Added the optimized version on top of this PR:
w0xlt@8d16914

For more context:
#1698 (comment)

Small supplementary update: I've created a corresponding Python implementation of the provided API functions based on secp256k1lab (https://github.com/theStack/secp256k1lab/blob/add_bip352_module_review_helper/src/secp256k1lab/bip352.py) (also linked in the PR description). The hope is that this makes reviewing this PR a bit easier by having a less noisy, "executable pseudo-code"-like description on what happens under the hood. The code passes the BIP352 test vectors and hence should be correct.

Added the optimized version on top of this PR: w0xlt@8d16914

For more context: #1698 (comment)

Thanks for rebasing on top of this PR, much appreciated! I will take a closer look within the next days.

Not providing prevouts_summary (de)serialization functionality yet in the API poses the risk that users try to do it anyway by treating the opaque object as "serialized". How to cope with that? Is adding a "don't do this" comment in API header sufficient?

Is there a reason for serializing prevouts_summary without light client functionality? If not, I think the don't do this comment is sufficient. Right now, in contrast to the docs of all other opaque objects, this is missing, however:

The exact representation of data inside the opaque data structures is implementation defined and not guaranteed to be portable between different platforms or versions.

@w0xlt, @jonasnick: Thanks for the reviews! I've addressed the suggested changes:

• in _recpient_scan_outputs: changed the type of k to uint32_t (comment above)
• in _recipient_create_label: added a scan key validity check (+added a test for that) (#1698 - comment)
• unified all mentions of "Silent Payments" to title case in the header API and example (#1698 - comment)
• fixed typo s/elemement/element/ (#1698 - review)
• in _recipient_scan_outputs: fixed comment in second label candidate (review above)
• extended the API header comment for the _prevouts_summary opaque data structure, to point out that the data structure is implementation defined (like docs of all other opaque structs) (comment above)

Nit: Not related to optimization, but the diff below removes some redundant public-key serialization code:

Given that this compressed-pubkey-serialization pattern shows up repeatedly also in other modules (ellswift, musig), I think it would make the most sense to add a general helper (e.g. in eckey{,_impl}.h), which could be done in an independent PR. I've opened issue #1773 to see if there is conceptual support for doing this.

Not providing prevouts_summary (de)serialization functionality yet in the API poses the risk that users try to do it anyway by treating the opaque object as "serialized". How to cope with that? Is adding a "don't do this" comment in API header sufficient?

Is there a reason for serializing prevouts_summary without light client functionality? If not, I think the don't do this comment is sufficient.

Good point, I can't think of a good reason for full nodes wanting to serialize prevouts_summary.

To address the open questions, I’ve reviewed the proposed changes by @w0xlt on 8d16914.

I'm going to focus more on the key aspects I extracted from the review and the merits of each change, rather on the big O improvement claims, because I didn't get that far.

These are multiple different changes rather than a single one, so to make the review easier I suggest to brake it in multiple commits. I would state on each of them the purpose and the real case scenario where the change would be relevant.

Also, I would use clearer names for the variables or at least document their purpose.

The changes I've identified so far are the following:

• Improve label lookup using hash table: I think this is implementation dependent and should be improved by the user rather than by the library itself.
  Examples, as part of the documentation, are a usage demonstration, although can point the user the best practices, I prefer clarity rather than performance on them. For example, on the rust-secp256k1 bindings, I used a HashMap for the example because it is a familiar standard structure for Rust users. If they would like to gain more performance there, they have other tools available to replace that structure by themselves.
• On secp256k1_silentpayments_recipient_sort_cmp, I understood the change: (r1->index < r2->index) ? -1 : (r1->index > r2->index) ? 1 : 0 is to make the unstable secp256k1_hsort implementation stable. Considering it only affects secp256k1_silentpayments_sender_create_outputs, which didn't receive more changes than a variable removal, what are the performance improvements there?
• SECP256K1_SP_SCAN_BATCH: I just learned about it (ge_set_gej_all) during this review, but seems that affine to jacobian conversion is needed for the comparison against labels, and is faster to do it in batches. I think the impact of this improvement will only affect the small subset of transaction with large amount of outputs. A good benchmark for this would be coinjoin transactions, although is not supported by BIP 352, a test case is doable.
• Double head: I'm not sure of the target of this change, I guess is to skip already found or scanned inputs, but couldn't figure out what is tracking each head.
• Binary tree search for x-only lookups: I think it explain itself, faster lookups on the not labeled case. This may have its merits, but I need to remove the other changes to have a clear answer.

In general I agree with @jonasnick that we should define a clear target to benchmark and improve. As I've said before, the base case should be a wallet with a single label for change.
For other improvements, I would try to match up plausible real world scenarios before making complex changes in the code base of the PR.
Finally, by looking at bench_silentpayments_full_tx_scan, the use_labels case is very simple. If we want to test performance improvements on the label lookup, I would start there.

In conclusion, from the proposed commit and the discussion around it, the only changes I've found clear enough to consider are:

• Tracking of found outputs: simple enough, small performance improvement for the usual case, better for larger transactions.

Thanks @nymius for reviewing the changes, addressing the main points, and proposing a simplification.

I’m currently splitting the optimization commit into smaller pieces to make it easier to review.
I’ll also take a closer look at your commit and run it against the benchmark files.

The only part of the discussion that still feels a bit ambiguous is the “base” or “usual” case.
From my understanding of #1698 (review), the concern is not about typical usage, but rather about an attacker crafting malicious transactions with many outputs, causing the scanning process to take hours.

So the goal of this optimization would be to mitigate that scenario, not the collaborative one.

I ran the examples/silentpayments_mixed_1.c file with simplified the version suggested by @nymius jonasnick@311b4eb . It shows slightly worse performance (see below), but the simpler approach may still be worth it

Without the secp256k1_silentpayments_recipient_sort_cmp stabilization, I got 82s for the complex version vs. 114s for the simpler one. Whether the 30s difference justifies the additional complexity is up for discussion — I don’t have a strong opinion.

Answering the questions: secp256k1_silentpayments_recipient_sort_cmp stabilization + heads speed up the non-adversarial case. The stable sort ensures that the transaction outputs are ordered sequentially by the index $k$.

The optimized receiver implementation relies on a heuristic (the head pointers) that assumes the next output it is looking for ($k+1$) is located immediately after the previous one ($k$).

• With Stable Sort: The scanner complexity is roughly O(N) (Linear).
• Without Stable Sort: The scanner complexity degrades to O(N²) (Quadratic).

@w0xlt, @nymius: Thanks for investigating this deeper. I've now also had a chance to look at the suggested optimizations and came to similar conclusions as stated in #1765 (comment). I particularly agree with the stated points that the changes should not increase complexity significantly and that the most important optimization candidate to consider for mitigating the worst-case scanning attack is "skip outputs that we have already found" (as previously stated by @jonasnick, see #1698 (comment) and jonasnick@311b4eb). I don't think stabilizing the sorting helps at all, since this is something that happens at the sender side, and we can't rely on the attacker using a specific implementation (even if they did, it's trivial for them to shuffle the outputs after creation).

For the proposed target to benchmark, I'm proposing the following modified example that exhibits the worst-case scanning time based on a labels cache with one entry (for change outputs), by creating a tx with 23255 outputs [1] all targeted for Bob: 1df4287

Shower-thought from this morning: what if we treat the tx_outputs input as actual list that we modify, and remove an entry if it is found? This would have a similar effect as the "track found outputs" idea, but without the need of dynamic memory allocation. It's a ~10-lines diff and seems to work fine: theStack@9aba470
It reduces the run-time of the proposed example above from ~10 minutes to roughly ~2 seconds on my machine.

Any thoughts on this? Maybe I'm still missing something.

[1] that's an upper bound of maximum outputs per block: floor(1000000/43) = 23255

Shower-thought from this morning: what if we treat the tx_outputs input as actual list that we modify, and remove an entry if it is found? This would have a similar effect as the "track found outputs" idea, but without the need of dynamic memory allocation. It's a ~10-lines diff and seems to work fine: theStack@9aba470 It reduces the run-time of the proposed example above from ~10 minutes to roughly ~2 seconds on my machine.

Any thoughts on this? Maybe I'm still missing something.

1df4287 is a good target.
These are the runtime times I've obtained testing against this target:

I had to increase stack size to be able to fit all N_OUTPUT size allocations in the example.

Initially I preferred the is_found allocation rather than the element shifts. But your solution seems to be more performant.

@theStack Yes — if we want to keep only the adversarial-scenario optimizations, we can drop sort stabilization and the extra heads.

I like your idea of avoiding dynamic memory allocation. That’s a very interesting direction. On my machine, the scan completes in about 0.4s, which feels like a good balance between simplicity and the optimization needed for the labeled case.

Below are the changes I had to make for your example to run on my machine and to record the scan time.

diff --git a/examples/silentpayments.c b/examples/silentpayments.c
index 5e71e73..d43332f 100644
--- a/examples/silentpayments.c
+++ b/examples/silentpayments.c
@@ -10,6 +10,7 @@
 #include <stdio.h>
 #include <stdlib.h>
 #include <string.h>
+#include <time.h>
 
 #include <secp256k1_extrakeys.h>
 #include <secp256k1_silentpayments.h>
@@ -112,15 +113,21 @@ const unsigned char* label_lookup(
     return NULL;
 }
 
+static secp256k1_xonly_pubkey tx_inputs[N_INPUTS];
+static const secp256k1_xonly_pubkey *tx_input_ptrs[N_INPUTS];
+static secp256k1_xonly_pubkey tx_outputs[N_OUTPUTS];
+static secp256k1_xonly_pubkey *tx_output_ptrs[N_OUTPUTS];
+static secp256k1_silentpayments_found_output found_outputs[N_OUTPUTS];
+static secp256k1_silentpayments_found_output *found_output_ptrs[N_OUTPUTS];
+static secp256k1_silentpayments_recipient recipients[N_OUTPUTS];
+static const secp256k1_silentpayments_recipient *recipient_ptrs[N_OUTPUTS];
+/* 2D array for holding multiple public key pairs. The second index, i.e., [2],
+ * is to represent the spend and scan public keys. */
+static unsigned char (*sp_addresses[N_OUTPUTS])[2][33];
+
 int main(void) {
     unsigned char randomize[32];
     unsigned char serialized_xonly[32];
-    secp256k1_xonly_pubkey tx_inputs[N_INPUTS];
-    const secp256k1_xonly_pubkey *tx_input_ptrs[N_INPUTS];
-    secp256k1_xonly_pubkey tx_outputs[N_OUTPUTS];
-    secp256k1_xonly_pubkey *tx_output_ptrs[N_OUTPUTS];
-    secp256k1_silentpayments_found_output found_outputs[N_OUTPUTS];
-    secp256k1_silentpayments_found_output *found_output_ptrs[N_OUTPUTS];
     secp256k1_silentpayments_prevouts_summary prevouts_summary;
     secp256k1_pubkey unlabeled_spend_pubkey;
     struct labels_cache bob_labels_cache;
@@ -209,11 +216,6 @@ int main(void) {
     {
         secp256k1_keypair sender_keypairs[N_INPUTS];
         const secp256k1_keypair *sender_keypair_ptrs[N_INPUTS];
-        secp256k1_silentpayments_recipient recipients[N_OUTPUTS];
-        const secp256k1_silentpayments_recipient *recipient_ptrs[N_OUTPUTS];
-        /* 2D array for holding multiple public key pairs. The second index, i.e., [2],
-         * is to represent the spend and scan public keys. */
-        unsigned char (*sp_addresses[N_OUTPUTS])[2][33];
         unsigned char seckey[32];
 
         printf("Sending...\n");
@@ -340,6 +342,9 @@ int main(void) {
              *        `secp256k1_silentpayments_recipient_prevouts_summary_create`
              *     2. Call `secp256k1_silentpayments_recipient_scan_outputs`
              */
+            clock_t start, end;
+            double cpu_time_used;
+
             ret = secp256k1_silentpayments_recipient_prevouts_summary_create(ctx,
                 &prevouts_summary,
                 smallest_outpoint,
@@ -356,14 +361,20 @@ int main(void) {
 
             /* Scan the transaction */
             n_found_outputs = 0;
+            
+            start = clock();
             ret = secp256k1_silentpayments_recipient_scan_outputs(ctx,
                 found_output_ptrs, &n_found_outputs,
-                (const secp256k1_xonly_pubkey * const *)tx_output_ptrs, N_OUTPUTS,
+                (const secp256k1_xonly_pubkey **)tx_output_ptrs, N_OUTPUTS,
                 bob_scan_key,
                 &prevouts_summary,
                 &unlabeled_spend_pubkey,
                 label_lookup, &bob_labels_cache /* NULL, NULL for no labels */
             );
+            end = clock();
+            cpu_time_used = ((double) (end - start)) / CLOCKS_PER_SEC;
+            printf("Bob's scan took %f seconds\n", cpu_time_used);
+            
             if (!ret) {
                 printf("This transaction is not valid for Silent Payments, skipping.\n");
                 return EXIT_SUCCESS;
@@ -435,7 +446,7 @@ int main(void) {
             n_found_outputs = 0;
             ret = secp256k1_silentpayments_recipient_scan_outputs(ctx,
                 found_output_ptrs, &n_found_outputs,
-                (const secp256k1_xonly_pubkey * const *)tx_output_ptrs, 1, /* dummy scan with one output (we only care about Bob) */
+                (const secp256k1_xonly_pubkey **)tx_output_ptrs, 1, /* dummy scan with one output (we only care about Bob) */
                 carol_scan_key,
                 &prevouts_summary,
                 &unlabeled_spend_pubkey,

@nymius, @w0xlt: Thanks once again for the quick feedback and for benchmarking! Shortly after my previous comment, I've been notified about yet another approach to tackle the worst-case scanning time attack (kudos to @furszy for bringing up the idea!), that I think is even more elegant: we can use the pointers in the tx_outputs list directly to track outputs by setting them to NULL if one has been found, and accordingly only treat them if they are non-NULL. With this, it's an only four lines of code change: theStack@2087f92. It kind of combines the previous two approaches of jonasnick@311b4eb and theStack@9aba470 (-> mark spent outputs, but not in a newly allocated array, but by modifying the tx_outputs input list, in order to avoid dynamic memory allocation), with very similar run-time results.

The only tiny drawback about these non-malloc approaches might be that something that is conceptually an "in" parameter is modified, which might be a bit unsound in a strict API design sense. On the other hand, it shouldn't matter for the user (I doubt that these lists passed in would ever be reused for anything else after by the callers), and we already do the same in the sending API for the recipients, so it's probably fine.

@theStack Yes — if we want to keep only the adversarial-scenario optimizations, we can drop sort stabilization and the extra heads.

The way I see it currently, code paths for non-adversarial scenarios with increasing k values would be hit so rarely in practice, that I'm sceptical that it's worth it put much effort into those optimizations. When scanning, the vast majority of transactions won't have any matches in the first place. Out of those few that do have a match, the vast majority will very likely again not contain any repeated recipient (IMHO it doesn't make that much sense to do that, unless the recipient explicitly asks "I want to receive my payment split up in multiple UTXOs, but still in a single tx"?), so in the bigger picture those optimizations wouldn't matter all that much, and I'd assume that the dominant factor should be by far all the (unavoidable) ECDH computations per transaction. But that's still more of a guess and it's still good to already have optimization ideas at hand if we need them in the future.

@theStack Thanks for continuing to refine the optimization. The deletion approach performs slightly better (0.40 s vs. 0.45 s), likely because deleting items shrinks the array and cuts the number of loop iterations by about 50% compared to nullifying them.

To summarize, the following table shows the proposed mitigations for the worst-case scanning attack so far, with benchmark results from my machine. The previous baseline commit with the worst-case example has been updated to include @w0xlt's changes, in order to work without stack size limit changes.
(EDIT: These benchmark results are based on a baseline that doesn't represent the worst-case and are thus not representative, as noticed by w0xlt below.)

The run-times of the fixes vary slightly (the removal approach "fix2" being the fastest, confirming #1765 (comment) above), but are all in the same ballpark. I don't think exact performance results matter much here, as the goal of the mitigation should be to IMHO roughly cut the run-time down from "minutes" to "seconds" (and remember, this is already for the absolute worst-case, one giant non-standard transaction filling out a whole block, and it can only slow down one specific SP recipient). Thus, I decided to pick the the simplest approach that avoids dynamic memory allocation, i.e. fix number 3 using NULL as marker in tx_outputs.

With that tackled, I believe that all of the open questions and TODOs are addressed now (updated the PR description accordingly). The latest force-push also includes a rebase on master (to include the CI fix #1771).

re-ACK 459a8f9

Rebased on master, in order to resolve a small merge conflict in the constant-time tests caused by #1865.

Also, since #1859 has now been merged, I took the opportunity to re-run the scanning benchmarks and updated the commit body e2ead23 with the results. As expected, the mul/sqr inlining sped up these scenarios noticeably as well (>7% on my machine 🚀 ), which is nicely visible in the range-diff (see git range-diff 459a8f9...eef27eb).

Thanks for the quick re-reviews, and sorry for force-pushing once again!

re-ACK eef27eb

Only benchmarks in the commit message of e2ead23 were changed since last ACK.

reACK eef27eb

A logistical note: I'm aware of all the ACKs. I plan to give this a hopefully (final) look very soon, so we can merge it.

Rebased on master and force-pushed with a fix in the "tests: add BIP-352 test vectors" commit to ensure that all test vectors with their entire subtest lists are exercised, see #1765 (comment) (thanks to @fjahr for pointing out this flaw).

The fix as a single commit can be found here (in case reviewers prefer that to re-review over git range-diff eef27eb...4aa1670)): theStack@f1f2909

reACK 4aa1670

Great find regarding the test vectors. Also very happy that running the full subtest lists did not expose any failures :)