Interpolation function of the ML sumcheck verifier corrected

CHANGELOG.md

@@ -19,6 +19,9 @@
 - [\#55](https://github.com/arkworks-rs/sumcheck/pull/55) Improve the interpolation performance and avoid unnecessary state clones.
 
 ### Bug fixes
+
+- [\#75](https://github.com/arkworks-rs/sumcheck/pull/75) Correct interpolation function in the verifier
+
 ## v0.4.0
 - Change dependency to version `0.4.0` of other arkworks-rs crates.
 

src/ml_sumcheck/protocol/verifier.rs

@@ -132,9 +132,10 @@ impl<F: Field> IPForMLSumcheck<F> {
     }
 }
 
-/// interpolate a uni-variate degree-`p_i.len()-1` polynomial and evaluate this
-/// polynomial at `eval_at`:
-///   \sum_{i=0}^len p_i * (\prod_{j!=i} (eval_at - j)/(i-j))
+/// interpolate the *unique* univariate polynomial of degree *at most*
+/// p_i.len()-1 passing through the y-values in p_i at x = 0,..., p_i.len()-1
+/// and evaluate this  polynomial at `eval_at`. In other words, efficiently compute
+///  \sum_{i=0}^{len p_i - 1} p_i[i] * (\prod_{j!=i} (eval_at - j)/(i-j))
 pub(crate) fn interpolate_uni_poly<F: Field>(p_i: &[F], eval_at: F) -> F {
     let len = p_i.len();
 
@@ -143,12 +144,24 @@ pub(crate) fn interpolate_uni_poly<F: Field>(p_i: &[F], eval_at: F) -> F {
     let mut prod = eval_at;
     evals.push(eval_at);
 
-    // `prod = \prod_{j} (eval_at - j)`
-    for e in 1..len {
-        let tmp = eval_at - F::from(e as u64);
+    //`prod = \prod_{j} (eval_at - j)`
+    // we return early if 0 <= eval_at <  len, i.e. if the desired value has been passed
+    let mut check = F::zero();
+    for i in 1..len {
+        if eval_at == check {
+            return p_i[i - 1];
+        }
+        check += F::one();
+
+        let tmp = eval_at - check;
         evals.push(tmp);
         prod *= tmp;
     }
+
+    if eval_at == check {
+        return p_i[len - 1];
+    }
+
     let mut res = F::zero();
     // we want to compute \prod (j!=i) (i-j) for a given i
     //
@@ -308,5 +321,13 @@ mod test {
         let query = F::rand(&mut prng);
 
         assert_eq!(poly.evaluate(&query), interpolate_uni_poly(&evals, query));
+
+        // test interpolation when we ask for the value at an x-cordinate
+        // we are already passing, i.e. in the range 0 <= x < len(values) - 1
+        let evals = vec![0, 1, 4, 9]
+            .into_iter()
+            .map(|i| F::from(i))
+            .collect::<Vec<F>>();
+        assert_eq!(interpolate_uni_poly(&evals, F::from(3)), F::from(9));
     }
 }