halo2/tmp/4-IsZero-Check-example-3.md

author: @Demian

references: https://learn.0xparc.org/materials/halo2

Goal
Overview
Iszero Chip
- structs
- impl IsZeroChip { ..
Example 3
- configure
- assign
- test
usage
References:

Goal

We want Prove that : f(a, b, c) = if a == b {c} else {a - b}

证明某人知道三个数字 a、b 和 c，使得当 a == b 时，输出为 c，否则输出为 a - b，而无需揭示a、b 和 c 的实际值。

how to describe it ? Firstly, let’s dive into the Iszero Chip

这一部分主要展示了如何复用另外一个电路来辅助电路设计

Overview

$ tree
├── fibonacci
│   └── example3.rs
├── is_zero.rs

Iszero Chip

structs

#![allow(unused)]
fn main() {
#[derive(Clone, Debug)]
pub struct IsZeroConfig<F> {
    pub value_inv: Column<Advice>,
    pub is_zero_expr: Expression<F>,
}

impl<F: FieldExt> IsZeroConfig<F> {
    pub fn expr(&self) -> Expression<F> {
        self.is_zero_expr.clone()
    }
}

pub struct IsZeroChip<F: FieldExt> {
    config: IsZeroConfig<F>,
}
}

impl IsZeroChip { ..

configure

传入参数：

q_enable, value 都接收一个闭包，将执行的时机和执行的具体内容进行解耦。也就是说，configure 方法可以决定何时执行 value，而调用者可以决定执行 value 时应该做什么
闭包可以捕获其环境，这意味着每次传入不同的闭包，configure 函数的行为都可能不同。这为函数调用者提供了更大的灵活性。

注意 AssignedCell 和 VirtualCells 的区别（读下 Source code）

#![allow(unused)]
fn main() {
impl<F: FieldExt> IsZeroChip<F> {
    pub fn construct(config: IsZeroConfig<F>) -> Self {
        IsZeroChip { config }
    }

    pub fn configure(
        meta: &mut ConstraintSystem<F>,
        q_enable: impl FnOnce(&mut VirtualCells<'_, F>) -> Expression<F>,
        value: impl FnOnce(&mut VirtualCells<'_, F>) -> Expression<F>,
        value_inv: Column<Advice>,
    ) -> IsZeroConfig<F> {
        let mut is_zero_expr = Expression::Constant(F::zero());

        meta.create_gate("is_zero", |meta| {
            //
            // valid | val |  val_inv |  1 - val * val_inv | val * (1 - val * val_inv)
            // ------+-----+----------+--------------------+-------------------
            //  yes  |  x  |    1/x   |        0           |   0
            //  no   |  x  |    0     |        1           |   x
            //  yes  |  0  |    0     |        1           |   0
            //  yes  |  0  |    y     |        1           |   0

            //
            let value = value(meta);
            let q_enable = q_enable(meta);
            let value_inv = meta.query_advice(value_inv, Rotation::cur());

            is_zero_expr = Expression::Constant(F::one()) - value.clone() * value_inv;
            vec![q_enable * value * is_zero_expr.clone()]  // gate's constraints
        });

        IsZeroConfig {
            value_inv,
            is_zero_expr,
        }
    }
}

configure defines the logic for the “is-zero” gate. It uses the following table to guide the logic:

valid | val |  val_inv |  1 - val * val_inv | val * (1 - val * val_inv)
------+-----+----------+--------------------+-------------------       
 yes  |  x  |    1/x   |        0           |   0                      
 no   |  x  |    0     |        1           |   x                      
 yes  |  0  |    0     |        1           |   0                      
 yes  |  0  |    y     |        1           |   0

第 1 / 3 / 4 行涉及到的约束不需要通过 q_enable 即可完成, 但是考虑第二行所涉及到的情况 :

如果 $P$ 是个 malicious Prover, 他提供了 val == x 和 val_inv == 0 , 此时仅靠 is_zero_expr 是无法分辨的 (这个 case 里 assign 函数会直接分配 self.config.value_inv i.e. $0$ 即认为这个值是 $0$
但是添加了 vec![q_enable * value * is_zero_expr.clone()] 约束就不一样了 , 约束强制要求 val * is_zero_expr i.e. val * ( 1 - val * val_inv) 必须为 0 , 从而解决了这种 malicious situation
如果 malicious $P$ 提供了这种 Witness, 将不会通过约束校验, 也就不会生成该 proof
只有 $P$ 提供了符合约束的 Witness, val_inv 才会被赋值给 val_inv column

The gate ensures that for valid rows:

If the $v a l u e \neq = 0$ , its inverse is computed such that their multiplication (val * val_inv) ’s results in 1.
If the $v a l u e == 0$ , its inverse can be any value, but the result of their multiplication should be 0.

The gate equation is q_enable * value * (1 - value * value_inv), which should be satisfied for the valid conditions.

assign(): This method is used to assign the inverse of a value (if it exists) or zero to the specified advice column in the circuit.

#![allow(unused)]
fn main() {
is_zero_expr = Expression::Constant(F::one()) - value.clone() * value_inv;
}

i.e. 1 - val * val_inv , like the table above :
- if val != 0 : is_zero_expr = 0
- if val == 0 : is_zero_expr = 1

vec![q_enable * value * is_zero_expr.clone()] is the gate’s constraint. it should be $0$

assign

#![allow(unused)]
fn main() {
pub fn assign(
	&self,
	region: &mut Region<'_, F>,
	offset: usize,
	value: Value<F>,
) -> Result<(), Error> {
	// value.invert()  OR  F::zero()
	let value_inv = value.map(|value| value.invert().unwrap_or(F::zero()));
	region.assign_advice(|| "value inv", self.config.value_inv, offset, || value_inv)?;
	Ok(())
}
}

在 IsZero 的验证过程中，将要验证的值（或输入值）分配到电路区域中，以便在电路中进行计算和约束的验证 :

如果要验证的值为零，assign 方法将为逆元分配一个特定的值（例如 F::zero()）
如果要验证的值不为零，value_inv columns 将被分配为 value.invert().unwrap_or(F::zero()) i.e. value.invert()

这些 IsZero 的 check 将被赋值到 value_inv column 并在其上得到体现

Example 3

welcome back, now we have the gadget IsZero , so we can constrain malicious $P$ ’s input

#![allow(unused)]
fn main() {
#[derive(Debug, Clone)]
struct FunctionConfig<F: FieldExt> {
    selector: Selector,
    a: Column<Advice>,
    b: Column<Advice>,
    c: Column<Advice>,
    a_equals_b: IsZeroConfig<F>,
    output: Column<Advice>,
}
#[derive(Debug, Clone)]
struct FunctionChip<F: FieldExt> {
    config: FunctionConfig<F>,
}
}

configure

Recap : f(a, b, c) = if a == b {c} else {a - b}

column : 除了常规的 a/b/c advice column, 还申请了 is_zero_advice_column
IsZeroChip : use crate::is_zero::{IsZeroChip, IsZeroConfig}; 使用了上面定义的 IsZero chip 来校验 $a == b$ 这个事情 (因为 a/b 都是 $P$ 提供的, 一个 malicious $P$ 有动机去提供 a=3 , b=4 然后 return c , 必须通过生成 proof 前的约束来限制 $P$ 的行为)
IsZeroChip::configure 返回 IsZeroConfig<F>

#![allow(unused)]
fn main() {
impl<F: FieldExt> FunctionChip<F> {
    pub fn construct(config: FunctionConfig<F>) -> Self { Self { config } }

    pub fn configure(meta: &mut ConstraintSystem<F>) -> FunctionConfig<F> {
        let selector = meta.selector();
        let a = meta.advice_column();
        let b = meta.advice_column();
        let c = meta.advice_column();
        let output = meta.advice_column();

        let is_zero_advice_column = meta.advice_column();
        
        let a_equals_b = IsZeroChip::configure(
            meta,
            |meta| meta.query_selector(selector),
            |meta| meta.query_advice(a, Rotation::cur()) - meta.query_advice(b, Rotation::cur()),
            is_zero_advice_column,
        );

        meta.create_gate("f(a, b, c) = if a == b {c} else {a - b}", |meta| {
            let s = meta.query_selector(selector);
            let a = meta.query_advice(a, Rotation::cur());
            let b = meta.query_advice(b, Rotation::cur());
            let c = meta.query_advice(c, Rotation::cur());
            let output = meta.query_advice(output, Rotation::cur());
            vec![
                s.clone() * (a_equals_b.expr() * (output.clone() - c)),
                s * (Expression::Constant(F::one()) - a_equals_b.expr()) * (output - (a - b)),
            ]
        });

        FunctionConfig {
            selector,
            a,
            b,
            c,
            a_equals_b,
            output,
        }
    }
}

assign

IsZeroChip::construct : 创建一个IsZeroChip实例
layouter.assign_region( :
1. is_zero_chip.assign(&mut region, 0, Value::known(a - b))?;
2. region.assign_advice(|| "output", self.config.output, 0, || Value::known(output))

#![allow(unused)]
fn main() {
pub fn assign(
  &self,
  mut layouter: impl Layouter<F>,
  a: F,  b: F,  c: F,
) -> Result<AssignedCell<F, F>, Error> {
  let is_zero_chip = IsZeroChip::construct(self.config.a_equals_b.clone());

  layouter.assign_region(
    || "f(a, b, c) = if a == b {c} else {a - b}",
    |mut region| {
      self.config.selector.enable(&mut region, 0)?;
      region.assign_advice(|| "a", self.config.a, 0, || Value::known(a))?;
      region.assign_advice(|| "b", self.config.b, 0, || Value::known(b))?;
      region.assign_advice(|| "c", self.config.c, 0, || Value::known(c))?;

      // 正式使用 IsZeroChip 子电路来检查 a - b 是否为零
      is_zero_chip.assign(&mut region, 0, Value::known(a - b))?;

      // Rust expr to calculate val.
      let output = if a == b { c } else { a - b };
      // assign to cell.
      region.assign_advice(|| "output", self.config.output, 0, || Value::known(output))
    },
  ) }
}

test

#![allow(unused)]
fn main() {
#[cfg(test)]
mod tests {
    use super::*;
    use halo2_proofs::{dev::MockProver, pasta::Fp};

    #[test]
    fn test_example3() {
        let circuit = FunctionCircuit {
            a: Fp::from(10),
            b: Fp::from(12),
            c: Fp::from(15),
        };

        let prover = MockProver::run(4, &circuit, vec![]).unwrap();
        prover.assert_satisfied();
    }
}
}

usage

cargo test -- --nocapture fibonacci::example3

# Draw
cargo test --release --all-features plot_fibo3

the white column is the instance column,
the pink one is the advice and
the purple one is the selector.
the green part shows the cells that have been assigned
- light green : selector not used.

World of Z2O-K7E