diff --git a/estimator/lwe_dual.py b/estimator/lwe_dual.py
index 6ce6957..0344f7f 100644
--- a/estimator/lwe_dual.py
+++ b/estimator/lwe_dual.py
@@ -60,15 +60,12 @@ def dual_reduce(
             )
 
         # Compute new secret distribution
-        if params.Xs.is_sparse:
+        if type(params.Xs) is SparseTernary:
             h = params.Xs.hamming_weight
             if not 0 <= h1 <= h:
                 raise OutOfBoundsError(f"Splitting weight {h1} must be between 0 and h={h}.")
-            # assuming the non-zero entries are uniform
-            p = h1 / 2
-            red_Xs = SparseTernary(params.n - zeta, h / 2 - p)
-            slv_Xs = SparseTernary(zeta, p)
-
+            # split the +1 and -1 entries in a balanced way.
+            slv_Xs, red_Xs = params.Xs.split_balanced(zeta, h1)
             if h1 == h:
                 # no reason to do lattice reduction if we assume
                 # that the hw on the reduction part is 0
@@ -176,7 +173,7 @@ def cost(
         Logging.log("dual", log_level, f"{repr(cost)}")
 
         rep = 1
-        if params.Xs.is_sparse:
+        if type(params.Xs) is SparseTernary:
             h = params.Xs.hamming_weight
             probability = RR(prob_drop(params.n, h, zeta, h1))
             rep = prob_amplify(success_probability, probability)
@@ -313,7 +310,7 @@ def f(beta):
             beta = cost["beta"]
 
         cost["zeta"] = zeta
-        if params.Xs.is_sparse:
+        if type(params.Xs) is SparseTernary:
             cost["h1"] = h1
         return cost
 
@@ -428,7 +425,7 @@ def __call__(
 
         params = params.normalize()
 
-        if params.Xs.is_sparse:
+        if type(params.Xs) is SparseTernary:
             Cost.register_impermanent(h1=False)
 
             def _optimize_blocksize(
diff --git a/estimator/nd.py b/estimator/nd.py
index 3f332c5..ab195b3 100644
--- a/estimator/nd.py
+++ b/estimator/nd.py
@@ -423,6 +423,27 @@ def resize(self, new_n):
         """
         return SparseTernary(new_n, self.p, self.m)
 
+    def split_balanced(self, new_n, new_hw=None):
+        """
+        Split the +1 and -1 entries in a balanced way, and return 2 SparseTernary distributions:
+        one of dimension `new_n` and the other of dimension `n - new_n`.
+
+        :param new_n: dimension of the first noise distribution
+        :param new_hw: hamming weight of the first noise distribution. If none, we take the most likely weight.
+        :return: tuple of (SparseTernary, SparseTernary)
+        """
+        n, hw = len(self), self.hamming_weight
+        if new_hw is None:
+            # Most likely split has same density: new_hw / new_n = hw / n.
+            new_hw = int(round(hw * new_n / n))
+
+        new_p = int(round((new_hw * self.p) / hw))
+        new_m = new_hw - new_p
+        return (
+            SparseTernary(new_n, new_p, new_m),
+            SparseTernary(n - new_n, self.p - new_p, self.m - new_m)
+        )
+
     @property
     def hamming_weight(self):
         return self.p + self.m