Spartan-ECDSA：最快的浏览器内 ZK secp256k1 ECDSA

Posted 2023-02-16 mutourend

1. 引言

前序博客：

• Efficient ZK ECDSA
• ECDSA VS Schnorr signature VS BLS signature
• Spartan: zkSNARKS without trusted setup学习笔记
• Spartan: zkSNARKS without trusted setup 源代码解析

Personae Labs开源的：

• https://github.com/personaelabs/spartan-ecdsa（Rust+TypeScript）

为当前最快的ZK ECDSA(sep256k1)。
Spartan-ECDSA证明ECDSA group membership的速度 要比 之前Personae Labs开源Efficient ZK ECDSA 快10倍。

Proving membership to a group of ECDSA public keys的性能情况为：

Benchmark	#
Constraints	8,076
Proving time in browser	4s
Proving time in Node.js	2s
Verification time in browser	1s
Verification time in Node.js	300ms
Proof size	16kb

• Measured on a M1 MacBook Pro with 80Mbps internet speed.
• Both proving and verification time in browser includes the time to download the circuit.

Spartan-ECDSA的核心技术为：

• 1）Secp256k1曲线+Secq256k1曲线（Secp256k1曲线的scalar域 为 Secq256k1曲线的基域）：因ECDSA验签过程中不仅包含${\mathbb{F}}_p$域内的运算，还包含${\mathbb{F}}_q$域内的运算。
• 2）Spartan证明系统：使用了基于Secq256k1曲线的Spartan分支。
• 3）Circom：Circom用于定义arithmetic circuit。采用了基于Secp256k1基域（即Secq256k1 scalar域）的Circom分支来进行算术化。
• 4）Nova-Scotia：为Circom R1CS编译器。fork了一个分支来将Circom circuits编译为Spartan可处理的二进制格式，并对Nova-Scotia做了小调整以兼容Secq256k1。

Personae Labs团队未来将进一步：

• 1）对Keccak加速
• 2）使链上验证成为可能：如可利用递归证明，降低单个证明的验证开销。

参考资料

[1] Twitter Introducing spartan-ecdsa
[2] ebfull hackmd Wait, what are these isomorphic elliptic curves doing on the secp256k1/secq256k1 cycle?
[3] Personae Labs 2023年2月博客 Introducing Spartan-ecdsa

附录：Secp256k1/Secq256k1曲线cycle

Secp256k1椭圆曲线$y^2=x^3+7$ defined over ${\mathbb{F}}_p$ with order $q$，而Secq256k1椭圆曲线$y^2=x^3+7$ defined over ${\mathbb{F}}_q$ with order $p$，二者形成了2-cycle curve，相应的SageMath脚本验证为：

p = 0xfffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f
q = 0xfffffffffffffffffffffffffffffffebaaedce6af48a03bbfd25e8cd0364141

# Secp256k1

P = GF(p)
aP = P(0x0000000000000000000000000000000000000000000000000000000000000000)
bP = P(0x0000000000000000000000000000000000000000000000000000000000000007)
Secp256k1 = EllipticCurve(P, (aP, bP))
Secp256k1.set_order(q)

# Secq256k1

Q = GF(q)
aQ = P(0x0000000000000000000000000000000000000000000000000000000000000000)
bQ = P(0x0000000000000000000000000000000000000000000000000000000000000007)
Secq256k1 = EllipticCurve(Q, (aQ, bQ))
Secq256k1.set_order(p)

print(
"Secp256k1 group order == Secq256k1 base field order:",
Secp256k1.order() == Secq256k1.base_field().cardinality()
)

print(
"Secp256k1 base field order == Secq256k1 group order:",
Secp256k1.base_field().cardinality() == Secq256k1.order()
)