I need to point out that Dory requires pairing, and therefore it cannot 
work with secp256k1?
Please circle back.
On Monday, July 22, 2024 at 9:16:18 AM UTC-5 Weiji Guo wrote:

> Greetings, bitcoin developers. I am writing to update you about the OP_ZKP 
>
> proposal I initiated last year. This time, it concerns which ZKP scheme to 
> use in 
>
> the underlying proving system. This email will contain the following four 
> parts:
>
>
> 1, background, mainly about the initial proposal. You may skip it if you 
> already 
>
> know about it.
>
> 2, high-level requirements for the ZKP scheme and the current priority 
> regarding 
>
> selection. Also, a brief explanation of precluding trusted setup and 
> FRI-based 
>
> schemes. 
>
> 3, ideas and open issues regarding Inner Product Argument.
>
> 4, what-ifs
>
>
> I should have studied all these further before sending this email, but I 
> also want to 
>
> have something to talk about during Bitcoin Nashville. During the conf, 
> you may 
>
> find me in the K14 booth for f2f talks.
>
>
> ———Background———
>
>
> For those who haven't heard about this idea, here is the link to the 
> earlier email 
>
> thread: 
> https://lists.linuxfoundation.org/pipermail/bitcoin-dev/2023-April/021592.html
>
>
> Before proposing any BIP, we must explore existing ZKP schemes and discuss 
>
> how to use them with OP_ZKP within applications. That's why I took a 
> detour to 
>
> develop potential applications to understand further how OP_ZKP could work 
> in 
>
> real-world applications. That work concluded about two months ago; since 
> then, I 
>
> have been working on OP_ZKP.
>
>
> ———High level requirements———
>
>
> 1, Security. The security assumptions should be minimal. Ideally, only the 
> ECDLP 
>
> assumption is taken, leading directly to the Inner Product Argument over 
>
> secp256k1. However, an open issue still blocks IPA from working in OP_ZKP 
>
> (details will follow soon). We might consider pairing in a transparent 
> setup setting, 
>
> but no trusted setup.
>
>
> 2, Small block size consumption. The proof should be both succinct and 
>
> concretely small. Being concretely small allows individual or small 
> numbers of 
>
> proofs to be posted on-chain without incurring too many costs. Otherwise, 
> they 
>
> will have to wait to be verified in a batch, should the scheme allow such. 
> Arguably, 
>
> waiting for batching is not a good idea, as the transactions will have to 
> be put on 
>
> hold, and there is no guarantee more will come. That said, we might 
> revisit this 
>
> choice should we run out of candidate schemes. 
>
>
> FRI-based proofs are considered too big for 100- to 128-bit provable 
> security. The 
>
> size is a few hundred kilobytes for a circuit of 2^20 size. Besides, it 
> does not allow 
>
> batched verification (see the following requirement). 
>
>
> 3, Batched verification is mandatory due to block size considerations. 
> Should the 
>
> situation allow, we could replace the proof data (as transaction 
> witnesses) of 
>
> thousands of OP_ZKP transactions with a single argument to save block 
> space 
>
> (and transaction costs). 
>
>
> It also saves verification time, even without the benefits of saving block 
> space. 
>
>
> 4, Small verification key for deployment. It seems natural but is not the 
> default 
>
> case for some schemes. 
>
>
> 5, Aggregated proving. This is optional as it happens off-chain. However, 
> it 
>
> effectively reduces proof size and verification time requirements for some 
> non-
>
> constant size proof schemes (we precluded trusted setup). Consider 
> aggregating 
>
> many sub-circuits together with a constant or log-sized argument. 
>
>
> 2^16 is a reasonable upper bound for a sub-circuit. Computing a block hash 
>
> takes about 60k constraints (Sha256 applied twice against an 80-byte block 
>
> header).
>
>
> But 2^16 is still too large for FRI-based proof. Each FRI-query should 
> cost 16^2 
>
> hashes, which is 8 kilobytes. There are dozens of queries to run to meet 
> the target 
>
> security (preferably 128-bit, without further security conjectures). 
> Interested 
>
> readers may refer to 
> https://a16zcrypto.com/posts/article/17-misconceptions-
>
> about-snarks/#section--11. 
>
>
> ———Inner Product Argument———
>
>
> Now, let's consider IPA-based solutions. IPA has a transparent setup, 
> requires 
>
> only ECDLP assumption, can work on the secp256k1 curve, has a relatively 
> small 
>
> proof size, and is capable of both batched verification and aggregated 
> proving. 
>
> We can use the aggregated proving technique to address the issue of linear 
> time 
>
> verification. The remaining open issue with IPA is that the size of the 
> verification 
>
> key is linear to the circuit size. 
>
>
> Let me expand on the last two issues. 
>
>
> The linear verification time of IPA comes from the need to recompute the 
>
> Pedersen commitment base in each round of reduction (could be postponed 
> but 
>
> still O(N)). We could adopt the aggregated proving technique to address 
> this 
>
> issue and effectively reduce the complexity of the actual verification 
> time. 
>
> Suppose there are M sub-circuits; each has a size bound of N (assuming N = 
>
> 2^16). We could have an argument to combine the M inner products into one. 
> This 
>
> argument is also made of IPA, thus having O(log(M)) size and O(M) 
> verification 
>
> time. Then the total proof size is O(log(M) + log(N)), and verification 
> time 
>
> becomes O(M)+O(N) rather than O(M*N) (there will be some extra costs per 
>
> aggregated proof, but let's skip it for now). This is how we could use IPA 
> to 
>
> achieve succinctness and to deal with huge circuits (we need this as we 
> might 
>
> recursively verify proofs of other schemes). 
>
>
> Linear verification key comes from the need for the verifier to use all 
> the circuit 
>
> constants. It is at least accurate for BP, BP+, and even Halo, and it used 
> to be 
>
> acceptable as the multi-scalar multiplication dominates the verification 
> costs. 
>
> However, it becomes an issue with aggregated proving in terms of both 
>
> deployment size and verification time. When M is large enough, the 
> aggregated 
>
> verification time to compute with these constants is non-trivial, even 
> compared to 
>
> the MSM costs. The more significant issue is with the circuit deployment. 
> We have 
>
> to save all these parameters on-chain. 
>
>
> To further expand on this, let me re-iterate the Sonic Arithmetization 
> process. It is 
>
> R1CS-based with some pre-processing. Suppose there are N multiplication 
> gates 
>
> such that:
>
>       A_i * B_i = C_i 
>
> for i in [0 - N-1], A_i, B_i, and C_i as field elements, and A, B, and C 
> are circuit 
>
> witnesses of N-element vectors. 
>
>
> There are also Q linear constraints to capture the addition gates, circuit 
> wiring, 
>
> and multiplied-by-constant constraints:
>
>       WL*A + WR*B + WO*C = K
>
> where WL, WR, and WO are Q*N matrixes of field elements, and K is a 
> Q-element 
>
> vector of field elements. WL, WR, WO, and K are circuit constants. 
>
>
> Although WL, WR, and WO are vastly sparse, they are at least O(N) size. 
> The 
>
> verifier will have to compute on them during verification after generating 
> a 
>
> challenge value in response to the prover's commitment to the witnesses. 
> This 
>
> O(N) size prevents us from deploying verification keys on-chain. 
>
>
> The natural idea is to commit to those constants somehow and to offload 
> the 
>
> commitment opening to the prover. Although the verifier still pays O(N) 
> time to 
>
> verify, the deployment size is constant, and the opening size is only 
> O(log(N)) 
>
> instead of O(N), assuming an IPA opening. However, this no longer holds 
> with 
>
> aggregated proving, as there is no guarantee that the aggregated constants 
>
> aWL, aWR, and aWO will be sparse. The complexity could become O(N^2). 
>
>
> The open issue here is to find a way to commit to WL, WR, and WO such that 
> 1) 
>
> the verification key could be tiny, 2) the proof size to open the 
> commitment is 
>
> acceptable, and 3) the commitments could be aggregated. If necessary, 
> modify 
>
> the arithmetization further, ideally without introducing new security 
> assumptions. 
>
> -- If we do, for example, introduce SXDH, then we can consider schemes 
> like 
>
> Dory, which has a transparent setup and O(log(N)) verifier time. 
>
>
> ———What-ifs———
>
>
> What if the above open issue could be resolved? The resulting proving 
> system 
>
> should work even for a Raspberry Pi 4 node. I ran some benchmarks on 
> Raspberry 
>
> Pi 4 for some elliptical curve operations. The general conclusion is that 
> it is about 
>
> ten times slower than my Mac Book Pro for a single thread. For a circuit 
> of 2^16 
>
> size, the verification time with MBP is doubt 200~300 ms (inferred by 
>
> benchmarking MSM of this size); therefore, it would be about 2~3 seconds 
> for 
>
> RP4. The aggregation might double or triple the time cost. 
>
>
> Now, for block verification, counted in batched verification, it should be 
>
> acceptable for RP4 to spend around 10 seconds verifying ALL OP_ZKP 
>
> transactions in a new block. Luckily, we won't have too many existing 
> blocks full 
>
> of OP_ZKP transactions any time soon. 
>
>
> For transaction forwarding, a simple technique of pool-then-batched 
> verification 
>
> could work for RP4. As its name suggests, an RP4 node could simply pool 
> any 
>
> incoming OP_ZKP transactions in memory when it is already busy verifying 
> some. 
>
> When it is done, the node retrieves all the pooled OP_ZKP transactions and 
>
> verifies them in a batch. This way, it only adds a few seconds of latency 
> to 
>
> forwarding any OP_ZKP transactions. 
>
>
> What if the open issue cannot be resolved? We might consider Dory. It is 
>
> transparent, requires pairing, and has logarithmic proof size but 
> concretely larger 
>
> than Bulletproofs (but Torus-based optimization might help here by a 
> factor of 3; 
>
> check out here: https://youtu.be/i-uap69_1uw?t=4044 and 
> https://eprint.iacr.org/
>
> 2007/429.pdf ). It also has logarithmic verification time, and batched 
> verification 
>
> allows for saving block space and verification time. The community might 
> need to 
>
> accept the SXDH assumption.
>
>
> That's it. Thanks for reading. Any comments are welcome. 
>
> And again, see you in the K14 booth, Nashville.
>
>
> Regards,
>
> Weiji
>
>
>

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