There is a huge gap between the upper and lower bounds on the share size of secret-sharing schemes for $n$ -party access structures; consistent with our current knowledge the optimal share size can be anywhere between polynomial and exponential in $n$ . For linear secret-sharing schemes, the share size for almost all $n$ -party access structures is exponential in $n$ . We would like to study larger classes of secret-sharing schemes with two goals: 1) prove lower bounds for larger classes of secret-sharing schemes; and 2) construct efficient secret-sharing schemes. Given this motivation, Paskin-Cherniavsky and Radune (ITC’20) introduced a new class of secret-sharing schemes in which the shares are generated by applying degree- $d$ polynomials to the secret and some random field elements. We define and study two additional classes of polynomial secret-sharing schemes: 1) schemes in which the reconstruction of the secret is done using polynomials; and 2) schemes in which both sharing and reconstruction are done by polynomials. Our main result is a construction of secret-sharing schemes and conditional disclosure of secrets protocols with quadratic sharing and reconstruction that are more efficient than linear secret-sharing schemes. To complement our results, we prove lower bounds on the share size for schemes with polynomial reconstruction. Finally, we give an evidence that schemes with polynomial sharing are probably stronger than schemes with polynomial reconstruction.There is a huge gap between the upper and lower bounds on the share size of secret-sharing schemes for$n$-party access structures; consistent with our current knowledge the optimal share size can be anywhere between polynomial and exponential in$n$. For linear secret-sharing schemes, the share size for almost all$n$-party access structures is exponential in$n$. We would like to study larger classes of secret-sharing schemes with two goals: 1) prove lower bounds for larger classes of secret-sharing schemes; and 2) construct efficient secret-sharing schemes. Given this motivation, Paskin-Cherniavsky and Radune (ITC’20) introduced a new class of secret-sharing schemes in which the shares are generated by applying degree-$d$polynomials to the secret and some random field elements. We define and study two additional classes of polynomial secret-sharing schemes: 1) schemes in which the reconstruction of the secret is done using polynomials; and 2) schemes in which both sharing and reconstruction are done by polynomials. Our main result is a construction of secret-sharing schemes and conditional disclosure of secrets protocols with quadratic sharing and reconstruction that are more efficient than linear secret-sharing schemes. To complement our results, we prove lower bounds on the share size for schemes with polynomial reconstruction. Finally, we give an evidence that schemes with polynomial sharing are probably stronger than schemes with polynomial reconstruction.

IEEE Transactions on Information TheoryJul 18, 20233 citations

Cryptanalysis and improvement of fair quantum secret sharing based on symmetric bivariate polynomial

Jiebin Gu, Feixiang Li, Shixin Zhu

Abstract As a fundamental component of quantum cryptography, quantum secret sharing (SS) faces a critical security challenge: dishonest participants may gain exclusive access to the shared secret through deceptive actions. To address this, recent advances in fair quantum SS have emerged, allowing equitable secret access among all participants. However, our analysis reveals security vulnerabilities in Bai et al ’s fair quantum SS scheme (Physica A, 2022). First, we show that Bai et al ’s scheme is vulnerable to collusion attack by m − 1 dishonest participants when m − 1 ⩾ t , where t is the threshold value. Second, we point that Bai et al ’s scheme is insecure under our defined attack model—the latent attack. In this work, a possible improved scheme is proposed to overcome the security flaws and halve the computational cost.Abstract As a fundamental component of quantum cryptography, quantum secret sharing (SS) faces a critical security challenge: dishonest participants may gain exclusive access to the shared secret through deceptive actions. To address this, recent advances in fair quantum SS have emerged, allowing equitable secret access among all participants. However, our analysis reveals security vulnerabilities in Bai et al ’s fair quantum SS scheme (Physica A, 2022). First, we show that Bai et al ’s scheme is vulnerable to collusion attack by m − 1 dishonest participants when m − 1 ⩾ t , where t is the threshold value. Second, we point that Bai et al ’s scheme is insecure under our defined attack model—the latent attack. In this work, a possible improved scheme is proposed to overcome the security flaws and halve the computational cost.

Journal of Physics A Mathematical and TheoreticalOct 29, 20251 citation

Strongly ideal secret sharing schemes

Steven J. Phillips, Nicholas C. K. Phillips

Journal of CryptologyOct 1, 199215 citations

Computational Quantum Secret Sharing

Alper Çakan, Vipul Goyal, Chen-Da Liu-Zhang +1 more

Quantum secret sharing (QSS) allows a dealer to distribute a secret quantum state among a set of parties in such a way that certain authorized subsets can reconstruct the secret, while unauthorized subsets obtain no information about it. Previous works on QSS for general access structures focused solely on the existence of perfectly secure schemes, and the share size of the known schemes is necessarily exponential even in cases where the access structure is computed by polynomial size monotone circuits. This stands in stark contrast to the classical setting, where polynomial-time computationally-secure secret sharing schemes have been long known for all access structures computed by polynomial-size monotone circuits under standard hardness assumptions, and one can even obtain shares which are much shorter than the secret (which is impossible with perfect security). While QSS was introduced over twenty years ago, previous works only considered information-theoretic privacy. In this work, we initiate the study of computationally-secure QSS and show that computational assumptions help significantly in building QSS schemes, just as in the classical case. We present a simple compiler and use it to obtain a large variety results: We construct polynomial-time computationally-secure QSS schemes under standard hardness assumptions for a rich class of access structures. This includes many access structures for which previous results in QSS necessarily required exponential share size. In fact, we can go even further: We construct QSS schemes for which the size of the quantum shares is significantly smaller than the size of the secret. As in the classical setting, this is impossible with perfect security. We also apply our compiler to obtain results beyond computational QSS. In the information-theoretic setting, we improve the share size of perfect QSS schemes for a large class of n-party access structures to 1.5^{n+o(n)}, improving upon best known schemes and matching the best known result for general access structures in the classical setting. Finally, among other things, we study the class of access structures which can be efficiently implemented when the quantum secret sharing scheme has access to a given number of copies of the secret, including all such functions in 𝖯 and NP.Quantum secret sharing (QSS) allows a dealer to distribute a secret quantum state among a set of parties in such a way that certain authorized subsets can reconstruct the secret, while unauthorized subsets obtain no information about it. Previous works on QSS for general access structures focused solely on the existence of perfectly secure schemes, and the share size of the known schemes is necessarily exponential even in cases where the access structure is computed by polynomial size monotone circuits. This stands in stark contrast to the classical setting, where polynomial-time computationally-secure secret sharing schemes have been long known for all access structures computed by polynomial-size monotone circuits under standard hardness assumptions, and one can even obtain shares which are much shorter than the secret (which is impossible with perfect security). While QSS was introduced over twenty years ago, previous works only considered information-theoretic privacy. In this work, we initiate the study of computationally-secure QSS and show that computational assumptions help significantly in building QSS schemes, just as in the classical case. We present a simple compiler and use it to obtain a large variety results: We construct polynomial-time computationally-secure QSS schemes under standard hardness assumptions for a rich class of access structures. This includes many access structures for which previous results in QSS necessarily required exponential share size. In fact, we can go even further: We construct QSS schemes for which the size of the quantum shares is significantly smaller than the size of the secret. As in the classical setting, this is impossible with perfect security. We also apply our compiler to obtain results beyond computational QSS. In the information-theoretic setting, we improve the share size of perfect QSS schemes for a large class of n-party access structures to 1.5^{n+o(n)}, improving upon best known schemes and matching the best known result for general access structures in the classical setting. Finally, among other things, we study the class of access structures which can be efficiently implemented when the quantum secret sharing scheme has access to a given number of copies of the secret, including all such functions in 𝖯 and NP.

Leibniz international proceedings in informaticsJan 1, 20230 citations

There is a huge gap between the upper and lower bounds on the share size of secret-sharing schemes for

n

-party access structures; consistent with our current knowledge the optimal share size can be anywhere between polynomial and exponential in

n

. For linear secret-sharing schemes, the share size for almost all

n

-party access structures is exponential in

n

. We would like to study larger classes of secret-sharing schemes with two goals: 1) prove lower bounds for larger classes of secret-sharing schemes; and 2) construct efficient secret-sharing schemes. Given this motivation, Paskin-Cherniavsky and Radune (ITC’20) introduced a new class of secret-sharing schemes in which the shares are generated by applying degree-

d

polynomials to the secret and some random field elements. We define and study two additional classes of polynomial secret-sharing schemes: 1) schemes in which the reconstruction of the secret is done using polynomials; and 2) schemes in which both sharing and reconstruction are done by polynomials. Our main result is a construction of secret-sharing schemes and conditional disclosure of secrets protocols with quadratic sharing and reconstruction that are more efficient than linear secret-sharing schemes. To complement our results, we prove lower bounds on the share size for schemes with polynomial reconstruction. Finally, we give an evidence that schemes with polynomial sharing are probably stronger than schemes with polynomial reconstruction.