@@ -1,10 +1,11 @@
 use base2k::{
-    Backend, FFT64, MatZnxDft, MatZnxDftToMut, MatZnxDftToRef, Module, Scratch, VecZnx, VecZnxDft, VecZnxDftOps, VecZnxDftToMut,
-    VecZnxDftToRef, VecZnxToMut, VecZnxToRef, ZnxInfos, ZnxZero,
+    Backend, FFT64, MatZnxDft, MatZnxDftToRef, Module, Scratch, VecZnx, VecZnxAlloc, VecZnxDft, VecZnxDftAlloc, VecZnxDftOps,
+    VecZnxDftToMut, VecZnxDftToRef, VecZnxToMut, VecZnxToRef, ZnxInfos, ZnxZero,
 };
 
 use crate::{
     grlwe::GRLWECt,
+    rgsw::RGSWCt,
     rlwe::{RLWECt, RLWECtDft},
     utils::derive_size,
 };
@@ -65,6 +66,36 @@ where
         VecZnxDft<A, B>: VecZnxDftToRef<B>;
 }
 
+pub trait ProdByScratchSpace {
+    fn prod_by_grlwe_scratch_space(module: &Module<FFT64>, lhs: usize, rhs: usize) -> usize;
+    fn prod_by_rgsw_scratch_space(module: &Module<FFT64>, lhs: usize, rhs: usize) -> usize;
+}
+
+pub trait ProdBy<D> {
+    fn prod_by_grlwe<R>(&mut self, module: &Module<FFT64>, rhs: &GRLWECt<R, FFT64>, scratch: &mut Scratch)
+    where
+        MatZnxDft<R, FFT64>: MatZnxDftToRef<FFT64>;
+
+    fn prod_by_rgsw<R>(&mut self, module: &Module<FFT64>, rhs: &RGSWCt<R, FFT64>, scratch: &mut Scratch)
+    where
+        MatZnxDft<R, FFT64>: MatZnxDftToRef<FFT64>;
+}
+
+pub trait FromProdByScratchSpace {
+    fn from_prod_by_grlwe_scratch_space(module: &Module<FFT64>, res_size: usize, lhs: usize, rhs: usize) -> usize;
+    fn from_prod_by_rgsw_scratch_space(module: &Module<FFT64>, res_size: usize, lhs: usize, rhs: usize) -> usize;
+}
+
+pub trait FromProdBy<D, L> {
+    fn from_prod_by_grlwe<R>(&mut self, module: &Module<FFT64>, lhs: &L, rhs: &GRLWECt<R, FFT64>, scratch: &mut Scratch)
+    where
+        MatZnxDft<R, FFT64>: MatZnxDftToRef<FFT64>;
+
+    fn from_prod_by_rgsw<R>(&mut self, module: &Module<FFT64>, lhs: &L, rhs: &RGSWCt<R, FFT64>, scratch: &mut Scratch)
+    where
+        MatZnxDft<R, FFT64>: MatZnxDftToRef<FFT64>;
+}
+
 pub(crate) trait MatZnxDftProducts<D, C>: Infos
 where
     MatZnxDft<C, FFT64>: MatZnxDftToRef<FFT64> + ZnxInfos,
@@ -75,6 +106,31 @@ MatZnxDft<C, FFT64>: MatZnxDftToRef<FFT64>,
         VecZnx<R>: VecZnxToMut,
         VecZnx<A>: VecZnxToRef;
 
+    fn mul_rlwe_scratch_space(module: &Module<FFT64>, res_size: usize, a_size: usize, grlwe_size: usize) -> usize;
+
+    fn mul_rlwe_inplace_scratch_space(module: &Module<FFT64>, res_size: usize, mat_size: usize) -> usize {
+        Self::mul_rlwe_scratch_space(module, res_size, res_size, mat_size)
+    }
+
+    fn mul_rlwe_dft_scratch_space(module: &Module<FFT64>, res_size: usize, a_size: usize, mat_size: usize) -> usize {
+        (Self::mul_rlwe_scratch_space(module, res_size, a_size, mat_size) | module.vec_znx_idft_tmp_bytes())
+            + module.bytes_of_vec_znx(2, a_size)
+            + module.bytes_of_vec_znx(2, res_size)
+    }
+
+    fn mul_rlwe_dft_inplace_scratch_space(module: &Module<FFT64>, res_size: usize, mat_size: usize) -> usize {
+        (Self::mul_rlwe_inplace_scratch_space(module, res_size, mat_size) | module.vec_znx_idft_tmp_bytes())
+            + module.bytes_of_vec_znx(2, res_size)
+    }
+
+    fn mul_mat_rlwe_scratch_space(module: &Module<FFT64>, res_size: usize, a_size: usize, mat_size: usize) -> usize {
+        Self::mul_rlwe_dft_inplace_scratch_space(module, res_size, mat_size) + module.bytes_of_vec_znx_dft(2, a_size)
+    }
+
+    fn mul_mat_rlwe_inplace_scratch_space(module: &Module<FFT64>, res_size: usize, mat_size: usize) -> usize {
+        Self::mul_rlwe_dft_inplace_scratch_space(module, res_size, mat_size) + module.bytes_of_vec_znx_dft(2, res_size)
+    }
+
     fn mul_rlwe_inplace<R>(&self, module: &Module<FFT64>, res: &mut RLWECt<R>, scratch: &mut Scratch)
     where
         MatZnxDft<C, FFT64>: MatZnxDftToRef<FFT64> + ZnxInfos,
@@ -132,7 +188,6 @@ }
 
     fn mul_rlwe_dft_inplace<R>(&self, module: &Module<FFT64>, res: &mut RLWECtDft<R, FFT64>, scratch: &mut Scratch)
     where
-        MatZnxDft<C, FFT64>: MatZnxDftToRef<FFT64> + ZnxInfos,
         VecZnxDft<R, FFT64>: VecZnxDftToRef<FFT64> + VecZnxDftToMut<FFT64>,
     {
         let log_base2k: usize = self.log_base2k();
@@ -160,11 +215,10 @@ module.vec_znx_dft(res, 0, &res_idft, 0);
         module.vec_znx_dft(res, 1, &res_idft, 1);
     }
 
-    fn mul_grlwe<R, A>(&self, module: &Module<FFT64>, res: &mut GRLWECt<R, FFT64>, a: &GRLWECt<A, FFT64>, scratch: &mut Scratch)
+    fn mul_mat_rlwe<R, A>(&self, module: &Module<FFT64>, res: &mut R, a: &A, scratch: &mut Scratch)
     where
-        MatZnxDft<C, FFT64>: MatZnxDftToRef<FFT64> + ZnxInfos,
-        MatZnxDft<R, FFT64>: MatZnxDftToMut<FFT64> + MatZnxDftToRef<FFT64> + ZnxInfos,
-        MatZnxDft<A, FFT64>: MatZnxDftToRef<FFT64> + ZnxInfos,
+        A: GetRow<FFT64> + Infos,
+        R: SetRow<FFT64> + Infos,
     {
         let (tmp_row_data, scratch1) = scratch.tmp_vec_znx_dft(module, 2, a.size());
 
@@ -176,22 +230,25 @@ };
 
         let min_rows: usize = res.rows().min(a.rows());
 
-        (0..min_rows).for_each(|row_i| {
-            a.get_row(module, row_i, &mut tmp_row);
-            self.mul_rlwe_dft_inplace(module, &mut tmp_row, scratch1);
-            res.set_row(module, row_i, &tmp_row);
+        (0..res.rows()).for_each(|row_i| {
+            (0..self.cols()).for_each(|col_j| {
+                a.get_row(module, row_i, col_j, &mut tmp_row);
+                self.mul_rlwe_dft_inplace(module, &mut tmp_row, scratch1);
+                res.set_row(module, row_i, col_j, &tmp_row);
+            });
         });
 
         tmp_row.data.zero();
 
         (min_rows..res.rows()).for_each(|row_i| {
-            res.set_row(module, row_i, &tmp_row);
-        })
+            (0..self.cols()).for_each(|col_j| {
+                res.set_row(module, row_i, col_j, &tmp_row);
+            });
+        });
     }
 
-    fn mul_grlwe_inplace<R>(&self, module: &Module<FFT64>, res: &mut R, scratch: &mut Scratch)
+    fn mul_mat_rlwe_inplace<R>(&self, module: &Module<FFT64>, res: &mut R, scratch: &mut Scratch)
     where
-        MatZnxDft<C, FFT64>: MatZnxDftToRef<FFT64> + ZnxInfos,
         R: GetRow<FFT64> + SetRow<FFT64> + Infos,
     {
         let (tmp_row_data, scratch1) = scratch.tmp_vec_znx_dft(module, 2, res.size());
@@ -202,12 +259,12 @@ log_base2k: res.log_base2k(),
             log_k: res.log_k(),
         };
 
-        (0..self.cols()).for_each(|col_j| {
-            (0..res.rows()).for_each(|row_i| {
+        (0..res.rows()).for_each(|row_i| {
+            (0..self.cols()).for_each(|col_j| {
                 res.get_row(module, row_i, col_j, &mut tmp_row);
                 self.mul_rlwe_dft_inplace(module, &mut tmp_row, scratch1);
                 res.set_row(module, row_i, col_j, &tmp_row);
             });
-        })
+        });
     }
 }

@@ -7,8 +7,9 @@ };
 use sampling::source::Source;
 
 use crate::{
-    elem::{GetRow, Infos, MatZnxDftProducts, SetRow},
+    elem::{FromProdBy, FromProdByScratchSpace, GetRow, Infos, MatZnxDftProducts, ProdBy, ProdByScratchSpace, SetRow},
     keys::SecretKeyDft,
+    rgsw::RGSWCt,
     rlwe::{RLWECt, RLWECtDft, RLWEPt},
     utils::derive_size,
 };
@@ -41,18 +42,6 @@ module.vmp_extract_row(res, self, row_i, 0);
     }
 }
 
-impl<C> GRLWECt<C, FFT64>
-where
-    MatZnxDft<C, FFT64>: MatZnxDftToMut<FFT64>,
-{
-    pub fn set_row<R>(&mut self, module: &Module<FFT64>, row_i: usize, a: &RLWECtDft<R, FFT64>)
-    where
-        VecZnxDft<R, FFT64>: VecZnxDftToRef<FFT64>,
-    {
-        module.vmp_prepare_row(self, row_i, 0, a);
-    }
-}
-
 impl<T, B: Backend> Infos for GRLWECt<T, B> {
     type Inner = MatZnxDft<T, B>;
 
@@ -94,36 +83,6 @@ + module.bytes_of_vec_znx(2, size)
             + module.bytes_of_vec_znx(1, size)
             + module.bytes_of_vec_znx_dft(2, size)
     }
-
-    pub fn mul_rlwe_scratch_space(module: &Module<FFT64>, res_size: usize, a_size: usize, grlwe_size: usize) -> usize {
-        module.bytes_of_vec_znx_dft(2, grlwe_size)
-            + (module.vec_znx_big_normalize_tmp_bytes()
-                | (module.vmp_apply_tmp_bytes(res_size, a_size, a_size, 1, 2, grlwe_size)
-                    + module.bytes_of_vec_znx_dft(1, a_size)))
-    }
-
-    pub fn mul_rlwe_inplace_scratch_space(module: &Module<FFT64>, res_size: usize, grlwe_size: usize) -> usize {
-        Self::mul_rlwe_scratch_space(module, res_size, res_size, grlwe_size)
-    }
-
-    pub fn mul_rlwe_dft_scratch_space(module: &Module<FFT64>, res_size: usize, a_size: usize, grlwe_size: usize) -> usize {
-        (Self::mul_rlwe_scratch_space(module, res_size, a_size, grlwe_size) | module.vec_znx_idft_tmp_bytes())
-            + module.bytes_of_vec_znx(2, a_size)
-            + module.bytes_of_vec_znx(2, res_size)
-    }
-
-    pub fn mul_rlwe_dft_inplace_scratch_space(module: &Module<FFT64>, res_size: usize, grlwe_size: usize) -> usize {
-        (Self::mul_rlwe_inplace_scratch_space(module, res_size, grlwe_size) | module.vec_znx_idft_tmp_bytes())
-            + module.bytes_of_vec_znx(2, res_size)
-    }
-
-    pub fn mul_grlwe_scratch_space(module: &Module<FFT64>, res_size: usize, a_size: usize, grlwe_size: usize) -> usize {
-        Self::mul_rlwe_dft_inplace_scratch_space(module, res_size, grlwe_size) + module.bytes_of_vec_znx_dft(2, a_size)
-    }
-
-    pub fn mul_grlwe_inplace_scratch_space(module: &Module<FFT64>, res_size: usize, a_size: usize, grlwe_size: usize) -> usize {
-        Self::mul_rlwe_dft_inplace_scratch_space(module, res_size, grlwe_size) + module.bytes_of_vec_znx_dft(2, a_size)
-    }
 }
 
 pub fn encrypt_grlwe_sk<C, P, S>(
@@ -209,67 +168,6 @@ encrypt_grlwe_sk(
             module, self, pt, sk_dft, source_xa, source_xe, sigma, bound, scratch,
         )
     }
-
-    pub fn mul_rlwe<R, A>(&self, module: &Module<FFT64>, res: &mut RLWECt<R>, a: &RLWECt<A>, scratch: &mut Scratch)
-    where
-        MatZnxDft<C, FFT64>: MatZnxDftToRef<FFT64>,
-        VecZnx<R>: VecZnxToMut,
-        VecZnx<A>: VecZnxToRef,
-    {
-        MatZnxDftProducts::mul_rlwe(self, module, res, a, scratch);
-    }
-
-    pub fn mul_rlwe_inplace<R>(&self, module: &Module<FFT64>, res: &mut RLWECt<R>, scratch: &mut Scratch)
-    where
-        MatZnxDft<C, FFT64>: MatZnxDftToRef<FFT64> + ZnxInfos,
-        VecZnx<R>: VecZnxToMut + VecZnxToRef,
-    {
-        MatZnxDftProducts::mul_rlwe_inplace(self, module, res, scratch);
-    }
-
-    pub fn mul_rlwe_dft<R, A>(
-        &self,
-        module: &Module<FFT64>,
-        res: &mut RLWECtDft<R, FFT64>,
-        a: &RLWECtDft<A, FFT64>,
-        scratch: &mut Scratch,
-    ) where
-        MatZnxDft<C, FFT64>: MatZnxDftToRef<FFT64> + ZnxInfos,
-        VecZnxDft<R, FFT64>: VecZnxDftToMut<FFT64> + VecZnxDftToRef<FFT64> + ZnxInfos,
-        VecZnxDft<A, FFT64>: VecZnxDftToRef<FFT64> + ZnxInfos,
-    {
-        MatZnxDftProducts::mul_rlwe_dft(self, module, res, a, scratch);
-    }
-
-    pub fn mul_rlwe_dft_inplace<R>(&self, module: &Module<FFT64>, res: &mut RLWECtDft<R, FFT64>, scratch: &mut Scratch)
-    where
-        MatZnxDft<C, FFT64>: MatZnxDftToRef<FFT64> + ZnxInfos,
-        VecZnxDft<R, FFT64>: VecZnxDftToRef<FFT64> + VecZnxDftToMut<FFT64>,
-    {
-        MatZnxDftProducts::mul_rlwe_dft_inplace(self, module, res, scratch);
-    }
-
-    pub fn mul_grlwe<R, A>(
-        &self,
-        module: &Module<FFT64>,
-        res: &mut GRLWECt<R, FFT64>,
-        a: &GRLWECt<A, FFT64>,
-        scratch: &mut Scratch,
-    ) where
-        MatZnxDft<C, FFT64>: MatZnxDftToRef<FFT64> + ZnxInfos,
-        MatZnxDft<R, FFT64>: MatZnxDftToMut<FFT64> + MatZnxDftToRef<FFT64> + ZnxInfos,
-        MatZnxDft<A, FFT64>: MatZnxDftToRef<FFT64> + ZnxInfos,
-    {
-        MatZnxDftProducts::mul_grlwe(self, module, res, a, scratch);
-    }
-
-    pub fn mul_grlwe_inplace<R>(&self, module: &Module<FFT64>, res: &mut R, scratch: &mut Scratch)
-    where
-        MatZnxDft<C, FFT64>: MatZnxDftToRef<FFT64> + ZnxInfos,
-        R: GetRow<FFT64> + SetRow<FFT64> + Infos,
-    {
-        MatZnxDftProducts::mul_grlwe_inplace(self, module, res, scratch);
-    }
 }
 
 impl<C> GetRow<FFT64> for GRLWECt<C, FFT64>
@@ -308,6 +206,13 @@ impl<C> MatZnxDftProducts<GRLWECt<C, FFT64>, C> for GRLWECt<C, FFT64>
 where
     MatZnxDft<C, FFT64>: MatZnxDftToRef<FFT64> + ZnxInfos,
 {
+    fn mul_rlwe_scratch_space(module: &Module<FFT64>, res_size: usize, a_size: usize, grlwe_size: usize) -> usize {
+        module.bytes_of_vec_znx_dft(2, grlwe_size)
+            + (module.vec_znx_big_normalize_tmp_bytes()
+                | (module.vmp_apply_tmp_bytes(res_size, a_size, a_size, 1, 2, grlwe_size)
+                    + module.bytes_of_vec_znx_dft(1, a_size)))
+    }
+
     fn mul_rlwe<R, A>(&self, module: &Module<FFT64>, res: &mut RLWECt<R>, a: &RLWECt<A>, scratch: &mut Scratch)
     where
         MatZnxDft<C, FFT64>: MatZnxDftToRef<FFT64>,
@@ -341,3 +246,80 @@ module.vec_znx_big_normalize(log_base2k, res, 0, &res_big, 0, scratch1);
         module.vec_znx_big_normalize(log_base2k, res, 1, &res_big, 1, scratch1);
     }
 }
+
+impl ProdByScratchSpace for GRLWECt<Vec<u8>, FFT64> {
+    fn prod_by_grlwe_scratch_space(module: &Module<FFT64>, lhs: usize, rhs: usize) -> usize {
+        <GRLWECt<Vec<u8>, FFT64> as MatZnxDftProducts<GRLWECt<Vec<u8>, FFT64>, Vec<u8>>>::mul_mat_rlwe_inplace_scratch_space(
+            module, lhs, rhs,
+        )
+    }
+
+    fn prod_by_rgsw_scratch_space(module: &Module<FFT64>, lhs: usize, rhs: usize) -> usize {
+        <RGSWCt<Vec<u8>, FFT64> as MatZnxDftProducts<RGSWCt<Vec<u8>, FFT64>, Vec<u8>>>::mul_mat_rlwe_inplace_scratch_space(
+            module, lhs, rhs,
+        )
+    }
+}
+
+impl FromProdByScratchSpace for GRLWECt<Vec<u8>, FFT64> {
+    fn from_prod_by_grlwe_scratch_space(module: &Module<FFT64>, res_size: usize, lhs: usize, rhs: usize) -> usize {
+        <GRLWECt<Vec<u8>, FFT64> as MatZnxDftProducts<GRLWECt<Vec<u8>, FFT64>, Vec<u8>>>::mul_mat_rlwe_scratch_space(
+            module, res_size, lhs, rhs,
+        )
+    }
+
+    fn from_prod_by_rgsw_scratch_space(module: &Module<FFT64>, res_size: usize, lhs: usize, rhs: usize) -> usize {
+        <RGSWCt<Vec<u8>, FFT64> as MatZnxDftProducts<RGSWCt<Vec<u8>, FFT64>, Vec<u8>>>::mul_mat_rlwe_scratch_space(
+            module, res_size, lhs, rhs,
+        )
+    }
+}
+
+impl<MUT> ProdBy<GRLWECt<MUT, FFT64>> for GRLWECt<MUT, FFT64>
+where
+    GRLWECt<MUT, FFT64>: GetRow<FFT64> + SetRow<FFT64> + Infos,
+{
+    fn prod_by_grlwe<R>(&mut self, module: &Module<FFT64>, rhs: &GRLWECt<R, FFT64>, scratch: &mut Scratch)
+    where
+        MatZnxDft<R, FFT64>: MatZnxDftToRef<FFT64>,
+    {
+        rhs.mul_mat_rlwe_inplace(module, self, scratch);
+    }
+
+    fn prod_by_rgsw<R>(&mut self, module: &Module<FFT64>, rhs: &RGSWCt<R, FFT64>, scratch: &mut Scratch)
+    where
+        MatZnxDft<R, FFT64>: MatZnxDftToRef<FFT64>,
+    {
+        rhs.mul_mat_rlwe_inplace(module, self, scratch);
+    }
+}
+
+impl<MUT, REF> FromProdBy<GRLWECt<MUT, FFT64>, GRLWECt<REF, FFT64>> for GRLWECt<MUT, FFT64>
+where
+    GRLWECt<MUT, FFT64>: GetRow<FFT64> + SetRow<FFT64> + Infos,
+    GRLWECt<REF, FFT64>: GetRow<FFT64> + Infos,
+{
+    fn from_prod_by_grlwe<R>(
+        &mut self,
+        module: &Module<FFT64>,
+        lhs: &GRLWECt<REF, FFT64>,
+        rhs: &GRLWECt<R, FFT64>,
+        scratch: &mut Scratch,
+    ) where
+        MatZnxDft<R, FFT64>: MatZnxDftToRef<FFT64>,
+    {
+        rhs.mul_mat_rlwe(module, self, lhs, scratch);
+    }
+
+    fn from_prod_by_rgsw<R>(
+        &mut self,
+        module: &Module<FFT64>,
+        lhs: &GRLWECt<REF, FFT64>,
+        rhs: &RGSWCt<R, FFT64>,
+        scratch: &mut Scratch,
+    ) where
+        MatZnxDft<R, FFT64>: MatZnxDftToRef<FFT64>,
+    {
+        rhs.mul_mat_rlwe(module, self, lhs, scratch);
+    }
+}

@@ -7,7 +7,7 @@ };
 use sampling::source::Source;
 
 use crate::{
-    elem::{GetRow, Infos, MatZnxDftProducts, SetRow},
+    elem::{FromProdBy, FromProdByScratchSpace, GetRow, Infos, MatZnxDftProducts, ProdBy, ProdByScratchSpace, SetRow},
     grlwe::GRLWECt,
     keys::SecretKeyDft,
     rlwe::{RLWECt, RLWECtDft, RLWEPt, encrypt_rlwe_sk},
@@ -71,43 +71,6 @@ + module.bytes_of_vec_znx(2, size)
             + module.bytes_of_vec_znx(1, size)
             + module.bytes_of_vec_znx_dft(2, size)
     }
-
-    pub fn mul_rlwe_scratch_space(module: &Module<FFT64>, res_size: usize, a_size: usize, rgsw_size: usize) -> usize {
-        module.bytes_of_vec_znx_dft(2, rgsw_size)
-            + ((module.bytes_of_vec_znx_dft(2, a_size) + module.vmp_apply_tmp_bytes(res_size, a_size, a_size, 2, 2, rgsw_size))
-                | module.vec_znx_big_normalize_tmp_bytes())
-    }
-
-    pub fn mul_rlwe_inplace_scratch_space(module: &Module<FFT64>, res_size: usize, rgsw_size: usize) -> usize {
-        Self::mul_rlwe_scratch_space(module, res_size, res_size, rgsw_size)
-    }
-
-    pub fn mul_rlwe_dft_scratch_space(module: &Module<FFT64>, res_size: usize, a_size: usize, grlwe_size: usize) -> usize {
-        (Self::mul_rlwe_scratch_space(module, res_size, a_size, grlwe_size) | module.vec_znx_idft_tmp_bytes())
-            + module.bytes_of_vec_znx(2, a_size)
-            + module.bytes_of_vec_znx(2, res_size)
-    }
-
-    pub fn mul_rlwe_dft_inplace_scratch_space(module: &Module<FFT64>, res_size: usize, grlwe_size: usize) -> usize {
-        (Self::mul_rlwe_inplace_scratch_space(module, res_size, grlwe_size) | module.vec_znx_idft_tmp_bytes())
-            + module.bytes_of_vec_znx(2, res_size)
-    }
-
-    pub fn mul_grlwe_scratch_space(module: &Module<FFT64>, res_size: usize, a_size: usize, grlwe_size: usize) -> usize {
-        Self::mul_rlwe_dft_inplace_scratch_space(module, res_size, grlwe_size) + module.bytes_of_vec_znx_dft(2, a_size)
-    }
-
-    pub fn mul_grlwe_inplace_scratch_space(module: &Module<FFT64>, res_size: usize, a_size: usize, grlwe_size: usize) -> usize {
-        Self::mul_rlwe_dft_inplace_scratch_space(module, res_size, grlwe_size) + module.bytes_of_vec_znx_dft(2, a_size)
-    }
-
-    pub fn mul_rgsw_scratch_space(module: &Module<FFT64>, res_size: usize, a_size: usize, grlwe_size: usize) -> usize {
-        Self::mul_rlwe_dft_inplace_scratch_space(module, res_size, grlwe_size) + module.bytes_of_vec_znx_dft(2, a_size)
-    }
-
-    pub fn mul_rgsw_inplace_scratch_space(module: &Module<FFT64>, res_size: usize, a_size: usize, grlwe_size: usize) -> usize {
-        Self::mul_rlwe_dft_inplace_scratch_space(module, res_size, grlwe_size) + module.bytes_of_vec_znx_dft(2, a_size)
-    }
 }
 
 pub fn encrypt_rgsw_sk<C, P, S>(
@@ -195,67 +158,6 @@ encrypt_rgsw_sk(
             module, self, pt, sk_dft, source_xa, source_xe, sigma, bound, scratch,
         )
     }
-
-    pub fn mul_rlwe<R, A>(&self, module: &Module<FFT64>, res: &mut RLWECt<R>, a: &RLWECt<A>, scratch: &mut Scratch)
-    where
-        MatZnxDft<C, FFT64>: MatZnxDftToRef<FFT64>,
-        VecZnx<R>: VecZnxToMut,
-        VecZnx<A>: VecZnxToRef,
-    {
-        MatZnxDftProducts::mul_rlwe(self, module, res, a, scratch);
-    }
-
-    pub fn mul_rlwe_inplace<R>(&self, module: &Module<FFT64>, res: &mut RLWECt<R>, scratch: &mut Scratch)
-    where
-        MatZnxDft<C, FFT64>: MatZnxDftToRef<FFT64> + ZnxInfos,
-        VecZnx<R>: VecZnxToMut + VecZnxToRef,
-    {
-        MatZnxDftProducts::mul_rlwe_inplace(self, module, res, scratch);
-    }
-
-    pub fn mul_rlwe_dft<R, A>(
-        &self,
-        module: &Module<FFT64>,
-        res: &mut RLWECtDft<R, FFT64>,
-        a: &RLWECtDft<A, FFT64>,
-        scratch: &mut Scratch,
-    ) where
-        MatZnxDft<C, FFT64>: MatZnxDftToRef<FFT64> + ZnxInfos,
-        VecZnxDft<R, FFT64>: VecZnxDftToMut<FFT64> + VecZnxDftToRef<FFT64> + ZnxInfos,
-        VecZnxDft<A, FFT64>: VecZnxDftToRef<FFT64> + ZnxInfos,
-    {
-        MatZnxDftProducts::mul_rlwe_dft(self, module, res, a, scratch);
-    }
-
-    pub fn mul_rlwe_dft_inplace<R>(&self, module: &Module<FFT64>, res: &mut RLWECtDft<R, FFT64>, scratch: &mut Scratch)
-    where
-        MatZnxDft<C, FFT64>: MatZnxDftToRef<FFT64> + ZnxInfos,
-        VecZnxDft<R, FFT64>: VecZnxDftToRef<FFT64> + VecZnxDftToMut<FFT64>,
-    {
-        MatZnxDftProducts::mul_rlwe_dft_inplace(self, module, res, scratch);
-    }
-
-    pub fn mul_grlwe<R, A>(
-        &self,
-        module: &Module<FFT64>,
-        res: &mut GRLWECt<R, FFT64>,
-        a: &GRLWECt<A, FFT64>,
-        scratch: &mut Scratch,
-    ) where
-        MatZnxDft<C, FFT64>: MatZnxDftToRef<FFT64> + ZnxInfos,
-        MatZnxDft<R, FFT64>: MatZnxDftToMut<FFT64> + MatZnxDftToRef<FFT64> + ZnxInfos,
-        MatZnxDft<A, FFT64>: MatZnxDftToRef<FFT64> + ZnxInfos,
-    {
-        MatZnxDftProducts::mul_grlwe(self, module, res, a, scratch);
-    }
-
-    pub fn mul_grlwe_inplace<R>(&self, module: &Module<FFT64>, res: &mut R, scratch: &mut Scratch)
-    where
-        MatZnxDft<C, FFT64>: MatZnxDftToRef<FFT64> + ZnxInfos,
-        R: GetRow<FFT64> + SetRow<FFT64> + Infos,
-    {
-        MatZnxDftProducts::mul_grlwe_inplace(self, module, res, scratch);
-    }
 }
 
 impl<C> GetRow<FFT64> for RGSWCt<C, FFT64>
@@ -286,6 +188,12 @@ impl<C> MatZnxDftProducts<RGSWCt<C, FFT64>, C> for RGSWCt<C, FFT64>
 where
     MatZnxDft<C, FFT64>: MatZnxDftToRef<FFT64> + ZnxInfos,
 {
+    fn mul_rlwe_scratch_space(module: &Module<FFT64>, res_size: usize, a_size: usize, rgsw_size: usize) -> usize {
+        module.bytes_of_vec_znx_dft(2, rgsw_size)
+            + ((module.bytes_of_vec_znx_dft(2, a_size) + module.vmp_apply_tmp_bytes(res_size, a_size, a_size, 2, 2, rgsw_size))
+                | module.vec_znx_big_normalize_tmp_bytes())
+    }
+
     fn mul_rlwe<R, A>(&self, module: &Module<FFT64>, res: &mut RLWECt<R>, a: &RLWECt<A>, scratch: &mut Scratch)
     where
         MatZnxDft<C, FFT64>: MatZnxDftToRef<FFT64>,
@@ -318,3 +226,80 @@ module.vec_znx_big_normalize(log_base2k, res, 0, &res_big, 0, scratch1);
         module.vec_znx_big_normalize(log_base2k, res, 1, &res_big, 1, scratch1);
     }
 }
+
+impl ProdByScratchSpace for RGSWCt<Vec<u8>, FFT64> {
+    fn prod_by_grlwe_scratch_space(module: &Module<FFT64>, lhs: usize, rhs: usize) -> usize {
+        <GRLWECt<Vec<u8>, FFT64> as MatZnxDftProducts<GRLWECt<Vec<u8>, FFT64>, Vec<u8>>>::mul_mat_rlwe_inplace_scratch_space(
+            module, lhs, rhs,
+        )
+    }
+
+    fn prod_by_rgsw_scratch_space(module: &Module<FFT64>, lhs: usize, rhs: usize) -> usize {
+        <RGSWCt<Vec<u8>, FFT64> as MatZnxDftProducts<RGSWCt<Vec<u8>, FFT64>, Vec<u8>>>::mul_mat_rlwe_inplace_scratch_space(
+            module, lhs, rhs,
+        )
+    }
+}
+
+impl FromProdByScratchSpace for RGSWCt<Vec<u8>, FFT64> {
+    fn from_prod_by_grlwe_scratch_space(module: &Module<FFT64>, res_size: usize, lhs: usize, rhs: usize) -> usize {
+        <GRLWECt<Vec<u8>, FFT64> as MatZnxDftProducts<GRLWECt<Vec<u8>, FFT64>, Vec<u8>>>::mul_mat_rlwe_scratch_space(
+            module, res_size, lhs, rhs,
+        )
+    }
+
+    fn from_prod_by_rgsw_scratch_space(module: &Module<FFT64>, res_size: usize, lhs: usize, rhs: usize) -> usize {
+        <RGSWCt<Vec<u8>, FFT64> as MatZnxDftProducts<RGSWCt<Vec<u8>, FFT64>, Vec<u8>>>::mul_mat_rlwe_scratch_space(
+            module, res_size, lhs, rhs,
+        )
+    }
+}
+
+impl<MUT> ProdBy<RGSWCt<MUT, FFT64>> for RGSWCt<MUT, FFT64>
+where
+    RGSWCt<MUT, FFT64>: GetRow<FFT64> + SetRow<FFT64> + Infos,
+{
+    fn prod_by_grlwe<R>(&mut self, module: &Module<FFT64>, rhs: &GRLWECt<R, FFT64>, scratch: &mut Scratch)
+    where
+        MatZnxDft<R, FFT64>: MatZnxDftToRef<FFT64>,
+    {
+        rhs.mul_mat_rlwe_inplace(module, self, scratch);
+    }
+
+    fn prod_by_rgsw<R>(&mut self, module: &Module<FFT64>, rhs: &RGSWCt<R, FFT64>, scratch: &mut Scratch)
+    where
+        MatZnxDft<R, FFT64>: MatZnxDftToRef<FFT64>,
+    {
+        rhs.mul_mat_rlwe_inplace(module, self, scratch);
+    }
+}
+
+impl<MUT, REF> FromProdBy<RGSWCt<MUT, FFT64>, RGSWCt<REF, FFT64>> for RGSWCt<MUT, FFT64>
+where
+    RGSWCt<MUT, FFT64>: GetRow<FFT64> + SetRow<FFT64> + Infos,
+    RGSWCt<REF, FFT64>: GetRow<FFT64> + Infos,
+{
+    fn from_prod_by_grlwe<R>(
+        &mut self,
+        module: &Module<FFT64>,
+        lhs: &RGSWCt<REF, FFT64>,
+        rhs: &GRLWECt<R, FFT64>,
+        scratch: &mut Scratch,
+    ) where
+        MatZnxDft<R, FFT64>: MatZnxDftToRef<FFT64>,
+    {
+        rhs.mul_mat_rlwe(module, self, lhs, scratch);
+    }
+
+    fn from_prod_by_rgsw<R>(
+        &mut self,
+        module: &Module<FFT64>,
+        lhs: &RGSWCt<REF, FFT64>,
+        rhs: &RGSWCt<R, FFT64>,
+        scratch: &mut Scratch,
+    ) where
+        MatZnxDft<R, FFT64>: MatZnxDftToRef<FFT64>,
+    {
+        rhs.mul_mat_rlwe(module, self, lhs, scratch);
+    }
+}

@@ -6,9 +6,10 @@ };
 use sampling::source::Source;
 
 use crate::{
-    elem::Infos,
+    elem::{FromProdBy, FromProdByScratchSpace, Infos, MatZnxDftProducts, ProdBy, ProdByScratchSpace},
     grlwe::GRLWECt,
     keys::{PublicKey, SecretDistribution, SecretKeyDft},
+    rgsw::RGSWCt,
     utils::derive_size,
 };
 
@@ -83,134 +84,70 @@ module.vec_znx_dft(res, 1, self, 1);
     }
 }
 
-pub struct RLWEPt<C> {
-    pub data: VecZnx<C>,
-    pub log_base2k: usize,
-    pub log_k: usize,
-}
-
-impl<T> Infos for RLWEPt<T> {
-    type Inner = VecZnx<T>;
-
-    fn inner(&self) -> &Self::Inner {
-        &self.data
+impl ProdByScratchSpace for RLWECt<Vec<u8>> {
+    fn prod_by_grlwe_scratch_space(module: &Module<FFT64>, lhs: usize, rhs: usize) -> usize {
+        <GRLWECt<Vec<u8>, FFT64> as MatZnxDftProducts<GRLWECt<Vec<u8>, FFT64>, Vec<u8>>>::mul_rlwe_inplace_scratch_space(
+            module, lhs, rhs,
+        )
     }
 
-    fn log_base2k(&self) -> usize {
-        self.log_base2k
-    }
-
-    fn log_k(&self) -> usize {
-        self.log_k
+    fn prod_by_rgsw_scratch_space(module: &Module<FFT64>, lhs: usize, rhs: usize) -> usize {
+        <RGSWCt<Vec<u8>, FFT64> as MatZnxDftProducts<RGSWCt<Vec<u8>, FFT64>, Vec<u8>>>::mul_rlwe_inplace_scratch_space(
+            module, lhs, rhs,
+        )
     }
 }
 
-impl<C> VecZnxToMut for RLWEPt<C>
-where
-    VecZnx<C>: VecZnxToMut,
-{
-    fn to_mut(&mut self) -> VecZnx<&mut [u8]> {
-        self.data.to_mut()
+impl FromProdByScratchSpace for RLWECt<Vec<u8>> {
+    fn from_prod_by_grlwe_scratch_space(module: &Module<FFT64>, res_size: usize, lhs: usize, rhs: usize) -> usize {
+        <GRLWECt<Vec<u8>, FFT64> as MatZnxDftProducts<GRLWECt<Vec<u8>, FFT64>, Vec<u8>>>::mul_rlwe_scratch_space(
+            module, res_size, lhs, rhs,
+        )
     }
-}
 
-impl<C> VecZnxToRef for RLWEPt<C>
-where
-    VecZnx<C>: VecZnxToRef,
-{
-    fn to_ref(&self) -> VecZnx<&[u8]> {
-        self.data.to_ref()
+    fn from_prod_by_rgsw_scratch_space(module: &Module<FFT64>, res_size: usize, lhs: usize, rhs: usize) -> usize {
+        <RGSWCt<Vec<u8>, FFT64> as MatZnxDftProducts<RGSWCt<Vec<u8>, FFT64>, Vec<u8>>>::mul_rlwe_scratch_space(
+            module, res_size, lhs, rhs,
+        )
     }
 }
 
-impl RLWEPt<Vec<u8>> {
-    pub fn new<B: Backend>(module: &Module<B>, log_base2k: usize, log_k: usize) -> Self {
-        Self {
-            data: module.new_vec_znx(1, derive_size(log_base2k, log_k)),
-            log_base2k: log_base2k,
-            log_k: log_k,
-        }
-    }
-}
-
-pub struct RLWECtDft<C, B: Backend> {
-    pub data: VecZnxDft<C, B>,
-    pub log_base2k: usize,
-    pub log_k: usize,
-}
-
-impl<B: Backend> RLWECtDft<Vec<u8>, B> {
-    pub fn new(module: &Module<B>, log_base2k: usize, log_k: usize) -> Self {
-        Self {
-            data: module.new_vec_znx_dft(2, derive_size(log_base2k, log_k)),
-            log_base2k: log_base2k,
-            log_k: log_k,
-        }
-    }
-}
-
-impl<T, B: Backend> Infos for RLWECtDft<T, B> {
-    type Inner = VecZnxDft<T, B>;
-
-    fn inner(&self) -> &Self::Inner {
-        &self.data
-    }
-
-    fn log_base2k(&self) -> usize {
-        self.log_base2k
-    }
-
-    fn log_k(&self) -> usize {
-        self.log_k
-    }
-}
-
-impl<C, B: Backend> VecZnxDftToMut<B> for RLWECtDft<C, B>
+impl<MUT> ProdBy<RLWECt<MUT>> for RLWECt<MUT>
 where
-    VecZnxDft<C, B>: VecZnxDftToMut<B>,
+    VecZnx<MUT>: VecZnxToMut + VecZnxToRef,
 {
-    fn to_mut(&mut self) -> VecZnxDft<&mut [u8], B> {
-        self.data.to_mut()
+    fn prod_by_grlwe<R>(&mut self, module: &Module<FFT64>, rhs: &GRLWECt<R, FFT64>, scratch: &mut Scratch)
+    where
+        MatZnxDft<R, FFT64>: MatZnxDftToRef<FFT64>,
+    {
+        rhs.mul_rlwe_inplace(module, self, scratch);
     }
-}
 
-impl<C, B: Backend> VecZnxDftToRef<B> for RLWECtDft<C, B>
-where
-    VecZnxDft<C, B>: VecZnxDftToRef<B>,
-{
-    fn to_ref(&self) -> VecZnxDft<&[u8], B> {
-        self.data.to_ref()
+    fn prod_by_rgsw<R>(&mut self, module: &Module<FFT64>, rhs: &RGSWCt<R, FFT64>, scratch: &mut Scratch)
+    where
+        MatZnxDft<R, FFT64>: MatZnxDftToRef<FFT64>,
+    {
+        rhs.mul_rlwe_inplace(module, self, scratch);
     }
 }
 
-impl<C> RLWECtDft<C, FFT64>
+impl<MUT, REF> FromProdBy<RLWECt<MUT>, RLWECt<REF>> for RLWECt<MUT>
 where
-    VecZnxDft<C, FFT64>: VecZnxDftToRef<FFT64>,
+    VecZnx<MUT>: VecZnxToMut + VecZnxToRef,
+    VecZnx<REF>: VecZnxToRef,
 {
-    #[allow(dead_code)]
-    pub(crate) fn idft_scratch_space(module: &Module<FFT64>, size: usize) -> usize {
-        module.bytes_of_vec_znx(2, size) + (module.vec_znx_big_normalize_tmp_bytes() | module.vec_znx_idft_tmp_bytes())
+    fn from_prod_by_grlwe<R>(&mut self, module: &Module<FFT64>, lhs: &RLWECt<REF>, rhs: &GRLWECt<R, FFT64>, scratch: &mut Scratch)
+    where
+        MatZnxDft<R, FFT64>: MatZnxDftToRef<FFT64>,
+    {
+        rhs.mul_rlwe(module, self, lhs, scratch);
     }
 
-    pub(crate) fn idft<R>(&self, module: &Module<FFT64>, res: &mut RLWECt<R>, scratch: &mut Scratch)
+    fn from_prod_by_rgsw<R>(&mut self, module: &Module<FFT64>, lhs: &RLWECt<REF>, rhs: &RGSWCt<R, FFT64>, scratch: &mut Scratch)
     where
-        VecZnx<R>: VecZnxToMut,
+        MatZnxDft<R, FFT64>: MatZnxDftToRef<FFT64>,
     {
-        #[cfg(debug_assertions)]
-        {
-            assert_eq!(self.cols(), 2);
-            assert_eq!(res.cols(), 2);
-            assert_eq!(self.log_base2k(), res.log_base2k())
-        }
-
-        let min_size: usize = self.size().min(res.size());
-
-        let (mut res_big, scratch1) = scratch.tmp_vec_znx_big(module, 2, min_size);
-
-        module.vec_znx_idft(&mut res_big, 0, &self.data, 0, scratch1);
-        module.vec_znx_idft(&mut res_big, 1, &self.data, 1, scratch1);
-        module.vec_znx_big_normalize(self.log_base2k(), res, 0, &res_big, 0, scratch1);
-        module.vec_znx_big_normalize(self.log_base2k(), res, 1, &res_big, 1, scratch1);
+        rhs.mul_rlwe(module, self, lhs, scratch);
     }
 }
 
@@ -390,6 +327,204 @@ )
     }
 }
 
+pub(crate) fn encrypt_rlwe_pk<C, P, S>(
+    module: &Module<FFT64>,
+    ct: &mut RLWECt<C>,
+    pt: Option<&RLWEPt<P>>,
+    pk: &PublicKey<S, FFT64>,
+    source_xu: &mut Source,
+    source_xe: &mut Source,
+    sigma: f64,
+    bound: f64,
+    scratch: &mut Scratch,
+) where
+    VecZnx<C>: VecZnxToMut + VecZnxToRef,
+    VecZnx<P>: VecZnxToRef,
+    VecZnxDft<S, FFT64>: VecZnxDftToRef<FFT64>,
+{
+    #[cfg(debug_assertions)]
+    {
+        assert_eq!(ct.log_base2k(), pk.log_base2k());
+        assert_eq!(ct.n(), module.n());
+        assert_eq!(pk.n(), module.n());
+        if let Some(pt) = pt {
+            assert_eq!(pt.log_base2k(), pk.log_base2k());
+            assert_eq!(pt.n(), module.n());
+        }
+    }
+
+    let log_base2k: usize = pk.log_base2k();
+    let size_pk: usize = pk.size();
+
+    // Generates u according to the underlying secret distribution.
+    let (mut u_dft, scratch_1) = scratch.tmp_scalar_znx_dft(module, 1);
+
+    {
+        let (mut u, _) = scratch_1.tmp_scalar_znx(module, 1);
+        match pk.dist {
+            SecretDistribution::NONE => panic!(
+                "invalid public key: SecretDistribution::NONE, ensure it has been correctly intialized through Self::generate"
+            ),
+            SecretDistribution::TernaryFixed(hw) => u.fill_ternary_hw(0, hw, source_xu),
+            SecretDistribution::TernaryProb(prob) => u.fill_ternary_prob(0, prob, source_xu),
+            SecretDistribution::ZERO => {}
+        }
+
+        module.svp_prepare(&mut u_dft, 0, &u, 0);
+    }
+
+    let (mut tmp_big, scratch_2) = scratch_1.tmp_vec_znx_big(module, 1, size_pk); // TODO optimize size (e.g. when encrypting at low homomorphic capacity)
+    let (mut tmp_dft, scratch_3) = scratch_2.tmp_vec_znx_dft(module, 1, size_pk); // TODO optimize size (e.g. when encrypting at low homomorphic capacity)
+
+    // ct[0] = pk[0] * u + m + e0
+    module.svp_apply(&mut tmp_dft, 0, &u_dft, 0, pk, 0);
+    module.vec_znx_idft_tmp_a(&mut tmp_big, 0, &mut tmp_dft, 0);
+    tmp_big.add_normal(log_base2k, 0, pk.log_k(), source_xe, sigma, bound);
+
+    if let Some(pt) = pt {
+        module.vec_znx_big_add_small_inplace(&mut tmp_big, 0, pt, 0);
+    }
+
+    module.vec_znx_big_normalize(log_base2k, ct, 0, &tmp_big, 0, scratch_3);
+
+    // ct[1] = pk[1] * u + e1
+    module.svp_apply(&mut tmp_dft, 0, &u_dft, 0, pk, 1);
+    module.vec_znx_idft_tmp_a(&mut tmp_big, 0, &mut tmp_dft, 0);
+    tmp_big.add_normal(log_base2k, 0, pk.log_k(), source_xe, sigma, bound);
+    module.vec_znx_big_normalize(log_base2k, ct, 1, &tmp_big, 0, scratch_3);
+}
+
+pub struct RLWEPt<C> {
+    pub data: VecZnx<C>,
+    pub log_base2k: usize,
+    pub log_k: usize,
+}
+
+impl<T> Infos for RLWEPt<T> {
+    type Inner = VecZnx<T>;
+
+    fn inner(&self) -> &Self::Inner {
+        &self.data
+    }
+
+    fn log_base2k(&self) -> usize {
+        self.log_base2k
+    }
+
+    fn log_k(&self) -> usize {
+        self.log_k
+    }
+}
+
+impl<C> VecZnxToMut for RLWEPt<C>
+where
+    VecZnx<C>: VecZnxToMut,
+{
+    fn to_mut(&mut self) -> VecZnx<&mut [u8]> {
+        self.data.to_mut()
+    }
+}
+
+impl<C> VecZnxToRef for RLWEPt<C>
+where
+    VecZnx<C>: VecZnxToRef,
+{
+    fn to_ref(&self) -> VecZnx<&[u8]> {
+        self.data.to_ref()
+    }
+}
+
+impl RLWEPt<Vec<u8>> {
+    pub fn new<B: Backend>(module: &Module<B>, log_base2k: usize, log_k: usize) -> Self {
+        Self {
+            data: module.new_vec_znx(1, derive_size(log_base2k, log_k)),
+            log_base2k: log_base2k,
+            log_k: log_k,
+        }
+    }
+}
+
+pub struct RLWECtDft<C, B: Backend> {
+    pub data: VecZnxDft<C, B>,
+    pub log_base2k: usize,
+    pub log_k: usize,
+}
+
+impl<B: Backend> RLWECtDft<Vec<u8>, B> {
+    pub fn new(module: &Module<B>, log_base2k: usize, log_k: usize) -> Self {
+        Self {
+            data: module.new_vec_znx_dft(2, derive_size(log_base2k, log_k)),
+            log_base2k: log_base2k,
+            log_k: log_k,
+        }
+    }
+}
+
+impl<T, B: Backend> Infos for RLWECtDft<T, B> {
+    type Inner = VecZnxDft<T, B>;
+
+    fn inner(&self) -> &Self::Inner {
+        &self.data
+    }
+
+    fn log_base2k(&self) -> usize {
+        self.log_base2k
+    }
+
+    fn log_k(&self) -> usize {
+        self.log_k
+    }
+}
+
+impl<C, B: Backend> VecZnxDftToMut<B> for RLWECtDft<C, B>
+where
+    VecZnxDft<C, B>: VecZnxDftToMut<B>,
+{
+    fn to_mut(&mut self) -> VecZnxDft<&mut [u8], B> {
+        self.data.to_mut()
+    }
+}
+
+impl<C, B: Backend> VecZnxDftToRef<B> for RLWECtDft<C, B>
+where
+    VecZnxDft<C, B>: VecZnxDftToRef<B>,
+{
+    fn to_ref(&self) -> VecZnxDft<&[u8], B> {
+        self.data.to_ref()
+    }
+}
+
+impl<C> RLWECtDft<C, FFT64>
+where
+    VecZnxDft<C, FFT64>: VecZnxDftToRef<FFT64>,
+{
+    #[allow(dead_code)]
+    pub(crate) fn idft_scratch_space(module: &Module<FFT64>, size: usize) -> usize {
+        module.bytes_of_vec_znx(2, size) + (module.vec_znx_big_normalize_tmp_bytes() | module.vec_znx_idft_tmp_bytes())
+    }
+
+    pub(crate) fn idft<R>(&self, module: &Module<FFT64>, res: &mut RLWECt<R>, scratch: &mut Scratch)
+    where
+        VecZnx<R>: VecZnxToMut,
+    {
+        #[cfg(debug_assertions)]
+        {
+            assert_eq!(self.cols(), 2);
+            assert_eq!(res.cols(), 2);
+            assert_eq!(self.log_base2k(), res.log_base2k())
+        }
+
+        let min_size: usize = self.size().min(res.size());
+
+        let (mut res_big, scratch1) = scratch.tmp_vec_znx_big(module, 2, min_size);
+
+        module.vec_znx_idft(&mut res_big, 0, &self.data, 0, scratch1);
+        module.vec_znx_idft(&mut res_big, 1, &self.data, 1, scratch1);
+        module.vec_znx_big_normalize(self.log_base2k(), res, 0, &res_big, 0, scratch1);
+        module.vec_znx_big_normalize(self.log_base2k(), res, 1, &res_big, 1, scratch1);
+    }
+}
+
 pub(crate) fn encrypt_zero_rlwe_dft_sk<C, S>(
     module: &Module<FFT64>,
     ct: &mut RLWECtDft<C, FFT64>,
@@ -528,79 +663,81 @@ ScalarZnxDft<S, FFT64>: ScalarZnxDftToRef<FFT64>,
     {
         decrypt_rlwe_dft(module, pt, self, sk_dft, scratch);
     }
+}
 
-    pub fn mul_grlwe_assign<A>(&mut self, module: &Module<FFT64>, a: &GRLWECt<A, FFT64>, scratch: &mut Scratch)
-    where
-        VecZnxDft<C, FFT64>: VecZnxDftToMut<FFT64> + VecZnxDftToRef<FFT64>,
-        MatZnxDft<A, FFT64>: MatZnxDftToRef<FFT64>,
-    {
-        a.mul_rlwe_dft_inplace(module, self, scratch);
+impl ProdByScratchSpace for RLWECtDft<Vec<u8>, FFT64> {
+    fn prod_by_grlwe_scratch_space(module: &Module<FFT64>, lhs: usize, rhs: usize) -> usize {
+        <GRLWECt<Vec<u8>, FFT64> as MatZnxDftProducts<GRLWECt<Vec<u8>, FFT64>, Vec<u8>>>::mul_rlwe_dft_inplace_scratch_space(
+            module, lhs, rhs,
+        )
+    }
+
+    fn prod_by_rgsw_scratch_space(module: &Module<FFT64>, lhs: usize, rhs: usize) -> usize {
+        <RGSWCt<Vec<u8>, FFT64> as MatZnxDftProducts<RGSWCt<Vec<u8>, FFT64>, Vec<u8>>>::mul_rlwe_dft_inplace_scratch_space(
+            module, lhs, rhs,
+        )
     }
 }
 
-pub(crate) fn encrypt_rlwe_pk<C, P, S>(
-    module: &Module<FFT64>,
-    ct: &mut RLWECt<C>,
-    pt: Option<&RLWEPt<P>>,
-    pk: &PublicKey<S, FFT64>,
-    source_xu: &mut Source,
-    source_xe: &mut Source,
-    sigma: f64,
-    bound: f64,
-    scratch: &mut Scratch,
-) where
-    VecZnx<C>: VecZnxToMut + VecZnxToRef,
-    VecZnx<P>: VecZnxToRef,
-    VecZnxDft<S, FFT64>: VecZnxDftToRef<FFT64>,
-{
-    #[cfg(debug_assertions)]
-    {
-        assert_eq!(ct.log_base2k(), pk.log_base2k());
-        assert_eq!(ct.n(), module.n());
-        assert_eq!(pk.n(), module.n());
-        if let Some(pt) = pt {
-            assert_eq!(pt.log_base2k(), pk.log_base2k());
-            assert_eq!(pt.n(), module.n());
-        }
+impl FromProdByScratchSpace for RLWECtDft<Vec<u8>, FFT64> {
+    fn from_prod_by_grlwe_scratch_space(module: &Module<FFT64>, res_size: usize, lhs: usize, rhs: usize) -> usize {
+        <GRLWECt<Vec<u8>, FFT64> as MatZnxDftProducts<GRLWECt<Vec<u8>, FFT64>, Vec<u8>>>::mul_rlwe_dft_scratch_space(
+            module, res_size, lhs, rhs,
+        )
     }
 
-    let log_base2k: usize = pk.log_base2k();
-    let size_pk: usize = pk.size();
+    fn from_prod_by_rgsw_scratch_space(module: &Module<FFT64>, res_size: usize, lhs: usize, rhs: usize) -> usize {
+        <RGSWCt<Vec<u8>, FFT64> as MatZnxDftProducts<RGSWCt<Vec<u8>, FFT64>, Vec<u8>>>::mul_rlwe_dft_scratch_space(
+            module, res_size, lhs, rhs,
+        )
+    }
+}
 
-    // Generates u according to the underlying secret distribution.
-    let (mut u_dft, scratch_1) = scratch.tmp_scalar_znx_dft(module, 1);
-
+impl<MUT> ProdBy<RLWECtDft<MUT, FFT64>> for RLWECtDft<MUT, FFT64>
+where
+    VecZnxDft<MUT, FFT64>: VecZnxDftToMut<FFT64> + VecZnxDftToRef<FFT64>,
+{
+    fn prod_by_grlwe<R>(&mut self, module: &Module<FFT64>, rhs: &GRLWECt<R, FFT64>, scratch: &mut Scratch)
+    where
+        MatZnxDft<R, FFT64>: MatZnxDftToRef<FFT64>,
     {
-        let (mut u, _) = scratch_1.tmp_scalar_znx(module, 1);
-        match pk.dist {
-            SecretDistribution::NONE => panic!(
-                "invalid public key: SecretDistribution::NONE, ensure it has been correctly intialized through Self::generate"
-            ),
-            SecretDistribution::TernaryFixed(hw) => u.fill_ternary_hw(0, hw, source_xu),
-            SecretDistribution::TernaryProb(prob) => u.fill_ternary_prob(0, prob, source_xu),
-            SecretDistribution::ZERO => {}
-        }
+        rhs.mul_rlwe_dft_inplace(module, self, scratch);
+    }
 
-        module.svp_prepare(&mut u_dft, 0, &u, 0);
+    fn prod_by_rgsw<R>(&mut self, module: &Module<FFT64>, rhs: &RGSWCt<R, FFT64>, scratch: &mut Scratch)
+    where
+        MatZnxDft<R, FFT64>: MatZnxDftToRef<FFT64>,
+    {
+        rhs.mul_rlwe_dft_inplace(module, self, scratch);
     }
+}
 
-    let (mut tmp_big, scratch_2) = scratch_1.tmp_vec_znx_big(module, 1, size_pk); // TODO optimize size (e.g. when encrypting at low homomorphic capacity)
-    let (mut tmp_dft, scratch_3) = scratch_2.tmp_vec_znx_dft(module, 1, size_pk); // TODO optimize size (e.g. when encrypting at low homomorphic capacity)
+impl<MUT, REF> FromProdBy<RLWECtDft<MUT, FFT64>, RLWECtDft<REF, FFT64>> for RLWECtDft<MUT, FFT64>
+where
+    VecZnxDft<MUT, FFT64>: VecZnxDftToMut<FFT64> + VecZnxDftToRef<FFT64>,
+    VecZnxDft<REF, FFT64>: VecZnxDftToRef<FFT64>,
+{
+    fn from_prod_by_grlwe<R>(
+        &mut self,
+        module: &Module<FFT64>,
+        lhs: &RLWECtDft<REF, FFT64>,
+        rhs: &GRLWECt<R, FFT64>,
+        scratch: &mut Scratch,
+    ) where
+        MatZnxDft<R, FFT64>: MatZnxDftToRef<FFT64>,
+    {
+        rhs.mul_rlwe_dft(module, self, lhs, scratch);
+    }
 
-    // ct[0] = pk[0] * u + m + e0
-    module.svp_apply(&mut tmp_dft, 0, &u_dft, 0, pk, 0);
-    module.vec_znx_idft_tmp_a(&mut tmp_big, 0, &mut tmp_dft, 0);
-    tmp_big.add_normal(log_base2k, 0, pk.log_k(), source_xe, sigma, bound);
-
-    if let Some(pt) = pt {
-        module.vec_znx_big_add_small_inplace(&mut tmp_big, 0, pt, 0);
+    fn from_prod_by_rgsw<R>(
+        &mut self,
+        module: &Module<FFT64>,
+        lhs: &RLWECtDft<REF, FFT64>,
+        rhs: &RGSWCt<R, FFT64>,
+        scratch: &mut Scratch,
+    ) where
+        MatZnxDft<R, FFT64>: MatZnxDftToRef<FFT64>,
+    {
+        rhs.mul_rlwe_dft(module, self, lhs, scratch);
     }
-
-    module.vec_znx_big_normalize(log_base2k, ct, 0, &tmp_big, 0, scratch_3);
-
-    // ct[1] = pk[1] * u + e1
-    module.svp_apply(&mut tmp_dft, 0, &u_dft, 0, pk, 1);
-    module.vec_znx_idft_tmp_a(&mut tmp_big, 0, &mut tmp_dft, 0);
-    tmp_big.add_normal(log_base2k, 0, pk.log_k(), source_xe, sigma, bound);
-    module.vec_znx_big_normalize(log_base2k, ct, 1, &tmp_big, 0, scratch_3);
 }

@@ -1,14 +1,14 @@
 #[cfg(test)]
 
 mod tests {
-    use base2k::{FFT64, FillUniform, Module, ScalarZnx, ScalarZnxAlloc, ScratchOwned, Stats, VecZnxOps};
+    use base2k::{FFT64, Module, ScalarZnx, ScalarZnxAlloc, ScratchOwned, Stats, VecZnxOps};
     use sampling::source::Source;
 
     use crate::{
-        elem::Infos,
+        elem::{FromProdBy, FromProdByScratchSpace, Infos, ProdBy, ProdByScratchSpace},
         grlwe::GRLWECt,
         keys::{SecretKey, SecretKeyDft},
-        rlwe::{RLWECt, RLWECtDft, RLWEPt},
+        rlwe::{RLWECtDft, RLWEPt},
         test_fft64::grlwe::noise_grlwe_rlwe_product,
     };
 
@@ -67,413 +67,7 @@ module.free();
     }
 
     #[test]
-    fn mul_rlwe() {
-        let module: Module<FFT64> = Module::<FFT64>::new(2048);
-        let log_base2k: usize = 12;
-        let log_k_grlwe: usize = 60;
-        let log_k_rlwe_in: usize = 45;
-        let log_k_rlwe_out: usize = 60;
-        let rows: usize = (log_k_rlwe_in + log_base2k - 1) / log_base2k;
-
-        let sigma: f64 = 3.2;
-        let bound: f64 = sigma * 6.0;
-
-        let mut ct_grlwe: GRLWECt<Vec<u8>, FFT64> = GRLWECt::new(&module, log_base2k, log_k_grlwe, rows);
-        let mut ct_rlwe_in: RLWECt<Vec<u8>> = RLWECt::new(&module, log_base2k, log_k_rlwe_in);
-        let mut ct_rlwe_out: RLWECt<Vec<u8>> = RLWECt::new(&module, log_base2k, log_k_rlwe_out);
-        let mut pt_want: RLWEPt<Vec<u8>> = RLWEPt::new(&module, log_base2k, log_k_rlwe_in);
-        let mut pt_have: RLWEPt<Vec<u8>> = RLWEPt::new(&module, log_base2k, log_k_rlwe_out);
-
-        let mut source_xs: Source = Source::new([0u8; 32]);
-        let mut source_xe: Source = Source::new([0u8; 32]);
-        let mut source_xa: Source = Source::new([0u8; 32]);
-
-        // Random input plaintext
-        pt_want
-            .data
-            .fill_uniform(log_base2k, 0, pt_want.size(), &mut source_xa);
-
-        let mut scratch: ScratchOwned = ScratchOwned::new(
-            GRLWECt::encrypt_sk_scratch_space(&module, ct_grlwe.size())
-                | RLWECt::decrypt_scratch_space(&module, ct_rlwe_out.size())
-                | RLWECt::encrypt_sk_scratch_space(&module, ct_rlwe_in.size())
-                | GRLWECt::mul_rlwe_scratch_space(
-                    &module,
-                    ct_rlwe_out.size(),
-                    ct_rlwe_in.size(),
-                    ct_grlwe.size(),
-                ),
-        );
-
-        let mut sk0: SecretKey<Vec<u8>> = SecretKey::new(&module);
-        sk0.fill_ternary_prob(0.5, &mut source_xs);
-
-        let mut sk0_dft: SecretKeyDft<Vec<u8>, FFT64> = SecretKeyDft::new(&module);
-        sk0_dft.dft(&module, &sk0);
-
-        let mut sk1: SecretKey<Vec<u8>> = SecretKey::new(&module);
-        sk1.fill_ternary_prob(0.5, &mut source_xs);
-
-        let mut sk1_dft: SecretKeyDft<Vec<u8>, FFT64> = SecretKeyDft::new(&module);
-        sk1_dft.dft(&module, &sk1);
-
-        ct_grlwe.encrypt_sk(
-            &module,
-            &sk0.data,
-            &sk1_dft,
-            &mut source_xa,
-            &mut source_xe,
-            sigma,
-            bound,
-            scratch.borrow(),
-        );
-
-        ct_rlwe_in.encrypt_sk(
-            &module,
-            Some(&pt_want),
-            &sk0_dft,
-            &mut source_xa,
-            &mut source_xe,
-            sigma,
-            bound,
-            scratch.borrow(),
-        );
-
-        ct_grlwe.mul_rlwe(&module, &mut ct_rlwe_out, &ct_rlwe_in, scratch.borrow());
-
-        ct_rlwe_out.decrypt(&module, &mut pt_have, &sk1_dft, scratch.borrow());
-
-        module.vec_znx_sub_ab_inplace(&mut pt_have, 0, &pt_want, 0);
-
-        let noise_have: f64 = pt_have.data.std(0, log_base2k).log2();
-        let noise_want: f64 = noise_grlwe_rlwe_product(
-            module.n() as f64,
-            log_base2k,
-            0.5,
-            0.5,
-            0f64,
-            sigma * sigma,
-            0f64,
-            log_k_rlwe_in,
-            log_k_grlwe,
-        );
-
-        assert!(
-            (noise_have - noise_want).abs() <= 0.1,
-            "{} {}",
-            noise_have,
-            noise_want
-        );
-
-        module.free();
-    }
-
-    #[test]
-    fn mul_rlwe_inplace() {
-        let module: Module<FFT64> = Module::<FFT64>::new(2048);
-        let log_base2k: usize = 12;
-        let log_k_grlwe: usize = 60;
-        let log_k_rlwe: usize = 45;
-        let rows: usize = (log_k_rlwe + log_base2k - 1) / log_base2k;
-
-        let sigma: f64 = 3.2;
-        let bound: f64 = sigma * 6.0;
-
-        let mut ct_grlwe: GRLWECt<Vec<u8>, FFT64> = GRLWECt::new(&module, log_base2k, log_k_grlwe, rows);
-        let mut ct_rlwe: RLWECt<Vec<u8>> = RLWECt::new(&module, log_base2k, log_k_rlwe);
-        let mut pt_want: RLWEPt<Vec<u8>> = RLWEPt::new(&module, log_base2k, log_k_rlwe);
-        let mut pt_have: RLWEPt<Vec<u8>> = RLWEPt::new(&module, log_base2k, log_k_rlwe);
-
-        let mut source_xs: Source = Source::new([0u8; 32]);
-        let mut source_xe: Source = Source::new([0u8; 32]);
-        let mut source_xa: Source = Source::new([0u8; 32]);
-
-        // Random input plaintext
-        pt_want
-            .data
-            .fill_uniform(log_base2k, 0, pt_want.size(), &mut source_xa);
-
-        let mut scratch: ScratchOwned = ScratchOwned::new(
-            GRLWECt::encrypt_sk_scratch_space(&module, ct_grlwe.size())
-                | RLWECt::decrypt_scratch_space(&module, ct_rlwe.size())
-                | RLWECt::encrypt_sk_scratch_space(&module, ct_rlwe.size())
-                | GRLWECt::mul_rlwe_scratch_space(&module, ct_rlwe.size(), ct_rlwe.size(), ct_grlwe.size()),
-        );
-
-        let mut sk0: SecretKey<Vec<u8>> = SecretKey::new(&module);
-        sk0.fill_ternary_prob(0.5, &mut source_xs);
-
-        let mut sk0_dft: SecretKeyDft<Vec<u8>, FFT64> = SecretKeyDft::new(&module);
-        sk0_dft.dft(&module, &sk0);
-
-        let mut sk1: SecretKey<Vec<u8>> = SecretKey::new(&module);
-        sk1.fill_ternary_prob(0.5, &mut source_xs);
-
-        let mut sk1_dft: SecretKeyDft<Vec<u8>, FFT64> = SecretKeyDft::new(&module);
-        sk1_dft.dft(&module, &sk1);
-
-        ct_grlwe.encrypt_sk(
-            &module,
-            &sk0.data,
-            &sk1_dft,
-            &mut source_xa,
-            &mut source_xe,
-            sigma,
-            bound,
-            scratch.borrow(),
-        );
-
-        ct_rlwe.encrypt_sk(
-            &module,
-            Some(&pt_want),
-            &sk0_dft,
-            &mut source_xa,
-            &mut source_xe,
-            sigma,
-            bound,
-            scratch.borrow(),
-        );
-
-        ct_grlwe.mul_rlwe_inplace(&module, &mut ct_rlwe, scratch.borrow());
-
-        ct_rlwe.decrypt(&module, &mut pt_have, &sk1_dft, scratch.borrow());
-
-        module.vec_znx_sub_ab_inplace(&mut pt_have, 0, &pt_want, 0);
-
-        let noise_have: f64 = pt_have.data.std(0, log_base2k).log2();
-        let noise_want: f64 = noise_grlwe_rlwe_product(
-            module.n() as f64,
-            log_base2k,
-            0.5,
-            0.5,
-            0f64,
-            sigma * sigma,
-            0f64,
-            log_k_rlwe,
-            log_k_grlwe,
-        );
-
-        assert!(
-            (noise_have - noise_want).abs() <= 0.1,
-            "{} {}",
-            noise_have,
-            noise_want
-        );
-
-        module.free();
-    }
-
-    #[test]
-    fn mul_rlwe_dft() {
-        let module: Module<FFT64> = Module::<FFT64>::new(2048);
-        let log_base2k: usize = 12;
-        let log_k_grlwe: usize = 60;
-        let log_k_rlwe_in: usize = 45;
-        let log_k_rlwe_out: usize = 60;
-        let rows: usize = (log_k_rlwe_in + log_base2k - 1) / log_base2k;
-
-        let sigma: f64 = 3.2;
-        let bound: f64 = sigma * 6.0;
-
-        let mut ct_grlwe: GRLWECt<Vec<u8>, FFT64> = GRLWECt::new(&module, log_base2k, log_k_grlwe, rows);
-        let mut ct_rlwe_in: RLWECt<Vec<u8>> = RLWECt::new(&module, log_base2k, log_k_rlwe_in);
-        let mut ct_rlwe_in_dft: RLWECtDft<Vec<u8>, FFT64> = RLWECtDft::new(&module, log_base2k, log_k_rlwe_in);
-        let mut ct_rlwe_out: RLWECt<Vec<u8>> = RLWECt::new(&module, log_base2k, log_k_rlwe_out);
-        let mut ct_rlwe_out_dft: RLWECtDft<Vec<u8>, FFT64> = RLWECtDft::new(&module, log_base2k, log_k_rlwe_out);
-        let mut pt_want: RLWEPt<Vec<u8>> = RLWEPt::new(&module, log_base2k, log_k_rlwe_in);
-        let mut pt_have: RLWEPt<Vec<u8>> = RLWEPt::new(&module, log_base2k, log_k_rlwe_out);
-
-        let mut source_xs: Source = Source::new([0u8; 32]);
-        let mut source_xe: Source = Source::new([0u8; 32]);
-        let mut source_xa: Source = Source::new([0u8; 32]);
-
-        // Random input plaintext
-        pt_want
-            .data
-            .fill_uniform(log_base2k, 0, pt_want.size(), &mut source_xa);
-
-        let mut scratch: ScratchOwned = ScratchOwned::new(
-            GRLWECt::encrypt_sk_scratch_space(&module, ct_grlwe.size())
-                | RLWECt::decrypt_scratch_space(&module, ct_rlwe_out.size())
-                | RLWECt::encrypt_sk_scratch_space(&module, ct_rlwe_in.size())
-                | GRLWECt::mul_rlwe_scratch_space(
-                    &module,
-                    ct_rlwe_out.size(),
-                    ct_rlwe_in.size(),
-                    ct_grlwe.size(),
-                ),
-        );
-
-        let mut sk0: SecretKey<Vec<u8>> = SecretKey::new(&module);
-        sk0.fill_ternary_prob(0.5, &mut source_xs);
-
-        let mut sk0_dft: SecretKeyDft<Vec<u8>, FFT64> = SecretKeyDft::new(&module);
-        sk0_dft.dft(&module, &sk0);
-
-        let mut sk1: SecretKey<Vec<u8>> = SecretKey::new(&module);
-        sk1.fill_ternary_prob(0.5, &mut source_xs);
-
-        let mut sk1_dft: SecretKeyDft<Vec<u8>, FFT64> = SecretKeyDft::new(&module);
-        sk1_dft.dft(&module, &sk1);
-
-        ct_grlwe.encrypt_sk(
-            &module,
-            &sk0.data,
-            &sk1_dft,
-            &mut source_xa,
-            &mut source_xe,
-            sigma,
-            bound,
-            scratch.borrow(),
-        );
-
-        ct_rlwe_in.encrypt_sk(
-            &module,
-            Some(&pt_want),
-            &sk0_dft,
-            &mut source_xa,
-            &mut source_xe,
-            sigma,
-            bound,
-            scratch.borrow(),
-        );
-
-        ct_rlwe_in.dft(&module, &mut ct_rlwe_in_dft);
-        ct_grlwe.mul_rlwe_dft(
-            &module,
-            &mut ct_rlwe_out_dft,
-            &ct_rlwe_in_dft,
-            scratch.borrow(),
-        );
-        ct_rlwe_out_dft.idft(&module, &mut ct_rlwe_out, scratch.borrow());
-
-        ct_rlwe_out.decrypt(&module, &mut pt_have, &sk1_dft, scratch.borrow());
-
-        module.vec_znx_sub_ab_inplace(&mut pt_have, 0, &pt_want, 0);
-
-        let noise_have: f64 = pt_have.data.std(0, log_base2k).log2();
-        let noise_want: f64 = noise_grlwe_rlwe_product(
-            module.n() as f64,
-            log_base2k,
-            0.5,
-            0.5,
-            0f64,
-            sigma * sigma,
-            0f64,
-            log_k_rlwe_in,
-            log_k_grlwe,
-        );
-
-        assert!(
-            (noise_have - noise_want).abs() <= 0.1,
-            "{} {}",
-            noise_have,
-            noise_want
-        );
-
-        module.free();
-    }
-
-    #[test]
-    fn mul_rlwe_dft_inplace() {
-        let module: Module<FFT64> = Module::<FFT64>::new(2048);
-        let log_base2k: usize = 12;
-        let log_k_grlwe: usize = 60;
-        let log_k_rlwe: usize = 45;
-        let rows: usize = (log_k_rlwe + log_base2k - 1) / log_base2k;
-
-        let sigma: f64 = 3.2;
-        let bound: f64 = sigma * 6.0;
-
-        let mut ct_grlwe: GRLWECt<Vec<u8>, FFT64> = GRLWECt::new(&module, log_base2k, log_k_grlwe, rows);
-        let mut ct_rlwe: RLWECt<Vec<u8>> = RLWECt::new(&module, log_base2k, log_k_rlwe);
-        let mut ct_rlwe_dft: RLWECtDft<Vec<u8>, FFT64> = RLWECtDft::new(&module, log_base2k, log_k_rlwe);
-        let mut pt_want: RLWEPt<Vec<u8>> = RLWEPt::new(&module, log_base2k, log_k_rlwe);
-        let mut pt_have: RLWEPt<Vec<u8>> = RLWEPt::new(&module, log_base2k, log_k_rlwe);
-
-        let mut source_xs: Source = Source::new([0u8; 32]);
-        let mut source_xe: Source = Source::new([0u8; 32]);
-        let mut source_xa: Source = Source::new([0u8; 32]);
-
-        // Random input plaintext
-        pt_want
-            .data
-            .fill_uniform(log_base2k, 0, pt_want.size(), &mut source_xa);
-
-        let mut scratch: ScratchOwned = ScratchOwned::new(
-            GRLWECt::encrypt_sk_scratch_space(&module, ct_grlwe.size())
-                | RLWECt::decrypt_scratch_space(&module, ct_rlwe.size())
-                | RLWECt::encrypt_sk_scratch_space(&module, ct_rlwe.size())
-                | GRLWECt::mul_rlwe_scratch_space(&module, ct_rlwe.size(), ct_rlwe.size(), ct_grlwe.size()),
-        );
-
-        let mut sk0: SecretKey<Vec<u8>> = SecretKey::new(&module);
-        sk0.fill_ternary_prob(0.5, &mut source_xs);
-
-        let mut sk0_dft: SecretKeyDft<Vec<u8>, FFT64> = SecretKeyDft::new(&module);
-        sk0_dft.dft(&module, &sk0);
-
-        let mut sk1: SecretKey<Vec<u8>> = SecretKey::new(&module);
-        sk1.fill_ternary_prob(0.5, &mut source_xs);
-
-        let mut sk1_dft: SecretKeyDft<Vec<u8>, FFT64> = SecretKeyDft::new(&module);
-        sk1_dft.dft(&module, &sk1);
-
-        ct_grlwe.encrypt_sk(
-            &module,
-            &sk0.data,
-            &sk1_dft,
-            &mut source_xa,
-            &mut source_xe,
-            sigma,
-            bound,
-            scratch.borrow(),
-        );
-
-        ct_rlwe.encrypt_sk(
-            &module,
-            Some(&pt_want),
-            &sk0_dft,
-            &mut source_xa,
-            &mut source_xe,
-            sigma,
-            bound,
-            scratch.borrow(),
-        );
-
-        ct_rlwe.dft(&module, &mut ct_rlwe_dft);
-        ct_grlwe.mul_rlwe_dft_inplace(&module, &mut ct_rlwe_dft, scratch.borrow());
-        ct_rlwe_dft.idft(&module, &mut ct_rlwe, scratch.borrow());
-
-        ct_rlwe.decrypt(&module, &mut pt_have, &sk1_dft, scratch.borrow());
-
-        module.vec_znx_sub_ab_inplace(&mut pt_have, 0, &pt_want, 0);
-
-        let noise_have: f64 = pt_have.data.std(0, log_base2k).log2();
-        let noise_want: f64 = noise_grlwe_rlwe_product(
-            module.n() as f64,
-            log_base2k,
-            0.5,
-            0.5,
-            0f64,
-            sigma * sigma,
-            0f64,
-            log_k_rlwe,
-            log_k_grlwe,
-        );
-
-        assert!(
-            (noise_have - noise_want).abs() <= 0.1,
-            "{} {}",
-            noise_have,
-            noise_want
-        );
-
-        module.free();
-    }
-
-    #[test]
-    fn mul_grlwe() {
+    fn from_prod_by_grlwe() {
         let module: Module<FFT64> = Module::<FFT64>::new(2048);
         let log_base2k: usize = 12;
         let log_k_grlwe: usize = 60;
@@ -493,7 +87,7 @@
         let mut scratch: ScratchOwned = ScratchOwned::new(
             GRLWECt::encrypt_sk_scratch_space(&module, ct_grlwe_s0s1.size())
                 | RLWECtDft::decrypt_scratch_space(&module, ct_grlwe_s0s2.size())
-                | GRLWECt::mul_grlwe_scratch_space(
+                | GRLWECt::from_prod_by_grlwe_scratch_space(
                     &module,
                     ct_grlwe_s0s2.size(),
                     ct_grlwe_s0s1.size(),
@@ -544,12 +138,7 @@ scratch.borrow(),
         );
 
         // GRLWE_{s1}(s0) (x) GRLWE_{s2}(s1) = GRLWE_{s2}(s0)
-        ct_grlwe_s1s2.mul_grlwe(
-            &module,
-            &mut ct_grlwe_s0s2,
-            &ct_grlwe_s0s1,
-            scratch.borrow(),
-        );
+        ct_grlwe_s0s2.from_prod_by_grlwe(&module, &ct_grlwe_s0s1, &ct_grlwe_s1s2, scratch.borrow());
 
         let mut ct_rlwe_dft_s0s2: RLWECtDft<Vec<u8>, FFT64> = RLWECtDft::new(&module, log_base2k, log_k_grlwe);
         let mut pt: RLWEPt<Vec<u8>> = RLWEPt::new(&module, log_base2k, log_k_grlwe);
@@ -584,7 +173,7 @@ module.free();
     }
 
     #[test]
-    fn mul_grlwe_inplace() {
+    fn prod_by_grlwe() {
         let module: Module<FFT64> = Module::<FFT64>::new(2048);
         let log_base2k: usize = 12;
         let log_k_grlwe: usize = 60;
@@ -603,12 +192,7 @@
         let mut scratch: ScratchOwned = ScratchOwned::new(
             GRLWECt::encrypt_sk_scratch_space(&module, ct_grlwe_s0s1.size())
                 | RLWECtDft::decrypt_scratch_space(&module, ct_grlwe_s0s1.size())
-                | GRLWECt::mul_grlwe_scratch_space(
-                    &module,
-                    ct_grlwe_s0s1.size(),
-                    ct_grlwe_s0s1.size(),
-                    ct_grlwe_s1s2.size(),
-                ),
+                | GRLWECt::prod_by_grlwe_scratch_space(&module, ct_grlwe_s0s1.size(), ct_grlwe_s1s2.size()),
         );
 
         let mut sk0: SecretKey<Vec<u8>> = SecretKey::new(&module);
@@ -654,7 +238,7 @@ scratch.borrow(),
         );
 
         // GRLWE_{s1}(s0) (x) GRLWE_{s2}(s1) = GRLWE_{s2}(s0)
-        ct_grlwe_s1s2.mul_grlwe_inplace(&module, &mut ct_grlwe_s0s1, scratch.borrow());
+        ct_grlwe_s0s1.prod_by_grlwe(&module, &ct_grlwe_s1s2, scratch.borrow());
 
         let ct_grlwe_s0s2: GRLWECt<Vec<u8>, FFT64> = ct_grlwe_s0s1;
 

@@ -1,3 +1,4 @@
 mod grlwe;
 mod rgsw;
 mod rlwe;
+mod rlwe_dft;

@@ -2,7 +2,7 @@ #[cfg(test)]
 mod tests {
     use base2k::{
         FFT64, Module, ScalarZnx, ScalarZnxAlloc, ScalarZnxDftOps, ScratchOwned, Stats, VecZnxBig, VecZnxBigAlloc, VecZnxBigOps,
-        VecZnxDft, VecZnxDftAlloc, VecZnxDftOps, VecZnxOps, VecZnxToMut, ZnxViewMut, ZnxZero,
+        VecZnxDft, VecZnxDftAlloc, VecZnxDftOps, VecZnxOps, ZnxZero,
     };
     use sampling::source::Source;
 
@@ -83,120 +83,6 @@
                 pt_want.data.zero();
             });
         });
-
-        module.free();
-    }
-
-    #[test]
-    fn mul_rlwe() {
-        let module: Module<FFT64> = Module::<FFT64>::new(2048);
-        let log_base2k: usize = 12;
-        let log_k_grlwe: usize = 60;
-        let log_k_rlwe_in: usize = 45;
-        let log_k_rlwe_out: usize = 60;
-        let rows: usize = (log_k_rlwe_in + log_base2k - 1) / log_base2k;
-
-        let sigma: f64 = 3.2;
-        let bound: f64 = sigma * 6.0;
-
-        let mut ct_rgsw: RGSWCt<Vec<u8>, FFT64> = RGSWCt::new(&module, log_base2k, log_k_grlwe, rows);
-        let mut ct_rlwe_in: RLWECt<Vec<u8>> = RLWECt::new(&module, log_base2k, log_k_rlwe_in);
-        let mut ct_rlwe_out: RLWECt<Vec<u8>> = RLWECt::new(&module, log_base2k, log_k_rlwe_out);
-        let mut pt_rgsw: ScalarZnx<Vec<u8>> = module.new_scalar_znx(1);
-        let mut pt_want: RLWEPt<Vec<u8>> = RLWEPt::new(&module, log_base2k, log_k_rlwe_in);
-        let mut pt_have: RLWEPt<Vec<u8>> = RLWEPt::new(&module, log_base2k, log_k_rlwe_out);
-
-        let mut source_xs: Source = Source::new([0u8; 32]);
-        let mut source_xe: Source = Source::new([0u8; 32]);
-        let mut source_xa: Source = Source::new([0u8; 32]);
-
-        // Random input plaintext
-        // pt_want
-        //    .data
-        //    .fill_uniform(log_base2k, 0, pt_want.size(), &mut source_xa);
-
-        pt_want.to_mut().at_mut(0, 0)[1] = 1;
-
-        let k: usize = 1;
-
-        pt_rgsw.raw_mut()[k] = 1; // X^{k}
-
-        let mut scratch: ScratchOwned = ScratchOwned::new(
-            RGSWCt::encrypt_sk_scratch_space(&module, ct_rgsw.size())
-                | RLWECt::decrypt_scratch_space(&module, ct_rlwe_out.size())
-                | RLWECt::encrypt_sk_scratch_space(&module, ct_rlwe_in.size())
-                | RGSWCt::mul_rlwe_scratch_space(
-                    &module,
-                    ct_rlwe_out.size(),
-                    ct_rlwe_in.size(),
-                    ct_rgsw.size(),
-                ),
-        );
-
-        let mut sk: SecretKey<Vec<u8>> = SecretKey::new(&module);
-        sk.fill_ternary_prob(0.5, &mut source_xs);
-
-        let mut sk_dft: SecretKeyDft<Vec<u8>, FFT64> = SecretKeyDft::new(&module);
-        sk_dft.dft(&module, &sk);
-
-        ct_rgsw.encrypt_sk(
-            &module,
-            &pt_rgsw,
-            &sk_dft,
-            &mut source_xa,
-            &mut source_xe,
-            sigma,
-            bound,
-            scratch.borrow(),
-        );
-
-        ct_rlwe_in.encrypt_sk(
-            &module,
-            Some(&pt_want),
-            &sk_dft,
-            &mut source_xa,
-            &mut source_xe,
-            sigma,
-            bound,
-            scratch.borrow(),
-        );
-
-        ct_rgsw.mul_rlwe(&module, &mut ct_rlwe_out, &ct_rlwe_in, scratch.borrow());
-
-        ct_rlwe_out.decrypt(&module, &mut pt_have, &sk_dft, scratch.borrow());
-
-        module.vec_znx_rotate_inplace(k as i64, &mut pt_want, 0);
-
-        module.vec_znx_sub_ab_inplace(&mut pt_have, 0, &pt_want, 0);
-
-        let noise_have: f64 = pt_have.data.std(0, log_base2k).log2();
-
-        let var_gct_err_lhs: f64 = sigma * sigma;
-        let var_gct_err_rhs: f64 = 0f64;
-
-        let var_msg: f64 = 1f64 / module.n() as f64; // X^{k}
-        let var_a0_err: f64 = sigma * sigma;
-        let var_a1_err: f64 = 1f64 / 12f64;
-
-        let noise_want: f64 = noise_rgsw_rlwe_product(
-            module.n() as f64,
-            log_base2k,
-            0.5,
-            var_msg,
-            var_a0_err,
-            var_a1_err,
-            var_gct_err_lhs,
-            var_gct_err_rhs,
-            log_k_rlwe_in,
-            log_k_grlwe,
-        );
-
-        assert!(
-            (noise_have - noise_want).abs() <= 0.1,
-            "{} {}",
-            noise_have,
-            noise_want
-        );
 
         module.free();
     }

@@ -1,13 +1,19 @@
 #[cfg(test)]
-mod tests {
-    use base2k::{Decoding, Encoding, FFT64, Module, ScratchOwned, Stats, VecZnxOps, ZnxZero};
+mod tests_rlwe {
+    use base2k::{
+        Decoding, Encoding, FFT64, FillUniform, Module, ScalarZnx, ScalarZnxAlloc, ScratchOwned, Stats, VecZnxOps, VecZnxToMut,
+        ZnxViewMut, ZnxZero,
+    };
     use itertools::izip;
     use sampling::source::Source;
 
     use crate::{
-        elem::Infos,
+        elem::{FromProdBy, FromProdByScratchSpace, Infos, ProdBy, ProdByScratchSpace},
+        grlwe::GRLWECt,
         keys::{PublicKey, SecretKey, SecretKeyDft},
+        rgsw::RGSWCt,
         rlwe::{RLWECt, RLWECtDft, RLWEPt},
+        test_fft64::{grlwe::noise_grlwe_rlwe_product, rgsw::noise_rgsw_rlwe_product},
     };
 
     #[test]
@@ -190,6 +196,425 @@
         module.vec_znx_sub_ab_inplace(&mut pt_want, 0, &pt_have, 0);
 
         assert!(((1.0f64 / 12.0).sqrt() - pt_want.data.std(0, log_base2k) * (log_k_ct as f64).exp2()).abs() < 0.2);
+
+        module.free();
+    }
+
+    #[test]
+    fn from_prod_by_grlwe() {
+        let module: Module<FFT64> = Module::<FFT64>::new(2048);
+        let log_base2k: usize = 12;
+        let log_k_grlwe: usize = 60;
+        let log_k_rlwe_in: usize = 45;
+        let log_k_rlwe_out: usize = 60;
+        let rows: usize = (log_k_rlwe_in + log_base2k - 1) / log_base2k;
+
+        let sigma: f64 = 3.2;
+        let bound: f64 = sigma * 6.0;
+
+        let mut ct_grlwe: GRLWECt<Vec<u8>, FFT64> = GRLWECt::new(&module, log_base2k, log_k_grlwe, rows);
+        let mut ct_rlwe_in: RLWECt<Vec<u8>> = RLWECt::new(&module, log_base2k, log_k_rlwe_in);
+        let mut ct_rlwe_out: RLWECt<Vec<u8>> = RLWECt::new(&module, log_base2k, log_k_rlwe_out);
+        let mut pt_want: RLWEPt<Vec<u8>> = RLWEPt::new(&module, log_base2k, log_k_rlwe_in);
+        let mut pt_have: RLWEPt<Vec<u8>> = RLWEPt::new(&module, log_base2k, log_k_rlwe_out);
+
+        let mut source_xs: Source = Source::new([0u8; 32]);
+        let mut source_xe: Source = Source::new([0u8; 32]);
+        let mut source_xa: Source = Source::new([0u8; 32]);
+
+        // Random input plaintext
+        pt_want
+            .data
+            .fill_uniform(log_base2k, 0, pt_want.size(), &mut source_xa);
+
+        let mut scratch: ScratchOwned = ScratchOwned::new(
+            GRLWECt::encrypt_sk_scratch_space(&module, ct_grlwe.size())
+                | RLWECt::decrypt_scratch_space(&module, ct_rlwe_out.size())
+                | RLWECt::encrypt_sk_scratch_space(&module, ct_rlwe_in.size())
+                | RLWECt::from_prod_by_grlwe_scratch_space(
+                    &module,
+                    ct_rlwe_out.size(),
+                    ct_rlwe_in.size(),
+                    ct_grlwe.size(),
+                ),
+        );
+
+        let mut sk0: SecretKey<Vec<u8>> = SecretKey::new(&module);
+        sk0.fill_ternary_prob(0.5, &mut source_xs);
+
+        let mut sk0_dft: SecretKeyDft<Vec<u8>, FFT64> = SecretKeyDft::new(&module);
+        sk0_dft.dft(&module, &sk0);
+
+        let mut sk1: SecretKey<Vec<u8>> = SecretKey::new(&module);
+        sk1.fill_ternary_prob(0.5, &mut source_xs);
+
+        let mut sk1_dft: SecretKeyDft<Vec<u8>, FFT64> = SecretKeyDft::new(&module);
+        sk1_dft.dft(&module, &sk1);
+
+        ct_grlwe.encrypt_sk(
+            &module,
+            &sk0.data,
+            &sk1_dft,
+            &mut source_xa,
+            &mut source_xe,
+            sigma,
+            bound,
+            scratch.borrow(),
+        );
+
+        ct_rlwe_in.encrypt_sk(
+            &module,
+            Some(&pt_want),
+            &sk0_dft,
+            &mut source_xa,
+            &mut source_xe,
+            sigma,
+            bound,
+            scratch.borrow(),
+        );
+
+        ct_rlwe_out.from_prod_by_grlwe(&module, &ct_rlwe_in, &ct_grlwe, scratch.borrow());
+
+        ct_rlwe_out.decrypt(&module, &mut pt_have, &sk1_dft, scratch.borrow());
+
+        module.vec_znx_sub_ab_inplace(&mut pt_have, 0, &pt_want, 0);
+
+        let noise_have: f64 = pt_have.data.std(0, log_base2k).log2();
+        let noise_want: f64 = noise_grlwe_rlwe_product(
+            module.n() as f64,
+            log_base2k,
+            0.5,
+            0.5,
+            0f64,
+            sigma * sigma,
+            0f64,
+            log_k_rlwe_in,
+            log_k_grlwe,
+        );
+
+        assert!(
+            (noise_have - noise_want).abs() <= 0.1,
+            "{} {}",
+            noise_have,
+            noise_want
+        );
+
+        module.free();
+    }
+
+    #[test]
+    fn prod_grlwe() {
+        let module: Module<FFT64> = Module::<FFT64>::new(2048);
+        let log_base2k: usize = 12;
+        let log_k_grlwe: usize = 60;
+        let log_k_rlwe: usize = 45;
+        let rows: usize = (log_k_rlwe + log_base2k - 1) / log_base2k;
+
+        let sigma: f64 = 3.2;
+        let bound: f64 = sigma * 6.0;
+
+        let mut ct_grlwe: GRLWECt<Vec<u8>, FFT64> = GRLWECt::new(&module, log_base2k, log_k_grlwe, rows);
+        let mut ct_rlwe: RLWECt<Vec<u8>> = RLWECt::new(&module, log_base2k, log_k_rlwe);
+        let mut pt_want: RLWEPt<Vec<u8>> = RLWEPt::new(&module, log_base2k, log_k_rlwe);
+        let mut pt_have: RLWEPt<Vec<u8>> = RLWEPt::new(&module, log_base2k, log_k_rlwe);
+
+        let mut source_xs: Source = Source::new([0u8; 32]);
+        let mut source_xe: Source = Source::new([0u8; 32]);
+        let mut source_xa: Source = Source::new([0u8; 32]);
+
+        // Random input plaintext
+        pt_want
+            .data
+            .fill_uniform(log_base2k, 0, pt_want.size(), &mut source_xa);
+
+        let mut scratch: ScratchOwned = ScratchOwned::new(
+            GRLWECt::encrypt_sk_scratch_space(&module, ct_grlwe.size())
+                | RLWECt::decrypt_scratch_space(&module, ct_rlwe.size())
+                | RLWECt::encrypt_sk_scratch_space(&module, ct_rlwe.size())
+                | RLWECt::prod_by_grlwe_scratch_space(&module, ct_rlwe.size(), ct_grlwe.size()),
+        );
+
+        let mut sk0: SecretKey<Vec<u8>> = SecretKey::new(&module);
+        sk0.fill_ternary_prob(0.5, &mut source_xs);
+
+        let mut sk0_dft: SecretKeyDft<Vec<u8>, FFT64> = SecretKeyDft::new(&module);
+        sk0_dft.dft(&module, &sk0);
+
+        let mut sk1: SecretKey<Vec<u8>> = SecretKey::new(&module);
+        sk1.fill_ternary_prob(0.5, &mut source_xs);
+
+        let mut sk1_dft: SecretKeyDft<Vec<u8>, FFT64> = SecretKeyDft::new(&module);
+        sk1_dft.dft(&module, &sk1);
+
+        ct_grlwe.encrypt_sk(
+            &module,
+            &sk0.data,
+            &sk1_dft,
+            &mut source_xa,
+            &mut source_xe,
+            sigma,
+            bound,
+            scratch.borrow(),
+        );
+
+        ct_rlwe.encrypt_sk(
+            &module,
+            Some(&pt_want),
+            &sk0_dft,
+            &mut source_xa,
+            &mut source_xe,
+            sigma,
+            bound,
+            scratch.borrow(),
+        );
+
+        ct_rlwe.prod_by_grlwe(&module, &ct_grlwe, scratch.borrow());
+
+        ct_rlwe.decrypt(&module, &mut pt_have, &sk1_dft, scratch.borrow());
+
+        module.vec_znx_sub_ab_inplace(&mut pt_have, 0, &pt_want, 0);
+
+        let noise_have: f64 = pt_have.data.std(0, log_base2k).log2();
+        let noise_want: f64 = noise_grlwe_rlwe_product(
+            module.n() as f64,
+            log_base2k,
+            0.5,
+            0.5,
+            0f64,
+            sigma * sigma,
+            0f64,
+            log_k_rlwe,
+            log_k_grlwe,
+        );
+
+        assert!(
+            (noise_have - noise_want).abs() <= 0.1,
+            "{} {}",
+            noise_have,
+            noise_want
+        );
+
+        module.free();
+    }
+
+    #[test]
+    fn from_prod_by_rgsw() {
+        let module: Module<FFT64> = Module::<FFT64>::new(2048);
+        let log_base2k: usize = 12;
+        let log_k_grlwe: usize = 60;
+        let log_k_rlwe_in: usize = 45;
+        let log_k_rlwe_out: usize = 60;
+        let rows: usize = (log_k_rlwe_in + log_base2k - 1) / log_base2k;
+
+        let sigma: f64 = 3.2;
+        let bound: f64 = sigma * 6.0;
+
+        let mut ct_rgsw: RGSWCt<Vec<u8>, FFT64> = RGSWCt::new(&module, log_base2k, log_k_grlwe, rows);
+        let mut ct_rlwe_in: RLWECt<Vec<u8>> = RLWECt::new(&module, log_base2k, log_k_rlwe_in);
+        let mut ct_rlwe_out: RLWECt<Vec<u8>> = RLWECt::new(&module, log_base2k, log_k_rlwe_out);
+        let mut pt_rgsw: ScalarZnx<Vec<u8>> = module.new_scalar_znx(1);
+        let mut pt_want: RLWEPt<Vec<u8>> = RLWEPt::new(&module, log_base2k, log_k_rlwe_in);
+        let mut pt_have: RLWEPt<Vec<u8>> = RLWEPt::new(&module, log_base2k, log_k_rlwe_out);
+
+        let mut source_xs: Source = Source::new([0u8; 32]);
+        let mut source_xe: Source = Source::new([0u8; 32]);
+        let mut source_xa: Source = Source::new([0u8; 32]);
+
+        // Random input plaintext
+        pt_want
+            .data
+            .fill_uniform(log_base2k, 0, pt_want.size(), &mut source_xa);
+
+        pt_want.to_mut().at_mut(0, 0)[1] = 1;
+
+        let k: usize = 1;
+
+        pt_rgsw.raw_mut()[k] = 1; // X^{k}
+
+        let mut scratch: ScratchOwned = ScratchOwned::new(
+            RGSWCt::encrypt_sk_scratch_space(&module, ct_rgsw.size())
+                | RLWECt::decrypt_scratch_space(&module, ct_rlwe_out.size())
+                | RLWECt::encrypt_sk_scratch_space(&module, ct_rlwe_in.size())
+                | RLWECt::from_prod_by_rgsw_scratch_space(
+                    &module,
+                    ct_rlwe_out.size(),
+                    ct_rlwe_in.size(),
+                    ct_rgsw.size(),
+                ),
+        );
+
+        let mut sk: SecretKey<Vec<u8>> = SecretKey::new(&module);
+        sk.fill_ternary_prob(0.5, &mut source_xs);
+
+        let mut sk_dft: SecretKeyDft<Vec<u8>, FFT64> = SecretKeyDft::new(&module);
+        sk_dft.dft(&module, &sk);
+
+        ct_rgsw.encrypt_sk(
+            &module,
+            &pt_rgsw,
+            &sk_dft,
+            &mut source_xa,
+            &mut source_xe,
+            sigma,
+            bound,
+            scratch.borrow(),
+        );
+
+        ct_rlwe_in.encrypt_sk(
+            &module,
+            Some(&pt_want),
+            &sk_dft,
+            &mut source_xa,
+            &mut source_xe,
+            sigma,
+            bound,
+            scratch.borrow(),
+        );
+
+        ct_rlwe_out.from_prod_by_rgsw(&module, &ct_rlwe_in, &ct_rgsw, scratch.borrow());
+
+        ct_rlwe_out.decrypt(&module, &mut pt_have, &sk_dft, scratch.borrow());
+
+        module.vec_znx_rotate_inplace(k as i64, &mut pt_want, 0);
+
+        module.vec_znx_sub_ab_inplace(&mut pt_have, 0, &pt_want, 0);
+
+        let noise_have: f64 = pt_have.data.std(0, log_base2k).log2();
+
+        let var_gct_err_lhs: f64 = sigma * sigma;
+        let var_gct_err_rhs: f64 = 0f64;
+
+        let var_msg: f64 = 1f64 / module.n() as f64; // X^{k}
+        let var_a0_err: f64 = sigma * sigma;
+        let var_a1_err: f64 = 1f64 / 12f64;
+
+        let noise_want: f64 = noise_rgsw_rlwe_product(
+            module.n() as f64,
+            log_base2k,
+            0.5,
+            var_msg,
+            var_a0_err,
+            var_a1_err,
+            var_gct_err_lhs,
+            var_gct_err_rhs,
+            log_k_rlwe_in,
+            log_k_grlwe,
+        );
+
+        assert!(
+            (noise_have - noise_want).abs() <= 0.1,
+            "{} {}",
+            noise_have,
+            noise_want
+        );
+
+        module.free();
+    }
+
+    #[test]
+    fn prod_by_rgsw() {
+        let module: Module<FFT64> = Module::<FFT64>::new(2048);
+        let log_base2k: usize = 12;
+        let log_k_grlwe: usize = 60;
+        let log_k_rlwe_in: usize = 45;
+        let log_k_rlwe_out: usize = 60;
+        let rows: usize = (log_k_rlwe_in + log_base2k - 1) / log_base2k;
+
+        let sigma: f64 = 3.2;
+        let bound: f64 = sigma * 6.0;
+
+        let mut ct_rgsw: RGSWCt<Vec<u8>, FFT64> = RGSWCt::new(&module, log_base2k, log_k_grlwe, rows);
+        let mut ct_rlwe: RLWECt<Vec<u8>> = RLWECt::new(&module, log_base2k, log_k_rlwe_in);
+        let mut pt_rgsw: ScalarZnx<Vec<u8>> = module.new_scalar_znx(1);
+        let mut pt_want: RLWEPt<Vec<u8>> = RLWEPt::new(&module, log_base2k, log_k_rlwe_in);
+        let mut pt_have: RLWEPt<Vec<u8>> = RLWEPt::new(&module, log_base2k, log_k_rlwe_out);
+
+        let mut source_xs: Source = Source::new([0u8; 32]);
+        let mut source_xe: Source = Source::new([0u8; 32]);
+        let mut source_xa: Source = Source::new([0u8; 32]);
+
+        // Random input plaintext
+        pt_want
+            .data
+            .fill_uniform(log_base2k, 0, pt_want.size(), &mut source_xa);
+
+        pt_want.to_mut().at_mut(0, 0)[1] = 1;
+
+        let k: usize = 1;
+
+        pt_rgsw.raw_mut()[k] = 1; // X^{k}
+
+        let mut scratch: ScratchOwned = ScratchOwned::new(
+            RGSWCt::encrypt_sk_scratch_space(&module, ct_rgsw.size())
+                | RLWECt::decrypt_scratch_space(&module, ct_rlwe.size())
+                | RLWECt::encrypt_sk_scratch_space(&module, ct_rlwe.size())
+                | RLWECt::prod_by_rgsw_scratch_space(&module, ct_rlwe.size(), ct_rgsw.size()),
+        );
+
+        let mut sk: SecretKey<Vec<u8>> = SecretKey::new(&module);
+        sk.fill_ternary_prob(0.5, &mut source_xs);
+
+        let mut sk_dft: SecretKeyDft<Vec<u8>, FFT64> = SecretKeyDft::new(&module);
+        sk_dft.dft(&module, &sk);
+
+        ct_rgsw.encrypt_sk(
+            &module,
+            &pt_rgsw,
+            &sk_dft,
+            &mut source_xa,
+            &mut source_xe,
+            sigma,
+            bound,
+            scratch.borrow(),
+        );
+
+        ct_rlwe.encrypt_sk(
+            &module,
+            Some(&pt_want),
+            &sk_dft,
+            &mut source_xa,
+            &mut source_xe,
+            sigma,
+            bound,
+            scratch.borrow(),
+        );
+
+        ct_rlwe.prod_by_rgsw(&module, &ct_rgsw, scratch.borrow());
+
+        ct_rlwe.decrypt(&module, &mut pt_have, &sk_dft, scratch.borrow());
+
+        module.vec_znx_rotate_inplace(k as i64, &mut pt_want, 0);
+
+        module.vec_znx_sub_ab_inplace(&mut pt_have, 0, &pt_want, 0);
+
+        let noise_have: f64 = pt_have.data.std(0, log_base2k).log2();
+
+        let var_gct_err_lhs: f64 = sigma * sigma;
+        let var_gct_err_rhs: f64 = 0f64;
+
+        let var_msg: f64 = 1f64 / module.n() as f64; // X^{k}
+        let var_a0_err: f64 = sigma * sigma;
+        let var_a1_err: f64 = 1f64 / 12f64;
+
+        let noise_want: f64 = noise_rgsw_rlwe_product(
+            module.n() as f64,
+            log_base2k,
+            0.5,
+            var_msg,
+            var_a0_err,
+            var_a1_err,
+            var_gct_err_lhs,
+            var_gct_err_rhs,
+            log_k_rlwe_in,
+            log_k_grlwe,
+        );
+
+        assert!(
+            (noise_have - noise_want).abs() <= 0.1,
+            "{} {}",
+            noise_have,
+            noise_want
+        );
 
         module.free();
     }

@@ -0,0 +1,216 @@
+#[cfg(test)]
+mod tests {
+    use crate::{
+        elem::{FromProdBy, FromProdByScratchSpace, Infos, ProdBy, ProdByScratchSpace},
+        grlwe::GRLWECt,
+        keys::{SecretKey, SecretKeyDft},
+        rlwe::{RLWECt, RLWECtDft, RLWEPt},
+        test_fft64::grlwe::noise_grlwe_rlwe_product,
+    };
+    use base2k::{FFT64, FillUniform, Module, ScratchOwned, Stats, VecZnxOps};
+    use sampling::source::Source;
+
+    #[test]
+    fn from_prod_by_grlwe() {
+        let module: Module<FFT64> = Module::<FFT64>::new(2048);
+        let log_base2k: usize = 12;
+        let log_k_grlwe: usize = 60;
+        let log_k_rlwe_in: usize = 45;
+        let log_k_rlwe_out: usize = 60;
+        let rows: usize = (log_k_rlwe_in + log_base2k - 1) / log_base2k;
+
+        let sigma: f64 = 3.2;
+        let bound: f64 = sigma * 6.0;
+
+        let mut ct_grlwe: GRLWECt<Vec<u8>, FFT64> = GRLWECt::new(&module, log_base2k, log_k_grlwe, rows);
+        let mut ct_rlwe_in: RLWECt<Vec<u8>> = RLWECt::new(&module, log_base2k, log_k_rlwe_in);
+        let mut ct_rlwe_in_dft: RLWECtDft<Vec<u8>, FFT64> = RLWECtDft::new(&module, log_base2k, log_k_rlwe_in);
+        let mut ct_rlwe_out: RLWECt<Vec<u8>> = RLWECt::new(&module, log_base2k, log_k_rlwe_out);
+        let mut ct_rlwe_out_dft: RLWECtDft<Vec<u8>, FFT64> = RLWECtDft::new(&module, log_base2k, log_k_rlwe_out);
+        let mut pt_want: RLWEPt<Vec<u8>> = RLWEPt::new(&module, log_base2k, log_k_rlwe_in);
+        let mut pt_have: RLWEPt<Vec<u8>> = RLWEPt::new(&module, log_base2k, log_k_rlwe_out);
+
+        let mut source_xs: Source = Source::new([0u8; 32]);
+        let mut source_xe: Source = Source::new([0u8; 32]);
+        let mut source_xa: Source = Source::new([0u8; 32]);
+
+        // Random input plaintext
+        pt_want
+            .data
+            .fill_uniform(log_base2k, 0, pt_want.size(), &mut source_xa);
+
+        let mut scratch: ScratchOwned = ScratchOwned::new(
+            GRLWECt::encrypt_sk_scratch_space(&module, ct_grlwe.size())
+                | RLWECt::decrypt_scratch_space(&module, ct_rlwe_out.size())
+                | RLWECt::encrypt_sk_scratch_space(&module, ct_rlwe_in.size())
+                | RLWECtDft::from_prod_by_grlwe_scratch_space(
+                    &module,
+                    ct_rlwe_out.size(),
+                    ct_rlwe_in.size(),
+                    ct_grlwe.size(),
+                ),
+        );
+
+        let mut sk0: SecretKey<Vec<u8>> = SecretKey::new(&module);
+        sk0.fill_ternary_prob(0.5, &mut source_xs);
+
+        let mut sk0_dft: SecretKeyDft<Vec<u8>, FFT64> = SecretKeyDft::new(&module);
+        sk0_dft.dft(&module, &sk0);
+
+        let mut sk1: SecretKey<Vec<u8>> = SecretKey::new(&module);
+        sk1.fill_ternary_prob(0.5, &mut source_xs);
+
+        let mut sk1_dft: SecretKeyDft<Vec<u8>, FFT64> = SecretKeyDft::new(&module);
+        sk1_dft.dft(&module, &sk1);
+
+        ct_grlwe.encrypt_sk(
+            &module,
+            &sk0.data,
+            &sk1_dft,
+            &mut source_xa,
+            &mut source_xe,
+            sigma,
+            bound,
+            scratch.borrow(),
+        );
+
+        ct_rlwe_in.encrypt_sk(
+            &module,
+            Some(&pt_want),
+            &sk0_dft,
+            &mut source_xa,
+            &mut source_xe,
+            sigma,
+            bound,
+            scratch.borrow(),
+        );
+
+        ct_rlwe_in.dft(&module, &mut ct_rlwe_in_dft);
+        ct_rlwe_out_dft.from_prod_by_grlwe(&module, &ct_rlwe_in_dft, &ct_grlwe, scratch.borrow());
+        ct_rlwe_out_dft.idft(&module, &mut ct_rlwe_out, scratch.borrow());
+
+        ct_rlwe_out.decrypt(&module, &mut pt_have, &sk1_dft, scratch.borrow());
+
+        module.vec_znx_sub_ab_inplace(&mut pt_have, 0, &pt_want, 0);
+
+        let noise_have: f64 = pt_have.data.std(0, log_base2k).log2();
+        let noise_want: f64 = noise_grlwe_rlwe_product(
+            module.n() as f64,
+            log_base2k,
+            0.5,
+            0.5,
+            0f64,
+            sigma * sigma,
+            0f64,
+            log_k_rlwe_in,
+            log_k_grlwe,
+        );
+
+        assert!(
+            (noise_have - noise_want).abs() <= 0.1,
+            "{} {}",
+            noise_have,
+            noise_want
+        );
+
+        module.free();
+    }
+
+    #[test]
+    fn prod_by_grlwe() {
+        let module: Module<FFT64> = Module::<FFT64>::new(2048);
+        let log_base2k: usize = 12;
+        let log_k_grlwe: usize = 60;
+        let log_k_rlwe: usize = 45;
+        let rows: usize = (log_k_rlwe + log_base2k - 1) / log_base2k;
+
+        let sigma: f64 = 3.2;
+        let bound: f64 = sigma * 6.0;
+
+        let mut ct_grlwe: GRLWECt<Vec<u8>, FFT64> = GRLWECt::new(&module, log_base2k, log_k_grlwe, rows);
+        let mut ct_rlwe: RLWECt<Vec<u8>> = RLWECt::new(&module, log_base2k, log_k_rlwe);
+        let mut ct_rlwe_dft: RLWECtDft<Vec<u8>, FFT64> = RLWECtDft::new(&module, log_base2k, log_k_rlwe);
+        let mut pt_want: RLWEPt<Vec<u8>> = RLWEPt::new(&module, log_base2k, log_k_rlwe);
+        let mut pt_have: RLWEPt<Vec<u8>> = RLWEPt::new(&module, log_base2k, log_k_rlwe);
+
+        let mut source_xs: Source = Source::new([0u8; 32]);
+        let mut source_xe: Source = Source::new([0u8; 32]);
+        let mut source_xa: Source = Source::new([0u8; 32]);
+
+        // Random input plaintext
+        pt_want
+            .data
+            .fill_uniform(log_base2k, 0, pt_want.size(), &mut source_xa);
+
+        let mut scratch: ScratchOwned = ScratchOwned::new(
+            GRLWECt::encrypt_sk_scratch_space(&module, ct_grlwe.size())
+                | RLWECt::decrypt_scratch_space(&module, ct_rlwe.size())
+                | RLWECt::encrypt_sk_scratch_space(&module, ct_rlwe.size())
+                | RLWECtDft::prod_by_grlwe_scratch_space(&module, ct_rlwe_dft.size(), ct_grlwe.size()),
+        );
+
+        let mut sk0: SecretKey<Vec<u8>> = SecretKey::new(&module);
+        sk0.fill_ternary_prob(0.5, &mut source_xs);
+
+        let mut sk0_dft: SecretKeyDft<Vec<u8>, FFT64> = SecretKeyDft::new(&module);
+        sk0_dft.dft(&module, &sk0);
+
+        let mut sk1: SecretKey<Vec<u8>> = SecretKey::new(&module);
+        sk1.fill_ternary_prob(0.5, &mut source_xs);
+
+        let mut sk1_dft: SecretKeyDft<Vec<u8>, FFT64> = SecretKeyDft::new(&module);
+        sk1_dft.dft(&module, &sk1);
+
+        ct_grlwe.encrypt_sk(
+            &module,
+            &sk0.data,
+            &sk1_dft,
+            &mut source_xa,
+            &mut source_xe,
+            sigma,
+            bound,
+            scratch.borrow(),
+        );
+
+        ct_rlwe.encrypt_sk(
+            &module,
+            Some(&pt_want),
+            &sk0_dft,
+            &mut source_xa,
+            &mut source_xe,
+            sigma,
+            bound,
+            scratch.borrow(),
+        );
+
+        ct_rlwe.dft(&module, &mut ct_rlwe_dft);
+        ct_rlwe_dft.prod_by_grlwe(&module, &ct_grlwe, scratch.borrow());
+        ct_rlwe_dft.idft(&module, &mut ct_rlwe, scratch.borrow());
+
+        ct_rlwe.decrypt(&module, &mut pt_have, &sk1_dft, scratch.borrow());
+
+        module.vec_znx_sub_ab_inplace(&mut pt_have, 0, &pt_want, 0);
+
+        let noise_have: f64 = pt_have.data.std(0, log_base2k).log2();
+        let noise_want: f64 = noise_grlwe_rlwe_product(
+            module.n() as f64,
+            log_base2k,
+            0.5,
+            0.5,
+            0f64,
+            sigma * sigma,
+            0f64,
+            log_k_rlwe,
+            log_k_grlwe,
+        );
+
+        assert!(
+            (noise_have - noise_want).abs() <= 0.1,
+            "{} {}",
+            noise_have,
+            noise_want
+        );
+
+        module.free();
+    }
+}