diff --git a/qualtran/bloqs/gf_poly_arithmetic/gf2_poly_mul.py b/qualtran/bloqs/gf_poly_arithmetic/gf2_poly_mul.py
new file mode 100644
index 000000000..680709896
--- /dev/null
+++ b/qualtran/bloqs/gf_poly_arithmetic/gf2_poly_mul.py
@@ -0,0 +1,88 @@
+#  Copyright 2025 Google LLC
+#
+#  Licensed under the Apache License, Version 2.0 (the "License");
+#  you may not use this file except in compliance with the License.
+#  You may obtain a copy of the License at
+#
+#      https://www.apache.org/licenses/LICENSE-2.0
+#
+#  Unless required by applicable law or agreed to in writing, software
+#  distributed under the License is distributed on an "AS IS" BASIS,
+#  WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+#  See the License for the specific language governing permissions and
+#  limitations under the License.
+from functools import cached_property
+from typing import Dict, Set, TYPE_CHECKING, Union
+import numpy as np
+
+import attrs
+
+from qualtran import (
+    Bloq,
+    bloq_example,
+    BloqDocSpec,
+    DecomposeTypeError,
+    QGFPoly,
+    Register,
+    Signature,
+    QGF,
+)
+from qualtran.bloqs.gf_arithmetic import GF2MulViaKaratsuba, GF2Addition
+from qualtran.bloqs.gf_poly_arithmetic.gf_poly_split_and_join import GFPolyJoin, GFPolySplit
+from qualtran.symbolics import is_symbolic
+import qualtran.bloqs.polynomials as qp
+if TYPE_CHECKING:
+    from qualtran import BloqBuilder, Soquet
+    from qualtran.resource_counting import BloqCountDictT, BloqCountT, SympySymbolAllocator
+    from qualtran.simulation.classical_sim import ClassicalValT
+
+
+@attrs.frozen
+class MultGFPolyByOnePlusXkViaKaratsuba(qp.MultiplyPolyByOnePlusXkViaKaratsuba):
+    qgf_poly: QGFPoly
+
+    @cached_property
+    def n(self):
+        return self.qgf_poly.degree + 1
+
+    @cached_property
+    def coef_dtype(self):
+        return self.qgf_poly.qgf
+    
+    def _add_coefs(self, bb: 'BloqBuilder', x: 'Soquet', y: 'Soquet'):
+        return bb.add(GF2Addition(self.coef_dtype.bitsize), x=x, y=y)
+
+    def _subtract_coefs(self, bb: 'BloqBuilder', x: 'Soquet', y: 'Soquet'):
+        return bb.add(GF2Addition(self.coef_dtype.bitsize), x=x, y=y)
+
+    def _mult_coefs(self, bb: 'BloqBuilder', x: 'Soquet', y: 'Soquet', z: 'Soquet'):
+        return bb.add(GF2MulViaKaratsuba(self.coef_dtype), x=x, y=y, z=z)
+
+    def _mult_poly(self, bb: 'BloqBuilder', f_x: np.ndarray['Soquet'], g_x: np.ndarray['Soquet'], h_x: np.ndarray['Soquet']):
+        return bb.add(MultGFPolyViaKaratsuba(self.qgf_poly), f=f_x, g=g_x, h=h_x)
+
+
+@attrs.frozen
+class MultGFPolyViaKaratsuba(qp.PolynomialMultiplicationViaKaratsuba):
+    qgf_poly: QGFPoly
+
+    @cached_property
+    def n(self):
+        return self.qgf_poly.degree + 1
+
+    @cached_property
+    def coef_dtype(self):
+        return self.qgf_poly.qgf
+    
+    def _add_coefs(self, bb: 'BloqBuilder', x: 'Soquet', y: 'Soquet'):
+        return bb.add(GF2Addition(self.coef_dtype.bitsize), x=x, y=y)
+
+    def _subtract_coefs(self, bb: 'BloqBuilder', x: 'Soquet', y: 'Soquet'):
+        return bb.add(GF2Addition(self.coef_dtype.bitsize), x=x, y=y)
+
+    def _mult_coefs(self, bb: 'BloqBuilder', x: 'Soquet', y: 'Soquet', z: 'Soquet'):
+        return bb.add(GF2MulViaKaratsuba(self.coef_dtype), x=x, y=y, z=z)
+
+    def _mult_poly(self, bb: 'BloqBuilder', m: int, f_x: np.ndarray['Soquet'], g_x: np.ndarray['Soquet'], h_x: np.ndarray['Soquet']):
+        return bb.add(MultGFPolyByOnePlusXkViaKaratsuba(attrs.evolve(self.qgf_poly, degree=m-1)), f=f_x, g=g_x, h=h_x)
+
diff --git a/qualtran/bloqs/polynomials/multiplication.py b/qualtran/bloqs/polynomials/multiplication.py
new file mode 100644
index 000000000..0db2c67a8
--- /dev/null
+++ b/qualtran/bloqs/polynomials/multiplication.py
@@ -0,0 +1,274 @@
+#  Copyright 2025 Google LLC
+#
+#  Licensed under the Apache License, Version 2.0 (the "License");
+#  you may not use this file except in compliance with the License.
+#  You may obtain a copy of the License at
+#
+#      https://www.apache.org/licenses/LICENSE-2.0
+#
+#  Unless required by applicable law or agreed to in writing, software
+#  distributed under the License is distributed on an "AS IS" BASIS,
+#  WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+#  See the License for the specific language governing permissions and
+#  limitations under the License.
+
+import abc
+from functools import cached_property
+from typing import Dict, Optional, Sequence, Set, TYPE_CHECKING, Union
+
+import attrs
+import galois
+import numpy as np
+import sympy
+from galois import GF, Poly
+
+from qualtran import (
+    Bloq,
+    bloq_example,
+    BloqDocSpec,
+    DecomposeTypeError,
+    QBit,
+    QGF,
+    Register,
+    Side,
+    Signature,
+    QDType
+)
+from qualtran.bloqs.basic_gates import CNOT, Toffoli
+from qualtran.symbolics import ceil, is_symbolic, log2, Shaped, SymbolicInt
+
+if TYPE_CHECKING:
+    from qualtran import BloqBuilder, Soquet, SoquetT
+    from qualtran.resource_counting import BloqCountDictT, BloqCountT, SympySymbolAllocator
+    from qualtran.simulation.classical_sim import ClassicalValT
+
+
+@attrs.frozen
+class MultiplyPolyByOnePlusXkViaKaratsuba(abc.ABC, Bloq):
+    r"""Out of place multiplication of $(1 + x^k) fg$
+
+    Applies the transformation
+    $$
+    \ket{f}\ket{g}\ket{h} \rightarrow \ket{f}{\ket{g}}\ket{h \oplus (1+x^k)fg}
+    $$
+
+    Note: While this construction follows Algorithm2 of https://arxiv.org/abs/1910.02849v2,
+    it has a slight modification. Namely that the original construction doesn't work in
+    some cases where $k < n$. However reversing the order of the first set of CNOTs (line 2)
+    makes the construction work for all $k \leq n+1$.
+
+    This construction abstracts Algorithm2 to work with polynomials of any field.
+
+    Args:
+        n: The degree of the polynomial ($2^n$ is the size of the galois field).
+        k: An integer specifing the shift $1 + x^k$ (or $1 + 2^k$ for galois fields.)
+
+    Registers:
+        f: The first polynomial.
+        g: The second polyonmial.
+        h: The target polynomial.
+
+    References:
+        [Space-efficient quantum multiplication of polynomials for binary finite fields with
+            sub-quadratic Toffoli gate count](https://arxiv.org/abs/1910.02849v2) Algorithm 2
+    """
+
+    k: SymbolicInt
+    
+    def __attrs_post_init__(self):
+        if is_symbolic(self.k):
+            return
+        assert self.k > 0
+
+    @cached_property
+    @abc.abstractmethod
+    def n(self):
+        pass
+
+    @cached_property
+    @abc.abstractmethod
+    def coef_dtype(self):
+        pass
+
+
+    @cached_property
+    def l(self):
+        return max(0, 2 * self.n - self.k - 1)
+
+    @cached_property
+    def signature(self) -> 'Signature':
+        return Signature(
+            [
+                Register('f', dtype=self.coef_dtype, shape=(self.n,)),
+                Register('g', dtype=self.coef_dtype, shape=(self.n,)),
+                Register('h', dtype=self.coef_dtype, shape=(2 * self.k + self.l,)),
+            ]
+        )
+
+    @abc.abstractmethod
+    def _add_coefs(self, bb: 'BloqBuilder', x: 'Soquet', y: 'Soquet'):
+        """"""
+
+    @abc.abstractmethod
+    def _subtract_coefs(self, bb: 'BloqBuilder', x: 'Soquet', y: 'Soquet'):
+        """"""
+
+    @abc.abstractmethod
+    def _mult_coefs(self, bb: 'BloqBuilder', x: 'Soquet', y: 'Soquet', z: 'Soquet'):
+        """"""
+
+    @abc.abstractmethod
+    def _mult_poly(self, bb: 'BloqBuilder', f_x: np.ndarray['Soquet'], g_x: np.ndarray['Soquet'], h_x: np.ndarray['Soquet']):
+        """"""
+
+    def build_composite_bloq(
+        self, bb: 'BloqBuilder', f: 'SoquetT', g: 'SoquetT', h: 'SoquetT'
+    ) -> Dict[str, 'SoquetT']:
+        n = self.n
+        k = self.k
+        l = self.l
+        if is_symbolic(n) or is_symbolic(k) or is_symbolic(l):
+            raise DecomposeTypeError(f"symbolic decomposition is not supported for {self}")
+        assert isinstance(f, np.ndarray)
+        assert isinstance(g, np.ndarray)
+        assert isinstance(h, np.ndarray)
+        original = len(h)
+        if n > 1:
+            # Note: This is the reverse order of what https://arxiv.org/abs/1910.02849v2 has.
+            # The is because the reverse order to makes the construction work for k <= n+1.
+            for i in reversed(range(l)):
+                h[2 * k + i], h[k + i] = self._add_coefs(bb, h[2 * k + i], h[k + i])
+            for i in range(k):
+                h[k + i], h[i] = self._add_coefs(bb, h[k + i], h[i])
+
+            f, g, h[k : 2 * k + l] = self._mult_poly(bb, 
+                f, g, h[k : 2 * k + l]
+            )
+            for i in range(k):
+                h[k + i], h[i] = self._subtract_coefs(bb, h[k + i], h[i])
+            for i in range(l):
+                h[2 * k + i], h[k + i] = self._subtract_coefs(bb, h[2 * k + i], h[k + i])
+        else:
+            h[k], h[0] = self._add_coefs(bb, h[k], h[0])
+            (f[0], g[0]), h[k] = self._mult_coefs(bb, f[0], g[0], h[k])
+            h[k], h[0] = self._subtract_coefs(h[k], h[0])
+
+        assert len(h) == original, f'{original=} {len(h)}'
+        return {'f': f, 'g': g, 'h': h}
+
+
+@attrs.frozen
+class PolynomialMultiplicationViaKaratsuba(Bloq):
+    r"""Out of place multiplication of binary polynomial multiplication.
+
+    Applies the transformation
+    $$
+    \ket{f}\ket{g}\ket{h} \rightarrow \ket{f}{\ket{g}}\ket{h \oplus fg}
+    $$
+
+    The multiplication cost of this construction is $n^{\log_2{3}}$.
+
+    This construction abstracts Algorithm3 to work with polynomials of any field.
+
+    Args:
+        n: The degree of the polynomial ($2^n$ is the size of the galois field).
+
+    Registers:
+        f: The first polynomial.
+        g: The second polyonmial.
+        h: The target polynomial.
+
+    References:
+        [Space-efficient quantum multiplication of polynomials for binary finite fields with
+            sub-quadratic Toffoli gate count](https://arxiv.org/abs/1910.02849v2) Algorithm 3
+    """
+
+    @cached_property
+    @abc.abstractmethod
+    def n(self):
+        pass
+
+    @cached_property
+    @abc.abstractmethod
+    def coef_dtype(self):
+        pass
+
+    @cached_property
+    def signature(self) -> 'Signature':
+        return Signature(
+            [
+                Register('f', dtype=self.coef_dtype, shape=(self.n,)),
+                Register('g', dtype=self.coef_dtype, shape=(self.n,)),
+                Register('h', dtype=self.coef_dtype, shape=(2 * self.n - 1,)),
+            ]
+        )
+
+    @property
+    def k(self) -> 'SymbolicInt':
+        if isinstance(self.n, int):
+            return (self.n + 1) >> 1
+        return sympy.ceiling(self.n / 2)
+
+
+    @abc.abstractmethod
+    def _add_coefs(self, bb: 'BloqBuilder', x: 'Soquet', y: 'Soquet'):
+        """"""
+
+    @abc.abstractmethod
+    def _subtract_coefs(self, bb: 'BloqBuilder', x: 'Soquet', y: 'Soquet'):
+        """"""
+
+    @abc.abstractmethod
+    def _mult_coefs(self, bb: 'BloqBuilder', x: 'Soquet', y: 'Soquet', z: 'Soquet'):
+        """"""
+
+    @abc.abstractmethod
+    def _mult_poly(self, bb: 'BloqBuilder', m: int, f_x: np.ndarray['Soquet'], g_x: np.ndarray['Soquet'], h_x: np.ndarray['Soquet']):
+        """"""
+
+    def build_composite_bloq(
+        self, bb: 'BloqBuilder', f: 'SoquetT', g: 'SoquetT', h: 'SoquetT'
+    ) -> Dict[str, 'SoquetT']:
+        k, n = self.k, self.n
+        if is_symbolic(n) or is_symbolic(k):
+            raise DecomposeTypeError(f"symbolic decomposition is not supported for {self}")
+        assert isinstance(f, np.ndarray)
+        assert isinstance(g, np.ndarray)
+        assert isinstance(h, np.ndarray)
+
+        if n == 1:
+            (f[0], g[0]), h[0] = self._mult_coefs(f[0], g[0], h[0])
+            return {'f': f, 'g': g, 'h': h}
+
+        f[:k], g[:k], h[: 3 * k - 1] = self._mult_poly(bb, k, f[:k], g[:k], h[: 3 * k - 1])
+        w = 2 * k + max(0, 2 * (n - k) - k - 1)
+        delta = k + w - len(h)
+        if delta > 0:
+            # This happens for some values (e.g. n=3) where we need to add extra qubits that will always endup in zero state.
+            aux = bb.split(bb.allocate(delta, self.coef_dtype))
+            h = np.concatenate([h, aux])
+            assert isinstance(h, np.ndarray)
+
+        f[k:n], g[k:n], h[k : k + w] = self._mult_poly(bb, n - k, f[k:n], g[k:n], h[k : k + w]
+        )
+        if delta > 0:
+            aux = h[-delta:]
+            h = h[:-delta]
+            bb.free(bb.join(aux))
+
+        for i in range(n - k):
+            f[k + i], f[i] = self._add_coefs(bb, f[k + i], f[i])
+        for i in range(n - k):
+            g[k + i], g[i] = self._add_coefs(bb, g[k + i], g[i])
+
+        f[:k], g[:k], h[k : 3 * k - 1] = bb.add(  # type: ignore[index]
+            attrs.evolve(self, n=k), f=f[:k], g=g[:k], h=h[k : 3 * k - 1]
+        )
+
+        for i in range(n - k):
+            g[k + i], g[i] = self._subtract_coefs(bb, g[k + i], g[i])
+
+        for i in range(n - k):
+            f[k + i], f[i] = self._subtract_coefs(bb, f[k + i], f[i])
+
+        return {'f': f, 'g': g, 'h': h}