From mboxrd@z Thu Jan 1 00:00:00 1970 Return-Path: Received: from smtp1.osuosl.org (smtp1.osuosl.org [140.211.166.138]) by lists.linuxfoundation.org (Postfix) with ESMTP id 952EAC0032 for ; Thu, 3 Aug 2023 13:23:10 +0000 (UTC) Received: from localhost (localhost [127.0.0.1]) by smtp1.osuosl.org (Postfix) with ESMTP id 6EBF780C80 for ; Thu, 3 Aug 2023 13:23:10 +0000 (UTC) DKIM-Filter: OpenDKIM Filter v2.11.0 smtp1.osuosl.org 6EBF780C80 Authentication-Results: smtp1.osuosl.org; dkim=pass (2048-bit key) header.d=protonmail.com header.i=@protonmail.com header.a=rsa-sha256 header.s=protonmail3 header.b=yY9SKWzS X-Virus-Scanned: amavisd-new at osuosl.org X-Spam-Flag: NO X-Spam-Score: -2.1 X-Spam-Level: X-Spam-Status: No, score=-2.1 tagged_above=-999 required=5 tests=[BAYES_00=-1.9, DKIM_SIGNED=0.1, DKIM_VALID=-0.1, DKIM_VALID_AU=-0.1, DKIM_VALID_EF=-0.1, FREEMAIL_FROM=0.001, HTML_MESSAGE=0.001, SPF_HELO_PASS=-0.001, SPF_PASS=-0.001] autolearn=ham autolearn_force=no Received: from smtp1.osuosl.org ([127.0.0.1]) by localhost (smtp1.osuosl.org [127.0.0.1]) (amavisd-new, port 10024) with ESMTP id eCwFpkOqAr-5 for ; Thu, 3 Aug 2023 13:23:07 +0000 (UTC) Received: from mail-41103.protonmail.ch (mail-41103.protonmail.ch [185.70.41.103]) by smtp1.osuosl.org (Postfix) with ESMTPS id C78F58309A for ; Thu, 3 Aug 2023 13:23:06 +0000 (UTC) DKIM-Filter: OpenDKIM Filter v2.11.0 smtp1.osuosl.org C78F58309A Date: Thu, 03 Aug 2023 13:22:35 +0000 Authentication-Results: mail-41103.protonmail.ch; dkim=pass (2048-bit key) header.d=protonmail.com header.i=@protonmail.com header.b="yY9SKWzS" DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=protonmail.com; s=protonmail3; t=1691068972; x=1691328172; bh=T2P86x9joMAGvkg1z7ZwWDru4s+BezYOvdkS91Qxk7s=; h=Date:To:From:Subject:Message-ID:Feedback-ID:From:To:Cc:Date: Subject:Reply-To:Feedback-ID:Message-ID:BIMI-Selector; b=yY9SKWzSgoJnTqhB1bh/FKE2wpy0kvZJhsIe9Gr+kR5y//wsSMUV5CfcNWWZBcQVd Yz1lNvZxbLgLmY1GDPbEaSDWIg7jEXHOqYBuagRP6eb+1OlvfGCLJuTMUy1Oiz+krm gnMs2NdC5oBFkQevxWV9jMhGmV21gbd9SvW5/Ytwy7YeD/UOejj75oLZPP3ToPxs6z DbiRnoaanABuBiSFQkNuM1tl5IoD4NKMgXtLXZuhYoq3k9d2MmbaSNkMEUe66WKdw5 RuXUiR4bIjQFp4W+EbPEIgh6EBzs81QxN8JjdC/UffGnx7nJLy/BDaP1zylUOvwD7+ 1rIJ2m1CmOnmg== To: "bitcoin-dev@lists.linuxfoundation.org" From: josibake Message-ID: Feedback-ID: 27113594:user:proton MIME-Version: 1.0 Content-Type: multipart/signed; protocol="application/pgp-signature"; micalg=pgp-sha256; boundary="------b16a3eb241b27fea2d8059d49031be2893dbed25c65af7ad33a02377783822df"; charset=utf-8 X-Mailman-Approved-At: Thu, 03 Aug 2023 14:11:54 +0000 Subject: [bitcoin-dev] BIP352: Silent Payments X-BeenThere: bitcoin-dev@lists.linuxfoundation.org X-Mailman-Version: 2.1.15 Precedence: list List-Id: Bitcoin Protocol Discussion List-Unsubscribe: , List-Archive: List-Post: List-Help: List-Subscribe: , X-List-Received-Date: Thu, 03 Aug 2023 13:23:10 -0000 This is an OpenPGP/MIME signed message (RFC 4880 and 3156) --------b16a3eb241b27fea2d8059d49031be2893dbed25c65af7ad33a02377783822df Content-Type: multipart/mixed;boundary=---------------------dd0699eccbf5001b6feec2601f5b49ec -----------------------dd0699eccbf5001b6feec2601f5b49ec Content-Type: multipart/alternative;boundary=---------------------80428cf9cc25328c0c96dc9c9ba5060a -----------------------80428cf9cc25328c0c96dc9c9ba5060a Content-Transfer-Encoding: quoted-printable Content-Type: text/plain;charset=utf-8 Hi all, Since first posting the BIP for Silent Payments back in June, we've gone t= hrough several rounds of review and revision (a big thanks to everyone who= has reviewed so far!). We've also updated the BIP with a python reference= implementation and wallet test vectors. To summarize a few of the major c= hanges: - Use 33-byte compressed keys instead of 32-byte x-only keys, except whe= n creating the outputs - Negate the private key if necessary when spending taproot outputs = - If the receiving wallet uses labels, add a step to also check the nega= ted output - Add a versioning scheme for silent payment addresses For those interested, the implementation for Bitcoin Core has been updated= to reflect these changes here: https://github.com/bitcoin/bitcoin/pull/27= 827 . For convenience, and in the hopes of soliciting another round of rev= iew, the full specification is posted below and the full text of the BIP i= ncluding the overview, test vectors and appendix on light client support c= an be found here: https://github.com/bitcoin/bips/pull/1458). Cheers! -- Ruben, Josie =3D=3D Specification =3D=3D -------------------- We use the following functions and conventions: - outpoint (36 bytes): the COutPoint of an input (32-byte txid, least si= gnificant byte first || 4-byte vout, least significant byte first)[6] - sortoutpoints(v): sorts a vector v of outpoints in ascending order by = doing a byte by byte comparison lexicographically. - ser32(i): serializes a 32-bit unsigned integer i as a 4-byte sequence,= most significant byte first. - ser256(p): serializes the integer p as a 32-byte sequence, most signif= icant byte first. - serP(P): serializes the coordinate pair P =3D (x,y) as a byte sequence= using SEC1's compressed form: (0x02 or 0x03) || ser256(x), where the head= er byte depends on the parity of the omitted y coordinate. For everything not defined above, we use the notation from BIP340. ### =3D=3D=3D Versions =3D=3D=3D This document defines Silent Payments v0. Version is communicated through = the address in the same way as Segwit addresses. Future upgrades to silent= payments will require a new version. As much as possible, future upgrades= should support receiving from older wallets (e.g. a silent payments v0 wa= llet can send to both v0 and v1 addresses). Any changes that break compati= bility with older silent payment versions should be a new BIP. Future Silent Payments versions will use the following versioning scheme: 0 1 2 3 4 5 6 7 Compatibility +0 q p z r y 9 x 8 =C2=A0=C2=A0 backwards compatible +8 g f 2 t v d w 0 +16 s 3 j n 5 4 k h +24 c e 6 m u a 7 - v31 (l) is reserved for a backwards incompatible change, if needed. For Si= lent Payments v0: - If the receiver's silent payment address version is: - v0: check that the data part is exactly 66-bytes. Otherwise, fail - v1 through v30: read the first 66-bytes of the data part and disca= rd the remaining bytes (if any) - v31: fail - Receiver addresses are always BIP341 taproot outputs[7] - The sender should sign with one of the sighash flags DEFAULT, ALL, SIN= GLE, NONE (ANYONECANPAY is unsafe). It is strongly recommended implementat= ions use SIGHASH_DEFAULT when applicable, or SIGHASH_ALL[8] - Inputs used to derive the shared secret are from the Inputs For Shared= Secret Derivation list ### =3D=3D=3D Scanning transactions =3D=3D=3D A transaction is a Silent Payments v0 transaction and MUST be scanned if a= nd only if all of the following are true: - The transaction contains at least one BIP341 taproot output - The transaction has at least one input from the Inputs For Shared Secr= et Derivation list - The transaction does not spend a new, undefined output type (e.g. SegW= it versions > 1)[9] Otherwise, skip the transaction. This is to ensure forward compatibility w= ith future versions of silent payments without requiring future versions t= o scan a transaction multiple times with different rule sets. ### =3D=3D=3D Address encoding =3D=3D=3D A silent payment address is constructed in the following manner: - Let Bscan, bscan =3D Receiver's scan public key and corresponding priv= ate key - Let Bspend, bspend =3D Receiver's spend public key and corresponding p= rivate key - Let Bm =3D Bspend + m=C2=B7G, where m an optional integer tweak for la= beling - In the case of m =3D 0, no label is applied and Bm =3D Bspend - The final address is a Bech32m encoding of: - The human-readable part "sp" for mainnet, "tsp" for testnets (e.g.= signet, testnet) - The data-part values: - The character "q", to represent a silent payment address of ve= rsion 0 - The 66 byte concatenation of the receiver's public keys, serP(= Bscan) || serP(Bm) Note: BIP173 imposes a 90 character limit for Bech32 strings, whereas a si= lent payment address requires at least 117 characters[10]. Additionally, s= ince higher versions may add to the data field, it is recommended implemen= tations use a limit of 1023 characters (see BIP173: Checksum design for mo= re details). ### =3D=3D=3D Outpoints hash =3D=3D=3D The sender and receiver MUST calculate an outpoints hash for the transacti= on in the following manner: - Collect each outpoint used as an input to the transaction - Sort the outpoints with sortoutpoints(outpoints)[11] - Let outpoints =3D outpoint0 || =E2=80=A6 || outpointn - Let outpoints_hash =3D sha256(outpoints) ### =3D=3D=3D Inputs For Shared Secret Derivation =3D=3D=3D While any UTXO with known output scripts can be used to fund the transacti= on, the sender and receiver MUST use inputs from the following list when d= eriving the shared secret: - P2TR - P2WPKH - P2SH-P2WPKH - P2PKH Inputs with conditional branches or multiple public keys (e.g. CHECKMULTIS= IG) are not included as this introduces malleability and would allow a sen= der to re-sign with a different set of public keys after the silent paymen= t output has been derived. This is not a concern when the sender controls = all of the inputs, but is an issue for CoinJoins and other collaborative p= rotocols, where a malicious participant can participate in deriving the si= lent payment address with one set of keys and then re-broadcast the transa= ction with signatures for a different set of public keys. P2TR can have hi= dden conditional branches (script path), but we work around this as descri= bed below. =3D=3D=3D=3D P2TR =3D=3D=3D=3D The sender MUST use the private key corresponding to the taproot output ke= y (i.e. the tweaked private key for a key path spend). This can be a singl= e private key or an aggregate key (e.g. taproot outputs using MuSig2 or FR= OST)[12]. If this key is not available, the output cannot be included as a= n input to the transaction. The receiver always uses the taproot output ke= y when scanning, regardless of whether the taproot output is using a key p= ath spend or a script path spend[13]. The one exception is script path spends that use NUMS point H as their int= ernal key (where H =3D lift_x(0x50929b74c1a04954b78b4b6035e97a5e078a5a0f28= ec96d547bfee9ace803ac0) which is constructed by taking the hash of the sta= ndard uncompressed encoding of the secp256k1 base point G as X coordinate,= see BIP341: Constructing and spending Taproot outputs for more details), = in which case the output will be skipped for the purposes of shared secret= derivation[14]. =3D=3D=3D=3D P2WPKH =3D=3D=3D=3D The sender performs the tweak using the private key for the output and the= receiver obtains the public key from the witness. =3D=3D=3D=3D P2SH-P2WPKH =3D=3D=3D=3D The sender performs the tweak using the private key for the nested P2WPKH = output and the receiver obtains the public key from the witness. =3D=3D=3D=3D P2PKH =3D=3D=3D=3D The sender performs the tweak using the private key for the output and SHO= ULD sign using the standard script template: =C2=A0 =C2=A0 scriptSig: The receiver obtains the public key from the scriptSig. The receiver MUST = parse the scriptSig for the public key, even if the scriptSig is non-stand= ard (e.g. OP_DROP ). This is to address th= e third-party malleability of P2PKH scriptSigs. ### =3D=3D=3D Sender =3D=3D=3D #### =3D=3D=3D=3D Selecting inputs =3D=3D=3D=3D The sending wallet performs coin selection as usual with the following res= trictions: - At least one input MUST be from the Inputs For Shared Secret Derivatio= n list - Exclude inputs with witness version > 1 (see Scanning transactions) - For each taproot output spent the sending wallet MUST have access to t= he private key corresponding to the taproot output key, unless H is used a= s the internal public key #### =3D=3D=3D=3D Creating outputs =3D=3D=3D=3D After the inputs have been selected, the sender can create one or more out= puts for one or more silent payment addresses in the following manner: - Collect the private keys for each input from the Inputs For Shared Sec= ret Derivation list - For each private key ai corresponding to a BIP341 taproot output, chec= k that the private key produces a point with an even y-value and negate th= e private key if not[15] - Let a =3D a0 + a1 + =E2=80=A6 an, where each ai has been negated if ne= cessary - Generate the outpoints_hash, using the method described above - Group receiver silent payment addresses by Bscan (e.g. each group cons= ists of one Bscan and one or more Bm) - For each group: - Let ecdh_shared_secret =3D outpoints_hash=C2=B7a=C2=B7Bscan - Let n =3D 0 - For each Bm in the group: - Let tn =3D sha256(serP(ecdh_shared_secret) || ser32(n)) - Let Pmn =3D Bm + tn=C2=B7G - Encode Pmn as a BIP341 taproot output - Optionally, repeat with n++ to create additional outputs for t= he current Bm - If no additional outputs are required, continue to the next Bm= with n++[16] - Optionally, if the sending wallet implements receiving silent paym= ents, it can create change outputs in the following manner: - Let Achange =3D Aspend + sha256(ser256(ascan))=C2=B7G - Let change_shared_secret =3D outpoints_hash=C2=B7a=C2=B7Ascan - Let n =3D 0 - For each change output desired: - Let cn =3D sha256(serP(change_shared_secret) || ser32(n)) - Let Cn =3D Achange + cn=C2=B7G - Encode Cn as a BIP341 taproot output - Repeat with n++ for additional change outputs ### =3D=3D=3D Receiver =3D=3D=3D #### =3D=3D=3D=3D Key Derivation =3D=3D=3D=3D Two keys are needed to create a silent payments address: the spend key and= the scan key. While these keys can be generated independently, wallet sof= tware SHOULD use BIP32 derivation[17] to ensure compatibility across walle= ts. A scan and spend key pair using BIP32 derivation are defined (taking inspi= ration from BIP44) in the following manner: =C2=A0 =C2=A0 =C2=A0Scan private key: m / purpose' / coin_type' / account'= / 1' / 0 =C2=A0 =C2=A0 Spend private key: m / purpose' / coin_type' / account' / 0'= / 0 Wallet software MUST use hardened derivation to ensure the master key is n= ot exposed in the event the scan private key is compromised. Purpose is a = constant set to 352 following the BIP43 recommendation. Refer to BIP43 and= BIP44 for more details. #### =3D=3D=3D=3D Scanning =3D=3D=3D=3D If each of the checks in Scanning transactions passes, the receiving walle= t must: - Generate the outpoints_hash, using the method described above - Let A =3D A0 + A1 + =E2=80=A6 An, where each Ai is the public key of a= n input from the Inputs For Shared Secret Derivation list - Let ecdh_shared_secret =3D outpoints_hash=C2=B7bscan=C2=B7A - Check for outputs: - Let outputs_to_check =3D the taproot output key from each unspent = taproot output in the transaction - Starting with n =3D 0: - Let tn =3D sha256(serP(ecdh_shared_secret) || ser32(n)) - Compute Pn =3D Bspend + tn=C2=B7G - For each output in outputs_to_check: - If Pn equals output: - Add Pn to the wallet - Remove output from outputs_to_check and rescan outputs= _to_check with n++ - Else, if the wallet has precomputed labels (including the = change label, if used)[18]: - Compute m=C2=B7G =3D output - Pn - Check if m=C2=B7G exists in the list of labels used by= the wallet - If a match is found: - Add the Pn + m=C2=B7G to the wallet - Remove output from outputs_to_check and rescan out= puts_to_check with n++ - If the label is not found, negate output and check aga= in - If no matches are found, stop #### =3D=3D=3D=3D Backup and Recovery =3D=3D=3D=3D Since each silent payment output address is derived independently, regular= backups are recommended. When recovering from a backup, the wallet will n= eed to scan since the last backup to detect new payments. If using a seed/seed phrase only style backup, the user can recover the wa= llet's unspent outputs from the UTXO set (i.e. only scanning transactions = with at least one unspent taproot output) and can recover the full wallet = history by scanning the blockchain starting from the wallet birthday. If a= wallet uses labels or generates its change addresses using the change lab= el, this information SHOULD be included in the backup. If the user does no= t know whether labels or the change label were used, it is strongly recomm= ended they always check for the change label when recovering from backup a= nd precompute a large number of labels (e.g. 100k labels) to use when re-s= canning. This ensures that the wallet can recover all funds from only a se= ed/seed phrase backup. =3D=3D Backward Compatibility =3D=3D ----------------------------- Silent payments introduces a new address format and protocol for sending a= nd as such is not compatible with older wallet software or wallets which h= ave not implemented the silent payments protocol. =3D=3D Rationale and References =3D=3D ------------------------------- 1. =C2=A0Why not use out-of-band notifications Out of band notifications = (e.g. using something other than the Bitcoin blockchain) have been propose= d as a way of addressing the privacy and cost concerns of using the Bitcoi= n blockchain as a messaging layer. This, however, simply moves the privacy= and cost concerns somewhere else and increases the risk of losing money d= ue to a notification not being reliably delivered, or even censored, and m= akes this notification data critical for backup to recover funds. 2. Why allow for more than one output? Allowing Alice to break her paymen= t to Bob into multiple amounts opens up a number of privacy improving tech= niques for Alice, making the transaction look like a CoinJoin or better hi= ding the change amount by splitting both the payment and change outputs in= to multiple amounts. It also allows for Alice and Carol to both have their= own unique output paying Bob in the event they are in a collaborative tra= nsaction and both paying Bob's silent payment address. 3. What about inputs without public keys? Inputs without public keys can = still be spent in the transaction but are simply ignored in the Silent Pay= ments protocol. 4. How does using all inputs help light clients? If Alice uses a random = input for the tweak, Bob necessarily has to have access to and check all t= ransaction inputs, which requires performing an ECC multiplication per inp= ut. If instead Alice performs the tweak with the sum of the input public k= eys, Bob only needs the summed 33 byte public key per transaction and only= does one ECC multiplication per transaction. Bob can then use BIP158 bloc= k filters to determine if any of the outputs exist in a block and thus avo= ids downloading transactions which don't belong to him. It is still an ope= n question as to how Bob can source the 32 bytes per transaction in a trus= tless manner, see Appendix A: Light Client Support for more details. 5. Why does using all inputs matter for CoinJoin? If Alice uses a random = input to create the output for Bob, this necessarily reveals to Bob which = input Alice has control of. If Alice is paying Bob as part of a CoinJoin, = this would reveal which input belongs to her, degrading the anonymity set = of the CoinJoin and giving Bob more information about Alice. If instead al= l inputs are used, Bob has no way of knowing which input(s) belong to Alic= e. This comes at the cost of increased complexity as the CoinJoin particip= ants now need to coordinate to create the silent payment output and would = need to use Blind-Diffie-Hellman to prevent the other participants from le= arning who Alice is paying. 6. Why are outpoints little-endian? Despite using big endian throughout t= he rest of the BIP, outpoints are sorted and hashed matching their transac= tion serialization, which is little-endian. This allows a wallet to parse = a serialized transaction for use in silent payments without needing to re-= order the bytes when compute the outpoint hash. Note: despite outpoints be= ing stored and serialized as little-endian, the transaction hash (txid) is= always displayed as big-endian. 7. Why only taproot outputs? Providing too much optionality for the proto= col makes it difficult to implement and can be at odds with the goal of pr= oviding the best privacy. Limiting to taproot outputs helps simplify the i= mplementation significantly while also putting users in the best eventual = anonymity set. 8. Why recommend SIGHASH_[DEFAULT|ALL]? Since the output address for the = receiver is derived from from the sum of the Inputs For Shared Secret Deri= vation public keys, the inputs must not change once the sender has signed = the transaction. If the inputs are allowed to change after the fact, the r= eceiver will not be able to calculate the shared secret needed to find and= spend the output. It is currently an open question on how a future versio= n of silent payments could be made to work with new sighash flags such as = SIGHASH_GROUP and SIGHASH_ANYPREVOUT. 9. Why skip transactions that spend unknown output scripts? Skipping tran= sactions that spend unknown output scripts allows us to have a clean upgra= de path for Silent Payments by avoiding the need to scan the same transact= ion multiple times with different rule sets. If a fancy new output type is= added in the future and Silent Payments v1 is released with support, we w= ould want to avoid having to first scan the transaction with the silent pa= yment v0 rules and then again with the silent payment v1 rules. Note: this= restriction only applies to the inputs of a transaction. 10. Why do silent payment addresses need at least 117 characters? A silen= t payment address is a bech32m encoding comprised of the following parts: = - HRP [2-3] - separator [1] - version [1-2] - payload, 66 bytes concatenated pubkeys [ceil(66*8/5)] - checksum [6] = For a silent payments v0 address, this results in a 117 character addr= ess when using a 3 character HRP. Future versions of silent payment addres= ses may add to the payload, which is why a 1023 character limit is suggest= ed. 11. Why are outpoints sorted before hashing? This way the silent payment = otuput does not need to be recalculated if the wallet changes the order of= inputs, e.g. at signing time or during an RBF bump. 12. Are key aggregation techniques like FROST and MuSig2 supported? Any t= aproot output able to do a key path spend is supported. While a full speci= fication of how to do this securely is outside the scope of this BIP, in t= heory any offline key aggregation technique can be used, such as FROST or = MuSig2. This would require participants to perform the ECDH step collabora= tively e.g. ECDH =3D a0=C2=B7Bscan + a1=C2=B7Bscan + ... + at=C2=B7Bscan a= nd P =3D Bspend + hash(outpoints_hash=C2=B7ECDH || 0)=C2=B7G. Additionally= , it may be necessary for the participants to provide a DLEQ proof to ensu= re they are not acting maliciously. 13. Why not skip all taproot script path spends? This causes malleability= issues for CoinJoins. If the silent payments protocol skipped taproot scr= ipt path spends, this would allow an attacker to join a CoinJoin round, pa= rticipate in deriving the silent payment address using the tweaked private= key for a key path spend, and then broadcast their own version of the tra= nsaction using the script path spend. If the receiver were to only conside= r key path spends, they would skip the attacker's script path spend input = when deriving the shared secret and not be able to find the funds. Additio= nally, there may be scenarios where a sender has access to the key path pr= ivate key but spends the output using the script path. 14. Why skip outputs with H as the internal taproot key? If use cases get= popularized where the taproot key path cannot be used, these outputs can = still be included without getting in the way of making a silent payment, p= rovided they specifically use H as their internal taproot key. 15. Why do taproot private keys need to be checked? Recall from BIP340 th= at each x-only public key has two corresponding private keys, d and n - d.= To maintain parity between sender and receiver, it is necessary to use th= e private key corresponding to the even y-value when performing the ECDH s= tep since the receiver will assume the even y-value when summing the tapro= ot x-only public keys. 16. Why not re-use tn? when paying different labels to the same receiver?= If paying the same entity but to two separate labeled addresses in the sa= me transaction without incrementing n, the two outputs would be Bspend + t= n=C2=B7G + i=C2=B7G and Bspend + tn=C2=B7G + j=C2=B7G. The attacker could = subtract the two values and observe that the distance between i and j is s= mall. This would allow them to deduce that this transaction is a silent pa= yment transaction and that a single entity received two outputs, but won't= tell them who the entity is. 17. Why use BIP32 hardened derivation? Using BIP32 derivation allows user= s to add silent payments to an existing master seed. It also ensures that = a user's silent payment funds are recoverable in any wallet which supports= BIP32 derivation. Using hardened derivation ensures that it is safe to ex= port the scan private key without exposing the master key or spend private= key. 18. ^Why precompute labels? Precomputing the labels is not strictly neces= sary: a wallet could track the max number of labels it has used (call it M= ) and scan for labels by adding m=C2=B7G to P0 for each label m up to M an= d comparing to the transaction outputs. This is more performant than preco= mputing the labels and checking via subtraction in cases where the number = of eligible outputs exceeds the number of labels in use. In practice this = will mainly apply to users that choose never to use labels, or users that = use a single label for generating silent payment change outputs. If using = a large number of labels, the wallet would need to add all possible labels= to each output. This ends up being n=C2=B7M additions, where n is the num= ber of outputs in the transaction and M is the number of labels in the wal= let. By precomputing the labels, the wallet only needs to compute m=C2=B7G= once when creating the labeled address and can determine if a label was u= sed via a lookup, rather than adding each label to each output. 19. Data for Appendix A These numbers are based on data from January 2023= until June 2023 (the last 6 months of data at time time of writing). See = Silent payments light client data for the full analysis. 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