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

Secret-Sharing Schemes: A Survey

Amos Beimel

Lecture notes in computer scienceJan 1, 2011687 citations

PERFECT SECRET SHARING SCHEMES

K.G. Parvatov

PRIKLADNAYa DISKRETNAYa MATEMATIKADec 1, 20084 citations

Advance Sharing of Quantum Shares for Quantum Secrets

Mamoru Shibata, Ryutaroh Matsumoto

Secret sharing is a cryptographic scheme to encode a secret to multiple shares being distributed to participants, so that only qualified sets of participants can restore the original secret from their shares. When we encode a secret by a secret sharing scheme and distribute shares, sometimes not all participants are accessible, and it is desirable to distribute shares to those participants before a secret information is determined. Secret sharing schemes for classical secrets have been known to be able to distribute some shares before a given secret. Lie et al. found a ((2, 3))-threshold secret sharing for quantum secrets can distribute some shares before a given secret. However, it is unknown whether distributing some shares before a given secret is possible with other access structures of secret sharing for quantum secrets. We propose a quantum secret sharing scheme for quantum secrets that can distribute some shares before a given secret with other access structures.Secret sharing is a cryptographic scheme to encode a secret to multiple shares being distributed to participants, so that only qualified sets of participants can restore the original secret from their shares. When we encode a secret by a secret sharing scheme and distribute shares, sometimes not all participants are accessible, and it is desirable to distribute shares to those participants before a secret information is determined. Secret sharing schemes for classical secrets have been known to be able to distribute some shares before a given secret. Lie et al. found a ((2, 3))-threshold secret sharing for quantum secrets can distribute some shares before a given secret. However, it is unknown whether distributing some shares before a given secret is possible with other access structures of secret sharing for quantum secrets. We propose a quantum secret sharing scheme for quantum secrets that can distribute some shares before a given secret with other access structures.

arXiv:2302.14448IEICE Transactions on Fundamentals of Electronics Communications and Computer SciencesNov 23, 20234 citations

Multi-Secret Sharing Schemes

Carlo Blundo, Alfredo De Santis, Giovanni Di Crescenzo +2 more

Lecture notes in computer scienceJan 1, 1994104 citations

Advance Sharing of Quantum Shares for Classical Secrets

Rina Miyajima, Ryutaroh Matsumoto

Secret sharing schemes for classical secrets can be classified into classical secret sharing schemes and quantum secret sharing schemes. Classical secret sharing has been known to be able to distribute some shares before a given secret. On the other hand, quantum mechanics extends the capabilities of secret sharing beyond those of classical secret sharing. We propose quantum secret sharing with the capabilities in designing of access structures more flexibly and realizing higher efficiency beyond those of classical secret sharing, that can distribute some shares before a given secret.Secret sharing schemes for classical secrets can be classified into classical secret sharing schemes and quantum secret sharing schemes. Classical secret sharing has been known to be able to distribute some shares before a given secret. On the other hand, quantum mechanics extends the capabilities of secret sharing beyond those of classical secret sharing. We propose quantum secret sharing with the capabilities in designing of access structures more flexibly and realizing higher efficiency beyond those of classical secret sharing, that can distribute some shares before a given secret.

arXiv:2208.01847IEEE AccessJan 1, 20224 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.