Hi Ruben, After sending that last night, I realized the solution I had to deprivatizing the sender wouldn't work because it had the same problem of even divisibility in modulo N. And my math was incomplete I think. Also Marco D'Agostini pointed out other errors. And all this assumes that a modulus operator is defined for elliptic curve points in a way that makes these valid, which I'm not sure is true. But here's another try anyway: X' = X + i*X*hash((i*X)%N) = X + x*I*hash((x*I)%N) item = {recipient: X' % N, sender: I%N} // As before. Test for each filter item: (item.recipient - X) % N == ( x*item.sender*hash((x*item.sender) % N) ) % N So to muse further about the properties of this, in a block full of taproot sends you might have an upper limit of something like 13,000 transactions. N=2^8 would I think mean an 18% collision rate (ie 20% false positive rate) because `(1-1/2^8)^13000 = 0.82...`. If we were to go with that, each item is 4 bytes (1 byte per point component?) which would mean a 52kb filter without collisions, and an average of 43kb with 18% collisions (which can be removed as dupes). Maybe Golomb-Rice coding could help here as well like it does in the usual compact block filters. And since each collision with an address a client is watching on means downloading a whole block they don't need, maybe 18% collisions is too high, and we want to choose N = 2^10 or something to get down to 2% collisions. In any case, all this could be wrong if ECC modulus doesn't work this way. But was interesting to think about anyway. On Wed, Mar 30, 2022 at 12:58 AM Billy wrote: > > the sender can get in trouble too if they send money > > Good point. > > > how well this can be optimized without resorting to reducing anonymity > > Complete shot in the dark, but I wonder if something akin to compact block > filters could be done to support this case. If, for example, the tweaked > key were defined without hashing, I think something like that could be done: > > X' = i*X*G + X = x*I*G + X > > Your compact-block-filter-like things could then store a set of each `item > = {recipient: X' % N, sender: I%N}`, and a light client would download > this data and do the following to detect a likely payment for each filter > item: > > item.recipient - X%N == x*item.sender*G > > You can then scale N to the proper tradeoff between filter size and false > positives. I suppose this might make it possible to deprivitize a tweaked > key by checking to see what non-tweaked keys evenly divide it. Perhaps > that's what hashing was being used to solve. What if we added the shared > diffie hellman secret modulo N to remove this correlation: > > X' = i*X*G + X + (i*X)%N = x*I*G + X + (x*I)%N > > Then for each `item = {recipient: X' % N, sender: I%N}`, we detect via > `item.recipient - X%N == x*item.sender*(1+G)`. Is my math right here? I'm > thinking this should work because (a+b%N)%N == (a%N + b%N)%N. > > > > On Tue, Mar 29, 2022 at 10:36 AM Ruben Somsen wrote: > >> Hi Billy, >> >> Thanks for taking a look. >> >> >Maybe it would have been more accurate to say no *extra* on chain >> overhead >> >> I can see how it can be misinterpreted. I updated the gist to be more >> specific. >> >> >primary benefit of this is privacy for the recipient >> >> Fair, but just wanted to note the sender can get in trouble too if they >> send money to e.g. blacklisted addresses. >> >> >there could be a standard that [...] reduces the anonymity set a bit >> >> This has occurred to me but I am reluctant to make that trade-off. It >> seems best to first see how well this can be optimized without resorting to >> reducing anonymity, and it's hard to analyze exactly how impactful the >> anonymity degradation is (I suspect it's worse than you think because it >> can help strengthen existing heuristics about output ownership). >> >> Cheers, >> Ruben >> >> >> >> On Tue, Mar 29, 2022 at 4:57 PM Billy wrote: >> >>> Hi Ruben, >>> >>> Very interesting protocol. This reminds me of how monero stealth >>> addresses work, which gives monero the same downsides regarding light >>> clients (among other things). I was a bit confused by the following: >>> >>> > without requiring any interaction or on-chain overhead >>> >>> After reading through, I have to assume it was rather misleading to say >>> "no on-chain overhead". This still requires an on-chain transaction to be >>> sent to the tweaked address, I believe. Maybe it would have been more >>> accurate to say no *extra* on chain overhead (over a normal transaction)? >>> >>> It seems the primary benefit of this is privacy for the recipient. To >>> that end, it seems like a pretty useful protocol. It's definitely a level >>> of privacy one would only care about if they might receive a lot money >>> related to that address. However of course someone might not know they'll >>> receive an amount of money they want to be private until they receive it. >>> So the inability to easily do this without a full node is slightly less >>> than ideal. But it's another good reason to run a full node. >>> >>> Perhaps there could be a standard that can identify tweaked address, >>> such that only those addresses can be downloaded and checked by light >>> clients. It reduces the anonymity set a bit, but it would probably still be >>> sufficient. >>> >>> >>> >>> On Mon, Mar 28, 2022, 10:29 Ruben Somsen via bitcoin-dev < >>> bitcoin-dev@lists.linuxfoundation.org> wrote: >>> >>>> Hi all, >>>> >>>> I'm publishing a new scheme for private non-interactive address >>>> generation without on-chain overhead. It has upsides as well as downsides, >>>> so I suspect the main discussion will revolve around whether this is worth >>>> pursuing or not. There is a list of open questions at the end. >>>> >>>> I added the full write-up in plain text below, though I recommend >>>> reading the gist for improved formatting and in order to benefit from >>>> potential future edits: >>>> https://gist.github.com/RubenSomsen/c43b79517e7cb701ebf77eec6dbb46b8 >>>> >>>> Cheers, >>>> Ruben >>>> >>>> >>>> >>>> Silent Payments >>>> >>>> Receive private payments from anyone on a single static address without >>>> requiring any interaction or on-chain overhead >>>> >>>> >>>> >>>> OVERVIEW >>>> >>>> >>>> The recipient generates a so-called silent payment address and makes it >>>> publicly known. The sender then takes a public key from one of their chosen >>>> inputs for the payment, and uses it to derive a shared secret that is then >>>> used to tweak the silent payment address. The recipient detects the payment >>>> by scanning every transaction in the blockchain. >>>> >>>> Compared to previous schemes[1], this scheme avoids using the Bitcoin >>>> blockchain as a messaging layer[2] and requires no interaction between >>>> sender and recipient[3] (other than needing to know the silent payment >>>> address). The main downsides are the scanning requirement, the lack of >>>> light client support, and the requirement to control your own input(s). An >>>> example use case would be private one-time donations. >>>> >>>> While most of the individual parts of this idea aren’t novel, the >>>> resulting protocol has never been seriously considered and may be >>>> reasonably viable, particularly if we limit ourselves to detecting only >>>> unspent payments by scanning the UTXO set. We’ll start by describing a >>>> basic scheme, and then introduce a few improvements. >>>> >>>> >>>> >>>> BASIC SCHEME >>>> >>>> >>>> The recipient publishes their silent payment address, a single 32 byte >>>> public key: >>>> X = x*G >>>> >>>> The sender picks an input containing a public key: >>>> I = i*G >>>> >>>> The sender tweaks the silent payment address with the public key of >>>> their input: >>>> X' = hash(i*X)*G + X >>>> >>>> Since i*X == x*I (Diffie-Hellman Key Exchange), the recipient can >>>> detect the payment by calculating hash(x*I)*G + X for each input key I in >>>> the blockchain and seeing if it matches an output in the corresponding >>>> transaction. >>>> >>>> >>>> >>>> IMPROVEMENTS >>>> >>>> >>>> UTXO set scanning >>>> >>>> If we forgo detection of historic transactions and only focus on the >>>> current balance, we can limit the protocol to only scanning the >>>> transactions that are part of the UTXO set when restoring from backup, >>>> which may be faster. >>>> >>>> Jonas Nick was kind enough to go through the numbers and run a >>>> benchmark of hash(x*I)*G + X on his 3.9GHz Intel® Core™ i7-7820HQ CPU, >>>> which took roughly 72 microseconds per calculation on a single core. The >>>> UTXO set currently has 80 million entries, the average transaction has 2.3 >>>> inputs, which puts us at 2.3*80000000*72/1000/1000/60 = 221 minutes for a >>>> single core (under 2 hours for two cores). >>>> >>>> What these numbers do not take into account is database lookups. We >>>> need to fetch the transaction of every UTXO, as well as every transaction >>>> for every subsequent input in order to extract the relevant public key, >>>> resulting in (1+2.3)*80000000 = 264 million lookups. How slow this is and >>>> what can be done to improve it is an open question. >>>> >>>> Once we’re at the tip, every new unspent output will have to be >>>> scanned. It’s theoretically possible to scan e.g. once a day and skip >>>> transactions with fully spent outputs, but that would probably not be worth >>>> the added complexity. If we only scan transactions with taproot outputs, we >>>> can further limit our efforts, but this advantage is expected to dissipate >>>> once taproot use becomes more common. >>>> >>>> >>>> Variant using all inputs >>>> >>>> Instead of tweaking the silent payment address with one input, we could >>>> instead tweak it with the combination of all input keys of a transaction. >>>> The benefit is that this further lowers the scanning cost, since now we >>>> only need to calculate one tweak per transaction, instead of one tweak per >>>> input, which is roughly half the work, though database lookups remain >>>> unaffected. >>>> >>>> The downside is that if you want to combine your inputs with those of >>>> others (i.e. coinjoin), every participant has to be willing to assist you >>>> in following the Silent Payment protocol in order to let you make your >>>> payment. There are also privacy considerations which are discussed in the >>>> “Preventing input linkage” section. >>>> >>>> Concretely, if there are three inputs (I1, I2, I3), the scheme becomes: >>>> hash(i1*X + i2*X + i3*X)*G + X == hash(x*(I1+I2+I3))*G + X. >>>> >>>> >>>> Scanning key >>>> >>>> We can extend the silent payment address with a scanning key, which >>>> allows for separation of detecting and spending payments. We redefine the >>>> silent payment address as the concatenation of X_scan, X_spend, and >>>> derivation becomes X' = hash(i*X_scan)*G + X_spend. This allows your >>>> internet-connected node to hold the private key of X_scan to detect >>>> incoming payments, while your hardware wallet controls X_spend to make >>>> payments. If X_scan is compromised, privacy is lost, but your funds are not. >>>> >>>> >>>> Address reuse prevention >>>> >>>> If the sender sends more than one payment, and the chosen input has the >>>> same key due to address reuse, then the recipient address will also be the >>>> same. To prevent this, we can hash the txid and index of the input, to >>>> ensure each address is unique, resulting in X' = hash(i*X,txid,index)*G + >>>> X. Note this would make light client support harder. >>>> >>>> >>>> >>>> NOTEWORTHY DETAILS >>>> >>>> >>>> Light clients >>>> >>>> Light clients cannot easily be supported due to the need for scanning. >>>> The best we could do is give up on address reuse prevention (so we don’t >>>> require the txid and index), only consider unspent taproot outputs, and >>>> download a standardized list of relevant input keys for each block over >>>> wifi each night when charging. These input keys can then be tweaked, and >>>> the results can be matched against compact block filters. Possible, but not >>>> simple. >>>> >>>> >>>> Effect on BIP32 HD keys >>>> >>>> One side-benefit of silent payments is that BIP32 HD keys[4] won’t be >>>> needed for address generation, since every address will automatically be >>>> unique. This also means we won’t have to deal with a gap limit. >>>> >>>> >>>> Different inputs >>>> >>>> While the simplest thing would be to only support one input type (e.g. >>>> taproot key spend), this would also mean only a subset of users can make >>>> payments to silent addresses, so this seems undesirable. The protocol >>>> should ideally support any input containing at least one public key, and >>>> simply pick the first key if more than one is present. >>>> >>>> Pay-to-(witness-)public-key-hash inputs actually end up being easiest >>>> to scan, since the public key is present in the input script, instead of >>>> the output script of the previous transaction (which requires one extra >>>> transaction lookup). >>>> >>>> >>>> Signature nonce instead of input key >>>> >>>> Another consideration was to tweak the silent payment address with the >>>> signature nonce[5], but unfortunately this breaks compatibility with MuSig2 >>>> and MuSig-DN, since in those schemes the signature nonce changes depending >>>> on the transaction hash. If we let the output address depend on the nonce, >>>> then the transaction hash will change, causing a circular reference. >>>> >>>> >>>> Sending wallet compatibility >>>> >>>> Any wallet that wants to support making silent payments needs to >>>> support a new address format, pick inputs for the payment, tweak the silent >>>> payment address using the private key of one of the chosen inputs, and then >>>> proceed to sign the transaction. The scanning requirement is not relevant >>>> to the sender, only the recipient. >>>> >>>> >>>> >>>> PREVENTING INPUT LINKAGE >>>> >>>> >>>> A potential weakness of Silent Payments is that the input is linked to >>>> the output. A coinjoin transaction with multiple inputs from other users >>>> can normally obfuscate the sender input from the recipient, but Silent >>>> Payments reveal that link. This weakness can be mitigated with the “variant >>>> using all inputs”, but this variant introduces a different weakness – you >>>> now require all other coinjoin users to tweak the silent payment address, >>>> which means you’re revealing the intended recipient to them. >>>> >>>> Luckily, a blinding scheme[6] exists that allows us to hide the silent >>>> payment address from the other participants. Concretely, let’s say there >>>> are two inputs, I1 and I2, and the latter one is ours. We add a secret >>>> blinding factor to the silent payment address, X + blinding_factor*G = X', >>>> then we receive X1' = i1*X' (together with a DLEQ to prove correctness, see >>>> full write-up[6]) from the owner of the first input and remove the blinding >>>> factor with X1' - blinding_factor*I1 = X1 (which is equal to i1*X). >>>> Finally, we calculate the tweaked address with hash(X1 + i2*X)*G + X. The >>>> recipient can simply recognize the payment with hash(x*(I1+I2))*G + X. Note >>>> that the owner of the first input cannot reconstruct the resulting address >>>> because they don’t know i2*X. >>>> >>>> The blinding protocol above solves our coinjoin privacy concerns (at >>>> the expense of more interaction complexity), but we’re left with one more >>>> issue – what if you want to make a silent payment, but you control none of >>>> the inputs (e.g. sending from an exchange)? In this scenario we can still >>>> utilize the blinding protocol, but now the third party sender can try to >>>> uncover the intended recipient by brute forcing their inputs on all known >>>> silent payment addresses (i.e. calculate hash(i*X)*G + X for every publicly >>>> known X). While this is computationally expensive, it’s by no means >>>> impossible. No solution is known at this time, so as it stands this is a >>>> limitation of the protocol – the sender must control one of the inputs in >>>> order to be fully private. >>>> >>>> >>>> >>>> COMPARISON >>>> >>>> >>>> These are the most important protocols that provide similar >>>> functionality with slightly different tradeoffs. All of them provide fresh >>>> address generation and are compatible with one-time seed backups. The main >>>> benefits of the protocols listed below are that there is no scanning >>>> requirement, better light client support, and they don’t require control >>>> over the inputs of the transaction. >>>> >>>> >>>> Payment code sharing >>>> >>>> This is BIP47[2]. An OP_RETURN message is sent on-chain to the >>>> recipient to establish a shared secret prior to making payments. Using the >>>> blockchain as a messaging layer like this is generally considered an >>>> inefficient use of on-chain resources. This concern can theoretically be >>>> alleviated by using other means of communicating, but data availability >>>> needs to be guaranteed to ensure the recipient doesn’t lose access to the >>>> funds. Another concern is that the input(s) used to establish the shared >>>> secret may leak privacy if not kept separate. >>>> >>>> >>>> Xpub sharing >>>> >>>> Upon first payment, hand out an xpub instead of an address in order to >>>> enable repeat payments. I believe Kixunil’s recently published scheme[3] is >>>> equivalent to this and could be implemented with relative ease. It’s >>>> unclear how practical this protocol is, as it assumes sender and recipient >>>> are able to interact once, yet subsequent interaction is impossible. >>>> >>>> >>>> Regular address sharing >>>> >>>> This is how Bitcoin is commonly used today and may therefore be >>>> obvious, but it does satisfy similar privacy requirements. The sender >>>> interacts with the recipient each time they want to make a payment, and >>>> requests a new address. The main downside is that it requires interaction >>>> for every single payment. >>>> >>>> >>>> >>>> OPEN QUESTIONS >>>> >>>> >>>> Exactly how slow are the required database lookups? Is there a better >>>> approach? >>>> >>>> Is there any way to make light client support more viable? >>>> >>>> What is preferred – single input tweaking (revealing an input to the >>>> recipient) or using all inputs (increased coinjoin complexity)? >>>> >>>> Are there any security issues with the proposed cryptography? >>>> >>>> In general, compared to alternatives, is this scheme worth the added >>>> complexity? >>>> >>>> >>>> >>>> ACKNOWLEDGEMENTS >>>> >>>> >>>> Thanks to Kixunil, Calvin Kim, and Jonas Nick, holihawt and Lloyd >>>> Fournier for their help/comments, as well as all the authors of previous >>>> schemes. Any mistakes are my own. >>>> >>>> >>>> >>>> REFERENCES >>>> >>>> >>>> [1] Stealth Payments, Peter Todd: >>>> https://github.com/genjix/bips/blob/master/bip-stealth.mediawiki ↩︎ >>>> >>>> [2] BIP47 payment codes, Justus Ranvier: >>>> https://github.com/bitcoin/bips/blob/master/bip-0047.mediawiki >>>> >>>> [3] Reusable taproot addresses, Kixunil: >>>> https://gist.github.com/Kixunil/0ddb3a9cdec33342b97431e438252c0a >>>> >>>> [4] BIP32 HD keys, Pieter Wuille: >>>> https://github.com/bitcoin/bips/blob/master/bip-0032.mediawiki >>>> >>>> [5] 2020-01-23 ##taproot-bip-review, starting at 18:25: >>>> https://gnusha.org/taproot-bip-review/2020-01-23.log >>>> >>>> [6] Blind Diffie-Hellman Key Exchange, David Wagner: >>>> https://gist.github.com/RubenSomsen/be7a4760dd4596d06963d67baf140406 >>>> _______________________________________________ >>>> bitcoin-dev mailing list >>>> bitcoin-dev@lists.linuxfoundation.org >>>> https://lists.linuxfoundation.org/mailman/listinfo/bitcoin-dev >>>> >>>