@@ -41,8 +41,8 @@ rgsw_by_rgsw_inplace, rlwe_by_rgsw, secret_key_encrypt_rgsw, IsTrivial, RgswCiphertext,
         RlweCiphertext, RlweSecret,
     },
     utils::{
-        fill_random_ternary_secret_with_hamming_weight, generate_prime, mod_exponent,
-        puncture_p_rng, Global, TryConvertFrom1, WithLocal,
+        encode_x_pow_si_with_emebedding_factor, fill_random_ternary_secret_with_hamming_weight,
+        generate_prime, mod_exponent, puncture_p_rng, Global, TryConvertFrom1, WithLocal,
     },
     Decryptor, Encoder, Encryptor, Matrix, MatrixEntity, MatrixMut, MultiPartyDecryptor, Row,
     RowEntity, RowMut, Secret,
@@ -527,6 +527,24 @@
     reduced_ct_i_out
 }
 
+/// Assigns user with user_id segement of LWE secret indices for which they
+/// generate RGSW(X^{s[i]}) as the leader (i.e. for RLWExRGSW). If returned
+/// tuple is (start, end), user's segment is [start, end)
+pub(super) fn interactive_mult_party_user_id_lwe_segment(
+    user_id: usize,
+    total_users: usize,
+    lwe_n: usize,
+) -> (usize, usize) {
+    let per_user = (lwe_n as f64 / total_users as f64)
+        .ceil()
+        .to_usize()
+        .unwrap();
+    (
+        per_user * user_id,
+        std::cmp::min(per_user * (user_id + 1), lwe_n),
+    )
+}
+
 impl<M: Matrix, NttOp, RlweModOp, LweModOp, SKey> BoolEvaluator<M, NttOp, RlweModOp, LweModOp, SKey>
 where
     M: MatrixEntity + MatrixMut,
@@ -800,6 +818,8 @@ }
 
     pub(super) fn multi_party_server_key_share<K: InteractiveMultiPartyClientKey<Element = i32>>(
         &self,
+        user_id: usize,
+        total_users: usize,
         cr_seed: &MultiPartyCrs<[u8; 32]>,
         collective_pk: &M,
         client_key: &K,
@@ -809,10 +829,7 @@ BoolParameters<M::MatElement>,
         MultiPartyCrs<[u8; 32]>,
     > {
         assert_eq!(self.parameters().variant(), &ParameterVariant::MultiParty);
-        // let user_id = 0;
-
-        // let user_segment_start = 0;
-        // let user_segment_end = 1;
+        assert!(user_id < total_users);
 
         let sk_rlwe = client_key.sk_rlwe();
         let sk_lwe = client_key.sk_lwe();
@@ -836,41 +853,85 @@ &sk_rlwe,
         );
 
         // rgsw ciphertexts of lwe secret elements
-        let rgsw_cts = DefaultSecureRng::with_local_mut(|rng| {
-            let rgsw_rgsw_decomposer = self
-                .pbs_info
-                .parameters
-                .rgsw_rgsw_decomposer::<DefaultDecomposer<M::MatElement>>();
-            let (rgrg_d_a, rgrg_d_b) = (
-                rgsw_rgsw_decomposer.0.decomposition_count(),
-                rgsw_rgsw_decomposer.1.decomposition_count(),
+        let (self_leader_rgsws, not_self_leader_rgsws) = DefaultSecureRng::with_local_mut(|rng| {
+            let mut self_leader_rgsw = vec![];
+            let mut not_self_leader_rgsws = vec![];
+
+            let (segment_start, segment_end) = interactive_mult_party_user_id_lwe_segment(
+                user_id,
+                total_users,
+                self.pbs_info().lwe_n(),
             );
-            let (rgrg_gadget_a, rgrg_gadget_b) = (
-                rgsw_rgsw_decomposer.0.gadget_vector(),
-                rgsw_rgsw_decomposer.1.gadget_vector(),
-            );
-            let rgsw_cts = sk_lwe
-                .iter()
-                .map(|si| {
-                    let mut m = M::R::zeros(ring_size);
-                    //TODO(Jay): It will be nice to have a function that returns polynomial
-                    // (monomial infact!) corresponding to secret element embedded in ring X^{2N+1}.
-                    // Save lots of mistakes where one forgest to emebed si in bigger ring.
-                    let si = *si * (self.pbs_info.embedding_factor as i32);
-                    if si < 0 {
-                        // X^{-si} = X^{2N-si} = -X^{N-si}, assuming abs(si) < N
-                        // (which it is given si is secret element)
-                        m.as_mut()[ring_size - (si.abs() as usize)] = rlwe_q.neg_one();
-                    } else {
-                        m.as_mut()[si as usize] = M::MatElement::one();
-                    }
+
+            // self LWE secret indices
+            {
+                // LWE secret indices for which user is the leader they need to send RGSW(m) for
+                // RLWE x RGSW multiplication
+                let rlrg_decomposer = self.pbs_info().rlwe_rgsw_decomposer();
+                let (rlrg_d_a, rlrg_d_b) = (
+                    rlrg_decomposer.a().decomposition_count(),
+                    rlrg_decomposer.b().decomposition_count(),
+                );
+                let (gadget_a, gadget_b) = (
+                    rlrg_decomposer.a().gadget_vector(),
+                    rlrg_decomposer.b().gadget_vector(),
+                );
+                for s_index in segment_start..segment_end {
+                    let mut out_rgsw = M::zeros(rlrg_d_a * 2 + rlrg_d_b * 2, ring_size);
+                    public_key_encrypt_rgsw(
+                        &mut out_rgsw,
+                        &encode_x_pow_si_with_emebedding_factor::<
+                            M::R,
+                            CiphertextModulus<M::MatElement>,
+                        >(
+                            sk_lwe[s_index],
+                            self.pbs_info().embedding_factor(),
+                            ring_size,
+                            self.pbs_info().rlwe_q(),
+                        )
+                        .as_ref(),
+                        collective_pk,
+                        &gadget_a,
+                        &gadget_b,
+                        rlweq_modop,
+                        rlweq_nttop,
+                        rng,
+                    );
+                    self_leader_rgsw.push(out_rgsw);
+                }
+            }
+
+            // not self LWE secret indices
+            {
+                // LWE secret indices for which user isn't the leader, they need to send RGSW(m)
+                // for RGSW x RGSW multiplcation
+                let rgsw_rgsw_decomposer = self
+                    .pbs_info
+                    .parameters
+                    .rgsw_rgsw_decomposer::<DefaultDecomposer<M::MatElement>>();
+                let (rgrg_d_a, rgrg_d_b) = (
+                    rgsw_rgsw_decomposer.a().decomposition_count(),
+                    rgsw_rgsw_decomposer.b().decomposition_count(),
+                );
+                let (rgrg_gadget_a, rgrg_gadget_b) = (
+                    rgsw_rgsw_decomposer.a().gadget_vector(),
+                    rgsw_rgsw_decomposer.b().gadget_vector(),
+                );
 
-                    // public key RGSW encryption has no part that can be seeded, unlike secret key
-                    // RGSW encryption where RLWE'_A(m) is seeded
+                for s_index in (0..segment_start).chain(segment_end..self.parameters().lwe_n().0) {
                     let mut out_rgsw = M::zeros(rgrg_d_a * 2 + rgrg_d_b * 2, ring_size);
                     public_key_encrypt_rgsw(
                         &mut out_rgsw,
-                        &m.as_ref(),
+                        &encode_x_pow_si_with_emebedding_factor::<
+                            M::R,
+                            CiphertextModulus<M::MatElement>,
+                        >(
+                            sk_lwe[s_index],
+                            self.pbs_info().embedding_factor(),
+                            ring_size,
+                            self.pbs_info().rlwe_q(),
+                        )
+                        .as_ref(),
                         collective_pk,
                         &rgrg_gadget_a,
                         &rgrg_gadget_b,
@@ -879,10 +940,11 @@ rlweq_nttop,
                         rng,
                     );
 
-                    out_rgsw
-                })
-                .collect_vec();
-            rgsw_cts
+                    not_self_leader_rgsws.push(out_rgsw);
+                }
+            }
+
+            (self_leader_rgsw, not_self_leader_rgsws)
         });
 
         // LWE Ksk
@@ -893,14 +955,173 @@ &sk_lwe,
         );
 
         CommonReferenceSeededMultiPartyServerKeyShare::new(
-            rgsw_cts,
+            self_leader_rgsws,
+            not_self_leader_rgsws,
             auto_keys,
             lwe_ksk,
             cr_seed.clone(),
             self.pbs_info.parameters.clone(),
+            user_id,
         )
     }
 
+    pub(super) fn aggregate_multi_party_server_key_shares<S>(
+        &self,
+        shares: &[CommonReferenceSeededMultiPartyServerKeyShare<
+            M,
+            BoolParameters<M::MatElement>,
+            MultiPartyCrs<S>,
+        >],
+    ) -> SeededMultiPartyServerKey<M, MultiPartyCrs<S>, BoolParameters<M::MatElement>>
+    where
+        S: PartialEq + Clone,
+        M: Clone,
+    {
+        assert_eq!(self.parameters().variant(), &ParameterVariant::MultiParty);
+        assert!(shares.len() > 0);
+
+        let total_users = shares.len();
+
+        let parameters = shares[0].parameters().clone();
+        let cr_seed = shares[0].cr_seed();
+
+        let rlwe_n = parameters.rlwe_n().0;
+        let g = parameters.g() as isize;
+        let rlwe_q = parameters.rlwe_q();
+        let lwe_q = parameters.lwe_q();
+
+        // sanity checks
+        shares.iter().skip(1).for_each(|s| {
+            assert!(s.parameters() == &parameters);
+            assert!(s.cr_seed() == cr_seed);
+        });
+
+        let rlweq_modop = &self.pbs_info.rlwe_modop;
+        let rlweq_nttop = &self.pbs_info.rlwe_nttop;
+
+        // auto keys
+        let mut auto_keys = HashMap::new();
+        let auto_elements_dlog = parameters.auto_element_dlogs();
+        for i in auto_elements_dlog.into_iter() {
+            let mut key = M::zeros(parameters.auto_decomposition_count().0, rlwe_n);
+
+            shares.iter().for_each(|s| {
+                let auto_key_share_i = s.auto_keys().get(&i).expect("Auto key {i} missing");
+                assert!(
+                    auto_key_share_i.dimension()
+                        == (parameters.auto_decomposition_count().0, rlwe_n)
+                );
+                izip!(key.iter_rows_mut(), auto_key_share_i.iter_rows()).for_each(
+                    |(partb_out, partb_share)| {
+                        rlweq_modop.elwise_add_mut(partb_out.as_mut(), partb_share.as_ref());
+                    },
+                );
+            });
+
+            auto_keys.insert(i, key);
+        }
+
+        // rgsw ciphertext (most expensive part!)
+        let rgsw_cts = {
+            let rgsw_by_rgsw_decomposer =
+                parameters.rgsw_rgsw_decomposer::<DefaultDecomposer<M::MatElement>>();
+            let rlwe_x_rgsw_decomposer = self.pbs_info().rlwe_rgsw_decomposer();
+            let rgsw_x_rgsw_dimension = (
+                rgsw_by_rgsw_decomposer.a().decomposition_count() * 2
+                    + rgsw_by_rgsw_decomposer.b().decomposition_count() * 2,
+                rlwe_n,
+            );
+            let rlwe_x_rgsw_dimension = (
+                rlwe_x_rgsw_decomposer.a().decomposition_count() * 2
+                    + rlwe_x_rgsw_decomposer.b().decomposition_count() * 2,
+                rlwe_n,
+            );
+            let mut rgsw_x_rgsw_scratch_mat = M::zeros(
+                std::cmp::max(
+                    rgsw_by_rgsw_decomposer.a().decomposition_count(),
+                    rgsw_by_rgsw_decomposer.b().decomposition_count(),
+                ) + rlwe_x_rgsw_dimension.0,
+                rlwe_n,
+            );
+
+            let shares_in_correct_order = (0..total_users)
+                .map(|i| shares.iter().find(|s| s.user_id() == i).unwrap())
+                .collect_vec();
+
+            let lwe_n = self.parameters().lwe_n().0;
+            let (users_segments, users_segments_sizes): (Vec<(usize, usize)>, Vec<usize>) = (0
+                ..total_users)
+                .map(|(user_id)| {
+                    let (start_index, end_index) =
+                        interactive_mult_party_user_id_lwe_segment(user_id, total_users, lwe_n);
+                    ((start_index, end_index), end_index - start_index)
+                })
+                .unzip();
+
+            let mut rgsw_cts = Vec::with_capacity(lwe_n);
+            users_segments
+                .iter()
+                .enumerate()
+                .for_each(|(user_id, user_segment)| {
+                    let share = shares_in_correct_order[user_id];
+                    for secret_index in user_segment.0..user_segment.1 {
+                        let mut rgsw_i =
+                            share.self_leader_rgsws()[secret_index - user_segment.0].clone();
+                        // assert already exists in RGSW x RGSW rountine
+                        assert!(rgsw_i.dimension() == rlwe_x_rgsw_dimension);
+
+                        // multiply leader's RGSW ct at `secret_index` with RGSW cts of other users
+                        // for lwe index `secret_index`
+                        (0..total_users)
+                            .filter(|i| i != &user_id)
+                            .for_each(|other_user_id| {
+                                let mut offset = 0;
+                                if other_user_id < user_id {
+                                    offset = users_segments_sizes[other_user_id];
+                                }
+
+                                let mut other_rgsw_i = shares_in_correct_order[other_user_id]
+                                    .not_self_leader_rgsws()
+                                    [secret_index.checked_sub(offset).unwrap()]
+                                .clone();
+                                // assert already exists in RGSW x RGSW rountine
+                                assert!(other_rgsw_i.dimension() == rgsw_x_rgsw_dimension);
+
+                                // send to evaluation domain for RGSwxRGSW mul
+                                other_rgsw_i
+                                    .iter_rows_mut()
+                                    .for_each(|r| rlweq_nttop.forward(r.as_mut()));
+
+                                rgsw_by_rgsw_inplace(
+                                    &mut rgsw_i,
+                                    rlwe_x_rgsw_decomposer.a().decomposition_count(),
+                                    rlwe_x_rgsw_decomposer.b().decomposition_count(),
+                                    &other_rgsw_i,
+                                    &rgsw_by_rgsw_decomposer,
+                                    &mut rgsw_x_rgsw_scratch_mat,
+                                    rlweq_nttop,
+                                    rlweq_modop,
+                                )
+                            });
+
+                        rgsw_cts.push(rgsw_i);
+                    }
+                });
+
+            rgsw_cts
+        };
+
+        // LWE ksks
+        let mut lwe_ksk = M::R::zeros(rlwe_n * parameters.lwe_decomposition_count().0);
+        let lweq_modop = &self.pbs_info.lwe_modop;
+        shares.iter().for_each(|si| {
+            assert!(si.lwe_ksk().as_ref().len() == rlwe_n * parameters.lwe_decomposition_count().0);
+            lweq_modop.elwise_add_mut(lwe_ksk.as_mut(), si.lwe_ksk().as_ref())
+        });
+
+        SeededMultiPartyServerKey::new(rgsw_cts, auto_keys, lwe_ksk, cr_seed.clone(), parameters)
+    }
+
     pub(super) fn aggregate_non_interactive_multi_party_key_share(
         &self,
         cr_seed: &NonInteractiveMultiPartyCrs<[u8; 32]>,
@@ -1351,6 +1572,8 @@ .skip(1)
                     .for_each(|user_i_rgsws| {
                         rgsw_by_rgsw_inplace(
                             &mut rgsw_i,
+                            rgsw_by_rgsw_decomposer.a().decomposition_count(),
+                            rgsw_by_rgsw_decomposer.b().decomposition_count(),
                             &user_i_rgsws[s_index],
                             &rgsw_by_rgsw_decomposer,
                             &mut scratch_matrix,
@@ -1834,125 +2057,6 @@ ) -> bool {
         let m = decrypt_lwe(lwe_ct, &client_key.sk_rlwe(), &self.pbs_info.rlwe_modop);
         self.pbs_info.rlwe_q().decode(m)
     }
-
-    pub(super) fn aggregate_multi_party_server_key_shares<S>(
-        &self,
-        shares: &[CommonReferenceSeededMultiPartyServerKeyShare<
-            M,
-            BoolParameters<M::MatElement>,
-            MultiPartyCrs<S>,
-        >],
-    ) -> SeededMultiPartyServerKey<M, MultiPartyCrs<S>, BoolParameters<M::MatElement>>
-    where
-        S: PartialEq + Clone,
-        M: Clone,
-    {
-        assert_eq!(self.parameters().variant(), &ParameterVariant::MultiParty);
-        assert!(shares.len() > 0);
-        let parameters = shares[0].parameters().clone();
-        let cr_seed = shares[0].cr_seed();
-
-        let rlwe_n = parameters.rlwe_n().0;
-        let g = parameters.g() as isize;
-        let rlwe_q = parameters.rlwe_q();
-        let lwe_q = parameters.lwe_q();
-
-        // sanity checks
-        shares.iter().skip(1).for_each(|s| {
-            assert!(s.parameters() == &parameters);
-            assert!(s.cr_seed() == cr_seed);
-        });
-
-        let rlweq_modop = &self.pbs_info.rlwe_modop;
-        let rlweq_nttop = &self.pbs_info.rlwe_nttop;
-
-        // auto keys
-        let mut auto_keys = HashMap::new();
-        let auto_elements_dlog = parameters.auto_element_dlogs();
-        for i in auto_elements_dlog.into_iter() {
-            let mut key = M::zeros(parameters.auto_decomposition_count().0, rlwe_n);
-
-            shares.iter().for_each(|s| {
-                let auto_key_share_i = s.auto_keys().get(&i).expect("Auto key {i} missing");
-                assert!(
-                    auto_key_share_i.dimension()
-                        == (parameters.auto_decomposition_count().0, rlwe_n)
-                );
-                izip!(key.iter_rows_mut(), auto_key_share_i.iter_rows()).for_each(
-                    |(partb_out, partb_share)| {
-                        rlweq_modop.elwise_add_mut(partb_out.as_mut(), partb_share.as_ref());
-                    },
-                );
-            });
-
-            auto_keys.insert(i, key);
-        }
-
-        // rgsw ciphertext (most expensive part!)
-        let lwe_n = parameters.lwe_n().0;
-        let rgsw_by_rgsw_decomposer =
-            parameters.rgsw_rgsw_decomposer::<DefaultDecomposer<M::MatElement>>();
-        let mut scratch_matrix = M::zeros(
-            std::cmp::max(
-                rgsw_by_rgsw_decomposer.a().decomposition_count(),
-                rgsw_by_rgsw_decomposer.b().decomposition_count(),
-            ) + (rgsw_by_rgsw_decomposer.a().decomposition_count() * 2
-                + rgsw_by_rgsw_decomposer.b().decomposition_count() * 2),
-            rlwe_n,
-        );
-
-        let mut tmp_rgsw =
-            RgswCiphertext::<M, _>::empty(rlwe_n, &rgsw_by_rgsw_decomposer, rlwe_q.clone()).data;
-        let rgsw_cts = (0..lwe_n).into_iter().map(|index| {
-            // copy over rgsw ciphertext for index^th secret element from first share and
-            // treat it as accumulating rgsw ciphertext
-            let mut rgsw_i = shares[0].rgsw_cts()[index].clone();
-
-            shares.iter().skip(1).for_each(|si| {
-                // copy over si's RGSW[index] ciphertext and send to evaluation domain
-                izip!(tmp_rgsw.iter_rows_mut(), si.rgsw_cts()[index].iter_rows()).for_each(
-                    |(to_ri, from_ri)| {
-                        to_ri.as_mut().copy_from_slice(from_ri.as_ref());
-                        rlweq_nttop.forward(to_ri.as_mut())
-                    },
-                );
-
-                rgsw_by_rgsw_inplace(
-                    &mut rgsw_i,
-                    &tmp_rgsw,
-                    &rgsw_by_rgsw_decomposer,
-                    &mut scratch_matrix,
-                    rlweq_nttop,
-                    rlweq_modop,
-                );
-            });
-
-            rgsw_i
-        });
-        // d_a and d_b may differ for RGSWxRGSW multiplication and RLWExRGSW
-        // multiplication. After this point RGSW ciphertexts will only be used for
-        // RLWExRGSW multiplication (in blind rotation). Thus we drop any additional
-        // RLWE ciphertexts in RGSW ciphertexts after RGSw x RGSW multiplication
-        let rgsw_cts = rgsw_cts
-            .map(|ct_i_in| {
-                trim_rgsw_ct_matrix_from_rgrg_to_rlrg(
-                    ct_i_in,
-                    self.parameters().rgsw_by_rgsw_decomposition_params(),
-                    self.parameters().rlwe_by_rgsw_decomposition_params(),
-                )
-            })
-            .collect_vec();
-
-        // LWE ksks
-        let mut lwe_ksk = M::R::zeros(rlwe_n * parameters.lwe_decomposition_count().0);
-        let lweq_modop = &self.pbs_info.lwe_modop;
-        shares.iter().for_each(|si| {
-            assert!(si.lwe_ksk().as_ref().len() == rlwe_n * parameters.lwe_decomposition_count().0);
-            lweq_modop.elwise_add_mut(lwe_ksk.as_mut(), si.lwe_ksk().as_ref())
-        });
-
-        SeededMultiPartyServerKey::new(rgsw_cts, auto_keys, lwe_ksk, cr_seed.clone(), parameters)
-    }
 }
 
 impl<M, NttOp, RlweModOp, LweModOp, Skey> BoolEvaluator<M, NttOp, RlweModOp, LweModOp, Skey>
@@ -2267,6 +2371,8 @@ *ideal_i = *ideal_i + s_i;
             });
         });
 
+        let mut rng = DefaultSecureRng::new();
+
         // check noise in freshly encrypted RLWE ciphertext (ie var_fresh)
         if false {
             let mut rng = DefaultSecureRng::new();
@@ -2316,9 +2422,6 @@ }
 
         if true {
             // Generate server key shares
-            let mut rng = DefaultSecureRng::new();
-            let mut pk_cr_seed = [0u8; 32];
-            rng.fill_bytes(&mut pk_cr_seed);
             let public_key_share = parties
                 .iter()
                 .map(|k| bool_evaluator.multi_party_public_key_share(&int_mp_seed, k))
@@ -2329,12 +2432,13 @@ DefaultSecureRng,
                 ModularOpsU64<CiphertextModulus<u64>>,
             >::from(public_key_share.as_slice());
 
-            let pbs_cr_seed = [0u8; 32];
-            rng.fill_bytes(&mut pk_cr_seed);
             let server_key_shares = parties
                 .iter()
-                .map(|k| {
+                .enumerate()
+                .map(|(user_id, k)| {
                     bool_evaluator.multi_party_server_key_share(
+                        user_id,
+                        no_of_parties,
                         &int_mp_seed,
                         collective_pk.key(),
                         k,
@@ -2351,13 +2455,12 @@ let mut check = Stats { samples: vec![] };
                 izip!(ideal_lwe_sk.iter(), seeded_server_key.rgsw_cts().iter()).for_each(
                     |(s_i, rgsw_ct_i)| {
                         // X^{s[i]}
-                        let mut m_si = vec![0u64; rlwe_n];
-                        let s_i = *s_i * (bool_evaluator.pbs_info.embedding_factor as i32);
-                        if s_i < 0 {
-                            m_si[rlwe_n - (s_i.abs() as usize)] = rlwe_q.neg_one();
-                        } else {
-                            m_si[s_i as usize] = 1;
-                        }
+                        let m_si = encode_x_pow_si_with_emebedding_factor::<Vec<u64>, _>(
+                            *s_i,
+                            bool_evaluator.pbs_info.embedding_factor,
+                            rlwe_n,
+                            rlwe_q,
+                        );
 
                         // RLWE'(-sm)
                         let mut neg_s_eval =

@@ -314,7 +314,8 @@ }
 
 /// CRS seeded Multi-party server key share
 pub struct CommonReferenceSeededMultiPartyServerKeyShare<M: Matrix, P, S> {
-    rgsw_cts: Vec<M>,
+    self_leader_rgsws: Vec<M>,
+    not_self_leader_rgsws: Vec<M>,
     /// Auto keys. Key corresponding to g^{k} is at index `k`. Key corresponding
     /// to -g is at 0
     auto_keys: HashMap<usize, M>,
@@ -322,22 +323,27 @@ lwe_ksk: M::R,
     /// Common reference seed
     cr_seed: S,
     parameters: P,
+    user_id: usize,
 }
 
 impl<M: Matrix, P, S> CommonReferenceSeededMultiPartyServerKeyShare<M, P, S> {
     pub(super) fn new(
-        rgsw_cts: Vec<M>,
+        self_leader_rgsws: Vec<M>,
+        not_self_leader_rgsws: Vec<M>,
         auto_keys: HashMap<usize, M>,
         lwe_ksk: M::R,
         cr_seed: S,
         parameters: P,
+        user_id: usize,
     ) -> Self {
         CommonReferenceSeededMultiPartyServerKeyShare {
-            rgsw_cts,
+            self_leader_rgsws,
+            not_self_leader_rgsws,
             auto_keys,
             lwe_ksk,
             cr_seed,
             parameters,
+            user_id,
         }
     }
 
@@ -353,12 +359,20 @@ pub(super) fn auto_keys(&self) -> &HashMap<usize, M> {
         &self.auto_keys
     }
 
-    pub(super) fn rgsw_cts(&self) -> &[M] {
-        &self.rgsw_cts
+    pub(crate) fn self_leader_rgsws(&self) -> &[M] {
+        &self.self_leader_rgsws
+    }
+
+    pub(super) fn not_self_leader_rgsws(&self) -> &[M] {
+        &self.not_self_leader_rgsws
     }
 
     pub(super) fn lwe_ksk(&self) -> &M::R {
         &self.lwe_ksk
+    }
+
+    pub(super) fn user_id(&self) -> usize {
+        self.user_id
     }
 }
 

@@ -58,6 +58,8 @@ }
 
 pub fn gen_mp_keys_phase2<R, ModOp>(
     ck: &ClientKey,
+    user_id: usize,
+    total_users: usize,
     pk: &PublicKey<Vec<Vec<u64>>, R, ModOp>,
 ) -> CommonReferenceSeededMultiPartyServerKeyShare<
     Vec<Vec<u64>>,
@@ -65,8 +67,13 @@ BoolParameters<u64>,
     MultiPartyCrs<[u8; 32]>,
 > {
     BoolEvaluator::with_local_mut(|e| {
-        let server_key_share =
-            e.multi_party_server_key_share(MultiPartyCrs::global(), pk.key(), ck);
+        let server_key_share = e.multi_party_server_key_share(
+            user_id,
+            total_users,
+            MultiPartyCrs::global(),
+            pk.key(),
+            ck,
+        );
         server_key_share
     })
 }
@@ -251,7 +258,11 @@ // collective pk
         let pk = aggregate_public_key_shares(&pk_shares);
 
         // round 2
-        let server_key_shares = cks.iter().map(|k| gen_mp_keys_phase2(k, &pk)).collect_vec();
+        let server_key_shares = cks
+            .iter()
+            .enumerate()
+            .map(|(user_id, k)| gen_mp_keys_phase2(k, user_id, parties, &pk))
+            .collect_vec();
 
         // server key
         let server_key = aggregate_server_key_shares(&server_key_shares);

@@ -13,6 +13,7 @@ parameters::{CiphertextModulus, SMALL_MP_BOOL_PARAMS},
         },
         evaluator::MultiPartyCrs,
         ntt::NttBackendU64,
+        parameters::OPTIMISED_SMALL_MP_BOOL_PARAMS,
         random::DefaultSecureRng,
     };
 
@@ -25,7 +26,7 @@ NttBackendU64,
             ModularOpsU64<CiphertextModulus<u64>>,
             ModulusPowerOf2<CiphertextModulus<u64>>,
             ShoupServerKeyEvaluationDomain<Vec<Vec<u64>>>,
-        >::new(SMALL_MP_BOOL_PARAMS);
+        >::new(OPTIMISED_SMALL_MP_BOOL_PARAMS);
 
         let parties = 2;
 
@@ -72,7 +73,10 @@
         // round 2
         let server_key_shares = cks
             .iter()
-            .map(|c| evaluator.multi_party_server_key_share(&cr_seed, &pk.key(), c))
+            .enumerate()
+            .map(|(index, c)| {
+                evaluator.multi_party_server_key_share(index, parties, &cr_seed, &pk.key(), c)
+            })
             .collect_vec();
 
         let server_key = evaluator.aggregate_multi_party_server_key_shares(&server_key_shares);
@@ -88,9 +92,6 @@ let mut m1 = true;
 
         let mut c_m0 = evaluator.pk_encrypt(pk.key(), m0);
         let mut c_m1 = evaluator.pk_encrypt(pk.key(), m1);
-
-        let true_el_encoded = evaluator.parameters().rlwe_q().true_el();
-        let false_el_encoded = evaluator.parameters().rlwe_q().false_el();
 
         // let mut stats = Stats::new();
 

@@ -494,14 +494,14 @@ rlwe_n: PolynomialSize(1 << 11),
     lwe_n: LweDimension(500),
     lwe_decomposer_params: (DecompostionLogBase(1), DecompositionCount(11)),
     rlrg_decomposer_params: (
-        DecompostionLogBase(24),
+        DecompostionLogBase(16),
         (DecompositionCount(1), DecompositionCount(1)),
     ),
     rgrg_decomposer_params: Some((
-        DecompostionLogBase(12),
-        (DecompositionCount(3), DecompositionCount(3)),
+        DecompostionLogBase(8),
+        (DecompositionCount(6), DecompositionCount(6)),
     )),
-    auto_decomposer_params: (DecompostionLogBase(20), DecompositionCount(1)),
+    auto_decomposer_params: (DecompostionLogBase(24), DecompositionCount(1)),
     non_interactive_ui_to_s_key_switch_decomposer: None,
     g: 5,
     w: 10,

@@ -1114,6 +1114,8 @@ &_sk_encrypt_rgsw(&m, s.values(), &decomposer, &mod_op, &ntt_op),
                 );
             rgsw_by_rgsw_inplace(
                 &mut rgsw_carrym,
+                decomposer.a().decomposition_count(),
+                decomposer.b().decomposition_count(),
                 &rgsw_m.data,
                 &decomposer,
                 &mut scratch_matrix,

@@ -546,14 +546,19 @@ /// - 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
+///
+/// ## Note:
+/// - We treat RGSW x RGSW as multiple RLWE x RGSW multiplications.  .
 pub(crate) fn rgsw_by_rgsw_inplace<
     Mmut: MatrixMut,
     D: RlweDecomposer<Element = Mmut::MatElement>,
     ModOp: VectorOps<Element = Mmut::MatElement>,
     NttOp: Ntt<Element = Mmut::MatElement>,
 >(
-    rgsw_0: &mut Mmut,
-    rgsw_1_eval: &Mmut,
+    rgsw0: &mut Mmut,
+    rgsw0_da: usize,
+    rgsw0_db: usize,
+    rgsw1_eval: &Mmut,
     decomposer: &D,
     scratch_matrix: &mut Mmut,
     ntt_op: &NttOp,
@@ -567,11 +572,12 @@ let decomposer_b = decomposer.b();
     let d_a = decomposer_a.decomposition_count();
     let d_b = decomposer_b.decomposition_count();
     let max_d = std::cmp::max(d_a, d_b);
-    let rgsw_rows = d_a * 2 + d_b * 2;
-    assert!(rgsw_0.dimension().0 == rgsw_rows);
-    let ring_size = rgsw_0.dimension().1;
-    assert!(rgsw_1_eval.dimension() == (rgsw_rows, ring_size));
-    assert!(scratch_matrix.fits(max_d + rgsw_rows, ring_size));
+    let rgsw1_rows = d_a * 2 + d_b * 2;
+    let rgsw0_rows = rgsw0_da * 2 + rgsw0_db * 2;
+    let ring_size = rgsw0.dimension().1;
+    assert!(rgsw0.dimension().0 == rgsw0_rows);
+    assert!(rgsw1_eval.dimension() == (rgsw1_rows, ring_size));
+    assert!(scratch_matrix.fits(max_d + rgsw0_rows, ring_size));
 
     let (decomp_r_space, rgsw_space) = scratch_matrix.split_at_row_mut(max_d);
 
@@ -579,18 +585,25 @@ // zero rgsw_space
     rgsw_space
         .iter_mut()
         .for_each(|ri| ri.as_mut().fill(Mmut::MatElement::zero()));
-    let (rlwe_dash_space_nsm, rlwe_dash_space_m) = rgsw_space.split_at_mut(d_a * 2);
+    let (rlwe_dash_space_nsm, rlwe_dash_space_m) = rgsw_space.split_at_mut(rgsw0_da * 2);
     let (rlwe_dash_space_nsm_parta, rlwe_dash_space_nsm_partb) =
-        rlwe_dash_space_nsm.split_at_mut(d_a);
-    let (rlwe_dash_space_m_parta, rlwe_dash_space_m_partb) = rlwe_dash_space_m.split_at_mut(d_b);
+        rlwe_dash_space_nsm.split_at_mut(rgsw0_da);
+    let (rlwe_dash_space_m_parta, rlwe_dash_space_m_partb) =
+        rlwe_dash_space_m.split_at_mut(rgsw0_db);
 
-    let (rgsw0_nsm, rgsw0_m) = rgsw_0.split_at_row(d_a * 2);
-    let (rgsw1_nsm, rgsw1_m) = rgsw_1_eval.split_at_row(d_a * 2);
+    let (rgsw0_nsm, rgsw0_m) = rgsw0.split_at_row(rgsw0_da * 2);
+    let (rgsw1_nsm, rgsw1_m) = rgsw1_eval.split_at_row(d_a * 2);
 
     // RGSW x RGSW
     izip!(
-        rgsw0_nsm.iter().take(d_a).chain(rgsw0_m.iter().take(d_b)),
-        rgsw0_nsm.iter().skip(d_a).chain(rgsw0_m.iter().skip(d_b)),
+        rgsw0_nsm
+            .iter()
+            .take(rgsw0_da)
+            .chain(rgsw0_m.iter().take(rgsw0_db)),
+        rgsw0_nsm
+            .iter()
+            .skip(rgsw0_da)
+            .chain(rgsw0_m.iter().skip(rgsw0_db)),
         rlwe_dash_space_nsm_parta
             .iter_mut()
             .chain(rlwe_dash_space_m_parta.iter_mut()),
@@ -599,7 +612,9 @@ .iter_mut()
             .chain(rlwe_dash_space_m_partb.iter_mut()),
     )
     .for_each(|(rlwe_a, rlwe_b, rlwe_out_a, rlwe_out_b)| {
-        // Part A
+        // RLWE(m0) x RGSW(m1)
+
+        // Part A: Decomp<RLWE(m0)[A]> \cdot RLWE'(-sm1)
         decompose_r(rlwe_a.as_ref(), decomp_r_space.as_mut(), decomposer_a);
         decomp_r_space
             .iter_mut()
@@ -618,7 +633,7 @@ &rgsw1_nsm[d_a..],
             mod_op,
         );
 
-        // Part B
+        // Part B: Decompose<RLWE(m0)[B]> \cdot RLWE'(m1)
         decompose_r(rlwe_b.as_ref(), decomp_r_space.as_mut(), decomposer_b);
         decomp_r_space
             .iter_mut()
@@ -639,11 +654,11 @@ );
     });
 
     // copy over RGSW(m0m1) into RGSW(m0)
-    izip!(rgsw_0.iter_rows_mut(), rgsw_space.iter())
+    izip!(rgsw0.iter_rows_mut(), rgsw_space.iter())
         .for_each(|(to_ri, from_ri)| to_ri.as_mut().copy_from_slice(from_ri.as_ref()));
 
     // send back to coefficient domain
-    rgsw_0
+    rgsw0
         .iter_rows_mut()
         .for_each(|ri| ntt_op.backward(ri.as_mut()));
 }

@@ -1,12 +1,12 @@
 use std::{fmt::Debug, usize, vec};
 
 use itertools::{izip, Itertools};
-use num_traits::{FromPrimitive, PrimInt, Signed};
+use num_traits::{FromPrimitive, One, PrimInt, Signed};
 
 use crate::{
     backend::Modulus,
     random::{RandomElementInModulus, RandomFill},
-    Matrix,
+    Matrix, Row, RowEntity, RowMut,
 };
 pub trait WithLocal {
     fn with_local<F, R>(func: F) -> R
@@ -188,6 +188,31 @@ // println!("")
     }
 
     return r;
+}
+
+/// Returns a polynomial X^{emebedding_factor * si} \mod {Z_Q / X^{N}+1}
+pub(crate) fn encode_x_pow_si_with_emebedding_factor<
+    R: RowEntity + RowMut,
+    M: Modulus<Element = R::Element>,
+>(
+    si: i32,
+    embedding_factor: usize,
+    ring_size: usize,
+    modulus: &M,
+) -> R
+where
+    R::Element: One,
+{
+    assert!((si.abs() as usize) < ring_size);
+    let mut m = R::zeros(ring_size);
+    let si = si * (embedding_factor as i32);
+    if si < 0 {
+        // X^{-si} = X^{2N-si} = -X^{N-si}, assuming abs(si) < N
+        m.as_mut()[ring_size - (si.abs() as usize)] = modulus.neg_one();
+    } else {
+        m.as_mut()[si as usize] = R::Element::one();
+    }
+    m
 }
 
 pub(crate) fn puncture_p_rng<S: Default + Copy, R: RandomFill<S>>(