System and method for securely sharing cryptographic material
Systems and methods described herein relate to techniques in which multiple parties each generate and exchange quantities that are based on a shared secret (e.g., powers of the shared secret) without exposing the shared secret. According to a protocol, two or more parties may exchange sets of elliptic curve points generated over polynomials that can be used, by each of the two or more parties, to determine a power of a shared secret. The protocol may be utilised as part of determining parameters for a smart contract that is broadcast to a blockchain network (e.g., Bitcoin). Based on the protocol, an additional party (e.g., a third party different from the two or more parties) may perform a computational task such as execution of the smart contract.
1. A computer-implemented method comprising:
determining, at a participant of a plurality of participants, a function usable to map finite field elements, wherein the function is provided to the plurality of participants;
generating, by each participant of the plurality of participants and based on the function, polynomials evaluated in a set of points;
distributing the polynomials, by each participant to each other participant of the plurality of participants, to determine a sum of the polynomials distributed to the plurality of participants;
determining a master polynomial from the sum of the polynomials distributed to the plurality of participants, wherein the master polynomial corresponds to a power of a secret; and
collectively generating, among the plurality of participants and based on the secret, a blockchain transaction.
2. The computer-implemented method according to claim 1 , wherein the set of points is a set of elliptic curve points.
3. The computer-implemented method according to claim 1 , wherein determining the master polynomial from the sum of the polynomials is based on a Lagrange interpolation.
4. The computer-implemented method according to claim 1 , further comprising determining, based at least in part on the power of the secret, a common reference string that is determinable by the plurality of participants.
5. The computer-implemented method according to claim 1 , further comprising receiving values of the polynomials from each participant of the plurality of participants, the values as evaluated in the set of points by each participant.
6. The computer-implemented method according to claim 5 , wherein off-chain communications are utilised in distribution of the values of the polynomials to each of the other participants.
7. The computer-implemented method according to claim 1 , wherein a result of the function as evaluated serves as a coefficient or exponent of a generator of a group.
8. The computer-implemented method according to claim 1 , wherein the plurality of participants is exactly two participants.
9. The computer-implemented method according to claim 1 , wherein the secret is determined based on the master polynomial evaluated at a particular point.
10. The computer-implemented method according to claim 9 , wherein the particular point is an intersection point.
11. The computer-implemented method according to claim 10 , wherein the set of points excludes the intersection point.
12. The computer-implemented method according to claim 1 , wherein the polynomials evaluated in the set of points are of at least order 2.
13. The computer-implemented method according to claim 1 , wherein each point of the set of points is a different point.
14. A system, comprising:
a processor; and
memory including executable instructions that, as a result of execution by the processor, cause the system to:
determine, at a participant of a plurality of participants, a function usable to map finite field elements, wherein the function is provided to the plurality of participants;
generate, by each participant of the plurality of participants and based on the function, polynomials evaluated in a set of points;
distribute the polynomials, by each participant to each other participant of the plurality of participants, to determine a sum of the polynomials distributed to the plurality of participants;
determine a master polynomial from the sum of the polynomials distributed to the plurality of participants, wherein the master polynomial corresponds to a power of a secret; and
collectively generate, among the plurality of participants and based on the secret, a blockchain transaction.
15. The system according to claim 14 , wherein the set of points is a set of elliptic curve points.
16. The system according to claim 14 , wherein determining the master polynomial from the sum of the polynomials is based on a Lagrange interpolation.
17. The system according to claim 14 , wherein the executable instructions, as a result of execution by the processor, further cause the system to determine, based at least in part on the power of the secret, a common reference string that is determinable by the plurality of participants.
18. A non-transitory computer-readable storage medium having stored thereon executable instructions that, as a result of being executed by a processor of a computer system, cause the computer system to at least:
determine, at a participant of a plurality of participants, a function usable to map finite field elements, wherein the function is provided to the plurality of participants;
generate, by each participant of the plurality of participants and based on the function, polynomials evaluated in a set of points;
distribute the polynomials, by each participant to each other participant of the plurality of participants, to determine a sum of the polynomials distributed to the plurality of participants;
determine a master polynomial from the sum of the polynomials distributed to the plurality of participants, wherein the master polynomial corresponds to a power of a secret; and
collectively generate, among the plurality of participants and based on the secret, a blockchain transaction.
19. The non-transitory computer-readable storage medium according to claim 18 , wherein the set of points is a set of elliptic curve points.
20. The non-transitory computer-readable storage medium according to claim 18 , wherein determining the master polynomial from the sum of the polynomials is based on a Lagrange interpolation.