Ethereum uses the secp256k1 public key cryptography and Solana uses the ed25519 public key cryptography.
How is that situation handled in the case of bridges? Let's say that we want to certify a block on Ethereum. That would imply submitting the block and the signatures of the Solana validators.
So, the ed25519 cryptography would have to be computed. However, that costs about 300k of gas for each validator which is very expensive.
How are the Solana/Ethereum bridges built?
From the Solana side, the secp256k1 program allows for verifying Ethereum signatures: https://solana.com/docs/clients/javascript-reference#secp256k1program
So when a user moves funds into an Eth smart contract, you can provide the user's signature to Solana to verify the movement.
Of course, there are much better designs than this. For example, you can see how Wormhole does this at https://docs.wormhole.com/wormhole/explore-wormhole/guardian
In the case of Wormhole, you can have all the guardians provide secp256k1 signatures, so they can be verified on both Solana and Ethereum. This way, Ethereum, never needs to know about ed25519 signatures.
