Digital signatures of messages using signature shares
A computer-implemented method of generating a share of a digital signature of a message, wherein a threshold number of different signature shares from respective participants of a group of participants are required to generate the digital signature, wherein each participant has a respective private key share, the method being performed by a first one of the participants and comprising: generating a first message-independent component and a first message-dependent component, wherein the message-independent component is generated based on a first private key share and wherein the message-dependent component is generated based on the message; causing the first message-independent component to be made available to a coordinator; and causing a first signature share to be made available to the coordinator for generating the signature based on at least the threshold number of signature shares, wherein the first signature share comprises at least the message-dependent component.
1 . A computer-implemented method of participating in a threshold signature scheme for generating a digital signature (r, s) of a message m, wherein the digital signature comprises an r value and an s value, wherein a threshold number of different shares of the s value generated by respective participants of a group of participants are required to generate the s value of the digital signature, wherein each participant has a respective private key share a i of a private key a corresponding to a public key P that is usable to validate the digital signature, wherein each participant has a respective ephemeral private key share k i of an ephemeral private key k, and wherein the method is performed by a first one of the participants and comprises:
prior to obtaining the message m, generating a first message-independent component of a first share s i of the s value, wherein the first message-independent component is generated as a function of a first private key share a i of the private key a but not as a function of the message m,
after obtaining the message m, generating a first message-dependent component of the first share s i of the s value, wherein the first message-dependent component is generated as a function of a cryptographic hash of the message m,
wherein the first message-independent component and/or the first message-dependent component is also generated as a function of a first ephemeral private key share k i of the ephemeral private key k,
and
causing a first signature share s i to be made available to the coordinator, wherein the first signature share s i is generated as a function of the first message-independent component and the first message-dependent component, and wherein the coordinator is configured to generate the signature based on each of at least the threshold number of respective signature shares s i , each respective signature share s; being generated by a respective participant and being generated as a function of a respective message-independent component and a respective message-dependent component, the respective message-independent component being generated by the respective participant as a function of the respective private key share a i of the private key a and the respective message-dependent component being generated by the respective participant as a function of the cryptographic hash of the message m, wherein the respective message-independent component and/or the respective message-dependent component is also generated as a function of a respective ephemeral private key share k i of the ephemeral private key k.
2 . The method of claim 1 , wherein the first private key share is generated using a joint secret sharing scheme for generating a share of a first private key.
3 . The method of claim 2 , wherein the first ephemeral private key share is generated using the joint secret sharing scheme for generating a share of the ephemeral private key.
4 . The method of claim 1 , wherein the first value is generated based on an inverse corresponding to the first ephemeral private key share.
5 . The method of claim 4 , wherein generating the inverse of the first ephemeral private key share comprises:
generating an intermediate value based on the ephemeral private key and a first blinding key; and
generating the inverse of the first ephemeral private key share based on an inverse of the intermediate value and a first blinding key share of the first blinding key.
6 . The method of claim 5 , wherein the first blinding key share is generated using a joint secret sharing scheme for generating a share of the first blinding key.
7 . The method of claim 5 , wherein the intermediate value is generated by:
generating a first multiplicative key share based on the first ephemeral private key share and the first blinding key share;
obtaining a respective multiplicative key share from at least the threshold number of participants; and
generating the intermediate value based on the first multiplicative key share and each of the respective multiplicative key shares.
8 . The method of claim 5 , wherein the second value is generated based on a second blinding key share of a second blinding key.
9 . The method of claim 8 , wherein the second blinding key share is generated using a joint secret sharing scheme for generating a share of the second blinding key.
10 . The method of claim 4 , wherein the second value is generated based on a first pre-signature share, wherein the first pre-signature share is generated based on:
a first intermediary share, the first intermediary share being generated based on the first private key share and the inverse corresponding to the first ephemeral private key share; and
a respective intermediary share obtained from at least the threshold number of participants.
11 . The method of claim 5 , wherein the second value is generated based on a first pre-signature share, wherein the first pre-signature share is generated based on:
a first intermediary share, the first intermediary share being generated based on the first private key share and the first blinding key share; and
a respective intermediary share obtained from at least the threshold number of participants.
12 . The method of claim 11 , wherein the first pre-signature share is generated based on the inverse corresponding to the first ephemeral private key share.
13 . The method of claim 10 , wherein the first intermediary share is generated based on a second blinding key share.
14 . The method of claim 10 , wherein the first pre-signature share is generated based on the shared value.
15 . The method of claim 14 , wherein the first intermediary share is generated based on the shared value.
16 . The method of claim 1 , wherein the message comprises at least part of a blockchain transaction.
17 . The method of claim 1 , wherein the threshold number of participants is less than a total number of participants in the group of participants.
18 . Computer equipment comprising:
memory comprising one or more memory units; and
processing apparatus comprising one or more processing units, wherein the memory stores code arranged to run on the processing apparatus, the code being configured so as when run on the processing apparatus, the processing apparatus performs a method of
participating in a threshold signature scheme for generating a digital signature (r, s) of a message m, wherein the digital signature comprises an r value and an s value, wherein a threshold number of different shares of the s value generated by respective participants of a group of participants are required to generate the s value of the digital signature, wherein each participant has a respective private key share a i of a private key a corresponding to a public key P that is usable to validate the digital signature, wherein each participant has a respective ephemeral private key share k i of an ephemeral private key k, and wherein the method is performed by a first one of the participants and comprises:
prior to obtaining the message m, generating a first message-independent component of a first share s i of the s value, wherein the first message-independent component is generated as a function of a first private key share a i of the private key a but not as a function of the message m,
after obtaining the message m, generating a first message-dependent component of the first share s i of the s value, wherein the first message-dependent component is generated as a function of the message m,
wherein the first message-independent component and/or the first message-dependent component is also generated as a function of a first ephemeral private key share k i of the ephemeral private key k,
and
causing a first signature share s i to be made available to the coordinator, wherein the first signature share s i is generated as a function of the first message-independent component and the first message-dependent component, and wherein the coordinator is configured to generate the signature based on each of at least the threshold number of respective signature shares s i , each respective signature share s i being generated by a respective participant and being generated as a function of a respective message-independent component and a respective message-dependent component, the respective message-independent component being generated by the respective participant as a function of the respective private key share a i of the private key a and the respective message-dependent component being generated by the respective participant as a function of the message m, wherein the respective message-independent component and/or the respective message-dependent component is also generated as a function of a respective ephemeral private key share k i of the ephemeral private key k.
19 . A computer program embodied on computer-readable storage and configured so as, when run on computer equipment, to perform a method of
participating in a threshold signature scheme for generating a digital signature (r, s) of a message m, wherein the digital signature comprises an r value and an s value, wherein a threshold number of different shares of the s value generated by respective participants of a group of participants are required to generate the s value of the digital signature, wherein each participant has a respective private key share a i of a private key a corresponding to a public key P that is usable to validate the digital signature, wherein each participant has a respective ephemeral private key share k i of an ephemeral private key k, and wherein the method is performed by a first one of the participants and comprises:
prior to obtaining the message m, generating a first message-independent component of a first share s i of the s value, wherein the first message-independent component is generated as a function of a first private key share a i of the private key a but not as a function of the message m,
after obtaining the message m, generating a first message-dependent component of the first share s i of the s value, wherein the first message-dependent component is generated as a function of the message m,
wherein the first message-independent component and/or the first message-dependent component is also generated as a function of a first ephemeral private key share k i of the ephemeral private key k,
and
causing a first signature share s i to be made available to the coordinator, wherein the first signature share s i is generated as a function of the first message-independent component and the first message-dependent component, and wherein the coordinator is configured to generate the signature based on each of at least the threshold number of respective signature shares s i , each respective signature share s i being generated by a respective participant and being generated as a function of a respective message-independent component and a respective message-dependent component, the respective message-independent component being generated by the respective participant as a function of the respective private key share a i of the private key a and the respective message-dependent component being generated by the respective participant as a function of the message m, wherein the respective message-independent component and/or the respective message-dependent component is also generated as a function of a respective ephemeral private key share k i of the ephemeral private key k.