Adapter Signatures and Their Application in Cross-Chain Atomic Swaps

With the development of Bitcoin Layer 2 scaling solutions, asset transfers between Bitcoin and Layer 2 networks are becoming increasingly frequent. This trend is driven by the higher scalability, lower transaction fees, and high throughput provided by Layer 2 technologies. The interoperability between Bitcoin and Layer 2 networks is becoming a key component of the cryptocurrency ecosystem.

Currently, there are three main solutions for cross-chain transactions between Bitcoin and Layer 2: centralized cross-chain transactions, BitVM cross-chain bridges, and cross-chain atomic swaps. These technologies have their own characteristics in terms of trust assumptions, security, convenience, transaction limits, etc., catering to different application needs.

Centralized cross-chain transactions are fast, with a simple matching process, but security depends on the reliability of centralized institutions. The BitVM cross-chain bridge introduces an optimistic challenge mechanism, which is technically complex and suitable for large transactions. Cross-chain atomic swaps are decentralized, censorship-resistant, and provide good privacy protection, making them a widely used high-frequency cross-chain trading solution in decentralized exchanges.

Cross-chain atomic swap technology mainly includes two types: one based on Hash Time Locked Contracts (HTLC) ( and the other based on adapter signatures. HTLC is simple to implement but has privacy issues. Adapter signatures can protect privacy well and are a lighter, lower-cost solution.

This article focuses on the principles of Schnorr/ECDSA adapter signatures and cross-chain atomic swaps, analyzing the random number security issues that exist, as well as the system heterogeneity and algorithm heterogeneity problems in cross-chain scenarios, and provides corresponding solutions. In addition, it also discusses the application of adapter signatures in non-interactive digital asset custody.

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Adapter Signatures and Cross-Chain Atomic Swaps

) Schnorr adapter signatures and atomic swaps

The basic principle of Schnorr adapter signatures is as follows:

1. Alice chooses a random number r and calculates R = rG
2. Alice computes c = H###R||P||m(
3. Alice calculates s' = r + cx
4. Alice sends )R,s'( to Bob
5. Bob verifies s'G = R + cP
6. Bob chooses y, calculates Y = yG
7. Bob calculates s = s' + y
8. Bob broadcasts )R,s,Y(
9. Network verification sG = R + cP + Y

![Analyzing Bitcoin and Layer2 Asset Cross-Chain Technology])https://img-cdn.gateio.im/webp-social/moments-d1dea371c4dd34fed51cbd1b2a93474e.webp(

) ECDSA adapter signature and atomic swap

The basic principle of ECDSA adapter signatures is as follows:

1. Alice chooses a random number r and calculates R = rG
2. Alice calculates c = H###R||P||m(
3. Alice calculates s' = r^)-1()H(m( + xR_x)
4. Alice sends )R,s'( to Bob
5. Bob verifies R = )H(m( + xR_x)G / s'
6. Bob chooses y, calculates Y = yG
7. Bob calculates s = s' + y
8. Bob broadcasts )R,s,Y(
9. Network Verification R = )H(m( + xR_x)G / s + Y / s

![Analyzing Bitcoin and Layer 2 Asset Cross-Chain Technology])https://img-cdn.gateio.im/webp-social/moments-c1f7fb81382024c7d717e75038db0cf1.webp(

Problems and Solutions

) Random Number Problems and Solutions

The adapter signature contains issues of random number leakage and reuse, which may lead to private key exposure. The solution is to use the RFC 6979 specification to deterministically generate random numbers from the private key and message using the HMAC function.

cross-chain scenario issues and solutions

1. The heterogeneous problem between UTXO and account model systems: Bitcoin uses the UTXO model, while Ethereum uses the account model, making it impossible to pre-sign refund transactions. The solution is to implement atomic swap logic using smart contracts on the Ethereum side.

2. Security of adapter signatures with the same curve and different algorithms: If Bitcoin and Layer2 use the same curve but different signing algorithms, the adapter signatures remain secure.

3. Different curve adapter signatures are insecure: if Bitcoin and Layer2 use different elliptic curves, adapter signatures cannot be used for cross-chain atomic swaps.

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Digital Asset Custody Application

Adapter signatures can be used to implement non-interactive digital asset custody. The main participants include the buyer, seller, and custodian. The custodian does not need to participate in the initialization process and only needs to release the secret when necessary.

The specific process is as follows:

1. Create an unsigned funding transaction
2. Alice and Bob each create adapter signatures and verifiable encryption.
3. Alice and Bob verify the validity of the ciphertext and sign the funding transaction.
4. In the event of a dispute, the custodian may decrypt and release the secret to the corresponding party.

Verifiable encryption is a key technology for achieving non-interactive custody, with two main implementation solutions: Purify and Juggling.

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![Analyzing Bitcoin and Layer2 Asset Cross-Chain Technology])https://img-cdn.gateio.im/webp-social/moments-e09f20bac2bd4f245bdfc3006427e45b.webp(

![Analyzing Bitcoin and Layer 2 Asset Cross-Chain Technology])https://img-cdn.gateio.im/webp-social/moments-9c382f3c2f6eb018947793ebaeed1729.webp(

Summary

This article provides a detailed introduction to the principles of Schnorr/ECDSA adapter signatures and cross-chain atomic swaps, analyzes the security issues involved and the challenges in cross-chain application scenarios, and proposes corresponding solutions. It also discusses the application of adapter signatures in digital asset custody. Adapter signatures provide an efficient, secure, and privacy-preserving new solution for cross-chain asset exchange, and are expected to play an important role in the decentralized finance sector.