First off: huge thanks to all the organizers and participants at ZK Hack & ZK Summit! Both events were fantastic; I'm already looking forward to next time.

Now that the jetlag is behind me, I wanted to share some zero-knowledge research threads that I'm excited to keep thinking about and talking about.

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This is perhaps the most important emerging idea in zk research right now. These schemes achieve Incrementally Verifiable Computation (IVC) by *combining instances that haven't been proven*. This contrasts with the more familiar approach of proving each instance and then rolling up the results.

Nova introduced a technique for combining R1CS instances. Supernova generalized Nova to support arbitrary instruction sets. Sangria uses the same technique in order to combine (relaxed) PLONK instances.

A natural research question arises: can we apply this approach to AIR-FRI STARKs? The answer isn't obvious, as this technique depends on a homomorphic encryption scheme such as KZG.

This idea, proposed by the team behind TritonVM, aims to reduce the verification time (and thus, the recursion time) in a STARK system by shifting work from the Verifier to the Prover. Specifically, the Prover can generate a SNARK proof in order to alleviate the need for the Verifier to evaluate the AIR constraints at the DEEP query point.

Some research questions:

- How does the computational cost of the inner SNARK verification compare to the cost of the AIR evaluation that it aims to replace?
- What commitment scheme should be used for this inner SNARK?
- What's the right scope for the SNARK verification? Should it include arithmetic or hashing operations in addition to the AIR constraint evaluation?

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In contexts where the Prover is repeatedly doing the same lookup, LogUp reduces the number of columns necessary in lookup tables by ~50%, relative to PLOOKUP. About 30% of the witness-generation in RISC Zero is spent generating columns for a PLOOKUP-based lookup argument; we're looking into switching to LogUp for future versions.

The premise here is that by making use of logarithmic derivatives, we can transform a grand product accumulation into a (cheaper) grand sum accumulation. The LogUp paper (linked above) frames this technique in terms of a multivariate IOP; the Tip5 paper describes a version of this technique that's better suited to the univariate context.

In Denver last month, Daniel Lubarov discussed the idea of using the finite field of order 2^{31} - 1 for building Plonky3. This field is particularly nice for handling finite field arithmetic, since 2^{31} = 1. The major obstacle is that the multiplicative group of this field isn't good for NTTs, since 2^{31} - 2 doesn't have many 2s in its prime factorization.

There are a number of options for how to make this work:

- changing from a FRI-based commitment scheme to one based on Orion or Brakedown
- using a mixed-radix version of FRI that doesn't depend on high 2-adicity
- doing NTTs in a quadratic extension field (note that the multiplicative group of the quadratic extension has order 2
^{62}- 2^{32}and therefore has 2-adicity of 2^{32})

Looking forward to seeing what the Polygon Zero team is brewing up on this front.

We'd love to hear what you're thinking about, in regards to these questions or otherwise. Continue the conversation on Discord at ZKHack or RISC Zero.