@@ -25,8 +25,8 @@ DefaultSecureRng, NewWithSeed, RandomFill, RandomFillGaussianInModulus,
         RandomFillUniformInModulus, RandomGaussianElementInModulus,
     },
     rgsw::{
-        decrypt_rlwe, galois_auto, generate_auto_map, public_key_encrypt_rgsw,
-        rgsw_by_rgsw_inplace, secret_key_encrypt_rgsw, seeded_auto_key_gen,
+        decrypt_rlwe, generate_auto_map, public_key_encrypt_rgsw, rgsw_by_rgsw_inplace, rlwe_auto,
+        secret_key_encrypt_rgsw, seeded_auto_key_gen,
     },
     utils::{
         encode_x_pow_si_with_emebedding_factor, fill_random_ternary_secret_with_hamming_weight,

@@ -178,7 +178,7 @@ use crate::{
         bool::{evaluator::BoolEncoding, keys::NonInteractiveMultiPartyClientKey},
         pbs::{sample_extract, PbsInfo, WithShoupRepr},
         random::{NewWithSeed, RandomFillUniformInModulus},
-        rgsw::{key_switch, seeded_secret_key_encrypt_rlwe},
+        rgsw::{rlwe_key_switch, seeded_secret_key_encrypt_rlwe},
         utils::TryConvertFrom1,
         Encryptor, KeySwitchWithId, Matrix, MatrixEntity, MatrixMut, RowEntity, RowMut,
     };
@@ -324,7 +324,7 @@ let ksk = server_key.ui_to_s_ksk(user_id);
                 let decomposer = e.ni_ui_to_s_ks_decomposer().as_ref().unwrap();
 
                 // perform key switch
-                key_switch(
+                rlwe_key_switch(
                     self,
                     ksk.as_ref(),
                     ksk.shoup_repr(),

@@ -10,7 +10,7 @@ decomposer::RlweDecomposer,
     lwe::{decrypt_lwe, lwe_key_switch},
     parameters::{BoolParameters, CiphertextModulus},
     random::{DefaultSecureRng, RandomFillUniformInModulus},
-    rgsw::{decrypt_rlwe, galois_auto, IsTrivial, RlweCiphertext},
+    rgsw::{decrypt_rlwe, rlwe_auto, IsTrivial, RlweCiphertext},
     utils::{encode_x_pow_si_with_emebedding_factor, tests::Stats, TryConvertFrom1},
     ArithmeticOps, ClientKey, Decomposer, MatrixEntity, MatrixMut, ModInit, Ntt, NttInit,
     RowEntity, RowMut, VectorOps,
@@ -283,7 +283,7 @@ let mut rlwe = RlweCiphertext::<_, DefaultSecureRng>::new_trivial(M::zeros(2, rlwe_n));
             rlwe.data.get_row_mut(0).copy_from_slice(m.as_ref());
             rlwe.set_not_trivial();
 
-            galois_auto(
+            rlwe_auto(
                 &mut rlwe,
                 server_key.galois_key_for_auto(*k),
                 &mut scratch_matrix,

@@ -9,14 +9,11 @@ ops::{Div, Neg, Sub},
 };
 
 use crate::{
-    backend::{ArithmeticOps, GetModulus, Modulus, VectorOps},
-    decomposer::{self, Decomposer, RlweDecomposer},
-    ntt::{self, Ntt, NttInit},
-    random::{
-        DefaultSecureRng, NewWithSeed, RandomElementInModulus, RandomFill,
-        RandomFillGaussianInModulus, RandomFillUniformInModulus,
-    },
-    utils::{fill_random_ternary_secret_with_hamming_weight, ToShoup, TryConvertFrom1, WithLocal},
+    backend::Modulus,
+    decomposer::{Decomposer, RlweDecomposer},
+    ntt::{Ntt, NttInit},
+    random::{DefaultSecureRng, NewWithSeed, RandomFillUniformInModulus},
+    utils::{fill_random_ternary_secret_with_hamming_weight, ToShoup, WithLocal},
     Matrix, MatrixEntity, MatrixMut, Row, RowEntity, RowMut, Secret,
 };
 
@@ -210,8 +207,6 @@ to_ri.as_mut().copy_from_slice(from_ri.as_ref());
         });
 
         // sample A polynomials of RLWE'(m) - RLWE'A(m)
-        // TODO(Jay): Do we want to be generic over RandomGenerator used here? I think
-        // not.
         let mut p_rng = R::new_with_seed(value.seed.clone());
         izip!(data.iter_rows_mut().skip(value.d_a * 2).take(value.d_b * 1))
             .for_each(|ri| p_rng.random_fill(&value.modulus, ri.as_mut()));
@@ -528,7 +523,7 @@ decrypt_rlwe, generate_auto_map, measure_noise, public_key_encrypt_rgsw,
             rlwe_public_key, secret_key_encrypt_rgsw, seeded_auto_key_gen,
             seeded_secret_key_encrypt_rlwe,
         },
-        runtime::{galois_auto, rgsw_by_rgsw_inplace, rlwe_by_rgsw},
+        runtime::{rgsw_by_rgsw_inplace, rlwe_auto, rlwe_by_rgsw},
         AutoKeyEvaluationDomain, RgswCiphertext, RgswCiphertextEvaluationDomain, RlweCiphertext,
         RlwePublicKey, RlweSecret, SeededAutoKey, SeededRgswCiphertext, SeededRlweCiphertext,
         SeededRlwePublicKey,
@@ -636,69 +631,6 @@ );
         seeded_rgsw_ct
     }
 
-    /// Prints noise in RGSW ciphertext RGSW(m).
-    ///
-    /// - rgsw_ct: RGSW ciphertext in coefficient domain
-    pub(crate) fn _measure_noise_rgsw<T: Modulus<Element = u64> + Clone>(
-        rgsw_ct: &[Vec<u64>],
-        m: &[u64],
-        s: &[i32],
-        decomposer: &(DefaultDecomposer<u64>, DefaultDecomposer<u64>),
-        q: &T,
-    ) {
-        let gadget_vector_a = decomposer.a().gadget_vector();
-        let gadget_vector_b = decomposer.b().gadget_vector();
-        let d_a = gadget_vector_a.len();
-        let d_b = gadget_vector_b.len();
-        let ring_size = s.len();
-        assert!(Matrix::dimension(&rgsw_ct) == (d_a * 2 + d_b * 2, ring_size));
-        assert!(m.len() == ring_size);
-
-        let mod_op = ModularOpsU64::new(q.clone());
-        let ntt_op = NttBackendU64::new(q, ring_size);
-
-        let mul_mod =
-            |a: &u64, b: &u64| ((*a as u128 * *b as u128) % q.q().unwrap() as u128) as u64;
-        let s_poly = Vec::<u64>::try_convert_from(s, q);
-        let mut neg_s = s_poly.clone();
-        mod_op.elwise_neg_mut(neg_s.as_mut());
-        let neg_sm0m1 = negacyclic_mul(&neg_s, &m, mul_mod, q.q().unwrap());
-
-        // RLWE(\beta^j -s * m)
-        for j in 0..d_a {
-            let ideal_m = {
-                // RLWE(\beta^j -s * m)
-                let mut beta_neg_sm0m1 = vec![0u64; ring_size as usize];
-                mod_op.elwise_scalar_mul(beta_neg_sm0m1.as_mut(), &neg_sm0m1, &gadget_vector_a[j]);
-                beta_neg_sm0m1
-            };
-
-            let mut rlwe = vec![vec![0u64; ring_size as usize]; 2];
-            rlwe[0].copy_from_slice(rgsw_ct.get_row_slice(j));
-            rlwe[1].copy_from_slice(rgsw_ct.get_row_slice(d_a + j));
-            let noise = measure_noise(&rlwe, &ideal_m, &ntt_op, &mod_op, s);
-
-            println!(r"Noise RLWE(\beta^{j} -sm0m1): {noise}");
-        }
-
-        // RLWE(\beta^j  m)
-        for j in 0..d_b {
-            let ideal_m = {
-                // RLWE(\beta^j  m)
-                let mut beta_m0m1 = vec![0u64; ring_size as usize];
-                mod_op.elwise_scalar_mul(beta_m0m1.as_mut(), &m, &gadget_vector_b[j]);
-                beta_m0m1
-            };
-
-            let mut rlwe = vec![vec![0u64; ring_size as usize]; 2];
-            rlwe[0].copy_from_slice(rgsw_ct.get_row_slice(d_a * 2 + j));
-            rlwe[1].copy_from_slice(rgsw_ct.get_row_slice(d_a * 2 + d_b + j));
-            let noise = measure_noise(&rlwe, &ideal_m, &ntt_op, &mod_op, s);
-
-            println!(r"Noise RLWE(\beta^{j} m0m1): {noise}");
-        }
-    }
-
     #[test]
     fn rlwe_encrypt_decryption() {
         let logq = 50;
@@ -742,9 +674,6 @@ .iter()
             .map(|v| (((*v as f64 * p as f64) / q as f64).round() as u64) % p)
             .collect_vec();
         assert_eq!(m0, m_back);
-
-        // let noise = measure_noise(&rlwe_in_ct, &encoded_m, &ntt_op, &mod_op,
-        // s.values()); println!("Noise: {noise}");
     }
 
     #[test]
@@ -999,7 +928,7 @@ };
 
         // normal galois auto
         {
-            galois_auto(
+            rlwe_auto(
                 &mut rlwe_m,
                 &auto_key.data,
                 &mut scratch_space,
@@ -1104,7 +1033,7 @@ decomposer.b().decomposition_count()
                 )
         ];
 
-        _measure_noise_rgsw(&rgsw_carrym, &carry_m, s.values(), &decomposer, &q);
+        // _measure_noise_rgsw(&rgsw_carrym, &carry_m, s.values(), &decomposer, &q);
 
         for i in 0..2 {
             let mut m = vec![0u64; ring_size as usize];
@@ -1127,7 +1056,8 @@
             // measure noise
             carry_m = negacyclic_mul(&carry_m, &m, mul_mod, q);
             println!("########### Noise RGSW(carrym) in {i}^th loop ###########");
-            _measure_noise_rgsw(&rgsw_carrym, &carry_m, s.values(), &decomposer, &q);
+            // _measure_noise_rgsw(&rgsw_carrym, &carry_m, s.values(),
+            // &decomposer, &q);
         }
         {
             // RLWE(m) x RGSW(carry_m)

@@ -10,7 +10,7 @@ };
 
 use super::IsTrivial;
 
-pub(crate) fn routine<R: RowMut, ModOp: VectorOps<Element = R::Element>>(
+pub(crate) fn poly_fma_routine<R: RowMut, ModOp: VectorOps<Element = R::Element>>(
     write_to_row: &mut [R::Element],
     matrix_a: &[R],
     matrix_b: &[R],
@@ -48,11 +48,18 @@ });
     }
 }
 
-/// Sends RLWE_{s}(X) -> RLWE_{s}(X^k) where k is some galois element
+/// Sends RLWE_{s(X)}(m(X)) -> RLWE_{s(X)}(m{X^k}) where k is some galois
+/// element
 ///
-/// - scratch_matrix: must have dimension at-least d+2 x ring_size. d rows to
-///   store decomposed polynomials and 2 for rlwe
-pub(crate) fn galois_auto<
+/// - rlwe_in: Input ciphertext RLWE_{s(X)}(m(X)).
+/// - ksk: Auto key switching key with polynomials in evaluation domain
+/// - auto_map_index: If automorphism sends i^th coefficient of m(X) to j^th
+///   coefficient of m(X^k) then auto_map_index[i] = j
+/// - auto_sign_index: With a = m(X)[i], if m(X^k)[auto_map_index[i]] = -a, then
+///   auto_sign_index[i] = false, else auto_sign_index[i] = true
+/// - scratch_matrix: must have dimension at-least d+2 x ring_size. `d` rows to
+///   store decomposed polynomials nad 2 rows to store out RLWE temporarily.
+pub(crate) fn rlwe_auto<
     MT: Matrix + IsTrivial + MatrixMut,
     Mmut: MatrixMut<MatElement = MT::MatElement>,
     ModOp: ArithmeticOps<Element = MT::MatElement> + VectorOps<Element = MT::MatElement>,
@@ -119,7 +126,7 @@ // RLWE(m^k) = a', b'; RLWE(m) = a, b
         // key switch: (a * RLWE'(s(X^k)))
         let (ksk_a, ksk_b) = ksk.split_at_row(d);
         // a' = decomp<a> * RLWE'_A(s(X^k))
-        routine(
+        poly_fma_routine(
             tmp_rlwe_out[0].as_mut(),
             scratch_matrix_d_ring,
             ksk_a,
@@ -127,7 +134,7 @@ mod_op,
         );
 
         // b' += decomp<a(X^k)> * RLWE'_B(s(X^k))
-        routine(
+        poly_fma_routine(
             tmp_rlwe_out[1].as_mut(),
             scratch_matrix_d_ring,
             ksk_b,
@@ -181,6 +188,12 @@ .get_row_mut(1)
         .copy_from_slice(tmp_rlwe_out[1].as_ref());
 }
 
+/// Sends RLWE_{s(X)}(m(X)) -> RLWE_{s(X)}(m{X^k}) where k is some galois
+/// element
+///
+/// This is same as `galois_auto` with the difference that alongside `ksk` with
+/// key switching polynomials in evaluation domain, shoup representation,
+/// `ksk_shoup`, of the polynomials in evaluation domain is also supplied.
 pub(crate) fn galois_auto_shoup<
     MT: Matrix + IsTrivial + MatrixMut,
     Mmut: MatrixMut<MatElement = MT::MatElement>,
@@ -309,14 +322,12 @@ .get_row_mut(1)
         .copy_from_slice(tmp_rlwe_out[1].as_ref());
 }
 
-/// Returns RLWE(m0m1) = RLWE(m0) x RGSW(m1). Mutates rlwe_in inplace to equal
-/// RLWE(m0m1)
+/// Inplace mutates RLWE(m0) to equal RLWE(m0m1) = RLWE(m0) x RGSW(m1).
 ///
 /// - rlwe_in: is RLWE(m0) with polynomials in coefficient domain
 /// - rgsw_in: is RGSW(m1) with polynomials in evaluation domain
-/// - scratch_matrix_d_ring: is a matrix with atleast max(d_a, d_b) rows and
-///   ring_size columns. It's used to store decomposed polynomials and out RLWE
-///   temoporarily
+/// - scratch_matrix: with dimension (max(d_a, d_b) + 2) x ring_size columns.
+///   It's used to store decomposed polynomials and out RLWE temporarily
 pub(crate) fn rlwe_by_rgsw<
     Mmut: MatrixMut,
     MT: Matrix<MatElement = Mmut::MatElement> + MatrixMut<MatElement = Mmut::MatElement> + IsTrivial,
@@ -365,14 +376,14 @@ .iter_mut()
             .take(d_a)
             .for_each(|r| ntt_op.forward(r.as_mut()));
         // a_out += decomp<a_in> \cdot RLWE_A'(-sm)
-        routine(
+        poly_fma_routine(
             scratch_rlwe_out[0].as_mut(),
             &scratch_matrix_d_ring[..d_a],
             &rlwe_dash_nsm[..d_a],
             mod_op,
         );
         // b_out += decomp<a_in> \cdot RLWE_B'(-sm)
-        routine(
+        poly_fma_routine(
             scratch_rlwe_out[1].as_mut(),
             &scratch_matrix_d_ring[..d_a],
             &rlwe_dash_nsm[d_a..],
@@ -390,14 +401,14 @@ .iter_mut()
         .take(d_b)
         .for_each(|r| ntt_op.forward(r.as_mut()));
     // a_out += decomp<b_in> \cdot RLWE_A'(m)
-    routine(
+    poly_fma_routine(
         scratch_rlwe_out[0].as_mut(),
         &scratch_matrix_d_ring[..d_b],
         &rlwe_dash_m[..d_b],
         mod_op,
     );
     // b_out += decomp<b_in> \cdot RLWE_B'(m)
-    routine(
+    poly_fma_routine(
         scratch_rlwe_out[1].as_mut(),
         &scratch_matrix_d_ring[..d_b],
         &rlwe_dash_m[d_b..],
@@ -418,6 +429,11 @@ .copy_from_slice(scratch_rlwe_out[1].as_mut());
     rlwe_in.set_not_trivial();
 }
 
+/// Inplace mutates RLWE(m0) to equal RLWE(m0m1) = RLWE(m0) x RGSW(m1).
+///
+/// Same as `rlwe_by_rgsw` with the difference that alongside `rgsw_in` with
+/// polynomials in evaluation domain, shoup representation of polynomials in
+/// evaluation domain, `rgsw_in_shoup`, is also supplied.
 pub(crate) fn rlwe_by_rgsw_shoup<
     Mmut: MatrixMut,
     MT: Matrix<MatElement = Mmut::MatElement> + MatrixMut<MatElement = Mmut::MatElement> + IsTrivial,
@@ -528,27 +544,32 @@ .copy_from_slice(scratch_rlwe_out[1].as_mut());
     rlwe_in.set_not_trivial();
 }
 
-/// Inplace mutates rlwe_0 to equal RGSW(m0m1) = RGSW(m0)xRGSW(m1)
-/// in evaluation domain
+/// Inplace mutates RGSW(m0) to equal RGSW(m0m1) = RGSW(m0)xRGSW(m1)
+///
+/// RGSW x RGSW product requires multiple RLWE x RGSW products. For example,
+/// Define
+///
+///     RGSW(m0) = [RLWE(-sm), RLWE(\beta -sm), ..., RLWE(\beta^{d-1} -sm)
+///                 RLWE(m), RLWE(\beta m), ..., RLWE(\beta^{d-1} m)]
+///     And RGSW(m1)
 ///
-/// Warning -
-/// Pass a fresh RGSW ciphertext as the second operand, i.e. as `rgsw_1`.
-/// This is to assure minimal error growth in the resulting RGSW ciphertext.
-/// RGSWxRGSW boils down to d_rgsw*2 RLWExRGSW multiplications. Hence, the noise
-/// growth in resulting ciphertext depends on the norm of second RGSW
-/// ciphertext, not the first. This is useful in cases where one is accumulating
-/// multiple RGSW ciphertexts into 1. In which case, pass the accumulating RGSW
-/// ciphertext as rlwe_0 (the one with higher noise) and subsequent RGSW
-/// ciphertexts, with lower noise, to be accumulated as second
-/// operand.
+/// Then RGSW(m0) x RGSW(m1) equals:
+///     RGSW(m0m1) = [
+///                     rlwe_x_rgsw(RLWE(-sm), RGSW(m1)),
+///                     ...,
+///                     rlwe_x_rgsw(RLWE(\beta^{d-1} -sm), RGSW(m1)),
+///                     rlwe_x_rgsw(RLWE(m), RGSW(m1)),
+///                     ...,
+///                     rlwe_x_rgsw(RLWE(\beta^{d-1} m), RGSW(m1)),
+///                  ]
 ///
-/// - rgsw_0: RGSW(m0)
-/// - rgsw_1_eval: RGSW(m1) in Evaluation domain
-/// - scratch_matrix_d_plus_rgsw_by_ring: scratch space matrix with rows
-///   (max(d_a, d_b) + d_a*2+d_b*2) and columns ring_size
+/// Since noise growth in RLWE x RGSW depends on noise in RGSW ciphertext, it is
+/// clear to observe from above that noise in resulting RGSW(m0m1) equals noise
+/// accumulated in a single RLWE x RGSW and depends on noise in RGSW(m1) (i.e.
+/// rgsw_1_eval)
 ///
-/// ## Note:
-/// - We treat RGSW x RGSW as multiple RLWE x RGSW multiplications.  .
+/// - rgsw_0: RGSW(m0) in coefficient domain
+/// - rgsw_1_eval: RGSW(m1) in evaluation domain
 pub(crate) fn rgsw_by_rgsw_inplace<
     Mmut: MatrixMut,
     D: RlweDecomposer<Element = Mmut::MatElement>,
@@ -620,13 +641,13 @@ decomp_r_space
             .iter_mut()
             .take(d_a)
             .for_each(|ri| ntt_op.forward(ri.as_mut()));
-        routine(
+        poly_fma_routine(
             rlwe_out_a.as_mut(),
             &decomp_r_space[..d_a],
             &rgsw1_nsm[..d_a],
             mod_op,
         );
-        routine(
+        poly_fma_routine(
             rlwe_out_b.as_mut(),
             &decomp_r_space[..d_a],
             &rgsw1_nsm[d_a..],
@@ -639,13 +660,13 @@ decomp_r_space
             .iter_mut()
             .take(d_b)
             .for_each(|ri| ntt_op.forward(ri.as_mut()));
-        routine(
+        poly_fma_routine(
             rlwe_out_a.as_mut(),
             &decomp_r_space[..d_b],
             &rgsw1_m[..d_b],
             mod_op,
         );
-        routine(
+        poly_fma_routine(
             rlwe_out_b.as_mut(),
             &decomp_r_space[..d_b],
             &rgsw1_m[d_b..],
@@ -663,7 +684,21 @@ .iter_rows_mut()
         .for_each(|ri| ntt_op.backward(ri.as_mut()));
 }
 
-pub(crate) fn key_switch<
+/// Key switches input RLWE_{s'}(m) -> RLWE_{s}(m)
+///
+/// Let RLWE_{s'}(m) = [a, b] s.t. m+e = b - as'
+///
+/// Given key switchin key Ksk(s' -> s) = RLWE'_{s}(s') = [RLWE_{s}(beta^i s')]
+/// = [a, a*s + e + beta^i s'] for i \in [0,d), key switching computes:
+/// 1. RLWE_{s}(-s'a) = \sum signed_decompose(-a)[i] RLWE_{s}(beta^i s')
+/// 2. RLWE_{s}(m) = (b, 0) + RLWE_{s}(-s'a)
+///
+/// - rlwe_in: Input rlwe ciphertext
+/// - ksk: Key switching key Ksk(s' -> s) with polynomials in evaluation domain
+/// - ksk_shoup: Key switching key Ksk(s' -> s) with polynomials in evaluation
+///   domain in shoup representation
+/// - decomposer: Decomposer used for key switching
+pub(crate) fn rlwe_key_switch<
     M: MatrixMut + MatrixEntity,
     ModOp: GetModulus<Element = M::MatElement> + ShoupMatrixFMA<M::R> + VectorOps<Element = M::MatElement>,
     NttOp: Ntt<Element = M::MatElement>,