From fe887a3131eb806cc4c3597331fc70bb61a60eb7 Mon Sep 17 00:00:00 2001 From: Insun35 Date: Sat, 20 Jul 2024 00:48:17 +0900 Subject: [PATCH 1/3] add ZKP and SNARK overview --- .../Zero Knowledge Proofs/ZK-SNARK.md | 71 ++++++++++++++++++ .../Zero Knowledge Proofs.md | 52 +++++++++++++ .../Zero Knowledge Proofs/images/snark(1).png | Bin 0 -> 15994 bytes .../Zero Knowledge Proofs/images/snark(2).png | Bin 0 -> 407144 bytes .../Zero Knowledge Proofs/images/zkp(1).png | Bin 0 -> 91096 bytes .../Zero Knowledge Proofs/images/zkp(2).png | Bin 0 -> 90836 bytes .../Zero Knowledge Proofs/images/zkp(3).png | Bin 0 -> 87114 bytes 7 files changed, 123 insertions(+) create mode 100644 content/Privacy Enhancing Technologies (PET)/Zero Knowledge Proofs/ZK-SNARK.md create mode 100644 content/Privacy Enhancing Technologies (PET)/Zero Knowledge Proofs/Zero Knowledge Proofs.md create mode 100644 content/Privacy Enhancing Technologies (PET)/Zero Knowledge Proofs/images/snark(1).png create mode 100644 content/Privacy Enhancing Technologies (PET)/Zero Knowledge Proofs/images/snark(2).png create mode 100644 content/Privacy Enhancing Technologies (PET)/Zero Knowledge Proofs/images/zkp(1).png create mode 100644 content/Privacy Enhancing Technologies (PET)/Zero Knowledge Proofs/images/zkp(2).png create mode 100644 content/Privacy Enhancing Technologies (PET)/Zero Knowledge Proofs/images/zkp(3).png diff --git a/content/Privacy Enhancing Technologies (PET)/Zero Knowledge Proofs/ZK-SNARK.md b/content/Privacy Enhancing Technologies (PET)/Zero Knowledge Proofs/ZK-SNARK.md new file mode 100644 index 0000000..807551d --- /dev/null +++ b/content/Privacy Enhancing Technologies (PET)/Zero Knowledge Proofs/ZK-SNARK.md @@ -0,0 +1,71 @@ +## Needs of the SNARK + +90년대에 성능이 느린 개인 컴퓨터 대신 성능이 좋은 슈퍼컴퓨터가 연산을 처리하도록 위임해야했는데, 이때 슈퍼컴퓨터가 올바른 연산을 수행했는지 검증하기 위해 proof system에 대한 필요성이 언급되었다. 하지만 개인 컴퓨터 능력의 향상으로 인해 연산을 위임할 필요성이 없어졌다. + +현재 다시 proving system이 주목받는 이유는 이더리움 L1과같은 연산이 느린 네트워크의 연산을 unreliable한 GPU에 연산을 위임할 때 해당 연산의 유효성을 판단하기 위함이다. + +이때 데이터의 익명성을 보장하는 것은 ZK의 역할이고, 빠르고 효율적으로 검증하는 것이 SNARK의 역할이다. + +zk-SNARK use cases + +- Scalability: Outsourcing computation → 여러 트렌젝션들을 오프체인에서 연산한 결과를 succinct한 proof로 전달하면 L1은 해당 proof를 검증하는 것으로 비용을 줄일 수 있다.(zkRollup) +- Privacy: Private transactions on public blockchain(Tornado cash, Zcash, Iron Fish, Aleo) + +## Arithmetic circuit +![[snark(1).png]] +**Arithmetic circuit :** **directed acyclic graph(DAG) where internal nodes(gates) have operators(+,-,*) with inputs** + +The arithmetic circuit can do what polynomial time computations can do + +→ Fix a finite field $𝔽 = \{0,...,p-1\}$ for some prime p>2. + +→ |C| = # gates in C + +Type of Circuit + +- Unstructured circuit: a circuit with arbitrary wires +- Structured circuit: input → M → M → … → M → output (M is often called a virtual machine, VM - one step of a processor) + +## Completeness and Soundness + +- Completeness: Proof가 참일 때 Verifier가 참이라는 것을 확신할 수 있는 성질 +- Soundness: Prover가 거짓 Proof를 생성하려고 시도할 때 유효한 proof를 생성할 수 없도록 하는 성질 → Extractor + +## NARK: Non-interactive Argument of Knowledge +Before making the proof succinct, we gotta make it non-interactive. Basically a NARK is applied to an arithmetic circuit C +![[snark(2).png]] +### Preprocessing NARK + +$$ (S, P, V) $$ + +Public arithmetic circuit: + +$$ C(x,w) \rightarrow 𝔽 $$ + +- $x:$ Public statement in $𝔽^n$ +- $w:$ Secrete witness in $𝔽^n$ + +**Preprocessing(setup):** + +$$ S(C) \rightarrow (pp, vp) $$ + +- Input: Description of the circuit +- Output: Public parameters for prover and verifier + +**Proving Algorithm → proof:** + +$$ P(pp,x,w) \rightarrow \pi $$ + +**Verifying Algorithm → accept or reject:** + +$$ V(vp, x, \pi) \rightarrow Boolean $$ + +## Types of preprocessing Setup + +- Trusted setup per circuit: $S(C; r)$ random $r$ must be kept secret from prover + +- Trusted but universal (updatable) setup: secrete $r$ is independent of $C$ + + $$ S = (S_{init}, S_{index}): \; S_{init}(\lambda;r) \rightarrow gp, \; S_{index}(gp, C) \rightarrow (pp, vp) $$ + +- Transparent setup: $S(C)$ does not use secrete data (no trusted setup) diff --git a/content/Privacy Enhancing Technologies (PET)/Zero Knowledge Proofs/Zero Knowledge Proofs.md b/content/Privacy Enhancing Technologies (PET)/Zero Knowledge Proofs/Zero Knowledge Proofs.md new file mode 100644 index 0000000..f84de48 --- /dev/null +++ b/content/Privacy Enhancing Technologies (PET)/Zero Knowledge Proofs/Zero Knowledge Proofs.md @@ -0,0 +1,52 @@ +## What is Zero-Knowledge Proof? + +영지식증명(Zero-Knowledge Proof)이란 특정 명제가 참/거짓이라는 사실을 증명할 때, 어떠한 정보도 노출하지 않고 그 명제를 증명하는 방법이다. + +다음 예시는 Alice가 Bob에게 자신의 두 개의 공의 색이 다르다는 것을 ZKP를 통해 증명하는 과정이다. +![[zkp(1).png]] +Alice는 빨간색과 초록색의 공을 가지고있지만 Bob은 적록색맹이라 두 공의 색을 구분하지 못한다. 이때, Alice는 두 공의 색깔 정보를 밥에게 노출시키지 않고 오직 “두 공의 색이 다르다”라는 명제만 증명하려고 한다.![[zkp(2).png]] +Bob은 빨간색공과 초록색공을 각각 오른손과 왼손에 들고 Alice가 볼 수 없는 방으로 가지고 들어가서, 그 공의 위치를 바꾼다. + +![[zkp(3).png]] + +그리고 Bob은 Alice에게 두 공의 위치가 바뀌었는지 물어본다. 그러면 Alice는 공의 색이 서로 달라져있으므로 쉽게 바뀌었다고 답할 수 있다. 그러나 Bob의 입장에서는 여전히 Alice가 의심스러울 수 있다. 왜냐하면 Bob은 공의 위치를 바꾸든지 안바꾸든지 두가지의 경우밖에 없기 때문에, Alice가 공의 위치가 바뀌었는지 맞출 수 있는 확률은 50%이기 때문이다. + +그런데 Bob이 이러한 시행을 10번을 시행하면 Alice가 10번 모두 맞출확률은 $1/2^{10}$로 0.1% 이하의 확률로 줄어들게된다. 20번정도만 시행해도 그것을 모두 맞추는 확률은 백만분의 일 이하로, 사실상 거의 불가능하다고 할 수 있다. 이런 방식으로 Alice는 Bob에게 두 공의 색깔을 알리지 않고 자신이 가진 두개의 공이 서로 다른 색깔이라는 사실을 증명할 수 있다. + +>용어정리 +>- Secrete: 노출되지않아야하는 정보(공의 색깔정보: 초록, 빨강) +>- Prover: Secrete을 알고있다고 증명하려는 사람(Alice) +>- Verifier: Prover가 Secrete을 알고있는지 확인하려는 사람(Bob) +>- Challenge: Prover가 Secrete을 알고있는지 검증하기 위해 Verifier가 내는 문제(공 바꾸기) + +## Needs of the SNARK + +90년대에 성능이 느린 개인 컴퓨터 대신 성능이 좋은 슈퍼컴퓨터가 연산을 처리하도록 위임해야했는데, 이때 슈퍼컴퓨터가 올바른 연산을 수행했는지 검증하기 위해 proof system에 대한 필요성이 언급되었다. 하지만 개인 컴퓨터 능력의 향상으로 인해 연산을 위임할 필요성이 없어졌다. + +현재 다시 proving system이 주목받는 이유는 이더리움 L1과같은 연산이 느린 네트워크의 연산을 unreliable한 GPU에 연산을 위임할 때 해당 연산의 유효성을 판단하기 위함이다. + +이때 데이터의 익명성을 보장하는 것을 ZKness 라고 하고, 빠르고 효율적으로 검증할 수 있는 성질을 Succinct라고 한다. + +zkSNARK use cases + +- Scalability: Outsourcing computation → 여러 트렌젝션들을 오프체인에서 연산한 결과를 succinct한 proof로 전달하면 L1은 해당 proof를 검증하는 것으로 비용을 줄일 수 있다.(zkRollup) +- Privacy: Private transactions on public blockchain(Tornado cash, Zcash, Iron Fish, Aleo) +- Compliance: Espresso, Raposa + +## 영지식증명의 활용 + +영지식증명은 여러 분야에서 사용할 수 있지만 현재 가장 활발하게 활용되는 분야는 블록체인이고 그 중에서 블록체인에서의 Privacy, 그리고 그것들의 Scaling에 많이 사용되고 있다. + +영지식증명을 Privacy에 적용하는 예시로는 Membership proof가 있는데, Merkle Tree에 등록되어있는 멤버를 leaf node의 정보를 제공하지 않고 해당 머클트리의 멤버임을 증명할 수 있다. 가장 대표적인 프로토콜로는 Semaphore가 있고 Merkle proof를 이용해 private transaction을 할 수 있는 Zcash 등이 있다. + +한편, Scaling에 영지식증명을 활용하는 대표적인 예로는 Rollup이 있다. 예를들어 ECDSA로 서명을 하는 블록체인에 zk-Rollup을 사용하면 일일히 서명을 검증할 필요 없이 블록단위로 한번에 검증이 가능하기 때문에 컴퓨팅 자원이 훨씬 적게 소모되고 연산도 빨라진다. + +이러한 방식으로 영지식증명을 활용해 여러 분야, 특히 블록체인에서 프라이버시를 더욱 안전하게 지키고 스케일업을 할 수 있다. + +### Index +- [[ZK-SNARK]] +- [[Quadratic Arithmetic Program]] +- [[Polynomial theorem]] +- [[Pinocchio protocol]] +- [[Groth16]] +- [[zkEVM]] \ No newline at end of file diff --git a/content/Privacy Enhancing Technologies (PET)/Zero Knowledge Proofs/images/snark(1).png b/content/Privacy Enhancing Technologies (PET)/Zero Knowledge 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z0?opla0~;Gcn5NkX!d4ja!w>tlKMa$@$k<+{uZ-11BrICBv|i6Vh~Awly|S5;6Rv@ z!$>qPQmxNPM9w?PlX4XA&;F^%-x}3aCDEdnfNUobqb^=t=ONdlIl+i1^DbRnPe;0o z)1E|BV`hfM1QbyI+h+dbp^rqLHNgf6<|0v(p>4O&cYNG!sch+QrBw-`Zx3h%S-<$ z?#U~3=`Zlh!)effqs(WN^V+~?fl67o+!x!wNhRNhTcd8k(E9bK2Mak}|LX zp%j|26t{KE#+M&XKBh(6dL7y~!W52(ki+lP%(CgqHV?DdVZ4lJi(?&}by) zamR)~&?j&;5nsJb#N(UL{zq1a5N~g~8Tr6jirVPglJYnu`r{Mqz#)-!mC2M11^>MH zapIC?R;-JC(>zv$I{j{7*P@U35fRMmp_n2jbr=Ww;kvPa4E@^MRn Date: Sat, 20 Jul 2024 00:48:53 +0900 Subject: [PATCH 2/3] add ZK-SNARK protocols --- .../Zero Knowledge Proofs/Groth16.md | 58 +++++++++ .../Pinocchio protocol.md | 82 ++++++++++++ .../Polynomial theorem.md | 48 +++++++ .../Quadratic Arithmetic Program.md | 121 ++++++++++++++++++ 4 files changed, 309 insertions(+) create mode 100644 content/Privacy Enhancing Technologies (PET)/Zero Knowledge Proofs/Groth16.md create mode 100644 content/Privacy Enhancing Technologies (PET)/Zero Knowledge Proofs/Pinocchio protocol.md create mode 100644 content/Privacy Enhancing Technologies (PET)/Zero Knowledge Proofs/Polynomial theorem.md create mode 100644 content/Privacy Enhancing Technologies (PET)/Zero Knowledge Proofs/Quadratic Arithmetic Program.md diff --git a/content/Privacy Enhancing Technologies (PET)/Zero Knowledge Proofs/Groth16.md b/content/Privacy Enhancing Technologies (PET)/Zero Knowledge Proofs/Groth16.md new file mode 100644 index 0000000..6dc59e7 --- /dev/null +++ b/content/Privacy Enhancing Technologies (PET)/Zero Knowledge Proofs/Groth16.md @@ -0,0 +1,58 @@ +## Intro + +_**Groth16**_은 Pinocchio protocol에 비해 성능을 향상시킨 시스템이다. [[Pinocchio protocol]]에 비해 훨씬 적은 수의 페어링으로도 검증이 가능하고 CRS도 더 짧다. 다만 두번의 Trusted setup을 해야한다는 치명적인 단점이 있다. + +## CRS + +- Toxic waste: $\alpha, \beta, \gamma, \delta, \tau \in F_p$ +- $Q(x) = A(x)*B(x)-C(x) =H(x)*Z(x)$ → R1CS로부터 생성된 QAP polynomial Q(x) +- $n$개의 constraints가 있고, $Z(x) = (x-1)(x-2)...(x-n)$ 이다. +- computation에 $m$개의 variable을 사용하고, 그중 public input은 $l$개이다. +- $L_i(x) = \beta *A_i(x) + \alpha *B_i(x) + C_i(x)$ + +**Proving Key:** + +$G_1$ elements: $g_1, g_1^\alpha, g_1^\beta, g_1^\delta, \{g_1^{\tau^i},g_1^{\alpha\tau^i},g_1^{\beta\tau^i}\}_{i=0}^{n-1}, \{g_1^{L_i(\tau)/\delta}\}_{i=l+1}^m, \{g^{\tau^iZ(x)/\delta}\}_{i=0}^{n-2}$ + +$G_2$ elements: $g_2, g_2^\beta, g_2^\delta, \{g_2^{\tau^i}\}_{i=0}^{n-1}$ + +**Verification Key:** + +$G_1$ elements: $g_1, \{g_1^{L_i(\tau)/\gamma}\}_0^l$ + +$G_2$ elements: $g_2, g_2^\gamma, g_2^\delta$ + +$G_T$ element: $g_1^\alpha * g_2^\beta$ + +## Proof Generation + +- witness vector $w = (1, w_1,...,w_m)$ +- 랜덤값 $r,s$ +- Proof는 세개의 점 $A, B, C$로 이루어진다. + +※ 아래 식에서 아래첨자 1 또는 2는 $G_1, G_2$를 의미한다. (e.g. $\alpha_1 = \alpha * g_1$) + +$A$는 $G_1$ 위의 점이고 다음식을 만족한다. + +$A_1 = \alpha_1 + w_0*A_0(\tau)_1 + w_1*A_1(\tau)_1+ ...+ w_m*A_m(\tau)_1+r*\delta_1$ + +$B$는 $G_2$ 위의 점이고 다음식을 만족한다. + +$B_2 = \beta_2 + w_0*B_0(\tau)_2 + w_1*B_1(\tau)_2 + ... + w_m*B_m(\tau)_2 +s*\delta_2$ + +$C$는 $G_1$ 위의 점이고 다음식을 만족한다. + +$C_1 = w_{l+1}*(L_{l+1}(\tau)/\delta)_1 +...+ w_m*(L_m(\tau)/\delta)_1 + H(\tau)*(Z(\tau)/\delta)_1 + s*A_1 + r*B_1-r_s*\delta_1$ + +## Verification + +아래 식이 성립하는지 확인함으로써 위에서 생성한 Proof를 검증할 수 있다. + +$A_1*B_2 = \alpha_1 * \beta_2 + (w_0*L_0(\tau)/\gamma+...+ w_l*L_l(\tau)/\gamma)_1 * \gamma_2 + C_1 *\delta_2$ + +이 식에서 $\alpha_1*\beta_2$는 이미 verification key 로 주어져있으므로 3개의 paring만 연산하면 된다. + +- 좌변 $A_B = (\alpha+A(\tau) +r*\delta)*(\beta+B(\tau)+s*\delta) = A(\tau)*B(\tau) + \alpha*\beta+\alpha*B(\tau)+ \beta*A(\tau) + s*\alpha*\delta+s*A(\tau)*\delta+ r*\beta*\delta+ r*B(\tau)*\delta +s*r*\delta*\delta$ +- 우변 $\alpha*\beta + L(\tau) + H(\tau)*Z(\tau) + s*\alpha*\delta + s*A(\tau)*\delta + s*r*\delta*\delta+ r*\beta*\delta + r*B(\tau)*\delta +s*_*r*_*\delta*\delta -s*_*r*_*\delta*\delta \\ = α*_*β + β*_*A(τ) + α*_*B(τ) + C(τ) + H(τ)*_*Z(τ) + s*_*α*_*δ + s*_*A(τ)*_*δ + s*_*r*_*δ *_*δ + r*_*β*_*δ + r*_*B(τ)* δ \\ = C(τ) + H(τ)*_*Z(τ) + α*_*β + α *_*B(τ) + β*_*A(τ) + s*_*α*_*δ + s*_*A(τ)*_*δ + r*_*β*_*δ + r*_*B(τ)*_* δ + s*_*r*_*δ *δ$ + +위 좌변과 우변을 전개한 결과에서 공통 인수들을 소거해보면 $A(\tau)*B(\tau) = C(\tau) +H(\tau)*Z(\tau)$ 임을 알 수 있고, 결론적으로 증명하고자 했던 식을 검증할 수 있게 된다. diff --git a/content/Privacy Enhancing Technologies (PET)/Zero Knowledge Proofs/Pinocchio protocol.md b/content/Privacy Enhancing Technologies (PET)/Zero Knowledge Proofs/Pinocchio protocol.md new file mode 100644 index 0000000..4a04095 --- /dev/null +++ b/content/Privacy Enhancing Technologies (PET)/Zero Knowledge Proofs/Pinocchio protocol.md @@ -0,0 +1,82 @@ +## Intro +이 아티클은 [[ZK-SNARK]] 시스템 중 가장 먼저 실용화된 **Pinocchio protocol**에서 proof를 생성하고 검증하는 기본적인 과정을 서술하였다. + +## Homomorphic Hidings + +영지식증명 시스템에서 Prover가 증명하는 것은 보통 특정 다항식을 알고있다는 것이다. 다만, 알고있다는 사실 외에 다항식에 대한 어떠한 정보도 노출되면 안된다. + + zk-SNARK에서 다항식을 이용하는 이유 [[Polynomial theorem#2. Schwartz-Zippel Lemma]]에 의해 최대 차수가 $d$인 서로 다른 두 다항식의 교점은 최대 $d$개이다. 즉, Prover와 Verifier가 서로 다른 다항식을 가지고 있을 때 두 다항식이 겹치는 점은 최대 $d$개인 것이다. 예를들어 최대차수가 $d$인 두 다항식의 정의역의 범위가 $1**Lagrange Interpolation :** For $x_i, y_i$ with $i \in [0, n)$ and $x_i \ne x_j \; \forall \; i\ne j$, there is a unique polynomial $f$ of degree less than $n$ such that $f(x_i) = y_i \; \forall \; i \in [0,n)$ +> +→ In general case, takes $O(n\log^2n)$ via Divide & Conqure + FFT + +Let’s make Interpolated polynomials of each variables: + +$$ U(x_i) = u_i,\; V(x_i) = v_i,\; W(x_i) = w_i $$ + +In $𝔽_p$, we can use the case of $x_i = \omega^i, \; \omega^n = 1$. (→ $O(n\log n)$ via FFT) + +We can make each polynomials $U, V,W$: + +$$ U(\omega^i)= u_i,\; V(\omega^i)= u_i,\; W(\omega^i)= w_i $$ + +Now, what we gotta prove is : $U(\omega^i)*V(\omega^i)-W(\omega^i) = (x^n-1)Q(x)$ + +$\because U(\omega^i)*V(\omega^i) = W(\omega^i)$ +$U(\omega^i)*V(\omega^i)-W(\omega^i) \equiv 0 \mod{ (x-\omega^0)(x-\omega^1)...(x-\omega^{n-1})}$ +$U(\omega^i)*V(\omega^i)-W(\omega^i) \equiv 0 \mod{(x^n-1)}$ +$U(\omega^i)*V(\omega^i)-W(\omega^i) = (x^n-1)Q(x)$ + +### 2. Schwartz-Zippel Lemma +$P \in 𝔽[x_1,...,x_n]$ is a non-zero polynomial with total degree $d$ over finite field $𝔽$. If $r_1,r_2, ..., r_n$ are i.i.d. randomly selected from $𝔽$, then +$$ P_r[P(r_1,r_2,...,r_n) = 0] \le {d\over |𝔽|} $$ + +Therefore, it suffices to select random $t \in 𝔽$ and check : +$$ U(t)*V(t)-W(t) = (t^n-1)Q(t) $$ +By the way, we don’t want interactions between the prover and verifier. So, the prover gotta give the $t$ value with polynomials at once, but the prover may modify the equation which can hold the $t$ value even if it is not the identical equation. + +The verifier can convince when the prover make the $t$ value with the hash of the polynomials. +→ The probability of cheating with hash value is negligible : Fiat-Shamir heuristic + +### 3. Polynomial Commitment Schemes (PCS) + +- Commit: $f \rightarrow C(f)$ (binding and hiding, so that the verifier cannot know the exact $f$ and prover cannot change the $f$) +- Prove: Generate proof $\pi$ such that $f(x) = y$ +- Verify: Verify $f(x) =y$ given $x,y,C(f), \pi$. + +### 4. Adding ZK + +Adding random points to interpolate $U,V,W$ on, or adding random multiples of $x^n-1$ works in PLONKish. It depends on which SNARK system to use. \ No newline at end of file diff --git a/content/Privacy Enhancing Technologies (PET)/Zero Knowledge Proofs/Quadratic Arithmetic Program.md b/content/Privacy Enhancing Technologies (PET)/Zero Knowledge Proofs/Quadratic Arithmetic Program.md new file mode 100644 index 0000000..9cc6af1 --- /dev/null +++ b/content/Privacy Enhancing Technologies (PET)/Zero Knowledge Proofs/Quadratic Arithmetic Program.md @@ -0,0 +1,121 @@ +[[ZK-SNARK]]는 특정 다항식을 알고있음을 비대화형 영지식증명방식으로 증명하는 것인데, 증명 과정에서 Quadratic Arithmetic Program을 사용해 증명에 필요한 제약사항들을 생성할 수 있다. + +## Computation + +우선 증명하려는 computational problem을 정의해야하는데, 다음과 같은 문제를 예시로 들어보자. +$$x^3 + x + 5 = 35$$ +Prover가 이 문제의 정답을 알고 있다는 것을 증명한다고 하자. (참고로 이 문제의 정답은 3으로, 3차방정식을 풀 수 있으면 쉽게 답을 구할 수 있는 문제이지만 이해를 위해 간단한 예시를 이용한다.) 이 문제를 코드로 나타내면 다음과 같다. +```python +def qeval(x): + y = x**3 + return x + y + 5 +``` + +## Arithmetic circuit +첫 번째로 Flattening 이라는 작업을 통해 복잡한 원본 코드(computational problem)를 한번의 연산(두개의 변수와 `+, -, *` 중에 하나의 연산자)으로 구성된 순차적인 조각들로 나누게 되는데, 각각의 조각들을 **gate**라고 한다. 위 예시를 circuit으로 변환하면 다음과 같다. + +## R1CS + +다음으로 위의 circuit을 rank-1 matrix로 나타내는 R1CS(Rank-1 Constraint System)로 변환해야한다. R1CS는 세개의 벡터 그룹 `(a, b, c)` 와 솔루션 벡터 `s`로 이루어진 방정식 `s・a * s・b - s・c = 0` 이다. + +각 벡터(a, b, c)에는 circuit에서 생성된 변수들을 원소로 할당한다. 위 예시에서는 `'~one', 'x', 'out', 'sym_1', 'y', 'sym_2'` 이렇게 6개의 원소로 변수를 매핑한다.(’~one’은 상수 1) + +그리고 솔루션 벡터에는 이 조건식을 만족시키는 assignment로 구성된다. + +`sym_1 = x * x` 를 예로 들면 `x * x - sym_1 = 0` 로 a * b - c = 0 꼴로 바꿀 수 있고 a = x, b = x, c = sym_1 이다. 따라서 다음과 같이 변환할 수 있다. + +```jsx +a = [0, 1, 0, 0, 0, 0] // x +b = [0, 1, 0, 0, 0, 0] // x +c = [0, 0, 0, 1, 0, 0] // sym_1 +``` + +예를들어 벡터를 `['~one', 'x', 'out', 'sym_1', 'y', 'sym_2']` 로 매핑하면 a는 1x에 해당하므로 [0, 1, 0, 0, 0, 0]으로 나타낼 수 있다. + +같은 방식으로 위의 gate들을 변환하면 다음과 같다: + +1. `sym_1 = x * x` + + ```jsx + a = [0, 1, 0, 0, 0, 0] // x + b = [0, 1, 0, 0, 0, 0] // x + c = [0, 0, 0, 1, 0, 0] // sym_1 + ``` + +2. `y = sym_1 * x` + + ```jsx + a = [0, 0, 0, 1, 0, 0] // sym_1 + b = [0, 1, 0, 0, 0, 0] // x + c = [0, 0, 0, 0, 1, 0] // y + ``` + +3. `sym_2 = y + x` + + ```jsx + a = [0, 1, 0, 0, 1, 0] // y + x + b = [1, 0, 0, 0, 0, 0] // 1 + c = [0, 0, 0, 0, 0, 1] // sym_2 + ``` + +4. `~out = sym_2 + 5` + + ```jsx + a = [5, 0, 0, 0, 0, 1] // sym_2 + 5 + b = [1, 0, 0, 0, 0, 0] // 1 + c = [0, 0, 1, 0, 0, 0] // ~out + ``` + + +최종적으로 R1CS는 다음과 같다. + +```jsx +**A // 'a' vectors +[0, 1, 0, 0, 0, 0] +[0, 0, 0, 1, 0, 0] +[0, 1, 0, 0, 1, 0] +[5, 0, 0, 0, 0, 1] + +**B // 'b' vectors +[0, 1, 0, 0, 0, 0] +[0, 1, 0, 0, 0, 0] +[1, 0, 0, 0, 0, 0] +[1, 0, 0, 0, 0, 0] + +**C // 'c' vectors +[0, 0, 0, 1, 0, 0] +[0, 0, 0, 0, 1, 0] +[0, 0, 0, 0, 0, 1] +[0, 0, 1, 0, 0, 0] +``` + +그리고 A,B,C 행렬에 대해 다음을 만족하는 $s$가 적절한 assignment가 된다. (○으로 표시된 연산은 Hadamard product이다.) + +$(A・s) ◦(B・s)-(C・s) = \begin{bmatrix}0\\0\\0\\0\end{bmatrix}$ + +$s$가 될 수 있는 경우의 수는 많지만 `~out` 이 35가 되게 하려면 $s = \begin{bmatrix}1\\3\\35\\9\\27\\30\end{bmatrix}$이다. + +## QAP + +위에서 살펴본 예시는 R1CS로도 간단한 연산이 가능하지만 circuit의 크기가 커지면 constraint의 개수도 그만큼 증가하게 되고 연산이 복잡해지게 된다. + +Quadratic Arithmetic Program(QAP)은 R1CS의 각각의 constraint들을 한번에 다룰수 있는 다항식을 생성한다. 즉, R1CS의 변수들로 매핑된 각각의 함수에 $n$을 대입하면 n번째 gate의 해당 값이 나오는 다항식들로 변환하는 것이다. R1CS를 QAP로 변환하면 다음과 같다. + +```python +A = [A_1(x), A_2(x), A_3(x), A_4(x), A_5(x), A_6(x)] +B = [B_1(x), B_2(x), B_3(x), B_4(x), B_5(x), B_6(x)] +C = [C_1(x), C_2(x), C_3(x), C_4(x), C_5(x), C_6(x)] +``` + +예를들어 $A_2(x)$ 라는 다항식에 3을 대입하면 두번째 값(`x`)의 세번째 gate의 값인 1이 나오게 되고 4를 대입하면 네번째 값인 0이 나오게 된다.($A_2(3) = 1, A_2(4)= 0$) + +따라서 $A・s = A(x), \; B・s = B(x), \; C・s =C(x)$ 라고 하면, $x = 1,2,3,4$ 일때 $A(x)*B(x) -C(x) = 0$ 을 만족한다. + +여기서 주목할 점은 $Q(x) = A(x)*B(x) -C(x)$ 라고 하면, $Q(x)$의 해가 1,2,3,4라는 것은 다항식$Q(x)$에 $Z(x) = (x-1)(x-2)(x-3)(x-4)$ 가 인수로 포함되어있다는 것이다. + +따라서 R1CS의 constraint들을 각각 확인하는 문제에서 $Z(x)|Q(x)$ 를 확인하는 것으로 훨씬 간결하게 바뀌었다. + +위의 일련의 과정들이 QAP를 통해 R1CS 형태의 생성하는 프로세스이다. + +>Lagrange interpolation +$n$개의 점을 지나는 $d Date: Sat, 20 Jul 2024 00:49:01 +0900 Subject: [PATCH 3/3] add zkEVM --- .../Zero Knowledge Proofs/images/zkevm(1).png | Bin 0 -> 311584 bytes .../Zero Knowledge Proofs/images/zkevm(2).png | Bin 0 -> 155724 bytes .../Zero Knowledge Proofs/images/zkevm(3).png | Bin 0 -> 204366 bytes .../Zero Knowledge Proofs/images/zkevm(4).png | Bin 0 -> 105417 bytes .../Zero Knowledge Proofs/zkEVM.md | 88 ++++++++++++++++++ 5 files changed, 88 insertions(+) create mode 100644 content/Privacy Enhancing Technologies (PET)/Zero Knowledge Proofs/images/zkevm(1).png create mode 100644 content/Privacy Enhancing Technologies (PET)/Zero Knowledge Proofs/images/zkevm(2).png create mode 100644 content/Privacy Enhancing Technologies (PET)/Zero Knowledge Proofs/images/zkevm(3).png create mode 100644 content/Privacy Enhancing Technologies (PET)/Zero Knowledge Proofs/images/zkevm(4).png create mode 100644 content/Privacy Enhancing Technologies (PET)/Zero Knowledge Proofs/zkEVM.md diff --git a/content/Privacy Enhancing Technologies (PET)/Zero Knowledge Proofs/images/zkevm(1).png b/content/Privacy Enhancing Technologies (PET)/Zero Knowledge Proofs/images/zkevm(1).png new file mode 100644 index 0000000000000000000000000000000000000000..e7ed3d78950fae53c71a490632aabdf2109a3778 GIT binary patch literal 311584 zcmeFahdbMU^ggVrEv*g-MXe$dd)KN}g2Y}mYHy0#dsV9lDU}$pYm^SFwDzVn6%?&m zdzPa1=1G6!xt>q>{sYgYS8^rSJCU6CIp@C5ea`FkP7#{wO4qO6xk^MtbR7s6gf zizek4jSt%2B_eMy&m;HJIpfHK6@mV=22zbNT*55wN1?aJ26hRHTUz6>Y0G&5g9t~ zZA;%~b6-6^s#)6T*&5S`G*qkczBQy_oYcjeWc-98>QnYCGDW_3>uj&q%ON+f6j+z4|!363(4}e>3n(Qj|^M=jVfCnLqWD zzR)jf;Vgh)nkY@JP?AfNA%}f2GBG^v`sSLvvBccppl}LcY3s)KEDM038|j0^SCt-1 z!8Z#Ju+Gk_R2XO1F23Pmxr<`yS?Hf5!@eXmcZi-Ws@~I8TFly>b8kE#*xwK$KzU^%MUv6ad5IRfZM965pm+5 zlM<0!xI=Ua|Lg+(E`5RS|9!4_ft~2$|2j@gL=@pbMDl;$qmF<4_wN<{{_k)8_apIZ z;{W>&Qq|WN|L^lF^#9(f{v#wA|3KynGxQ)Ly3O|QegUox`b$J4O9Y2L(Dl9WXLJbt z#>b@HO;_%AsoJ(P3e085d`m9A5c&c7q4xW_+{uI0TDdnQRGQ@gLpjnRtSX>X7Hkb+ z<_;cr%^19aNd7G?)hD&F|1fAhO?ot_%fl2R8kEr;7_#c0r0|sJ0x=2ct-G@4PtKii zo(bnDI7h)b3eHh*j)HR(oTK0z1?MO@N5MG?&QWlVf^!s{qu?9`=O{Qw!8r=fQE-le za}=DT;2Z_#C^$#KISS5EaE^j=6r7{r90lhnI7h)b3eHh*j)HR(oTK0z1?MO@N5MG? 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지식을 가진 독자들을 대상으로 작성되었다. + +## Motivation![[zkevm(1).png]] +여전히 블록체인 트릴레마는 L1 블록체인의 풀리지않는 숙제이다. L1 의 확장성을 위한 솔루션으로 여러가지 방안들이 제시되었는데, 이더리움의 경우 스케일링 솔루션으로는 L2, 그 중에서도 ZK 롤업을 Execution layer로써 사용하고자 하는 움직임이 활발해지고있다. + +ZK 롤업과 옵티미스틱 롤업의 유일한 차이는 Validity proof 제출의 유무이다. Optimistic Rollup의 경우 “Presumption of innocent”를 전제로 L2에서 어떤 증명값도 제출하지 않고, 수동적으로 Challenge를 받는 방식이다. 반면 ZK 롤업의 경우 L2 데이터를 L1 블록에 올릴때마다 해당 데이터에 대한 Validity proof를 같이 제출한다. + +ZK 롤업에도 여러가지 형태가 있는데 이 아티클에서는 Universal ZK 롤업을 중점적으로 살펴볼 예정이다. Universal 롤업은 어떤 특정 기능만 가능한 App-specific 롤업(가령 송금의 기능만 있는 롤업)과 달리 zkEVM 또는 zkVM을 사용해 Proof를 생성하는 General purpose 롤업을 말한다. + +## [[ZK-SNARK]] +![[zkevm(2).png]] +### ZK-SNARK Structure +zkEVM에서 생성되는 Validity Proof는 ZK-SNARK라는 증명방식을 사용한다. ZK-SNARK는 Zero Knowledge Succinct Non-interactive Argument of Knowledge의 약자로 ZK의 속성인 Hiding과, SNARK의 속성인 Non-interactive를 모두 만족하면서 간결한 검증을 할 수 있는 증명방식이다. + +ZK-SNARK는 흔히 엔지니어링에서 사용하는 Front-end와 Back-end라는 용어를 사용하는데 널리 알려진 개념과는 다른 형태이다. ZK-SNARK의 work flow는 다음과 같다. + +**Setup** +합의된 형태를 통해 Prover와 Verifier가 증명과 검증을 위해 각각 가지고있는 파라미터 + +**Prover** +1. 증명하고자 하는 Function을 Arithmetic circuit의 형태로 바꾼다. +2. Arithmetic circuit을 Oracle(주로 다항식)의 형태로 바꾼다. +3. Oracle을 암호학적인 컴파일을 통해 특정 형태의 Commitment 값으로 Verifier에게 제출한다. + +**Verifier** +Prover로부터 받은 증명값을 검증로직으로 연산 후 Accept or Reject + +### ZK-SNARK Traits +ZK-SNARK에는 여러가지 스킴들이 존재하는데 각 스킴들마다의 장단점이 있기 때문에, zkEVM의 형태와 trade-off를 잘 고려하여 proving scheme을 선택해야한다. + +고려해야할 특성은 크게 3가지가 있다. + +1. **Setup:** + + Trusted setup per circuit - Circuit 별로 매번 다른 Trusted setup이 필요 (e.g. Groth16) + + Universal Trusted Setup - 한번의 Trusted setup으로 모든 Circuit에 사용 가능 (e.g. KZG) + + Transparent Setup: Trusted party가 필요하지 않은 Transparent setup (e.g. FRI) + +2. **Proving cost:** Prover가 proof를 생성하기 위한 complexity + +3. **Verifying cost:** ZK 롤업의 Verify는 보통 온체인에서 이루어지기 때문에 가스비와 직결되는 특성 + + Proof size - Circuit의 크기 대비 증명값의 크기 (e.g. KZG - O(1), IPA - O(log N)) + + Verifier time - Verifier proof 검증 로직의 time complexity (e.g. KZG - O(1), IPA - O(N)) + + +### Different types of proving schemes + +Proving scheme들은 사용하는 암호학적 가정들에 따라 나눌 수 있다. 각각의 기반 테크닉에 따라 Setup의 형태나 코스트가 달라진다. + +1. Pairing based: 타원곡선의 Bilinear pairing을 이용한 스킴 (e.g. KZG) +2. Discrete log based: 타원곡선의 이산로그문제(DLP)를 이용한 스킴 (e.g. Bulletproofs) +3. Hash based: 해시를 사용하는 스킴 (e.g. FRI) + +## Architecture of zkEVM![[zkevm(3).png]] + +### zkEVM proving data +zkEVM은 모든 트랜젝션들을 올바르게 실행시켰는지, 즉 EVM의 Execution Trace가 모두 올바른지를 증명하는 Circuit이다. 여기서 증명해야하는 것들은 EVM word가 256비트인지, 옵코드의 실행결과가 모두 올바른지, 스택과 메모리에서 값이 올바르게 읽고 쓰여졌는지 등 EVM 실행에 대한 모든 validity이다. + +zkEVM Prover가 가지는 input 데이터는 다음과 같다: +- **Input**: 이더리움의 기존 World state `t`, 새로운 World state `t+1`, L2 트랜젝션 +- **Witness**: Execution trace + +이 데이터들을 이용해 모든 Execution trace의 Codehash, Gas usage, R/W of PC & Stack pointer, Opcode 등 모든 실행 유효성을 증명한다. + +### zkEVM structure +zkEVM의 구조는 프로젝트마다 모두 상이하지만, 이런 거대한 크기의 circuit의 증명을 가능하게 한 요인 중 하나는 Lookup이다. Lookup은 circuit 내부에서 직접 연산을 처리하지 않고 미리 프리컴파일 되어있는 테이블에서 값이 “존재”하는지만을 증명하는 획기적인 방법이다. + +대부분의 zkEVM들은 기본적으로 EVM trace가 올바른지 증명하기 위한 EVM circuit이 있고 증명하고자 하는 값들이 프리컴파일된 여러 Lookup 테이블들이 존재한다. 가령 이더리움에서 사용하는 해시함수인 Keccak의 경우 Circuit 내부에서 연산하려면 코스트가 커지는데 해시값을 프리컴파일한 Keccak table에서 lookup하는 방식으로 constraints를 획기적으로 줄인다. + +따라서 zkEVM은 보통 Execution trace를 검증하기 위한 메인 EVM circuit과 Lookup을 위한 Auxiliary circuit들로 구성되어있다. + +zkEVM에서 중요한 또다른 미션은 여러개의 Proof들을 어떻게 줄이느냐인데, 이는 사용하는 proving scheme에 따라 다양한 어프로치들이 있다. ZCash에서 사용하는 Halo2(IPA)의 경우 Pasta curve를 사용한 recursion으로 proof를 accumulation하고, Scroll의 경우 Halo2(KZG)를 사용하는데 cycling이 불가능한 BN254를 사용하기 때문에 proof들을 합치기 위한 또다른 Aggregation circuit이 존재한다. + +## Types of zkEVM + +![[zkevm(4).png]] + +비탈릭 부테린이 제시한 zkEVM의 분류체계는 총 네가지로 나눌 수 있다. +1. Type1: 이더리움 컨센서스 레이어까지 모두 호환되는 이더리움 호환 zkEVM. 이더리움의 보안과 모든 인프라를 그대로 상속받을 수 있는 장점이 있는 반면, 그만큼 proving 코스트가 매우 크다. 현재 대표적으로 Taiko가 Type1 zkEVM을 개발 중이다. +2. Type2: EVM과 호환되는 EVM 호환 zkEVM. 이더리움의 MPT를 ZK-friendly structure인 ZK-Trie로 변경하는 경우와 같이 ZK를 위한 변경이 있지만 여전히 EVM레벨에서는 동등한 호환성을 가지고있다. 대표적인 프로젝트로 스크롤이 있다. +3. Type3: ZK-unfriendly Opcode나 구조들을 없앤 형태로 이더리움 디앱들이 그대로 온보딩하지 못하고 약간의 수정이 필요한 허들이 존재한다. +4. Type4: 자체적인 ZK-friendly 언어와 Virtual Machine을 사용하는 경우로 이더리움과 호환이 되지 않지만 다른 타입에 비해 proof 생성이 훨씬 빠르다는 장점이 있다. 대표적으로 StarkNet과 zkSync가 있다. \ No newline at end of file

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