[curves] Threshold ECDSA for Bitcoin
Mike Hamburg
mike at shiftleft.org
Sun Mar 30 23:34:00 PDT 2014
On Mar 30, 2014, at 11:19 PM, Trevor Perrin <trevp at trevp.net> wrote:
> On Fri, Mar 28, 2014 at 3:59 PM, Michael Hamburg <mike at shiftleft.org> wrote:
>> Out of curiosity, what's wrong with the following "obvious" protocol for threshold Schnorr?
>>
>> The signers have a polynomial share x_i of x. All the signers in the signing group know who is signing right now, and they know that x = sum a_i x_i, and they know the a_i. If weeding out bad participants is desired, then each signer's share [x_i]G of the public key is known to the other group members.
>>
>> Each signer computes R_i = [k_i]G for a random nonce k_i. They broadcast commitments to these choices, then broadcast revelations.
>>
>> Each signer computes R = sum [a_i] R_i, so that effectively r = sum a_i k_i; and c = Hash(R,m).
>>
>> Each signer creates and broadcasts a mini-sig s_i = c x_i + k_i. The signature is (R, s = sum a_i s_i). Since k = sum a_i k_i and x = sum a_i x_i, we have s = cx + k as desired.
>
>
> Are you describing this, or something close to it?
>
> http://cacr.uwaterloo.ca/techreports/2001/corr2001-13.ps
>
>
> Comparing against the threshold ECDSA of:
>
> http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.67.9913
> http://www.cs.princeton.edu/~stevenag/bitcoin_threshold_signatures.pdf
>
> The ECDSA protocol seems similar, except with extra steps to deal with
> the inversion.
>
> So is it accurate to say that both threshold ECDSA and threshold
> Schnorr can be done "well" (efficiently, robustly, good security
> proofs), but the Schnorr version is simpler / less communication?
>
> Trevor
I think the dealing phase of what I'm describing is simpler, because I always assume that exactly [threshold] many participants are trying to sign and not more. Also my design doesn't have that full proof, and there might be a hangup somewhere in trying to produce one.
And yeah, I think it's accurate to say that both can be thresholded, but Schnorr is simpler.
-- Mike
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