Redistribution of secret sharings
Method for redistribution of a (n,t)-secret sharing of a secrets from a set of dealers to a set of receivers. The method comprises performing by each of at least the threshold number t of dealers, —creating a (n′,t′)-secret sharing of its respective secret share, —creating a set of ciphertexts comprising for each receiver one encrypted sub-share of the n1 secret sub-shares of its respective secret share and being encrypted with respect to a public encryption key of the respective receiver. The public encryption key is a key of a public-key encryption scheme. —generating a non-interactive zero-knowledge proof that the set of ciphertexts jointly contain a redistribution of its secret share—broadcasting a dealing to the set of receivers, the dealing comprising the set of ciphertexts for the set of receivers and the corresponding non-interactive zero-knowledge proof.
1 . A computer-implemented method for redistribution of a (n,t)-secret sharing of a secret from a set of dealers to a set of receivers, the (n,t)-secret sharing comprising n secret shares, wherein a threshold number t≤n of the secret shares is sufficient to reconstruct the secret, the method comprising:
performing, by each of at least the threshold number t of dealers, the steps of:
creating a (n′,t′)-secret sharing of its respective secret share, the (n′,t′)-secret sharing comprising n′secret sub-shares of its respective secret share, wherein a threshold number t′≤n′ of the secret sub-shares is sufficient to reconstruct the secret share;
creating a set of ciphertexts, the set of ciphertexts comprising for each receiver of the set of receivers one encrypted sub-share of the n′secret sub-shares of its respective secret share, the encrypted secret sub-share being encrypted with respect to a public encryption key of a public-key encryption scheme of the respective receiver;
generating a non-interactive zero-knowledge proof that the set of ciphertexts jointly contain a redistribution of its secret share;
broadcasting a dealing to the set of receivers, the dealing comprising the set of ciphertexts for the set of receivers and the corresponding non-interactive zero-knowledge proof;
using a public-key encryption scheme with forward secrecy for encrypting the secret sub-shares for the receivers,
wherein the encryption scheme is configured to perform a chunked encryption of the sub-shares, wherein each sub-share is split into a plurality of plaintexts each representing a chunk; and
generating, by the set of dealers, a further non-interactive zero-knowledge proof, the further zero-knowledge proof being configured to prove that the chunked encryption is correct.
2 . The computer-implemented method according to claim 1 , further comprising:
checking, by a predefined set of parties, the non-interactive zero-knowledge proofs of the dealings; and
once a sufficient number t of proofs have been successfully checked,
decrypting, by each of the receivers, its encrypted sub-shares across the dealings; and
combining, by each of the receivers, the decrypted sub-shares to a new secret share of the secret.
3 . The computer-implemented method according to claim 2 , further comprising:
creating, by any party, from the associated public key material created by the set of dealers new associated public key material associated to the new secret sharing and the new secret shares of the set of receivers.
4 . The computer-implemented method according to claim 2 , further comprising performing the decrypting of the encrypted sub-shares in a separate process, on a virtual hardware security module, or on a hardware security module.
5 . The computer-implemented method according to claim 1 , wherein the zero-knowledge proofs are publicly verifiable zero-knowledge proofs such that any party can verify them.
6 . The computer-implemented method according to claim 1 , wherein the secret sharing is a linear secret sharing.
7 . The computer-implemented method according to claim 1 , further comprising:
creating, by each dealer, associated public key material associated to its secret share and its secret sharing of said secret share; and
adding the associated public key material to the dealing.
8 . The computer-implemented method according to claim 7 , wherein the associated public key material comprises:
a public group element related to the secret share of each dealer; and/or
public group elements which are related to the secret sharing created by each dealer.
9 . The computer-implemented method according to claim 7 , wherein the creating of the new associated public key material comprises:
deriving the new associated public key material from the associated public key material created by the set of dealers, the new associated public key material comprising
public share verification keys which are associated to the new secret shares of the receivers;
a public verification key related to the secret key; and/or
elements from which the public share verification keys of the receivers and/or the public verification key may be derived.
10 . The computer-implemented method according to claim 1 , wherein the secret is a secret key corresponding to a public verification key of a public key signature scheme.
11 . The computer-implemented method according to claim 1 , wherein the public-key encryption scheme uses chosen ciphertext attack security for the encrypting the secret sub-shares for the set of receivers.
12 . The computer-implemented method according to claim 1 ,
wherein the encryption scheme with forward secrecy is a binary tree encryption scheme.
13 . The computer-implemented method according to claim 1 ,
wherein the public-key encryption scheme is a multi-receiver encryption scheme, the multi-receiver encryption scheme being configured to encrypt the sub-shares of a dealer to the public encryption keys of the set of receivers.
14 . The computer-implemented method according to claim 13 ,
wherein the multi-receiver encryption scheme uses shared randomness to encrypt the sub-shares of the set of dealers to the public encryption keys of the set of receivers.
15 . A distributed network comprising a plurality of nodes, wherein the network is configured to perform a computer-implemented method for redistribution of a (n,t)-secret sharing of a secret from a set of dealers to a set of receivers, the (n,t)-secret sharing comprising n secret shares, wherein a threshold number t≤n of the secret shares is sufficient to reconstruct the secret s, the method comprising:
performing, by each of at least the threshold number t of dealers, the steps of:
creating a (n′,t′)-secret sharing of its respective secret share, the (n′,t′)-secret sharing comprising n′secret sub-shares of its respective secret share, wherein a threshold number t′≤n′ of the secret sub-shares is sufficient to reconstruct the secret share;
creating a set of ciphertexts, the set of ciphertexts comprising for each receiver of the set of receivers one encrypted sub-share of the n′secret sub-shares of its respective secret share, the encrypted secret sub-share being encrypted with respect to a public encryption key of a public-key encryption scheme of the respective receiver;
generating a non-interactive zero-knowledge proof that the set of ciphertexts jointly contain a redistribution of its secret share; and
broadcasting a dealing to the set of receivers, the dealing comprising the set of ciphertexts for the set of receivers and the corresponding non-interactive zero-knowledge proof;
using a public-key encryption scheme with forward secrecy for encrypting the secret sub-shares for the receivers,
wherein the encryption scheme is configured to perform a chunked encryption of the sub-shares, wherein each sub-share is split into a plurality of plaintexts each representing a chunk; and
generating, by the set of dealers, a further non-interactive zero-knowledge proof, the further zero-knowledge proof being configured to prove that the chunked encryption is correct.
16 . A node for a distributed network, the node comprising:
a hardware processor; and a memory, wherein the network comprises a plurality of nodes including the node, and the network is configured to perform a computer-implemented method for redistribution of a (n,t)-secret sharing of a secret from a set of dealers to a set of receivers, the (n,t)-secret sharing comprising n secret shares, wherein a threshold number t≤n of the secret shares is sufficient to reconstruct the secret s, the method comprising:
performing, by each of at least the threshold number t of dealers, the steps of:
creating a (n′,t′)-secret sharing of its respective secret share, the (n′,t′)-secret sharing comprising n′secret sub-shares of its respective secret share, wherein a threshold number t′≤n′ of the secret sub-shares is sufficient to reconstruct the secret share;
creating a set of ciphertexts, the set of ciphertexts comprising for each receiver of the set of receivers one encrypted sub-share of the n′secret sub-shares of its respective secret share, the encrypted secret sub-share being encrypted with respect to a public encryption key of a public-key encryption scheme of the respective receiver;
generating a non-interactive zero-knowledge proof that the set of ciphertexts jointly contain a redistribution of its secret share; and
broadcasting a dealing to the set of receivers, the dealing comprising the set of ciphertexts for the set of receivers and the corresponding non-interactive zero-knowledge proof;
using a public-key encryption scheme with forward secrecy for encrypting the secret sub-shares for the receivers,
wherein the encryption scheme is configured to perform a chunked encryption of the sub-shares, wherein each sub-share is split into a plurality of plaintexts each representing a chunk; and
generating, by the set of dealers, a further non-interactive zero-knowledge proof, the further zero-knowledge proof being configured to prove that the chunked encryption is correct.
17 . A non-transitory computer readable medium storing a computer program product for redistribution of a (n,t)-secret sharing of a secret from a set of dealers to a set of receivers, the (n,t)-secret sharing comprising n secret shares, wherein a threshold number t≤n of the secret shares is sufficient to reconstruct the secret, the computer program product comprising program instructions executable by one or more nodes of a distributed network to cause the one or more nodes to perform a method comprising:
performing, by each of at least the threshold number t of dealers, the steps of:
creating a (n′,t′)-secret sharing of its respective secret share, the (n′,t′)-secret sharing comprising n′secret sub-shares of its respective secret share, wherein a threshold number t′≤n′ of the secret sub-shares is sufficient to reconstruct the secret share;
creating a set of ciphertexts, the set of ciphertexts comprising for each receiver of the set of receivers one encrypted sub-share of the n′secret sub-shares of its respective secret share, the encrypted secret sub-share being encrypted with respect to a public encryption key of a public-key encryption scheme of the respective receiver;
generating a non-interactive zero-knowledge proof that the set of ciphertexts jointly contain a redistribution of its secret share;
broadcasting a dealing to the set of receivers, the dealing comprising the set of ciphertexts for the set of receivers and the corresponding non-interactive zero-knowledge proof;
using a public-key encryption scheme with forward secrecy for encrypting the secret sub-shares for the receivers,
wherein the encryption scheme is configured to perform a chunked encryption of the sub-shares, wherein each sub-share is split into a plurality of plaintexts each representing a chunk; and
generating, by the set of dealers, a further non-interactive zero-knowledge proof, the further zero-knowledge proof being configured to prove that the chunked encryption is correct.