Arrabiata: prepare constraints for cross-terms computation

arrabbiata/src/columns.rs

@@ -8,6 +8,8 @@ use std::{
 };
 use strum_macros::{EnumCount as EnumCountMacro, EnumIter};
 
+use crate::NUMBER_OF_COLUMNS;
+
 /// This enum represents the different gadgets that can be used in the circuit.
 /// The selectors are defined at setup time, can take only the values `0` or
 /// `1` and are public.
@@ -36,6 +38,27 @@ pub enum Column {
     X(usize),
 }
 
+/// Convert a column to a usize. This is used by the library [mvpoly] when we
+/// need to compute the cross-terms.
+/// For now, only the private inputs and the public inputs are converted,
+/// because there might not need to treat the selectors in the polynomial while
+/// computing the cross-terms (FIXME: check this later, but pretty sure it's the
+/// case).
+///
+/// Also, the [mvpoly::monomials] implementation of the trait [mvpoly::MVPoly]
+/// will be used, and the mapping here is consistent with the one expected by
+/// this implementation, i.e. we simply map to an increasing number starting at
+/// 0, without any gap.
+impl From<Column> for usize {
+    fn from(val: Column) -> usize {
+        match val {
+            Column::X(i) => i,
+            Column::PublicInput(i) => NUMBER_OF_COLUMNS + i,
+            Column::Selector(_) => unimplemented!("Selectors are not supported. This method is supposed to be called only to compute the cross-term and an optimisation is in progress to avoid the inclusion of the selectors in the multi-variate polynomial."),
+        }
+    }
+}
+
 pub struct Challenges<F: Field> {
     /// Challenge used to aggregate the constraints
     pub alpha: F,

arrabbiata/src/main.rs

@@ -101,6 +101,12 @@ pub fn main() {
         // FIXME:
         // Compute the accumulator for the permutation argument
 
+        // FIXME:
+        // Commit to the accumulator and absorb the commitment
+
+        // FIXME:
+        // Coin challenge α for combining the constraints
+
         // FIXME:
         // Compute the cross-terms
 

mvpoly/src/lib.rs

@@ -254,6 +254,45 @@ pub trait MVPoly<F: PrimeField, const N: usize, const D: usize>:
         u2: F,
     ) -> HashMap<usize, F>;
 
+    /// Compute the cross-terms of the given polynomial, scaled by the given
+    /// scalar.
+    ///
+    /// More explicitly, given a polynomial `P(X1, ..., Xn)` and a scalar α, the
+    /// method computes the the cross-terms of the polynomial `Q(X1, ..., Xn, α)
+    /// = α * P(X1, ..., Xn)`. For this reason, the method takes as input the
+    /// two different scalars `scalar1` and `scalar2` as we are considering the
+    /// scaling factor as a variable.
+    ///
+    /// This method is particularly useful when you need to compute a
+    /// (possibly random) combinaison of polynomials `P1(X1, ..., Xn), ...,
+    /// Pm(X1, ..., Xn)`, like when computing a quotient polynomial in the PlonK
+    /// PIOP, as the result is the sum of individual "scaled" polynomials:
+    /// ```text
+    /// Q(X_1, ..., X_n, α_1, ..., α_m) =
+    ///   α_1 P1(X_1, ..., X_n) +
+    ///   ...
+    ///   α_m Pm(X_1, ..., X_n) +
+    /// ```
+    ///
+    /// The polynomial must not necessarily be homogeneous. For this reason, the
+    /// values `u1` and `u2` represents the extra variable that is used to make
+    /// the polynomial homogeneous.
+    ///
+    /// The homogeneous degree is supposed to be the one defined by the type of
+    /// the polynomial `P`, i.e. `D`.
+    ///
+    /// The output is a map of `D` values that represents the cross-terms
+    /// for each power of `r`.
+    fn compute_cross_terms_scaled(
+        &self,
+        eval1: &[F; N],
+        eval2: &[F; N],
+        u1: F,
+        u2: F,
+        scalar1: F,
+        scalar2: F,
+    ) -> HashMap<usize, F>;
+
     /// Modify the monomial in the polynomial to the new value `coeff`.
     fn modify_monomial(&mut self, exponents: [usize; N], coeff: F);
 

mvpoly/src/monomials.rs

@@ -1,3 +1,8 @@
+use crate::{
+    prime,
+    utils::{compute_indices_nested_loop, naive_prime_factors, PrimeNumberGenerator},
+    MVPoly,
+};
 use ark_ff::{One, PrimeField, Zero};
 use kimchi::circuits::{expr::Variable, gate::CurrOrNext};
 use num_integer::binomial;
@@ -8,12 +13,6 @@ use std::{
     ops::{Add, Mul, Neg, Sub},
 };
 
-use crate::{
-    prime,
-    utils::{compute_indices_nested_loop, naive_prime_factors, PrimeNumberGenerator},
-    MVPoly,
-};
-
 /// Represents a multivariate polynomial in `N` variables with coefficients in
 /// `F`. The polynomial is represented as a sparse polynomial, where each
 /// monomial is represented by a vector of `N` exponents.
@@ -472,6 +471,41 @@ impl<const N: usize, const D: usize, F: PrimeField> MVPoly<F, N, D> for Sparse<F
         cross_terms_by_powers_of_r
     }
 
+    fn compute_cross_terms_scaled(
+        &self,
+        eval1: &[F; N],
+        eval2: &[F; N],
+        u1: F,
+        u2: F,
+        scalar1: F,
+        scalar2: F,
+    ) -> HashMap<usize, F> {
+        assert!(
+            D >= 2,
+            "The degree of the polynomial must be greater than 2"
+        );
+        let cross_terms = self.compute_cross_terms(eval1, eval2, u1, u2);
+
+        let mut res: HashMap<usize, F> = HashMap::new();
+        cross_terms.iter().for_each(|(power_r, coeff)| {
+            res.insert(*power_r, *coeff * scalar1);
+        });
+        cross_terms.iter().for_each(|(power_r, coeff)| {
+            res.entry(*power_r + 1)
+                .and_modify(|e| *e += *coeff * scalar2)
+                .or_insert(*coeff * scalar2);
+        });
+        let eval1_hom = self.homogeneous_eval(eval1, u1);
+        res.entry(1)
+            .and_modify(|e| *e += eval1_hom * scalar2)
+            .or_insert(eval1_hom * scalar2);
+        let eval2_hom = self.homogeneous_eval(eval2, u2);
+        res.entry(D)
+            .and_modify(|e| *e += eval2_hom * scalar1)
+            .or_insert(eval2_hom * scalar1);
+        res
+    }
+
     fn modify_monomial(&mut self, exponents: [usize; N], coeff: F) {
         self.monomials
             .entry(exponents)
@@ -520,12 +554,19 @@ impl<F: PrimeField, const N: usize, const D: usize> From<F> for Sparse<F, N, D>
     }
 }
 
-impl<F: PrimeField, const N: usize, const D: usize, const M: usize> From<Sparse<F, N, D>>
-    for Result<Sparse<F, M, D>, String>
+impl<F: PrimeField, const N: usize, const D: usize, const M: usize, const D_PRIME: usize>
+    From<Sparse<F, N, D>> for Result<Sparse<F, M, D_PRIME>, String>
 {
-    fn from(poly: Sparse<F, N, D>) -> Result<Sparse<F, M, D>, String> {
+    fn from(poly: Sparse<F, N, D>) -> Result<Sparse<F, M, D_PRIME>, String> {
         if M < N {
-            return Err("The number of variables must be greater than N".to_string());
+            return Err(format!(
+                "The final number of variables {M} must be greater than {N}"
+            ));
+        }
+        if D_PRIME < D {
+            return Err(format!(
+                "The final degree {D_PRIME} must be greater than initial degree {D}"
+            ));
         }
         let mut monomials = HashMap::new();
         poly.monomials.iter().for_each(|(exponents, coeff)| {

mvpoly/src/prime.rs

@@ -432,6 +432,18 @@ impl<F: PrimeField, const N: usize, const D: usize> MVPoly<F, N, D> for Dense<F,
         unimplemented!()
     }
 
+    fn compute_cross_terms_scaled(
+        &self,
+        _eval1: &[F; N],
+        _eval2: &[F; N],
+        _u1: F,
+        _u2: F,
+        _scalar1: F,
+        _scalar2: F,
+    ) -> HashMap<usize, F> {
+        unimplemented!()
+    }
+
     fn modify_monomial(&mut self, exponents: [usize; N], coeff: F) {
         let mut prime_gen = PrimeNumberGenerator::new();
         let primes = prime_gen.get_first_nth_primes(N);

mvpoly/tests/monomials.rs

@@ -712,3 +712,95 @@ fn test_from_expr_ec_addition() {
         assert_eq!(eval, exp_eval);
     }
 }
+
+#[test]
+fn test_cross_terms_fixed_polynomial_and_eval_homogeneous_degree_3() {
+    // X
+    let x = {
+        // We say it is of degree 2 for the cross-term computation
+        let mut x = Sparse::<Fp, 1, 2>::zero();
+        x.add_monomial([1], Fp::one());
+        x
+    };
+    // X * Y
+    let scaled_x = {
+        let scaling_var = {
+            let mut v = Sparse::<Fp, 2, 2>::zero();
+            v.add_monomial([0, 1], Fp::one());
+            v
+        };
+        let x: Sparse<Fp, 2, 2> = {
+            let x: Result<Sparse<Fp, 2, 2>, String> = x.clone().into();
+            x.unwrap()
+        };
+        x.clone() * scaling_var
+    };
+    // x1 = 42, α1 = 1
+    // x2 = 42, α2 = 2
+    let eval1: [Fp; 2] = [Fp::from(42), Fp::one()];
+    let eval2: [Fp; 2] = [Fp::from(42), Fp::one() + Fp::one()];
+    let u1 = Fp::one();
+    let u2 = Fp::one() + Fp::one();
+    let scalar1 = eval1[1];
+    let scalar2 = eval2[1];
+
+    let cross_terms_scaled_p1 = {
+        // When computing the cross-terms, the method supposes that the polynomial
+        // is of degree D - 1.
+        // We do suppose we homogenize to degree 3.
+        let scaled_x: Sparse<Fp, 2, 3> = {
+            let p: Result<Sparse<Fp, 2, 3>, String> = scaled_x.clone().into();
+            p.unwrap()
+        };
+        scaled_x.compute_cross_terms(&eval1, &eval2, u1, u2)
+    };
+    let cross_terms = {
+        let x: Sparse<Fp, 1, 2> = {
+            let x: Result<Sparse<Fp, 1, 2>, String> = x.clone().into();
+            x.unwrap()
+        };
+        x.compute_cross_terms_scaled(
+            &eval1[0..1].try_into().unwrap(),
+            &eval2[0..1].try_into().unwrap(),
+            u1,
+            u2,
+            scalar1,
+            scalar2,
+        )
+    };
+    assert_eq!(cross_terms, cross_terms_scaled_p1);
+}
+
+#[test]
+fn test_cross_terms_scaled() {
+    let mut rng = o1_utils::tests::make_test_rng(None);
+    let p1 = unsafe { Sparse::<Fp, 4, 2>::random(&mut rng, None) };
+    let scaled_p1 = {
+        // Scaling variable is U. We do this by adding a new variable.
+        let scaling_variable: Sparse<Fp, 5, 3> = {
+            let mut p: Sparse<Fp, 5, 3> = Sparse::<Fp, 5, 3>::zero();
+            p.add_monomial([0, 0, 0, 0, 1], Fp::one());
+            p
+        };
+        // Simply transforming p1 in the expected degree and with the right
+        // number of variables
+        let p1 = {
+            let p1: Result<Sparse<Fp, 5, 3>, String> = p1.clone().into();
+            p1.unwrap()
+        };
+        scaling_variable.clone() * p1.clone()
+    };
+    let random_eval1: [Fp; 5] = std::array::from_fn(|_| Fp::rand(&mut rng));
+    let random_eval2: [Fp; 5] = std::array::from_fn(|_| Fp::rand(&mut rng));
+    let scalar1 = random_eval1[4];
+    let scalar2 = random_eval2[4];
+    let u1 = Fp::rand(&mut rng);
+    let u2 = Fp::rand(&mut rng);
+    let cross_terms = {
+        let eval1: [Fp; 4] = random_eval1[0..4].try_into().unwrap();
+        let eval2: [Fp; 4] = random_eval2[0..4].try_into().unwrap();
+        p1.compute_cross_terms_scaled(&eval1, &eval2, u1, u2, scalar1, scalar2)
+    };
+    let scaled_cross_terms = scaled_p1.compute_cross_terms(&random_eval1, &random_eval2, u1, u2);
+    assert_eq!(cross_terms, scaled_cross_terms);
+}