Abstract: Systems and methods are described for using equivalent secret values across different elliptic curves. For example, a transferring party may wish to exchange a first asset on a first blockchain with a recipient for a second asset on a second blockchain. After exchanging sets of public keys with a recipient, a transferring party may generate a zero-knowledge proof and public keys associated with a selected bitstring. The recipient may then verify the proof, which shows that private keys associated with the public keys associated with the bitstring are both derived from the bitstring without revealing the bitstring itself. Once validity of the private keys has been established, the transferring party may publish a second signature to claim the second asset. The published second signature may then be used to publish a first signature (generated using the selected bitstring) on the first blockchain to claim the first asset.

Abstract: Systems and methods are described for transferring and verifying the transfer of an asset from a limited-participant side chain back to a main blockchain. A public difference, associated with a secret difference, is determined as a difference between a main blockchain address and the public offline key of a transferring participant. The public difference is used, along with each participant public online key, to generate a ring signature key for each participant. A ring signature is then generated over the ring signature keys, based on the public online keys and a set of uniform random scalars (each associated with a participant public online key). The main blockchain address, a first coefficient from the ring signature, and the uniform random scalars are then published. When verified, the published ring signature shows that the transferring participant has control of the main blockchain address and the private offline key.