Subject: [bitcoin-dev] Silent Payments – Non-interactive private payments with no on-chain overhead

From: Ruben Somsen <[email-masked]>
Date: Mon, 28 Mar 2022 17:27:56 +0200

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