The goal of this protocol is for Bob to get Alice to perform a Diffie-Hellman key exchange blindly, such that when the unblinded value is returned, Alice recognizes it as her own, but can’t distinguish it from others (i.e. similar to a blind signature).
Alice:
A = a*G
return A
Bob:
Y = hash_to_curve(secret_message)
r = random blinding factor
B'= Y + r*G
return B'
Alice:
C' = a*B'
(= a*Y + a*r*G)
return C'
Bob:
C = C' - r*A
(= C' - a*r*G)
(= a*Y)
return C, secret_message
Alice:
Y = hash_to_curve(secret_message)
C == a*Y
If true, C must have originated from Alice
I unearthed this protocol from a seemingly long forgotten cypherpunk mailing list post by David Wagner from 1996 (edit: perhaps not as forgotten as I thought, as Lucre is an implementation of it). It was devised as an alternative to RSA blinding in order to get around the now-expired patent by David Chaum. As in all ecash protocols, the secret_message is remembered by Alice in order to prevent double spends.
One benefit of this scheme is that it's relatively straightforward to perform in a threshold setting (only requires curve multiplication). One downside is that validation is more involved than simply checking a signature, as this step requries repeating the Diffie-Hellman Key Exchange.
The protocol also has one additional weakness that can be addressed. Bob can't be certain that C' was correctly generated and thus corresponds to a*B' . Alice can resolve this by also supplying a discrete log equality proof (DLEQ), showing that a in A = a*G is equal to a in C' = a*B'. This equality can be proven with a relatively simple Schnorr signature, as described below.
(These steps occur once Alice returns C')
Alice:
r = random nonce
R1 = r*G
R2 = r*B'
e = hash(R1,R2,A,C')
s = r + e*a
return e, s
Bob:
R1 = s*G - e*A
R2 = s*B'- e*C'
e == hash(R1,R2,A,C')
If true, a in A = a*G must be equal to a in C' = a*B'
Thanks to Eric Sirion, Andrew Poelstra, and Adam Gibson for their helpful comments.