Size of ring signatures

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Size of ring signatures

Postby RichardK » Tue Mar 17, 2015 6:10 pm

Hello.

I see you've answered to Adam Back's tweet about the size of ring signatures and you were going to look into this.

Image

Did you have a chance? I think it is an interesting topic since ring signatures are basically the main feature of CryptoNote. Will this version of ring signatures have a place in CryptoNote structure?
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Re: Size of ring signatures

Postby *tech_star* » Sun Apr 12, 2015 4:23 pm

From Bitcointalk discussion:
The traceable ring signature used in cryptonote https://cryptonote.org/whitepaper.pdf looks like:

KEYGEN: P_i=x_i*G, I_i=x_i*H(P_i)

SIGN: as signer j; random s_i, w_i

(I relabeled q_i as s_i to be more standard, and relabeled the signer s as signer j)

IF i=j THEN L_i=s_i*G ELSE L_i=s_i*G+w_i*P_i
IF i=j THEN R_i=s_i*H(P_i) ELSE R_i=s_i*H(P_i)+w_i*I_j

c=h(m,L_1,...,L_n,R_1,...,R_n)

IF i=j THEN c_i=c-sum_{i!=j}(c_i) ELSE c_i=w_i
IF i=j THEN r_i=w_i-c_i*x_i ELSE r_i=w_i

\sigma = (m,I_j,c_1,...,c_n,r_1,...,r_n)

VERIFY:

L_i'=r_i*G+c_i*P_i
R_i'=r_i*H(P_i)+c_i*I_j
sum_{1..n}( c_j ) =? h(m,L_1',...,L_n',R_1',...,R_n')

LINK: reject duplicate I_j values.

where H(.) is a hash2curve function (taking a value in Zn and deterministically mapping it to a curve point), and h(.) is a hash function with a hash output size very close to n the order of the curve, ie h(.)=SHA256(.) mod n.

Towards finding a more compact ring signature I'd been trying to find a way to make c_i into a CPRNG generated sequence as they are basically arbitrary, though they must be bound to the rest of the signature (non-malleable) so that you can compute at most n-1 existential signature forgeries without knowing any private keys.

I found this paper "1-out-of-n Signatures from a Variety of Keys" by Abe, Ohkubo and Suzuki http://citeseerx.ist.psu.edu/viewdoc/do ... 1&type=pdf section 5.1 shows a way to do it. I show here how to add traceability to it in a way that makes it compatible with crypto note:

KEYGEN: P_i=x_i*G, I_i=x_i*H(P_i)

SIGN: as signer j; \alpha = random, \forall_{i!=j} s_i = random

c_{j+1} = h(P_1,...,P_n,\alpha*G,\alpha*H(P_j))
c_{j+2} = h(P_1,...,P_n,s_{j+1}*G+c_{j+1}*P_{j+1},s_{j+1}*H(P_{j+1})+c_{j+1}*I_j)
...
c_j = h(P_1,...,P_n,s_{j-1}*G+c_{j-1}*P_{j-1},s_{j-1}*H(P_{j-1})+c_{j-1}*I_j)

so that defines c_1,...,c_n with j values taken mod l some number of signers. Next find the s_j value:

Now \alpha*G = s_j*G+c_j*P_j so \alpha = s_j+c_j*x_j so s_j = \alpha - c_j*x_j mod n.

Similarly \alpha*H(P_j) = s_j*H(P_j)+c_j*I_j so \alpha works there too.

\sigma = (m,I_j,c_1,s_1,...,s_n)

VERIFY:

\forall_{i=1..n} compute e_i=s_i*G+c_i*P_i and E_i=s_i*H(P_i)+c_i*I_j and c_{i+1}=h(P_1,...,P_n,e_i,E_i)

check c_{n+1}=c_1

LINK: reject duplicate I_j values.

This alternate linkable ring signature tends to 1/2 the size of the crypto note ring signature as the signature is 3+n values vs 2+2n values.

Adam


As far as I see ring signatures size is still linearly dependent on n. So yes, Adam Back really found the way to make ring signatures smaller (and it's great!). But it's only two times smaller, that's not critical for the technology.
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