Adapter Signatures and Their Applications in Cross-Chain Atomic Swaps

With the rapid development of Bitcoin Layer 2 scaling solutions, the frequency of cross-chain asset transfers between Bitcoin and Layer 2 networks has significantly increased. This trend is driven by the higher scalability, lower transaction fees, and high throughput provided by Layer 2 technology. These advancements facilitate more efficient and cost-effective transactions, thereby promoting the wider adoption and integration of Bitcoin in various applications. Consequently, the interoperability between Bitcoin and Layer 2 networks is becoming a key component of the cryptocurrency ecosystem, driving innovation and providing users with a more diverse and powerful set of financial tools.

There are three typical solutions for cross-chain transactions between Bitcoin and Layer 2, namely centralized cross-chain transactions, BitVM cross-chain bridge, and cross-chain atomic swaps. These three technologies differ in terms of trust assumptions, security, convenience, transaction limits, etc., and can meet different application needs.

The advantages of centralized cross-chain trading lie in its speed and the relative ease of the matching process, as centralized institutions can quickly confirm and process transactions. However, the security of this method completely depends on the reliability and reputation of the centralized institution. If the centralized institution encounters technical failures, malicious attacks, or defaults, the users' funds face a higher risk. Additionally, centralized cross-chain trading may also leak user privacy, requiring users to consider this method carefully when choosing it.

The BitVM cross-chain bridge technology is relatively complex. This technology introduces an optimistic challenge mechanism, making it relatively complicated. Moreover, the optimistic challenge mechanism involves a large number of challenge and response transactions, resulting in higher transaction fees. Therefore, the BitVM cross-chain bridge is only suitable for ultra-large transactions, similar to the issuance of U, leading to a lower frequency of use.

Cross-chain atomic swaps are a type of contract that enables decentralized cryptocurrency trading. Atomic swaps must involve two parties, and no third party can interrupt or interfere with the swapping process. This means that the technology is decentralized, censorship-resistant, offers better privacy protection, and can achieve high-frequency cross-chain transactions, thus being widely used in decentralized exchanges.

Cross-chain atomic swap technology mainly includes Hash Time Lock and Adapter Signature. Cross-chain atomic swaps based on Hash Time Lock (HTLC) allow two users to conduct time-restricted cryptocurrency transactions, meaning the recipient must submit cryptographic proof to the contract within a specified time, or the funds will be returned to the sender. Cross-chain atomic swaps based on Adapter Signature replace the "secret hash" swaps that rely on on-chain scripts, including time locks and hash locks. Since such scripts are not involved, on-chain space usage is reduced, making atomic swaps based on Adapter Signature lighter and cheaper.

There are security issues with the random number in adapter signatures and the problems of system heterogeneity and algorithm heterogeneity in cross-chain scenarios. Random number leakage or reuse can lead to private key leakage. The solution is to use RFC 6979, which eliminates the need to generate random numbers by deterministically deriving the random number from the private key and the message to be signed.

In cross-chain scenarios, the heterogeneous problem of UTXO and account model systems can be solved by designing Dapp applications similar to Tornado Cash on the Bitlayer side, providing privacy services for transactions on the Bitlayer side in atomic swaps between BTC and Bitlayer. When using the same curve but different algorithms, adapter signatures are secure. However, when the curves are different, adapter signatures are not secure.

Adapter signatures can also be applied to digital asset custody, enabling non-interactive threshold digital asset custody. This method has the advantage of being non-interactive, but its flexibility is not as high as that of threshold Schnorr signatures.

Overall, the adapter signature provides an effective solution for cross-chain atomic swaps, but various factors need to be considered in practical applications, including security, privacy, and system compatibility.