Learning new things!
Following the first post: Circle STARK: Understanding Circle Group’s Operation as Rotation
The goal of this post is to explain 2-adic FFT on circle domain.
In STARK, there are two cases where FFT is used:
The goal of this blog is to explain circle group without complex mathematical definitions. However, this approach is not rigorous, it serves more like a intuition of understanding the geometry of a circle group. Before diving into the blog, may you need some pre-readings like Exploring circle STARKs or Yet …
There are different ways to describe the GKR algorithm
from Libra paper:
\(\tilde{V}_i(g) = \sum\limits_{x,y \in \{0,1\}^{s_{i+1}}} f_i(x,y)\)
\(= \sum\limits_{x,y \in \{0,1\}^{s_{i+1}}} \left( \widetilde{add}_{i+1}(g,x,y)(\tilde{V}_{i+1}(x) + \tilde{V}_{i+1}(y))\right.\)
\(\left. + \tilde{mult}_{i+1}(g,x,y)(\tilde{V}_{i+1}(x)\tilde{V}_{i+1}(y)) \right)\)
from hyrax paper:
$$\widetilde{V_0}(q’,q)=\sum\limits_{h’\in \{0,1\}^{b_N}}\sum\limits_{h_L\in \{0,1\}^{b_G}}\sum\limits_{h_R\in \{0,1\}^{b_G}}P_{q’,q,1}(h’,h_L,h_R)$$
$$=\sum\limits_{h’\in \{0,1\}^{b_N}}\sum\limits_{h_L\in \{0,1\}^{b_G}}\sum\limits_{h_R\in \{0,1\}^{b_G}}\widetilde{eq_1}(q’,h’)$$
$$\cdot [\widetilde{mul_1}(q,h_L,h_R)(\widetilde{V_1}(h’,h_L)\times \widetilde{V_1}(h’,h_R) )$$
$$+\widetilde{add_1}(q,h_L,h_R)(\widetilde{V_1}(h’,h_L)+ \widetilde{V_1}(h’,h_R) )]$$…
Prover is \(\mathcal{O}(C)\)
Verifier is \(\mathcal{O}(d\log C)\) for d-depth log-space uniform circuit.
note: in sumcheck/GKR context, the size of the circuit usually is described as \(2^l\), namely, the inputs of the circuit is \(2^l\), for a linear prover in this setting, has complexity \(\mathcal{O}(2^l)\), a logarithmic verifier has \(\mathcal{O}(l)\) …
R1CS instance \(x=(\mathbb{F},A,B,C,io,m,n)\)
let \(Z=(io,1,w)\), then we have for example
\(\underbrace{\begin{bmatrix}0&1&1&0\\0&0&1&0\\…&…&…&..\\…&…&…&..\end{bmatrix}}_{A}\begin{pmatrix}w_1\\1\\w_2\\w_3\end{pmatrix}\circ \underbrace{\begin{bmatrix}0&1&0&0\\0&0&1&0\\…&…&…&..\\…&…&…&..\end{bmatrix}}_{B}\begin{pmatrix}w_1\\1\\w_2\\w_3\end{pmatrix}\)
\(=\underbrace{\begin{bmatrix}1&0&0&0\\0&0&0&1\\…&…&…&..\\…&…&…&..\end{bmatrix}}_{C}\begin{pmatrix}w_1\\1\\w_2\\w_3\end{pmatrix}\)
This express the constraints:
$$(1+w_2) \cdot 1=w_1$$
$$w_2 \cdot w_2=w_3$$
(here \(w_1\) is public input/output )
In order to use the sum-check protocol, we need to …
Using the following example to go through GKR protocol
this blogs follows the example in Spartan 预备知识：GKR with ZK Argument
zero knowledge version of GKR, Hyrax approach.
Take example circuit in figure 1, following the Arithmetization described in post, a step by step description of GKR arithmetication of a concrete example.
\(q’\in \{0,1\}^{b_N}\): The index of one of the \(N\) identical copies of the base circuit \(C_0\) within \(C\).…
To understand the attack on Nova, I figure out I need to understand the role of the hash function and the validation check of hash function in Nova, from the first paper where the “cycle of curves” is not introduced, the circuit is illustrated as
to …
After understanding of the folding scheme of relaxed R1CS, of which the key idea is you can “fold” two proofs to be one, with this ability of relaxed R1CS in mind, how can one build a recursive proof from scratch? in this note I will describe several attempts to build …