Prover code for permutation argument #485
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wborgeaud
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Feb 14, 2022
nbgl
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Feb 14, 2022
This was referenced Feb 14, 2022
wborgeaud
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Feb 15, 2022
starky/src/prover.rs
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| /// Compute a single Z polynomial. | ||
| fn compute_z_poly<F: Field>( | ||
| instance: PermutationInstance<F>, | ||
| trace_poly_values: &[PolynomialValues<F>], | ||
| ) -> PolynomialValues<F> { | ||
| let PermutationInstance { pair, challenge } = instance; | ||
| let PermutationPair { | ||
| lhs_columns, | ||
| rhs_columns, | ||
| } = pair; | ||
| let PermutationChallenge { beta, gamma } = challenge; | ||
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| let degree = trace_poly_values[0].len(); | ||
| let mut reduced_lhs = PolynomialValues::constant(gamma, degree); | ||
| let mut reduced_rhs = PolynomialValues::constant(gamma, degree); | ||
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| let both_cols = lhs_columns.iter().zip_eq(rhs_columns); | ||
| for ((lhs, rhs), weight) in both_cols.zip(beta.powers()) { | ||
| reduced_lhs.add_assign_scaled(&trace_poly_values[*lhs], weight); | ||
| reduced_rhs.add_assign_scaled(&trace_poly_values[*rhs], weight); | ||
| } | ||
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| // Compute the quotients. | ||
| let reduced_rhs_inverses = F::batch_multiplicative_inverse(&reduced_rhs.values); | ||
| let mut quotients = reduced_lhs.values; | ||
| batch_multiply_inplace(&mut quotients, &reduced_rhs_inverses); | ||
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| // Compute Z, which contains partial products of the quotients. | ||
| let mut partial_products = Vec::with_capacity(degree); | ||
| let mut acc = F::ONE; | ||
| for q in quotients { | ||
| partial_products.push(acc); | ||
| acc *= q; | ||
| } | ||
| PolynomialValues::new(partial_products) | ||
| } |
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Maybe I'm missing something, but it seems like this could be done over batches of quotient_degree_factor instances.
If I understand correctly, this does (in the case of two lhs columns l_0, l_1 and two rhs columns r_0, r_1):
Z(gx) = Z(x) * (gamma + l_0 + beta * l_1) / (gamma + r_0 + beta * r_1)
=>
Z(gx) * (gamma + r_0 + beta * r_1) - Z(x) * (gamma + l_0 + beta * l_1) = 0.
This last constraint is of degree 2, whereas the vanishing polynomial can be of degree up to 3 (with rate_bits=1).
So it seems like having Z take batches of quotient_degree_factor instances would yield a constraint of degree constraint_degree, and would result in fewer Z polynomials.
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Ah yes, that was my intention but I forgot to actually implement the batching logic here. I'll just merge the non-batch version for now, as I'm not sure when I'll have time to finish this and want to minimize conflicts, but I added a TODO and I'll come back to this when I have a chance.
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It's slightly complex because we batch `constraint_degree - 1` permutation arguments into a single `Z` polynomial. This is a slight generalization of the [technique](https://zcash.github.io/halo2/design/proving-system/lookup.html) described in the Halo2 book. Without this batching, we would simply have `num_challenges` random challenges (betas and gammas). With this batching, however, we need to use different randomness for each permutation argument within the same batch. Hence we end up generating `batch_size * num_challenges` challenges for all permutation arguments.
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It's slightly complex because we batch
constraint_degree - 1permutation arguments into a singleZpolynomial. This is a slight generalization of the technique described in the Halo2 book.Without this batching, we would simply have
num_challengesrandom challenges (betas and gammas). With this batching, however, we need to use different randomness for each permutation argument within the same batch. Hence we end up generatingbatch_size * num_challengeschallenges for all permutation arguments.