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Masniff.Hurricane wrote:

Wex, have you already seen this http://www.reddit.com/r/Bitcoin/comment ... em/cq4b52u ?

wex wrote:http://users.ics.aalto.fi/knyberg/program.txt

15

> - Title: Just A Little Bit More

> - Authors:

Nigel Smart (University of Bristol)

Yuval Yarom (University of Adelaide)

Joop van de Pol (University of Bristol)

> - Quick abstract (200 characters including spaces)

We exploit a property of many standard elliptic curves to reduce the number of signatures

needed to be observed and demonstrate how we break ECDSA on a secp256k1 curve using only 25 signatures.

As far as I know ECDSA on a secp256k1 curve is used in Bitcoin (https://en.bitcoin.it/wiki/Secp256k1). So the question is, Bitcoin is getting hacked?

Wex, have you already seen this http://www.reddit.com/r/Bitcoin/comment ... em/cq4b52u ?

Thanks. But still I don't understand why can't Bitcoin and CryptoNote be hacked with this kind of attacks?

Statistics: Posted by wex — Fri Apr 17, 2015 11:44 am

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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.

Statistics: Posted by *tech_star* — Sun Apr 12, 2015 4:23 pm

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