# Zero-Knowledge Puzzles

• paying to a public key on an elliptic curve different from the curve secp256k1 used in Bitcoin today
• paying to a group of public keys, which can be spent if one knows any of the private key, without revealing which one.

# ∑ Protocols

1. Peggy computes a commitment A using a random number a. She shares A with Victor, but doesn’t reveal a.
2. Victor generates a random number e as challenge and shares it with Peggy.
3. Peggy uses a and e to compute an answer z and sends back to Victor.

# Examples of ZK Puzzles

## Pay to a Generic Public Key (P2GPK)

• It can use curve with higher security, such as secp521r1, than the hardcoded curve secp256k1. This can be desirable if a large amount of bitcoins is controlled by a single key for decades. This also means Bitcoin can upgrade to more secure signature scheme without breaking changes, by implementing it using existing opcodes³.
• It can reuse compatible keys from elsewhere. For example, PGP supports elliptic curve keys and bitcoins can be sent to PGP keys, even if they are based other curves.

## Composition

• Pay to Group Privately (P2GP): anyone of a group of key owners can spend the funds, without disclosing which one redeemed, using proof from OR composition. This is a generalization of 1-of-n multisig, but more private. For example, Peggy proves she knows the private key of public key Y or Z, i.e., she knows x such that
• Pay to Threshold Group Privately (P2TGP): any m of n members in a group can collectively redeem the UTXO without revealing which m members, using proof from AND and OR composition. This generalizes P2GP and m-of-n multisig. For example, a 2-of-3 ZK puzzle requires

# Acknowledgements

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## More from sCrypt

sCrypt Inc (https://scrypt.io) is a company with a mission to provide integrated on-chain smart contracting solutions using the original BitCoin Protocol on BSV

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## sCrypt

sCrypt Inc (https://scrypt.io) is a company with a mission to provide integrated on-chain smart contracting solutions using the original BitCoin Protocol on BSV