# Encryption scheme that requires fewer keys for decryption than for encryption

I want to attempt creating a decentralized dead man's switch, so I'm looking for an asymmetric encryption scheme, with the following properties (ideally), or something close to them:

1. There is a group of n people, each with their own public and private key.
2. Any one person can, at any time, encrypt a message using all public keys.
3. The catch that I'm looking for is that you wouldn't need all n private keys to decrypt the message, but only some number k < n.
4. It MUST be unfeasible to decrypt the message even if you have k-1 keys, but very easy with k keys.
5. Once k-1 receivers have comitted to the decryption, it MUST be impossible for the kth one to complete the decryption privately and obtain the message without publishing it. I intend to use it in an Ethereum contract, so there might be a way to avoid the impossibility of fair trade without a trusted party, using the blockchain.
6. (optional) The sender can choose the number k arbitrarily during encryption.
7. (optional) None of the receivers need to reveal their actual private keys during decryption.

Layered RSA (or any other asymmetric cipher for that matter), wouldn't do. It a) doesn't have property 5, and b) requires encrypting your message in $$\binom{n}{k}$$ different ways, which is unfeasible for large numbers.

Does such a thing exist? Is it even theoretically possible?
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