}⟩,$$

$$1(A)_{p}=\left\{\begin{array}[]{ll}1&\textrm{if there exists }i\in A\textrm{ with }b(p)_{i}=1,\\ 0&\textrm{otherwise}.\end{array}\right.$$

$$\begin{array}[]{lc}\textrm{minimize}&\langle\vec{h},\vec{y}\rangle,\\ \textrm{subject to}&\langle a(-1,A),\vec{y}\rangle\geq 0,\forall A\in\mathrm{A}_{Q},\\ &\langle-a(-1,A),\vec{y}\rangle\geq L,\forall A\in\mathrm{A}_{F},\\ &\vec{y}\geq 0,\end{array}$$

$$\varphi (\vec{z})=(\underset{\gamma (1)\text{ times}}{\underbrace{z_1,\dots ,z_1}},\dots ,\underset{\gamma (m)\text{ times}}{\underbrace{z_m,\dots ,z_m}})\in {\mathbf{F}}_q^{m^{\prime }}.$$

Full text

{summary}

The multiple assignment scheme is to assign one or more shares to single participant so that any kind of access structure can be realized by classical secret sharing schemes. We propose its quantum version including ramp secret sharing schemes. Then we propose an integer optimization approach to minimize the average share size.

1 Introduction

Secret sharing (SS) [1] is a cryptographic scheme to encode a secret to multiple shares being distributed to participants, so that only qualified sets of participants can reconstruct the original secret from their shares. Traditionally both secret and shares were classical information (bits). Several authors, e.g. [2, 3, 4] extended the traditional SS to quantum one so that a quantum secret can be encoded to quantum shares.

A set of participants is called forbidden if the set has absolutely no information about the secret. A secret sharing scheme is called perfect [5] if every set of participants is always qualified or forbidden. If a set is neither qualified or forbidden in a secret sharing scheme, the scheme is said to be ramp or non-perfect. A merit of the ramp schemes is to reduce share size (the number of bits or qubits) while keeping the secret size [6, 7, 8].

Traditionally, the access structure called the threshold structure has been the most focused one, e.g. [1, 2, 7], where a set of participants is qualified if and only if the number of participants is $\geq t$. A scheme with a threshold structure is called a threshold scheme. A well-known method to realize an arbitrary access structure is the multiple assignment scheme proposed by Shamir [1] and named by Ito et al. [9]. On the other hand, Smith [4] showed how to realize an arbitrary access structure in quantum perfect secret sharing schemes, while nobody has shown a construction of quantum ramp schemes with arbitrary access structures.

The multiple assignment scheme assigns multiple shares of a threshold scheme to single participants, and a single share can be assigned to multiple participants. It is not straight forward to adapt the multiple assignment scheme, as the no-cloning theorem [10] prevents us from making multiple copies of a single quantum share. The first purpose of this paper is to propose a quantum version of the multiple assignment scheme.

For a given size of secret, it is important to reduce the size of shares. A demerit of multiple assignment scheme in [9] was lack of consideration of share size. Later, Iwamoto et al. [11] proposed an integer optimization approach to minimize the worst-case or the average share size of multiple assignment scheme. The second purpose of this paper is to adapt Iwamoto et al.’s integer optimization problem to our proposed quantum setting.

2 Review of Previous Research Results

By a classical secret sharing scheme, we mean that its secret and its shares are classical information, while by a quantum secret sharing scheme, its secret and its shares are quantum information. For a set $T$, $2^{T}$ denotes its power set $\{T_{0}\mid T_{0}$ is a subset of $T\}$, and we have $|2^{T}|=2^{|T|}$.

Firstly, we review the multiple assignment scheme named by Ito et al. [9] and originally proposed by Shamir [1]. The multiple assignment scheme construct a classical secret sharing scheme with $n$ participants from that with $m$ participants. It is a map $\Phi$ from $\{1$, …, $n\}$ to $2^{\{1,\ldots,m\}}$. Let $W_{1}$, …, $W_{m}$ be the shares of the original secret sharing scheme. The new secret sharing scheme constructed by $\Phi$ distributes $\{W_{j}\mid j\in\Phi(i)\}$ to the $i$-th participants. For example, suppose that $m=3$, $n=2$, $\Phi(1)=\{1,2\}$, and $\Phi(2)=\{2,3\}$. Then, in the new constructed secret sharing scheme, the first participant receives $\{W_{1}$, $W_{2}\}$ as his/her share, and the second one receives $\{W_{2}$, $W_{3}\}$ as his/her share. This method works fine with the classical information. But its straightforward extension to the quantum information is impossible, because the quantum no-cloning theorem [10] prevents us from distributing the same $W_{2}$ to both first and second participants. To avoid this impossibility, we will focus the relation between two shares $\{W_{1}$, $W_{2}\}$ and $\{W_{2}$, $W_{3}\}$, which can be expressed by linear codes, and will propose to transfer the relation to the quantum setting.

A classical secret sharing is said to be linear if any linear combination of shares expresses the corresponding linear combination of secrets [12]. Let $\mathbf{F}_{q}$ be a finite field with $q$ elements. It was shown that any linear classical secret sharing scheme can be expressed [13, Proposition 1] by a pair of linear codes $C_{2}\subset C_{1}\subset\mathbf{F}_{q}^{n}$ as follows, provided that the linearity is considered over $\mathbf{F}_{q}$.

For a classical secret sharing scheme corresponding to $C_{2}\subset C_{1}$, the set of secrets is the factor space

[TABLE]

Therefore a secret $S\in C_{1}/C_{2}$ is a subset of $C_{1}$. For a given secret $S\in C_{1}/C_{2}$, a vector $X=(X_{1}$, …, $X_{n})$ is chosen uniformly randomly from $S$. A subset of $\{X_{1}$, …, $X_{n}\}$ is distributed to each participant as his/her share. For practical use of secret sharing schemes, it is indispensable to have a criterion by which one can identify qualified or forbidden sets of shares. Let $I\subset\{1$, …, $n\}$, and a set of participants collectively have $\{X_{i}\mid i\in I\}$ as their shares. Let $P_{I}$ be the projection map sending $(x_{1}$, …, $x_{n})\in\mathbf{F}_{q}^{n}$ to $(x_{i})_{i\in I}$, and $P_{I}(C_{1})=\{P_{I}(\vec{x})\mid\vec{x}\in C_{1}\}$. It was shown in [14] that the set of shares expressed by $I$ is qualified iff

[TABLE]

and is forbidden iff

[TABLE]

It is known that most of quantum ramp secret sharing schemes can also be described by a pair of linear codes $C_{2}\subset C_{1}\subset\mathbf{F}_{q}^{n}$ as follows [15]. Let $L=\dim C_{1}-\dim C_{2}$, then the dimension of (pure state) quantum secret is $q^{L}$ and its orthonormal basis can be chosen as $\{|\vec{s}\rangle\mid\vec{s}\in\mathbf{F}_{q}^{L}\}$. The linear space of all the possible quantum shares is $q^{n}$-dimensional, and its orthonormal basis can be chosen as $\{|\vec{x}\rangle\mid\vec{x}\in\mathbf{F}_{q}^{n}\}$. We fix an $\mathbf{F}_{q}$-linear map $f$ from $\mathbf{F}_{q}^{L}$ to the factor linear space $C_{1}/C_{2}$, and a quantum secret $|\vec{s}\rangle$ is encoded to

[TABLE]

which is the same as the encoding procedure of the CSS quantum error-correcting codes [16, 17]. Equation (3) can be regarded as a quantum state of $n$ particles having dimension $q$. In this paper qudit refers to a quantum object that is represented by $q$-dimensional complex linear space. Each participant receives a non-overlapping subset of the $n$ particles of Eq. (3) as his/her quantum share.

As well as the classical case, we need a criterion to tell if a set of shares is qualified or forbidden. Recall that a share set is qualified if and only if its complement is forbidden [2, 7] when the quantum secret sharing scheme is a pure-state scheme, which encode a pure-state secret to a pure-state shares [2]. It was also shown [2] that it is sufficient to consider pure-state schemes. Let $I\subset\{1$, …, $n\}$, and a set of participants collectively have $I$ as their shares, that is, the set of participants has the $i$-th quantum particle among $n$ particles, each of which has dimension $q$, if and only if $i\in I$. The share set $I$ is qualified if and only if $I$ is qualified and $\overline{I}$ is forbidden in the classical secret sharing scheme constructed from $C_{1}\supset C_{2}$, where $\overline{I}=\{1$, …, $n\}$. In other words, $I$ is qualified if and only if both Eq. (2) with $I$ substituted by $\overline{I}$ and Eq. (1) hold.

3 Proposed Method to Construct a Quantum Ramp Secret Sharing

Scheme with a General Access Structure

Suppose that the number of participants is $n$. Let $q$ be a prime power as before, and the dimension of quantum secret is assumed to be $q^{L}$ for a positive integer $L$. Let $\mathcal{A}_{Q}\subset 2^{\{1,\ldots,n\}}$ be the family of qualified sets given as the requirement for a quantum secret sharing scheme to be constructed. Since we have restricted ourselves to the pure-state schemes, the family of forbidden sets must be $\mathcal{A}_{F}=2^{\{1,\ldots,n\}}\setminus\mathcal{A}_{Q}$. It is also assumed that $\mathcal{A}_{Q}$ satisfies the monotonicity condition [5], that is, if $A\in\mathcal{A}_{Q}$ and $A\subseteq B\in 2^{\{1,\ldots,n\}}$ then $B\in\mathcal{A}_{Q}$. The monotonicity condition of $\mathcal{A}_{Q}$ implies the monotonicity condition of $\mathcal{A}_{F}$ in the reverse order, that is, if $B\in\mathcal{A}_{F}$ and $B\supseteq A\in 2^{\{1,\ldots,n\}}$ then $A\in\mathcal{A}_{F}$.

We introduce some notations from [11]. Let $\vec{y}=(t$, $x_{1}$, …, $x_{2^{n}-1})$. Later $t$ becomes the design parameter of the underlying threshold ramp quantum secret sharing scheme. Specifically, the underlying ramp secret sharing allows reconstruction of the secret only from $t$ or more shares. Let $b(p)_{i}$ as the $i$-th bit of the binary representation of a positive integer $p$. For a set $A\subset\{1$, …, $n\}$, define

[TABLE]

Define $2^{n}$-dimensional vector $a(\ell,A)=(\ell,1(A)_{1}$, …, $1(A)_{2^{n}-1})$. Let $h_{p}$ be the number of $1$’s in the binary representation of a positive integer $p$, and $\vec{h}=(h_{0}$, $h_{1}$, …, $h_{2^{n}-1})$. As $\textrm{IP}^{R2}_{\widetilde{\rho}}$ in [11], we solve the following integer optimization problem:

[TABLE]

where $\langle\cdot,\cdot\rangle$ denotes the inner product of two vectors. Since the above integer optimization problem is a relaxed version of the original $\textrm{IP}^{R2}_{\widetilde{\rho}}$ in [11], by Theorem 25 of [11] there must be at least one solution $\vec{y}$ to our integer optimization problem. By following [11], one can construct a classical ramp secret sharing scheme with $n$ participants, the qualified set $\mathcal{A}_{Q}$, the forbidden set $\mathcal{A}_{F}$ and the classical secret consisting of $L$ symbols in $\mathbf{F}_{q}$. Since their construction produces a classical linear secret sharing scheme, it can be described by a nested pair of linear codes $C_{2}\subset C_{1}\subset\mathbf{F}_{q}^{m}$, where $m=x_{1}+\cdots+x_{2^{n}-1}$ determined by a solution $\vec{y}=(t$, $x_{1}$, …, $x_{2^{n}-1})$ of the above integer optimization problem. In the construction method [11], $(t,L,m)$ classical ramp secret sharing scheme is the underlying secret sharing scheme used in construction of the desired secret sharing scheme. Observe that the constructed classical ramp secret sharing scheme has the minimum average share size, as proved in [11]. Recall that in the constructed classical secret sharing scheme expressed as $C_{2}\subset C_{1}$, a participant receives some components of $(z_{1}$, …, $z_{m})\in C_{1}$ as his/her share. Let $V_{i}=\{j\mid$ the $i$-th share contains $z_{j}\}$ $\subset\{1$, …, $m\}$.

To avoid violation of the quantum no-cloning theorem, we modify the coding theoretic expression $C_{1}/C_{2}$ and $V_{i}$. Note that only expression is modified and the secret sharing scheme itself is not modified. Let $\gamma(j)=|\{i\mid j\in V_{i}\}|$, and $m^{\prime}=\gamma(1)+\cdots+\gamma(n)$. For $\vec{z}=(z_{1}$, …, $z_{m})\in C_{1}$, define

[TABLE]

Let $C^{\prime}_{1}=\phi(C_{1})$ and $C^{\prime}_{2}=\phi(C_{2})$. Define $V^{\prime}_{i}\subset\{1$, …, $m^{\prime}\}$ such that $\{z_{\ell}\mid\ell\in V_{i}\}=\{\phi(\vec{z})_{j}\mid j\in V^{\prime}_{i}\}$ and $V^{\prime}_{i}\cap V^{\prime}_{i^{\prime}}=\emptyset$ for $i\neq i^{\prime}$, where $\vec{z}=(z_{1}$, …, $z_{m})\in C_{1}$. By the disjointedness $V^{\prime}_{i}\cap V^{\prime}_{i^{\prime}}=\emptyset$ we can avoid the violation of the quantum no-cloning theorem. The above change of notations makes no change in the actual operation of the constructed classical secret sharing scheme. Then the same constructed secret sharing scheme can also be described by the code pair $C^{\prime}_{2}\subset C^{\prime}_{1}$. We also have $|V^{\prime}_{1}|+|V^{\prime}_{2}|+\cdots+|V^{\prime}_{n}|=m^{\prime}$.

We construct the ramp quantum secret sharing scheme from $C^{\prime}_{2}\subset C^{\prime}_{1}\subset\mathbf{F}_{q}^{m^{\prime}}$ in which the $i$-th participants receives the $j$-th qudit among $m^{\prime}$ qudits if and only if $w_{j}\in V^{\prime}_{i}$, where $(w_{1}$, …, $w_{m^{\prime}})\in C^{\prime}_{1}$ and $m^{\prime}$ qudits are defined from the quantum secret as Eq. (3). Then the constructed quantum secret sharing scheme has $\mathcal{A}_{Q}$ as its qualified set, $\mathcal{A}_{F}$ as its forbidden set, and $L$-qudit in its quantum share, and the average share size is generally small.

Acknowledgment

The author would like to thank Prof. Mitsugu Iwamoto for helpful discussion, and an anonymous reviewer to identify the critical error in the initial manuscript. This research is partly supported by the National Institute of Information and Communications Technology, Japan, and by the Japan Society for the Promotion of Science Grant Nos. 23246071 and 26289116.