diff --git a/src/error.rs b/src/error.rs
index 9e27319..a3a24cd 100644
--- a/src/error.rs
+++ b/src/error.rs
@@ -9,6 +9,8 @@ pub enum Error {
     NotEnoughShares,
     #[fail(display = "Signature shares contain a duplicated index")]
     DuplicateEntry,
+    #[fail(display = "The degree is too high for the coefficients to be indexed by usize.")]
+    DegreeTooHigh,
     #[fail(
         display = "Failed to `mlock` {} bytes starting at address: {}",
         n_bytes,
diff --git a/src/lib.rs b/src/lib.rs
index 8d48461..9bdc274 100644
--- a/src/lib.rs
+++ b/src/lib.rs
@@ -863,6 +863,14 @@ mod tests {
         assert_eq!(None, sk_bob.decrypt(&fake_ciphertext));
     }
 
+    #[test]
+    fn test_random_extreme_thresholds() {
+        let mut rng = rand::thread_rng();
+        let sks = SecretKeySet::random(0, &mut rng);
+        assert_eq!(0, sks.threshold());
+        assert!(SecretKeySet::try_random(usize::max_value(), &mut rng).is_err());
+    }
+
     #[test]
     fn test_threshold_enc() {
         let mut rng = rand::thread_rng();
diff --git a/src/poly.rs b/src/poly.rs
index 165d284..0760554 100644
--- a/src/poly.rs
+++ b/src/poly.rs
@@ -26,7 +26,7 @@ use pairing::bls12_381::{Fr, G1Affine, G1};
 use pairing::{CurveAffine, CurveProjective, Field};
 use rand::Rng;
 
-use error::Result;
+use error::{Error, Result};
 use into_fr::IntoFr;
 use secret::{clear_fr, ContainsSecret, MemRange, Safe};
 
@@ -357,8 +357,12 @@ impl Poly {
     ///
     /// # Errors
     ///
-    /// Returns an `Error::MlockFailed` if we have reached the systems's locked memory limit.
+    /// Returns an `Error::MlockFailed` if we have reached the systems's locked memory limit or if
+    /// the degree is `usize::max_value()`.
     pub fn try_random<R: Rng>(degree: usize, rng: &mut R) -> Result<Self> {
+        if degree == usize::max_value() {
+            return Err(Error::DegreeTooHigh);
+        }
         let coeff: Vec<Fr> = (0..=degree).map(|_| rng.gen()).collect();
         Poly::try_from(coeff)
     }
@@ -744,9 +748,12 @@ impl BivarPoly {
     ///
     /// Returns an `Error::MlockFailed` if we have reached the systems's locked memory limit.
     pub fn try_random<R: Rng>(degree: usize, rng: &mut R) -> Result<Self> {
+        let len = coeff_pos(degree, degree)
+            .and_then(|l| l.checked_add(1))
+            .ok_or(Error::DegreeTooHigh)?;
         let poly = BivarPoly {
             degree,
-            coeff: (0..coeff_pos(degree + 1, 0)).map(|_| rng.gen()).collect(),
+            coeff: (0..len).map(|_| rng.gen()).collect(),
         };
         poly.mlock_secret()?;
         Ok(poly)
@@ -765,7 +772,8 @@ impl BivarPoly {
         let mut result = Fr::zero();
         for (i, x_pow_i) in x_pow.into_iter().enumerate() {
             for (j, y_pow_j) in y_pow.iter().enumerate() {
-                let mut summand = self.coeff[coeff_pos(i, j)];
+                let index = coeff_pos(i, j).expect("polynomial degree too high");
+                let mut summand = self.coeff[index];
                 summand.mul_assign(&x_pow_i);
                 summand.mul_assign(y_pow_j);
                 result.add_assign(&summand);
@@ -797,7 +805,8 @@ impl BivarPoly {
                 // TODO: clear these secrets from the stack.
                 let mut result = Fr::zero();
                 for (j, x_pow_j) in x_pow.iter().enumerate() {
-                    let mut summand = self.coeff[coeff_pos(i, j)];
+                    let index = coeff_pos(i, j).expect("polynomial degree too high");
+                    let mut summand = self.coeff[index];
                     summand.mul_assign(x_pow_j);
                     result.add_assign(&summand);
                 }
@@ -832,13 +841,12 @@ impl BivarPoly {
 }
 
 /// A commitment to a symmetric bivariate polynomial.
-#[derive(Debug, Clone, Serialize, Deserialize, Eq, PartialEq)]
+#[derive(Debug, Clone, Eq, PartialEq)]
 pub struct BivarCommitment {
     /// The polynomial's degree in each of the two variables.
-    degree: usize,
+    pub(crate) degree: usize,
     /// The commitments to the coefficients.
-    #[serde(with = "super::serde_impl::projective_vec")]
-    coeff: Vec<G1>,
+    pub(crate) coeff: Vec<G1>,
 }
 
 impl Hash for BivarCommitment {
@@ -864,7 +872,8 @@ impl BivarCommitment {
         let mut result = G1::zero();
         for (i, x_pow_i) in x_pow.into_iter().enumerate() {
             for (j, y_pow_j) in y_pow.iter().enumerate() {
-                let mut summand = self.coeff[coeff_pos(i, j)];
+                let index = coeff_pos(i, j).expect("polynomial degree too high");
+                let mut summand = self.coeff[index];
                 summand.mul_assign(x_pow_i);
                 summand.mul_assign(*y_pow_j);
                 result.add_assign(&summand);
@@ -880,7 +889,8 @@ impl BivarCommitment {
             .map(|i| {
                 let mut result = G1::zero();
                 for (j, x_pow_j) in x_pow.iter().enumerate() {
-                    let mut summand = self.coeff[coeff_pos(i, j)];
+                    let index = coeff_pos(i, j).expect("polynomial degree too high");
+                    let mut summand = self.coeff[index];
                     summand.mul_assign(*x_pow_j);
                     result.add_assign(&summand);
                 }
@@ -907,14 +917,12 @@ fn powers<T: IntoFr>(into_x: T, degree: usize) -> Vec<Fr> {
 }
 
 /// Returns the position of coefficient `(i, j)` in the vector describing a symmetric bivariate
-/// polynomial.
-fn coeff_pos(i: usize, j: usize) -> usize {
+/// polynomial. If `i` or `j` are too large to represent the position as a `usize`, `None` is
+/// returned.
+pub(crate) fn coeff_pos(i: usize, j: usize) -> Option<usize> {
     // Since the polynomial is symmetric, we can order such that `j >= i`.
-    if j >= i {
-        j * (j + 1) / 2 + i
-    } else {
-        i * (i + 1) / 2 + j
-    }
+    let (j, i) = if j >= i { (j, i) } else { (i, j) };
+    i.checked_add(j.checked_mul(j.checked_add(1)?)? / 2)
 }
 
 #[cfg(test)]
@@ -932,7 +940,7 @@ mod tests {
         let mut i = 0;
         let mut j = 0;
         for n in 0..100 {
-            assert_eq!(n, coeff_pos(i, j));
+            assert_eq!(Some(n), coeff_pos(i, j));
             if i >= j {
                 j += 1;
                 i = 0;
@@ -940,6 +948,8 @@ mod tests {
                 i += 1;
             }
         }
+        let too_large = 1 << (0usize.count_zeros() / 2);
+        assert_eq!(None, coeff_pos(0, too_large));
     }
 
     #[test]
diff --git a/src/serde_impl.rs b/src/serde_impl.rs
index 0078101..ae1f5fd 100644
--- a/src/serde_impl.rs
+++ b/src/serde_impl.rs
@@ -1,3 +1,45 @@
+use std::borrow::Cow;
+
+use pairing::bls12_381::G1;
+use serde::de::Error as DeserializeError;
+use serde::{Deserialize, Deserializer, Serialize, Serializer};
+
+use poly::{coeff_pos, BivarCommitment};
+
+const ERR_DEG: &str = "commitment degree does not match coefficients";
+
+/// A type with the same content as `BivarCommitment`, but that has not been validated yet.
+#[derive(Serialize, Deserialize)]
+struct WireBivarCommitment<'a> {
+    /// The polynomial's degree in each of the two variables.
+    degree: usize,
+    /// The commitments to the coefficients.
+    #[serde(with = "projective_vec")]
+    coeff: Cow<'a, [G1]>,
+}
+
+impl Serialize for BivarCommitment {
+    fn serialize<S: Serializer>(&self, s: S) -> Result<S::Ok, S::Error> {
+        WireBivarCommitment {
+            degree: self.degree,
+            coeff: Cow::Borrowed(&self.coeff),
+        }.serialize(s)
+    }
+}
+
+impl<'de> Deserialize<'de> for BivarCommitment {
+    fn deserialize<D: Deserializer<'de>>(d: D) -> Result<Self, D::Error> {
+        let WireBivarCommitment { degree, coeff } = Deserialize::deserialize(d)?;
+        if coeff_pos(degree, degree).and_then(|l| l.checked_add(1)) != Some(coeff.len()) {
+            return Err(D::Error::custom(ERR_DEG));
+        }
+        Ok(BivarCommitment {
+            degree,
+            coeff: coeff.into(),
+        })
+    }
+}
+
 /// Serialization and deserialization of a group element's compressed representation.
 pub mod projective {
     use pairing::{CurveAffine, CurveProjective, EncodedPoint};
@@ -34,6 +76,7 @@ pub mod projective {
 /// Serialization and deserialization of vectors of projective curve elements.
 pub mod projective_vec {
     use std::borrow::Borrow;
+    use std::iter::FromIterator;
     use std::marker::PhantomData;
 
     use pairing::CurveProjective;
@@ -62,19 +105,21 @@ pub mod projective_vec {
         }
     }
 
-    pub fn serialize<S, C>(vec: &[C], s: S) -> Result<S::Ok, S::Error>
+    pub fn serialize<S, C, T>(vec: T, s: S) -> Result<S::Ok, S::Error>
     where
         S: Serializer,
         C: CurveProjective,
+        T: AsRef<[C]>,
     {
-        let wrap_vec: Vec<CurveWrap<C, &C>> = vec.iter().map(CurveWrap::new).collect();
+        let wrap_vec: Vec<CurveWrap<C, &C>> = vec.as_ref().iter().map(CurveWrap::new).collect();
         wrap_vec.serialize(s)
     }
 
-    pub fn deserialize<'de, D, C>(d: D) -> Result<Vec<C>, D::Error>
+    pub fn deserialize<'de, D, C, T>(d: D) -> Result<T, D::Error>
     where
         D: Deserializer<'de>,
         C: CurveProjective,
+        T: FromIterator<C>,
     {
         let wrap_vec = <Vec<CurveWrap<C, C>>>::deserialize(d)?;
         Ok(wrap_vec.into_iter().map(|CurveWrap(c, _)| c).collect())
@@ -156,6 +201,8 @@ mod tests {
     use pairing::Engine;
     use rand::{self, Rng};
 
+    use poly::BivarPoly;
+
     #[derive(Debug, Serialize, Deserialize)]
     pub struct Vecs<E: Engine> {
         #[serde(with = "super::projective_vec")]
@@ -181,4 +228,16 @@ mod tests {
         let de_vecs = bincode::deserialize(&ser_vecs).expect("deserialize vecs");
         assert_eq!(vecs, de_vecs);
     }
+
+    #[test]
+    fn bivar_commitment() {
+        let mut rng = rand::thread_rng();
+        for deg in 1..8 {
+            let poly = BivarPoly::random(deg, &mut rng);
+            let comm = poly.commitment();
+            let ser_comm = bincode::serialize(&comm).expect("serialize commitment");
+            let de_comm = bincode::deserialize(&ser_comm).expect("deserialize commitment");
+            assert_eq!(comm, de_comm);
+        }
+    }
 }