Topic: Anonymity in the Mini-Blockchain scheme - page 2. (Read 5480 times)

Yes, I was talking about the advantage relative to Bitcoin. I know the account tree in my scheme will be at least an order of magnitude bigger than in the MBC, probably more. I tried to maintain the accounts in the scheme, but it was simply not possible to have a high degree of privacy with reusable accounts.

Well there are no unspent outputs in the MBC scheme which is why I think your scheme is closer to Bitcoin. If I were to send 1000 micro-transactions to the same address using Cryptonite the account tree would only grow larger on the first transaction, but if the address was already in the account tree before I sent the first tx then the tree wouldn't grow at all. In your scheme the tree would grow for each of those 1000 transactions because you're recording unspent outputs rather than a balance sheet like Cryptonite. And that is the main reason why Bitcoin will never achieve the level of scalability that Cryptonite has, despite what some other people might claim.

Yeah I've thought about it but it's obviously not going to be the easiest thing in the world. It's also very hard to decide when and how the maintenance fees should be subtracted from address balances even if you know what the maintenance fee should be. There always seem to be serious difficulties involved regardless of how the problem is approached.

You understood it perfectly, the account tree has the unspent outputs of all transactions. The advantage for me of separating the unspent outputs from the transactions is that the transactions are much larger in size (in this scheme) than the unspent outputs, so there is a considerable saving. The minimum output value was borrowed from Bitcoin and it seems to be working well for them. An interesting variation could be forcing the minimum output value to be a multiple of the transaction fee, for example, if you include a transaction fee of X then all outputs must be at least 3X.
I remember that you wanted to have maintenance fees in Cryptonite to control dust but you were having problems with the actual mechanism of deciding the value, but having the stakeholders voting on it seems to be a good solution. Nxt is going to implement (or has already implemented?) voting by stakeholders. It is probably worth seeing how they do it, maybe it is applicable to Cryptonite.

Very nice work once again. I like the idea of using temporary addresses for every transaction, it seems to be a powerful way of increasing privacy. If I'm understanding your proposal correctly, the temporary addresses created by transactions are essentially what you store in the account tree, or you could also say that the account tree essentially contains all the unspent outputs from transactions (aka the receipts). What you have proposed is more similar to Bitcoin than the MBC scheme because it revolves around unspent outputs and brings back concepts like change and coin minting without inputs.

At some point in the future Bitcoin developers may figure out a way to compress all the unspent outputs into a compact structure which can be shared among peers much like in your scheme. At the end of the day your scheme might not end up being much more scalable than Bitcoin. But since your scheme, like the MBC scheme, doesn't use tx scripting, that allows you to implement all these clever mathematical tricks to produce a high level of anonymity and privacy while still remaining much more scalable than Bitcoin in its current state and all current anon coins.

Obviously it wont be more scalable than the MBC scheme but the original scheme doesn't provide the remarkably high level of anonymity that your scheme does and anonymity always comes at a cost. I also noticed that your new solution to preventing small value outputs (aka dust) is to enforce a minimum value output. Micro-transactions are much more costly in your scheme compared to the MBC scheme so I can see why you would gravitate towards that solution but it still seems like a rather poor solution to me, especially if the minimum value were to remain static over time.

A better solution might be to simply have miners enforce a minimum transaction fee. I've never really been a fan of the Proof-of-Stake mechanism but one reason I think PoS can be truly useful is for allowing stakeholders to vote on the minimum transaction fee and other important values such as the maximum block size. The problem of pruning dust really is one of the hardest things to solve I think. Sacrificing micro-transactions for more scalability is probably the best solution for your scheme but there are definitely better solutions for the MBC scheme.

I am still getting around it, but sure, I will pm you if I find some part of it questionable. Yet it may take me a lot of time and courage to come up with a critique of your model, since I am not a cryptographer, but rather a number theorist. Therefore I hope Adam come on board sooner than later
About a lighter version, I guess no. You may need a more accessible and lighter presentation of your model for general audience later, when and if your technicalities get approved by the mentioned experts and is ready to be implemented.

Thank you. I would also like to have their opinion, especially Adam Back since I borrowed a lot from his homomorphic value scheme (https://bitcointalksearch.org/topic/m.3294618). But every opinion is welcomed, I just want the paper to be reviewed.
Bybitcoin, if you have any doubt you can PM me and will respond ASAP. I know the paper ended up being too "dense". Maybe I should write a "lighter" version?

Very nice.. I hope to have Adam Back and Gregory Maxwell's opinion about this new one as well. Meanwhile I will try to digest the content by myself!

As promised, the new paper. https://drive.google.com/file/d/0B21vncLoIlIydUVGcjdDak1pRGc/view?usp=sharing

No, I apologize for the long absence. I was occupied during January and February. Also, I was not satisfied with the scheme so I spent the last month rewriting it. I will publish it here next week.

What happened to this? Has it been proved to be impossible? 

You would need to prove plaintext and random value equivalence between EC pedersen (additively homomorphic) and regular Elgamal (multiplicatively homomorphic). I don't think you can do it with discrete log equivalence protocol. Especially with only 2 points/values.

Well presuming we're talking compressed points, thats 256-bits per point or value, then I think doing what I said should be 1x 256-bit point homomorphic value, 2x 256-bit elgamal encryption, a proof of discrete log equivalence signature (2x 256-bit) so 4 values.  4 vs 6, net saving?  Maybe I missed a point not sure without writing out the protcool.  And the CRT scheme while interesting is kind of shiny and new and slowish to decrypt.  It looks ok to me, in terms of crypto-conservatism; but I think this elgamal equivalence proof etc is even more conservative.  (And so is the schoenmaker's range proof IMO).

Adam

I haven't thought of that. But it's going to be more expensive than the CRT Elgamal scheme. A CRT Elgamal cyphertext would be 6 EC points. Your idea would be 3 EC points (1 from the commitment + 2 from Elgamal) plus the size of the ZK proof (probably going to be 3 EC points + 3 256-bit integers?). And it still would have the problem of requiring that users connect every 7 days to the network.

There are variants of schnorr proof of knowledge (DSA is a variant of schnorr) where you can prove that encrypted values are the same by combining with Elgamal.  So I think you should be able to prove that the sender sent the recipient in a way decryptable by their advertised public key, an encrypted value which matches the (non-decryptable) second encryption.  eg if you look at Brands is a more complicate version but there are a few survey papers showing all the common things you can easily prove using schnorr variants.

ie so prove that the plaintext under the encryption would result in the recipient knowing a value that would allow it to spend the coin.  in your labelling make a schnorr-related proof that y=y' and r=r'.

I did think of this going back a few weeks in my comments on your original scheme but maybe neglected to say it.

Adam

Digicoin: I'm going to wait a bit more because I'm trying to find a range proof more efficient than Schoenmaker's bit-by-bit proof. Also, I would like to see if I could introduce multi-input transactions (as proposed in Adam's "Ringcoin" homomorphic scheme) into this scheme.

Adam: The url is working fine in my browser but I will change it.
I need it to be decryptable because you don't know if the sender of the transaction will send the right value and random value. In your homomorphic scheme this isn't a problem because you could simply ignore the transaction but this scheme runs on top of the mini-blockchain which actually has accounts. Suppose you have an account with balance x and corresponding Pedersen commitment xG+vH. Then I send you a transaction with value y (it can even be zero) and random value r,so yG+rH, but I send to you encrypted any other values (let's say y' and r'). The two commitments will be added and your balance will be (x+y)G+(v+r)H. Now you can't open the commitment of your own balance so, you can't make transactions because you won't be able to produce the required ZK proofs. Finally I can send a message telling you to pay me z bitcoins or I won't tell you the real values.
Also, the mini-blockchain only stores transactions for a limited time (in cryptonite's case it's 7 days) so if someone receives a transaction and doesn't connect to the network in 7 days, he won't see the transaction and will no longer know its own balance.

[The url is b0rken for the additively homomorphic Elgamal variant you mentioned.  Should be http://ecewp.ece.wpi.edu/wordpress/crypto/files/2012/10/main.pdf]

Their additive Elgamal variant seems interesting, but I am not sure why you need it to be simultaneously decryptable and additive: instead you can have additive simply with vG+xH for two EC bases G and G, value v, and random value x, and presuming the owner knows his own decryption, he can do the addition, and communicate the value & new x value to the recipient using ECIES or normal Elgamal?

Adam

When will your paper be updated with your new findings?

Ok, found a good elliptic curve additively homomorphic encryption scheme. Here's the paper, http://ecewp.ece.wpi.edu/wordpress/crypto/files/2012/10/main.pdf. What the authors did was using regular elliptic curve Elgamal to encrypt several times the same message modulo different (small) pairwise coprime numbers. To decrypt one simply brute-forces the decryption of each plaintext (which should be easy since the moduli are small)  and then, using the chinese remainder theorem, one can recover the original message.

In more detail:
Setup-> Choose a elliptic curve prime field F_p, a base point P in F_p and a random integer x in [1;p-1]. Calculate point Q=x*P. Assume that the message space is Z_s and that sEncryption-> Choose random integers r_1,...,r_t in [1;p-1]. Calculate m_i= m mod d_i for all i in {1,..,t}. The ciphertext is the t tuples {(A_i,B_i)=(r_i*P, r_i*Q+m_i*P), for all i in {1,...,t}}.
Decryption-> Calculate C_i= B_i - x*A_i=m_i*P for all i. Then calculate m_i= log_P(C_i) for all i. Finally use the extended Euclidean algorithm to solve the system of congruences {m=m_1 mod d_1,...,m=m_t mod d_t}.

Assuming the message space is 2^51 (like bitcoin), we would need three 17-bit moduli. The public key would be a bit longer than regular Elgamal and the ciphertexts would have 3x the size but it would be fast to decrypt and more efficient than Paillier encryption.

Adam, I reviewed the proof and you are absolutely right. For some reason I thought that wrap-around wasn't a problem and that I could prove y=x_1+...+x_n over the integers. The thing is that Groth's proof can prove that a vector is the Hadamard product of two other vectors over the integers or prove that a value is the inner product of two vectors but only over the integers modulo n. Strangely he treats both cases as interchangeable and never mentions this critical difference.
Anyway, I need to add range proofs for all input/output amounts and then it's enough to check if Com(y)/(Com(x_1)*...*Com(x_n))=Com(0). I will probably use Berry Schoenmaker's range proof with your improvements.
Also, I have been thinking of ways to substitute the Paillier's encryption for another encryption scheme. First, I can't trust the sender of a transaction to send the value and the nonce of the transaction (how you did in your homomorphic value scheme) because if he sends the wrong value the receiver no longer can use his account. Worse, a malicious user could do that on purpose to ask for ransom or to disrupt the network.
The other suggestion (from gmaxwell) was to use elliptic curve Elgamal. I did some research and found that a 3 GHz Pentium 4 processor can do an EC addition on a NIST 256-bit prime field curve in 8.3*10^-8 seconds. Extrapolating from that, it can crack a 24 bit EC discrete log on average in 0.7 seconds. So, if I reduce the maximum amount per transaction to 2^24 units, I could use EC Elgamal instead of Paillier. As a bonus, the range proof size would be cut in half. So it may be a viable idea.


My issue wasn't with storing the balance. The problem is if a transaction is deleted before the receiver connects to the network. Then the receiver has no way of finding the transaction amount.

You're referring to another scheme committed transactions http://www.mail-archive.com/[email protected]/msg02184.html and detail on https://bitcointalksearch.org/topic/m.2157994, your description is correct.  

There is an unsolved problem with it though - how to prevent hostile miners censoring the keys instead (by pretending they never saw them)... there needs to be a second stage validation to say ok, this is the disclosed key and the signatures are good & values add up.  You cant directly impose consensus on not disclosing that because it creates a DoS risk - that someone really doesnt reveal the key at all, or keeps it to themselves, like a selfish-mining type of attack.

The homomorphic encrypted transaction values is different, just encrypting the value.  So the payments are still as traceable as without them, just the values are hidden from the miners and from the ledger.  Miners and anyone else can still verify that the amounts add up, just they cant tell how much they actually are.

Adam