Title: Flattening Nested 2-of-2 Of a 1-of-1 And a k-of-n
Note
It is possible to flatten the below into a single-layer quorum signing group:
- a 2-of-2 composed of:
- a normal single signer
- a k-of-n quorum signer
This is done by simply requiring that the “single signer” participant holds multiple shares in a larger non-nested k-of-n group.
To determine the flattened k-of-n and the number of shares the single signer has:
flattened_k = n + 1
flattened_n = 2 * n - k + 1
single_signer_shares = n - k + 1
Here are a few concrete examples:
- Example 1
- Suppose we have:
- Single Signer Ursula
- 2-of-3 Signers Alice, Bob, Carol
- This flattens to 4 of 5 with Ursula holding 2 shares:
flattened_k = 3 + 1 = 4flattened_n = 2 * 3 - 2 + 1 = 5single_signer_shares = 3 - 2 + 1 = 2
- So we have Ursula1, Ursula2, Alice, Bob, and Carol.
- Ursula1 and Ursula2, plus any 2 of the 3 Alice, Bob, and Carol, can form a quorum in the 4-of-5.
- The group Alice, Bob, and Carol is just 3, and cannot form a quorum of 4 to overpower Ursula.
- Suppose we have:
- Example 2
- Suppose we have:
- Single Signer Ursula
- 2-of-5 Signers Alice, Bob, Carol, Dave, Evans
- This flattens to 6 of 9 with Ursula holding 4 shares:
flattened_k = 5 + 1 = 6flattened_n = 2 * 5 - 2 + 1 = 9single_signer_shares = 5 - 2 + 1 = 4
- So we have Ursula1, Ursula2, Ursula3, Ursula4, Alice,
Bob, Carol, Dave, Evans.
- Ursula1, Ursula2, Ursula3, Ursula4, plus any 2 of Alice, Bob, Carol, Dave, and Evans, can form a quorum in the 6-of-9.
- The group Alice, Bob, Carol, Dave, and Evans is just 5, and cannot form a quorum of 6 to overpower Ursula.
- Suppose we have:
- Example 3
- Suppose we have:
- Single Signer Ursula
- 6-of-7 Signers Alice, Bob, Carol, Dave, Evans, Fergus, Greg
- This flattens to 8 of 9 with Ursula holding 2 shares:
flattened_k = 7 + 1 = 8flattened_n = 2 * 7 - 6 + 1 = 9single_signer_shares = 7 - 6 + 1 = 2
- So we have Ursula1, Ursula2, Alice, Bob, Carol, Dave,
Evans, Fergus, Greg.
- Ursula1 and Ursula2, plus any 6 of Alice, Bob, Carol, Evans, Fergus, and Greg, can form a quorum in the 8-of-9.
- The group Alice, Bob, Carol, Dave, Evans, Fergus, and Greg is just 7, and cannot form a quorum of 8 to overpower Ursula.
- Suppose we have:
Derivation
The key here is flattened_k = n + 1.
This assures us that the group of n participants cannot
overpower the priveleged single signer, thus always
requiring the participation of the priveleged single
signer, as in the original 2-of-2 of the single signer
plus the k-of-n quorum signers.
From there, we need to ensure that the group of n
participants all just retain having one share in the
group, but the priveleged single signer needs to fill in
more than one share.
As the original is k, then the priveleged single
signer logically has to get the difference between
the flattened_k and k, or in other words,
single_signer_shares = flattened_k - k = n + 1 - k = n - k + 1.
single_signer_shares cannot be less than that as
then even k of the quorum signers plus the single
signer would not even achieve flattened_k.
If it were more than that, then the priveleged single
signer could overpower the quorum signers by choosing
less than k of the quorum signers to achieve
flattened_k.
There are still n participants in the quorum signing
group, and the priveleged single signer has
single_signer_shares, so adding them together gives us
the total new number of shares for the flattened group:
flattened_n = n + single_signer_shares = n + n - k + 1 = 2 * n - k + 1.
Applications
- Currently we have no proof that FROST-in-MuSig is safe.
However, in many actual applications where such nesting
may be desirable, one side is often a single signer.
- For example, a paranoid user might want to have k-of-n of their signing devices, and otherwise connect to the Lightning Network, which uses 2-of-2 channels, with some LSP as the other signer in the channel.
- For MultiChannel, Ursula is a priveleged single signer that wants to use the same liquidity with the flexibility to offer outgoing HTLCs/PTLCs/MultiPTLCs amongst N LSPs, with improved availability by allowing one or more of the LSPs to go down when Ursula wants to make a payment.
- Statechain BS can have its security “improved” by having
the statechain operator be a k-of-n with the current user
being a priveleged single signer.
- Resharing inside a 2-of-2 of a k-of-n is actually
cryptographically dubious as novel cryptography.
However, resharing a flat k-of-n is not as novel.
- There are a lot of broken VSS schemes still though. But at least you are not adding even more brokenness by putting a 2-of-2 on top of a k-of-n and then resharing.
- Against this, no amount of invoking the ghost of TEE can protect end users against recovery of old backups of supposedly-deleted keys. Literally my first job was an underpaid third-world engineer at just-above-minimum-wage (minumum wage for third-worlders, not first-worlders) reverse-engineering ICs: opening them up, taking pictures from microscopes, carefully removing a metal layer, taking pictures again, etc. until you reach the polysilicon and doped silicon layers, and then figuring out the overall circuit so that the parent company could check if patents were violated and sue. Yes, you can see burnt fuses that way, it is just another metal layer. For reference, Intel SGX stores the per-chip hardware attestation key as burnt fuses on the integrated circuit, and Intel hands over possession of the CPU integrated circuit to buyers, who can now open up the IC and extract the hardware attestation key to attest to anything they want and Intel will give its CA to support that “remote” hardware attestation key. Possession is eleven points of the law, and they say there are but twelve. Just use a hardware wallet, because in that case, you can hire the same team of underpaid third-world reverse-engineers to see if the wallet has any hidden private keys that the manufacturer embedded. Use real-world cut-and-choose (buy 2 HW wallets, mark one “heads” the other “tails”, flip a coin, have the reverse-engineers work on the losing wallet, if they find nothing wrong, that is good evidence you can rely on the winning wallet).
- Resharing inside a 2-of-2 of a k-of-n is actually
cryptographically dubious as novel cryptography.
However, resharing a flat k-of-n is not as novel.
