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 encrypting amounts and asset types of a verifiable transaction on a blockchain ledger. For each asset, an asset tag is blinded, multiplied by the amount of the asset, and the product is blinded again to create an encrypted amount of the asset. Both encrypted amount of the asset and a corresponding generated output value are within a value range, and the sum of the encrypted input value and the encrypted output value equals zero. Rangeproofs for each of the encrypted output values are associated with a different public key. Each public key is signed with a ring signature based on a public key of a recipient. A second ring signature is used to verify each asset tag, where the private key of the second ring signature for each asset is a difference between a first blinding value and an output coefficient.

Abstract: Systems and methods are described for encrypting an amount transacted on a blockchain ledger, while preserving the transaction's ability to be verified. A blinding amount is added to an input value, and an output value is generated and encrypted. Both the input value and the output value are within a value range, where a sum of any two values within the range does not exceed an overflow threshold. The sum of the encrypted input value and the encrypted output value may equal zero. Rangeproofs associated with each of the input value and the output value are generated. The rangeproofs prove that the input value and the output value fall within the value range, and each rangeproof may be associated with a different public key. Each public key may be signed with a ring signature based on a public key of a recipient in the transaction.

Abstract: Systems and methods are described for transferring an asset from a parent chain to a sidechain. A simplified payment verification (SPV) proof associated with the parent chain asset may be generated. The SPV proof may include a threshold level of work. The SPV proof associated with the parent chain asset may be validated, and a sidechain asset corresponding to the parent chain asset may be generated. If no reorganization proof is detected, the sidechain asset is released. To redeem the sidechain asset in the parent chain, a SPV proof associated with the sidechain asset may be generated. The parent chain may validate the SPV proof associated with the sidechain asset. The parent chain asset associated with the sidechain asset may be held for a second predetermined contest period. The parent chain asset may then be released if no reorganization proof associated with the sidechain asset is detected.

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.