Adapter Signatures and Their Application in Cross-Chain Atomic Swaps

With the rapid development of Bitcoin Layer2 scaling technology, cross-chain asset transfers between Bitcoin and Layer2 networks have become increasingly frequent. This trend is primarily attributed to the higher scalability, lower transaction fees, and greater throughput provided by Layer2 technology. These advancements facilitate more efficient and cost-effective transactions, thereby promoting the widespread adoption and integration of Bitcoin in various applications. Consequently, the interoperability between Bitcoin and Layer2 networks is becoming a key component of the cryptocurrency ecosystem, driving innovation and offering users more diverse and powerful financial tools.

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

The advantages of centralized cross-chain trading are its fast speed and simple operation, but its security completely relies on the reliability of centralized institutions, which poses a higher risk. The BitVM cross-chain bridge introduces an optimistic challenge mechanism, which is relatively complex with higher transaction fees, mainly suitable for large transactions. Cross-chain atomic swaps are a decentralized, uncensorable, and privacy-protecting technology that can achieve high-frequency cross-chain trading and is widely used in decentralized exchanges.

The cross-chain atomic swap technology mainly includes two schemes: the Hash Time-Locked Contract (HTLC) based on ( and the adapter signature. The HTLC scheme has privacy leakage issues, while the adapter signature-based scheme can effectively address this problem. This article will focus on the adapter signature and its application in cross-chain atomic swaps.

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 generates a random number r and calculates R = r * G
2. Alice calculates the adapter Y = y * G, where y is the adapter secret
3. Alice calculates c = H###R + Y, m(
4. Alice calculates s' = r + c * x
5. Alice sends the pre-signed )R, s'( to Bob
6. Bob verifies e)G, s' * G( ?= e)P, R + Y + c * P(
7. Bob calculates s = s' + y after obtaining y
8. Bob broadcasts )R, s( completes the transaction

The cross-chain atomic swap process based on Schnorr adapter signatures is as follows:

1. Alice generates a random number r_A and calculates R_A = r_A * G
2. Alice computes Y_A = y_A * G, where y_A is the adapter secret.
3. Alice calculates the pre-signed )R_A, s'_A( and sends it to Bob
4. Bob verifies Alice's pre-signature
5. Bob repeats steps 1-3, generates his own pre-signed )R_B, s'_B( and sends it to Alice.
6. Alice verifies Bob's pre-signed message
7. Alice and Bob exchange adapters Y_A and Y_B
8. Alice uses y_B to complete Bob's signature, and Bob uses y_A to complete Alice's signature.
9. Alice and Bob each broadcast the complete signatures to complete the transaction.

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) ECDSA adapter signature and atomic swap

The basic principle of ECDSA adapter signatures is as follows:

1. Alice generates a random number k and calculates R = k * G.
2. Alice calculates the adapter Y = y * G, where y is the adapter secret.
3. Alice calculates s' = k^###-1( * )H(m( + x * R_x)
4. Alice sends the pre-signed )R, s'( to Bob
5. Bob verifies R ?= )s'^(-1( * H)m() * G + )s'^(-1( * R_x * s') * P
6. Bob calculates s = s' + y after obtaining y.
7. Bob broadcasts )R, s( transaction completed

The cross-chain atomic swap process based on ECDSA adapter signatures is similar to the Schnorr scheme.

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Problems and Solutions

) Random number problem and solution

There is a security risk of random number leakage and reuse in the adapter signature, which may lead to private key leakage. The solution is to adopt the RFC 6979 specification to generate random numbers in a deterministic manner:

k = SHA256###sk, msg, counter(

This ensures the uniqueness and reproducibility of random numbers while avoiding the risks posed by weak random number generators.

) 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, which leads to the inability to pre-sign refund transactions. The solution is to implement atomic swap logic using smart contracts on the Ethereum side.

2. Security of the same curve with different algorithms: When using the same elliptic curve but different signature algorithms ### such as Schnorr and ECDSA (, the adapter signature scheme remains secure.

3. The insecurity of different curves: If two systems use different elliptic curves, then adapter signatures cannot be directly used for cross-chain atomic swaps.

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Digital Asset Custody Application

Non-interactive digital asset custody can be realized based on adapter signatures:

1. Alice and Bob create a funding transaction with a 2-of-2 MuSig output.
2. Alice and Bob generate adapter signatures respectively and exchange verification.
3. Both parties sign and broadcast the funding transaction.
4. In case of a dispute, the custodian may decrypt and provide the adapter secret to one party.
5. The party that obtains the secret can complete the adapter signature and broadcast the settlement transaction.

This solution does not require the involvement of a custodian for initialization, offering non-interactive advantages. Verifiable encryption technologies ) such as Purify or Juggling ( can be used to achieve secure transmission of adapter secrets.

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Summary

Adapter signature technology provides an efficient and privacy-preserving solution for cross-chain atomic swaps. By reasonable design, it can overcome issues such as random number security and system heterogeneity. In addition, adapter signatures can also be extended to non-interactive digital asset custody scenarios. With the growing demand for cross-chain solutions, adapter signature technology is expected to play an important role in blockchain interoperability.

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