diff --git a/README.md b/README.md
index de52435..348b00b 100644
--- a/README.md
+++ b/README.md
@@ -3,7 +3,7 @@
 > [!WARNING]  
 > This is a research project and hasn't been audited. Use in production at your own risk.
 
-Circuit for proving the correct encryption under BFV fully homomorphic encryption scheme. Note that this can be also generalized to any RLWE-based algorithms. Paper -> https://eprint.iacr.org/2024/594.
+Circuit for proving the correct encryption under BFV fully homomorphic encryption scheme. Note that this can be also generalized to any RLWE-based algorithms. Paper -> https://eprint.iacr.org/2024/594. The implementation manages the public inputs of the circuit [in a slightly different way](https://github.com/privacy-scaling-explorations/greco/pull/30) than the one described in the paper.
 
 ### Generate Parameters
 
@@ -51,7 +51,7 @@ By default, the benchmarks are run on the input data generated by the python scr
 
 To run the benchmarks on different parameters you have to:
 
-1. Modify the constants being imported. For example `use crate::constants::sk_enc_constants_8192_4x55_65537::...`
+1. Modify the constants being imported at the top of the file. For example `use crate::constants::sk_enc_constants_8192_4x55_65537::...`
 2. Modify the `file_path` within `bench_sk_enc_full_prover()` function to point to the new input data file. For example `let file_path = "src/data/sk_enc_8192_4x55_65537.json";`
 
 Given the same parameters, the benchmarks will run the prover and verifier for different values of `k`, where `2**k` is the maximum number of rows in the advice columns. After setting a specific `k`, halo2-lib will configure the circuit accordingly. With less rows at disposal, the configuration will contain more columns, and viceversa. 
@@ -83,19 +83,19 @@ bfv params: "src/data/sk_enc_1024_1x27_65537"
 +----+--------------------+--------------------+-----------------------+-------------------------+
 | K  | VK Generation Time | PK Generation Time | Proof Generation Time | Proof Verification Time |
 +----+--------------------+--------------------+-----------------------+-------------------------+
-| 12 | 459.975208ms       | 97.593417ms        | 776.78975ms           | 5.264583ms              |
+| 12 | 490.086041ms       | 104.025833ms       | 805.957292ms          | 5.473208ms              |
 +----+--------------------+--------------------+-----------------------+-------------------------+
-| 13 | 376.945083ms       | 82.856458ms        | 685.505709ms          | 3.661792ms              |
+| 13 | 425.17925ms        | 86.270958ms        | 786.931167ms          | 3.958292ms              |
 +----+--------------------+--------------------+-----------------------+-------------------------+
-| 14 | 327.682125ms       | 83.545375ms        | 771.287958ms          | 2.984834ms              |
+| 14 | 420.792916ms       | 93.239208ms        | 836.020791ms          | 3.149542ms              |
 +----+--------------------+--------------------+-----------------------+-------------------------+
-| 15 | 353.973334ms       | 98.673167ms        | 927.688292ms          | 2.498542ms              |
+| 15 | 405.360125ms       | 99.7695ms          | 955.489416ms          | 2.868708ms              |
 +----+--------------------+--------------------+-----------------------+-------------------------+
-| 16 | 562.989334ms       | 135.40525ms        | 1.479128167s          | 2.474292ms              |
+| 16 | 620.653458ms       | 141.8055ms         | 1.640361125s          | 2.588709ms              |
 +----+--------------------+--------------------+-----------------------+-------------------------+
-| 17 | 771.174084ms       | 230.993791ms       | 2.5907405s            | 2.771833ms              |
+| 17 | 907.310042ms       | 235.223167ms       | 2.67054475s           | 2.668542ms              |
 +----+--------------------+--------------------+-----------------------+-------------------------+
-| 18 | 1.448189792s       | 388.711083ms       | 4.966078959s          | 2.502875ms              |
+| 18 | 1.641826417s       | 428.383166ms       | 5.120378208s          | 2.77925ms               |
 +----+--------------------+--------------------+-----------------------+-------------------------+
 ```
 
@@ -112,19 +112,19 @@ bfv params: "src/data/sk_enc_2048_1x53_65537"
 +----+--------------------+--------------------+-----------------------+-------------------------+
 | K  | VK Generation Time | PK Generation Time | Proof Generation Time | Proof Verification Time |
 +----+--------------------+--------------------+-----------------------+-------------------------+
-| 12 | 984.314167ms       | 209.913583ms       | 1.597872292s          | 8.236375ms              |
+| 12 | 1.02548225s        | 220.633042ms       | 1.671973959s          | 8.7885ms                |
 +----+--------------------+--------------------+-----------------------+-------------------------+
-| 13 | 878.817375ms       | 193.876416ms       | 1.4118565s            | 5.399292ms              |
+| 13 | 937.555209ms       | 200.602167ms       | 1.494908333s          | 5.504542ms              |
 +----+--------------------+--------------------+-----------------------+-------------------------+
-| 14 | 771.242375ms       | 186.790416ms       | 1.386890583s          | 3.735833ms              |
+| 14 | 826.149709ms       | 198.48025ms        | 1.475256583s          | 4.139916ms              |
 +----+--------------------+--------------------+-----------------------+-------------------------+
-| 15 | 669.985041ms       | 190.327167ms       | 1.395613667s          | 3.198541ms              |
+| 15 | 780.143541ms       | 201.493042ms       | 1.532136625s          | 3.031834ms              |
 +----+--------------------+--------------------+-----------------------+-------------------------+
-| 16 | 849.207834ms       | 219.047875ms       | 1.786423708s          | 2.649333ms              |
+| 16 | 820.142333ms       | 232.910583ms       | 1.97258525s           | 2.8235ms                |
 +----+--------------------+--------------------+-----------------------+-------------------------+
-| 17 | 978.258416ms       | 288.281ms          | 2.821929417s          | 2.700958ms              |
+| 17 | 1.155686625s       | 311.328625ms       | 3.081958s             | 2.587541ms              |
 +----+--------------------+--------------------+-----------------------+-------------------------+
-| 18 | 1.477394209s       | 452.682833ms       | 5.152408292s          | 2.576083ms              |
+| 18 | 1.702295666s       | 499.721792ms       | 5.291279833s          | 2.588834ms              |
 +----+--------------------+--------------------+-----------------------+-------------------------+
 ```
 
@@ -142,19 +142,19 @@ bfv params: "src/data/sk_enc_4096_2x55_65537"
 +----+--------------------+--------------------+-----------------------+-------------------------+
 | K  | VK Generation Time | PK Generation Time | Proof Generation Time | Proof Verification Time |
 +----+--------------------+--------------------+-----------------------+-------------------------+
-| 12 | 3.195995042s       | 759.2565ms         | 5.046092875s          | 20.572291ms             |
+| 12 | 3.348614875s       | 774.583583ms       | 5.161994167s          | 21.526084ms             |
 +----+--------------------+--------------------+-----------------------+-------------------------+
-| 13 | 2.791800458s       | 649.398458ms       | 4.3928835s            | 12.524709ms             |
+| 13 | 2.906377042s       | 683.615625ms       | 4.391677667s          | 13.025541ms             |
 +----+--------------------+--------------------+-----------------------+-------------------------+
-| 14 | 2.357667417s       | 631.544333ms       | 4.036068458s          | 7.709ms                 |
+| 14 | 2.4261365s         | 645.739291ms       | 4.09474375s           | 8.119833ms              |
 +----+--------------------+--------------------+-----------------------+-------------------------+
-| 15 | 2.1610035s         | 579.862292ms       | 3.474897667s          | 5.023917ms              |
+| 15 | 2.269133833s       | 602.751583ms       | 3.734384125s          | 5.128375ms              |
 +----+--------------------+--------------------+-----------------------+-------------------------+
-| 16 | 1.986164917s       | 590.40925ms        | 3.76638225s           | 3.719708ms              |
+| 16 | 2.161939667s       | 614.730875ms       | 3.86856775s           | 3.715334ms              |
 +----+--------------------+--------------------+-----------------------+-------------------------+
-| 17 | 1.684586375s       | 571.34275ms        | 3.785945209s          | 2.992958ms              |
+| 17 | 1.983521625s       | 617.253042ms       | 4.26015725s           | 2.987833ms              |
 +----+--------------------+--------------------+-----------------------+-------------------------+
-| 18 | 2.234584458s       | 745.59725ms        | 6.14576425s           | 2.646542ms              |
+| 18 | 2.683152334s       | 781.056125ms       | 6.43829325s           | 3.073709ms              |
 +----+--------------------+--------------------+-----------------------+-------------------------+
 ```
 
@@ -174,19 +174,19 @@ bfv params: "src/data/sk_enc_8192_4x55_65537"
 +----+--------------------+--------------------+-----------------------+-------------------------+
 | K  | VK Generation Time | PK Generation Time | Proof Generation Time | Proof Verification Time |
 +----+--------------------+--------------------+-----------------------+-------------------------+
-| 12 | 12.015320708s      | 3.287664166s       | 17.841099166s         | 76.339042ms             |
+| 12 | 12.260128667s      | 3.215819459s       | 18.129232625s         | 81.15875ms              |
 +----+--------------------+--------------------+-----------------------+-------------------------+
-| 13 | 10.122673209s      | 2.693322459s       | 14.74348625s          | 37.897917ms             |
+| 13 | 10.736324042s      | 2.808556291s       | 15.87686625s          | 39.417833ms             |
 +----+--------------------+--------------------+-----------------------+-------------------------+
-| 14 | 8.418134917s       | 2.40066075s        | 12.849173459s         | 20.831916ms             |
+| 14 | 9.037218833s       | 2.484174917s       | 13.582268708s         | 22.246125ms             |
 +----+--------------------+--------------------+-----------------------+-------------------------+
-| 15 | 6.549789916s       | 2.019510917s       | 9.986622458s          | 11.373042ms             |
+| 15 | 6.576586083s       | 2.077855666s       | 10.57232875s          | 10.488208ms             |
 +----+--------------------+--------------------+-----------------------+-------------------------+
-| 16 | 5.803876875s       | 1.875661917s       | 9.325606542s          | 7.393ms                 |
+| 16 | 6.135039375s       | 1.919404291s       | 9.59165725s           | 7.775375ms              |
 +----+--------------------+--------------------+-----------------------+-------------------------+
-| 17 | 5.049139291s       | 1.819083667s       | 8.978745s             | 4.181208ms              |
+| 17 | 5.278453s          | 1.899231666s       | 9.79360175s           | 4.564875ms              |
 +----+--------------------+--------------------+-----------------------+-------------------------+
-| 18 | 4.78612025s        | 1.866191166s       | 10.295328167s         | 3.325291ms              |
+| 18 | 5.676761667s       | 2.186588542s       | 11.418666042s         | 3.698959ms              |
 +----+--------------------+--------------------+-----------------------+-------------------------+
 ```
 
@@ -210,19 +210,17 @@ bfv params: "src/data/sk_enc_16384_8x54_65537"
 +----+--------------------+--------------------+-----------------------+-------------------------+
 | K  | VK Generation Time | PK Generation Time | Proof Generation Time | Proof Verification Time |
 +----+--------------------+--------------------+-----------------------+-------------------------+
-| 13 | 42.00265175s       | 14.450244625s      | 56.991641875s         | 158.753791ms            |
+| 13 | 43.515304875s      | 14.407916667s      | 58.255132625s         | 166.599041ms            |
 +----+--------------------+--------------------+-----------------------+-------------------------+
-| 14 | 35.048573583s      | 11.7795115s        | 48.7445865s           | 71.56725ms              |
+| 14 | 35.551011334s      | 12.041587959s      | 50.729567334s         | 76.108667ms             |
 +----+--------------------+--------------------+-----------------------+-------------------------+
-| 15 | 25.851191458s      | 9.141273625s       | 37.561299208s         | 32.032583ms             |
+| 15 | 28.063048833s      | 9.476986125s       | 40.625269333s         | 34.402ms                |
 +----+--------------------+--------------------+-----------------------+-------------------------+
-| 16 | 22.46793575s       | 8.425404167s       | 34.270651s            | 17.28725ms              |
+| 16 | 24.613478458s      | 8.841772291s       | 36.525364292s         | 19.868333ms             |
 +----+--------------------+--------------------+-----------------------+-------------------------+
-| 17 | 20.218826084s      | 8.164861959s       | 32.454668959s         | 10.2455ms               |
+| 17 | 20.710349792s      | 8.064435916s       | 33.025659s            | 10.015ms                |
 +----+--------------------+--------------------+-----------------------+-------------------------+
-| 18 | 17.590890541s      | 7.103390958s       | 29.432529166s         | 6.9695ms                |
-+----+--------------------+--------------------+-----------------------+-------------------------+
-| 19 | 15.177152s         | 6.799758542s       | 30.783108166s         | 3.855708ms              |
+| 18 | 17.944376875s      | 7.127217625s       | 31.521722708s         | 7.063ms                 |
 +----+--------------------+--------------------+-----------------------+-------------------------+
 ```
 
@@ -252,16 +250,16 @@ bfv params: "src/data/sk_enc_32768_15x59_65537"
 +----+--------------------+--------------------+-----------------------+-------------------------+
 | K  | VK Generation Time | PK Generation Time | Proof Generation Time | Proof Verification Time |
 +----+--------------------+--------------------+-----------------------+-------------------------+
-| 14 | 149.354869083s     | 65.239957417s      | 177.298666333s        | 337.225833ms            |
+| 14 | 152.501748625s     | 68.303257667s      | 188.803432959s        | 361.9965ms              |
 +----+--------------------+--------------------+-----------------------+-------------------------+
-| 15 | 115.596876125s     | 51.89580625s       | 154.238312375s        | 147.297625ms            |
+| 15 | 126.365190875s     | 53.02766575s       | 163.305771833s        | 154.303292ms            |
 +----+--------------------+--------------------+-----------------------+-------------------------+
-| 16 | 94.846151459s      | 40.290301042s      | 130.1495495s          | 61.721333ms             |
+| 16 | 102.026992625s     | 41.624153458s      | 137.814063958s        | 66.756875ms             |
 +----+--------------------+--------------------+-----------------------+-------------------------+
-| 17 | 77.865249s         | 34.1876465s        | 111.415501042s        | 26.904083ms             |
+| 17 | 83.088430792s      | 35.14210775s       | 119.241555167s        | 29.228375ms             |
 +----+--------------------+--------------------+-----------------------+-------------------------+
-| 18 | 64.948204541s      | 29.328586666s      | 102.15197475s         | 14.061625ms             |
+| 18 | 69.2703075s        | 30.016031167s      | 105.986900125s        | 13.839083ms             |
 +----+--------------------+--------------------+-----------------------+-------------------------+
-| 19 | 62.038049125s      | 29.776692208s      | 104.779942208s        | 8.43725ms               |
+| 19 | 65.224209125s      | 29.8882395s        | 109.52214475s         | 8.5535ms                |
 +----+--------------------+--------------------+-----------------------+-------------------------+
 ```
diff --git a/scripts/circuit_sk.py b/scripts/circuit_sk.py
index 859b7e1..7667d51 100644
--- a/scripts/circuit_sk.py
+++ b/scripts/circuit_sk.py
@@ -144,7 +144,6 @@ def main(args):
 
     # initiate counters for the number of advice cells needed for each constraint phase
     phase_0_assignment_advice_cell_count = 0
-    phase_1_assignment_advice_cell_count = 0
     phase_1_range_check_advice_cell_count = 0
     phase_1_eval_at_gamma_constraint_advice_cell_count = 0
     phase_1_encryption_constraint_advice_cell_count = 0
@@ -164,6 +163,8 @@ def main(args):
 
     r1is_assigned = []
     r2is_assigned = []
+    ais_assigned = []
+    ct0is_assigned = []
 
     for i in range(len(ctis)):
         r1i_assigned = assign_to_circuit(r1is[i], p)
@@ -174,42 +175,45 @@ def main(args):
         phase_0_assignment_advice_cell_count += len(r1i_assigned.coefficients)
         phase_0_assignment_advice_cell_count += len(r2i_assigned.coefficients)
 
+        ai_assigned = assign_to_circuit(ais[i], p)
+        ct0i_assigned = assign_to_circuit(ct0is[i], p)
+        ais_assigned.append(ai_assigned)
+        ct0is_assigned.append(ct0i_assigned)
+
+        phase_0_assignment_advice_cell_count += len(ai_assigned.coefficients)
+        phase_0_assignment_advice_cell_count += len(ct0i_assigned.coefficients)
+
     # For the sake of simplicity, we generate a random challenge here
     gamma = randint(0, 1000)
 
     '''
-    CIRCUIT - PHASE 1 - ASSIGNMENT
+    CIRCUIT - PHASE 1
     '''
 
     # Every assigned value must be an element of the field Zp. Negative coefficients `-z` are assigned as `p - z`
-    ais_at_gamma_assigned = []
-    ct0is_at_gamma_assigned = []
+    # ais_at_gamma_assigned = []
+    # ct0is_at_gamma_assigned = []
     qi_constants = []
     k0i_constants = []
 
     for i in range(len(ctis)):
-        ai_at_gamma = ais[i].evaluate(gamma)
-        ai_at_gamma_assigned = assign_to_circuit(Polynomial([ai_at_gamma]), p).coefficients[0]
-        ais_at_gamma_assigned.append(ai_at_gamma_assigned)
+        # ai_at_gamma = ais[i].evaluate(gamma)
+        # ai_at_gamma_assigned = assign_to_circuit(Polynomial([ai_at_gamma]), p).coefficients[0]
+        # ais_at_gamma_assigned.append(ai_at_gamma_assigned)
 
-        phase_1_assignment_advice_cell_count += 1
+        # phase_1_assignment_advice_cell_count += 1
 
-        ct0i_at_gamma = ctis[i][0].evaluate(gamma)
-        ct0i_at_gamma_assigned = assign_to_circuit(Polynomial([ct0i_at_gamma]), p).coefficients[0]
-        ct0is_at_gamma_assigned.append(ct0i_at_gamma_assigned)
+        # ct0i_at_gamma = ctis[i][0].evaluate(gamma)
+        # ct0i_at_gamma_assigned = assign_to_circuit(Polynomial([ct0i_at_gamma]), p).coefficients[0]
+        # ct0is_at_gamma_assigned.append(ct0i_at_gamma_assigned)
 
-        phase_1_assignment_advice_cell_count += 1
+        # phase_1_assignment_advice_cell_count += 1
 
         qi_constants.append(qis[i])
 
         k0i_constant = assign_to_circuit(Polynomial([k0is[i]]), p).coefficients[0]
         k0i_constants.append(k0i_constant)
 
-    cyclo_at_gamma = cyclo.evaluate(gamma)
-    cyclo_at_gamma_assigned = assign_to_circuit(Polynomial([cyclo_at_gamma]), p).coefficients[0]
-
-    phase_1_assignment_advice_cell_count += 1
-
     '''
     CIRCUIT - PHASE 1 - RANGE CHECK
     '''
@@ -258,6 +262,15 @@ def main(args):
     k1_at_gamma_assigned = k1_assigned.evaluate(gamma)
     phase_1_eval_at_gamma_constraint_advice_cell_count += len(k1_assigned.coefficients) * 2 - 1
 
+    cyclo_at_gamma = cyclo.evaluate(gamma)
+    cyclo_at_gamma_assigned = assign_to_circuit(Polynomial([cyclo_at_gamma]), p).coefficients[0]
+    n_bits_N = n.bit_length()
+
+    # This corresponds to `load_rlc_cache`
+    phase_1_eval_at_gamma_constraint_advice_cell_count += (n_bits_N - 1) * 4
+    # This corresponds to `add`
+    phase_1_eval_at_gamma_constraint_advice_cell_count += 4
+
     for i in range(len(ctis)):
         # sanity check. The coefficients of ct0i should be in the range [-(qi-1)/2, (qi-1)/2]
         bound = int((qis[i] - 1) / 2)
@@ -337,12 +350,18 @@ def main(args):
         phase_1_eval_at_gamma_constraint_advice_cell_count += len(r1is_assigned[i].coefficients) * 2 - 1
         phase_1_eval_at_gamma_constraint_advice_cell_count += len(r2is_assigned[i].coefficients) * 2 - 1
 
+        ai_gamma_assigned = ais_assigned[i].evaluate(gamma)
+        ct0i_gamma_assigned = ct0is_assigned[i].evaluate(gamma)
+
+        phase_1_eval_at_gamma_constraint_advice_cell_count += len(ais_assigned[i].coefficients) * 2 - 1
+        phase_1_eval_at_gamma_constraint_advice_cell_count += len(ct0is_assigned[i].coefficients) * 2 - 1
+
         '''
         CIRCUIT - PHASE 1 - CORRECT ENCRYPTION CONSTRAINT
         '''
 
-        lhs = ct0is_at_gamma_assigned[i]
-        rhs = (ais_at_gamma_assigned[i] * s_at_gamma_assigned + e_at_gamma_assigned + (k1_at_gamma_assigned * k0i_constants[i]) + (r1i_gamma_assigned * qi_constants[i]) + (r2i_gamma_assigned * cyclo_at_gamma_assigned))
+        lhs = ct0i_gamma_assigned
+        rhs = (ai_gamma_assigned * s_at_gamma_assigned + e_at_gamma_assigned + (k1_at_gamma_assigned * k0i_constants[i]) + (r1i_gamma_assigned * qi_constants[i]) + (r2i_gamma_assigned * cyclo_at_gamma_assigned))
         phase_1_encryption_constraint_advice_cell_count += 16
 
         assert lhs % p == rhs % p
@@ -355,22 +374,20 @@ def main(args):
         cyclo_at_gamma_assigned_expected = assign_to_circuit(Polynomial([cyclo_at_gamma]), p).coefficients[0]
         assert cyclo_at_gamma_assigned == cyclo_at_gamma_assigned_expected
 
-        ai_gamma = ais[i].evaluate(gamma)
-        ai_gamma_assigned_expected = assign_to_circuit(Polynomial([ai_gamma]), p).coefficients[0]
-        assert ais_at_gamma_assigned[i] == ai_gamma_assigned_expected
+        ai_gamma_assigned_expected = ais_assigned[i].evaluate(gamma)
+        assert ai_gamma_assigned == ai_gamma_assigned_expected
 
-        ct0i_gamma = ctis[i][0].evaluate(gamma)
-        ct0i_gamma_assigned_expected = assign_to_circuit(Polynomial([ct0i_gamma]), p).coefficients[0]
-        assert ct0is_at_gamma_assigned[i] == ct0i_gamma_assigned_expected
+        ct0i_gamma_assigned_expected = ct0is_assigned[i].evaluate(gamma)
+        assert ct0i_gamma_assigned == ct0i_gamma_assigned_expected
 
         assert qis[i] == qi_constants[i]
 
         k0i_assigned_expected = assign_to_circuit(Polynomial([k0is[i]]), p).coefficients[0]
         assert k0i_constants[i] == k0i_assigned_expected
 
-    total_advice_cell_count = phase_0_assignment_advice_cell_count + phase_1_assignment_advice_cell_count + phase_1_range_check_advice_cell_count + phase_1_eval_at_gamma_constraint_advice_cell_count + phase_1_encryption_constraint_advice_cell_count
+    total_advice_cell_count = phase_0_assignment_advice_cell_count + phase_1_range_check_advice_cell_count + phase_1_eval_at_gamma_constraint_advice_cell_count + phase_1_encryption_constraint_advice_cell_count
 
-    print_advice_cells_info(total_advice_cell_count, phase_0_assignment_advice_cell_count, phase_1_assignment_advice_cell_count, phase_1_range_check_advice_cell_count, phase_1_eval_at_gamma_constraint_advice_cell_count, phase_1_encryption_constraint_advice_cell_count)
+    print_advice_cells_info(total_advice_cell_count, phase_0_assignment_advice_cell_count, phase_1_range_check_advice_cell_count, phase_1_eval_at_gamma_constraint_advice_cell_count, phase_1_encryption_constraint_advice_cell_count)
     # ais and ct0is need to be parsed such that their coefficients are in the range [0, p - 1]
     # we don't call them assigned because they are never assigned to the circuit
     ais_in_p = [assign_to_circuit(ai, p) for ai in ais]
diff --git a/scripts/utils.py b/scripts/utils.py
index bd6226e..c73b492 100644
--- a/scripts/utils.py
+++ b/scripts/utils.py
@@ -55,16 +55,13 @@ def count_advice_cells_needed_for_poly_range_check(poly: Polynomial, bound: int,
 
     return count
 
-def print_advice_cells_info(total_advice_cell_count, phase_0_count, phase_1_assignment_count, phase_1_range_check_count, phase_1_eval_at_gamma_count, phase_1_encryption_constraint_count):
+def print_advice_cells_info(total_advice_cell_count, phase_0_count, phase_1_range_check_count, phase_1_eval_at_gamma_count, phase_1_encryption_constraint_count):
     print("Halo2 Circuit Profile:")
     print(f"Total Advice Cells Needed: {total_advice_cell_count}")
     
     print("\nPhase 0 - Assignment:")
     print(f" - Count: {phase_0_count}, Percentage: {(phase_0_count / total_advice_cell_count) * 100:.2f}%")
 
-    print("\nPhase 1 - Assignment:")
-    print(f" - Count: {phase_1_assignment_count}, Percentage: {(phase_1_assignment_count / total_advice_cell_count) * 100:.2f}%")
-
     print("\nPhase 1 - Range Check:")
     print(f" - Count: {phase_1_range_check_count}, Percentage: {(phase_1_range_check_count / total_advice_cell_count) * 100:.2f}%")
 
diff --git a/src/data/sk_enc_4096_2x55_65537.json b/src/data/sk_enc_4096_2x55_65537.json
index 98d1da8..c6c9db5 100644
--- a/src/data/sk_enc_4096_2x55_65537.json
+++ b/src/data/sk_enc_4096_2x55_65537.json
@@ -1 +1 @@
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"21888242871839275222246405745257275088548364400416034343698199373165270909407", "21888242871839275222246405745257275088548364400416034343698197500672313312895", "11218753896255340", "21888242871839275222246405745257275088548364400416034343698191024823119054211", "5748593710632213", "21888242871839275222246405745257275088548364400416034343698190728209610070030", "21888242871839275222246405745257275088548364400416034343698196292193321364593", "21888242871839275222246405745257275088548364400416034343698203781251853271307", "21888242871839275222246405745257275088548364400416034343698191468895423112500", "21888242871839275222246405745257275088548364400416034343698193720391470511833", "3927715249313530", "21888242871839275222246405745257275088548364400416034343698192410131470614950", "21888242871839275222246405745257275088548364400416034343698201808852267729518", "3428009085644187", "21888242871839275222246405745257275088548364400416034343698201752539688699345", "21888242871839275222246405745257275088548364400416034343698199218844166935500", "21888242871839275222246405745257275088548364400416034343698198066223491827849", "12768556953060907", "21888242871839275222246405745257275088548364400416034343698194745091870645034", "6535893409358843", "21888242871839275222246405745257275088548364400416034343698197990224010151340", "21888242871839275222246405745257275088548364400416034343698202714733703794643", "21888242871839275222246405745257275088548364400416034343698196047591766933617", "75173623470187", "21888242871839275222246405745257275088548364400416034343698193574407378859641", "3607566632712260", "21888242871839275222246405745257275088548364400416034343698192440855243846174", "21888242871839275222246405745257275088548364400416034343698193936918720542503", "8487181164749757", "21888242871839275222246405745257275088548364400416034343698200187386989804900", "11870861277053896", "4080212671401962", "10538127081919969", "10187822570508468", "4228620937584671", "21888242871839275222246405745257275088548364400416034343698197216197687828234", "21888242871839275222246405745257275088548364400416034343698192800725357395077", "7038453698460889", "4592177634842867", "21888242871839275222246405745257275088548364400416034343698203185668411269869", "13085671072223171", "21888242871839275222246405745257275088548364400416034343698203828596655398606", "2354817440901081", "21888242871839275222246405745257275088548364400416034343698202375322575385020", "8811675935240474", "21888242871839275222246405745257275088548364400416034343698196366258765349085", "21888242871839275222246405745257275088548364400416034343698190792565106719347", "11492444954183870", "2280217705709698", "21888242871839275222246405745257275088548364400416034343698203951606704338062", "11908836224920538", "4884761776745823", "21888242871839275222246405745257275088548364400416034343698194976332280997682", "8821963981472519", "8875290677540229", "21888242871839275222246405745257275088548364400416034343698203354245843567323", "1177830225609183", "9646099303499108", "11077874594416349", "21888242871839275222246405745257275088548364400416034343698203919348554306112", "3995586445002873", "13090658495278054", "12004663998052941", "1541386037917458", "21888242871839275222246405745257275088548364400416034343698194674531970343115", "21888242871839275222246405745257275088548364400416034343698192441543369018671", "9280116859690750", "698665966019605", "21888242871839275222246405745257275088548364400416034343698203954598714909736", "21888242871839275222246405745257275088548364400416034343698193517871196484575", "3369929152927563", "21888242871839275222246405745257275088548364400416034343698191945409828089706", "5426118338834424", "21888242871839275222246405745257275088548364400416034343698196333498065338528", "21888242871839275222246405745257275088548364400416034343698200840648833078755", "21888242871839275222246405745257275088548364400416034343698194701254590213996", "3282985244739015", "5740785935213936", "21888242871839275222246405745257275088548364400416034343698198208824157350198", "21888242871839275222246405745257275088548364400416034343698201369458111672423", "21888242871839275222246405745257275088548364400416034343698195973807045464261", "21888242871839275222246405745257275088548364400416034343698199238671319112661", "21888242871839275222246405745257275088548364400416034343698204100177085098688", "4421654317310063"]]}
\ No newline at end of file
diff --git a/src/sk_encryption_circuit.rs b/src/sk_encryption_circuit.rs
index 2915929..0f612f3 100644
--- a/src/sk_encryption_circuit.rs
+++ b/src/sk_encryption_circuit.rs
@@ -22,7 +22,7 @@ pub fn test_params() -> RlcCircuitParams {
             num_fixed: 1,
             num_lookup_advice_per_phase: vec![0, 1],
             lookup_bits: Some(8),
-            num_instance_columns: 0,
+            num_instance_columns: 1,
         },
         num_rlc_columns: 1,
     }
@@ -57,8 +57,8 @@ pub struct Payload<F: ScalarField> {
     k1_assigned: PolyAssigned<F>,
     r2is_assigned: Vec<PolyAssigned<F>>,
     r1is_assigned: Vec<PolyAssigned<F>>,
-    ais: Vec<Vec<String>>,
-    ct0is: Vec<Vec<String>>,
+    ais_assigned: Vec<PolyAssigned<F>>,
+    ct0is_assigned: Vec<PolyAssigned<F>>,
 }
 
 impl<F: ScalarField> RlcCircuitInstructions<F> for BfvSkEncryptionCircuit {
@@ -69,6 +69,7 @@ impl<F: ScalarField> RlcCircuitInstructions<F> for BfvSkEncryptionCircuit {
     /// In this phase, the polynomials for each matrix $S_i$ are assigned to the circuit. Namely:
     /// * polynomials `s`, `e`, `k1` are assigned to the witness table. This has to be done only once as these polynomial are common to each $S_i$ matrix
     /// * polynomials `r1i`, `r2i` are assigned to the witness table for each $S_i$ matrix
+    /// * polynomials `ai`, `ct0i` are assigned to the witness table for each $S_i$ matrix and exposed as public inputs
 
     /// Witness values are element of the finite field $\mod{p}$. Negative coefficients $-z$ are assigned as field elements $p - z$.
 
@@ -80,6 +81,8 @@ impl<F: ScalarField> RlcCircuitInstructions<F> for BfvSkEncryptionCircuit {
     ) -> Self::FirstPhasePayload {
         let ctx = builder.base.main(0);
 
+        let mut public_inputs = vec![];
+
         let s = Poly::<F>::new(self.s.clone());
         let s_assigned = PolyAssigned::new(ctx, s);
 
@@ -91,6 +94,8 @@ impl<F: ScalarField> RlcCircuitInstructions<F> for BfvSkEncryptionCircuit {
 
         let mut r2is_assigned = vec![];
         let mut r1is_assigned = vec![];
+        let mut ais_assigned = vec![];
+        let mut ct0is_assigned = vec![];
 
         for z in 0..self.ct0is.len() {
             let r2i = Poly::<F>::new(self.r2is[z].clone());
@@ -100,16 +105,36 @@ impl<F: ScalarField> RlcCircuitInstructions<F> for BfvSkEncryptionCircuit {
             let r1i = Poly::<F>::new(self.r1is[z].clone());
             let r1i_assigned = PolyAssigned::new(ctx, r1i);
             r1is_assigned.push(r1i_assigned);
+
+            let ai = Poly::<F>::new(self.ais[z].clone());
+            let ai_assigned = PolyAssigned::new(ctx, ai);
+            ais_assigned.push(ai_assigned);
+
+            let ct0i = Poly::<F>::new(self.ct0is[z].clone());
+            let ct0i_assigned = PolyAssigned::new(ctx, ct0i);
+            ct0is_assigned.push(ct0i_assigned);
+        }
+
+        // EXPOSE PUBLIC INPUTS ais and ct0is
+        for ai in ais_assigned.iter() {
+            public_inputs.push(ai.assigned_coefficients[0]);
+        }
+
+        // push each ct0i polynomial to the public inputs
+        for ct0i in ct0is_assigned.iter() {
+            public_inputs.push(ct0i.assigned_coefficients[0]);
         }
 
+        builder.base.assigned_instances[0] = public_inputs;
+
         Payload {
             s_assigned,
             e_assigned,
             k1_assigned,
             r2is_assigned,
             r1is_assigned,
-            ais: self.ais.clone(),
-            ct0is: self.ct0is.clone(),
+            ais_assigned,
+            ct0is_assigned,
         }
     }
 
@@ -120,14 +145,6 @@ impl<F: ScalarField> RlcCircuitInstructions<F> for BfvSkEncryptionCircuit {
     /// - The coefficients of $S_i$ are in the expected range.  
     /// - $U_i(\gamma) \times S_i(\gamma) =Ct_{0,i}(\gamma)$
 
-    /// ##### Assignment
-    /// * Assign evaluations to the circuit: `ai(gamma)`, `ct0i(gamma)` for each $U_i$ matrix
-    /// * Assign `cyclo(gamma)` to the circuit. This has to be done only once as the cyclotomic polynomial is common to each $U_i$ matrix
-    /// * Expose `ai(gamma)`, `ct0i(gamma)` for each $U_i$ matrix
-    /// * Expose `cyclo(gamma)` as public input
-
-    /// Since the polynomials `cyclo`, `ai` and `ct0i` are known to the verifier, the evaluation at $\gamma$ doesn't need to be constrained inside the circuit. Instead, this can be safely be performed (and verified) outside the circuit.
-
     /// ##### Range Check
 
     /// The coefficients of the private polynomials from each $i$-th matrix $S_i$ are checked to be in the correct range
@@ -138,10 +155,10 @@ impl<F: ScalarField> RlcCircuitInstructions<F> for BfvSkEncryptionCircuit {
 
     /// ##### Evaluation at $\gamma$ Constraint
 
-    /// Contrary to the polynomials `cyclo`, `ai` and `ct0i`, the polynomials belonging to each $S_i$ matrix  are not known by the verifier. Therefore, their evaluation at $\gamma$ must be constrained inside the circuit.
-
     /// * Constrain the evaluation of the polynomials `s`, `e`, `k1` at $\gamma$. This has to be done only once as these polynomial are common to each $S_i$ matrix
     /// * Constrain the evaluation of the polynomials `r1i`, `r2i` at $\gamma$ for each $S_i$ matrix
+    /// * Constrain the evaluation of the polynomials `ai` and `ct0i` at $\gamma$ for each $U_i$ matrix
+    /// * Constrain the evaluation of the polynomials `cyclo` at $\gamma$ . This has to be done only once as the cyclotomic polynomial is common to each $U_i$ matrix
 
     /// ##### Correct Encryption Constraint
 
@@ -161,31 +178,17 @@ impl<F: ScalarField> RlcCircuitInstructions<F> for BfvSkEncryptionCircuit {
             k1_assigned,
             r2is_assigned,
             r1is_assigned,
-            ais,
-            ct0is,
+            ais_assigned,
+            ct0is_assigned,
         } = payload;
 
-        // ASSIGNMENT
-
         let (ctx_gate, ctx_rlc) = builder.rlc_ctx_pair();
-        let gamma = *rlc.gamma();
+        let gate = range.gate();
 
-        let mut ais_at_gamma_assigned = vec![];
-        let mut ct0is_at_gamma_assigned = vec![];
         let mut qi_constants = vec![];
         let mut k0i_constants = vec![];
 
-        for z in 0..ct0is.len() {
-            let ai = Poly::<F>::new(ais[z].clone());
-            let ai_at_gamma = ai.eval(gamma);
-            let ai_at_gamma_assigned = ctx_gate.load_witness(ai_at_gamma);
-            ais_at_gamma_assigned.push(ai_at_gamma_assigned);
-
-            let ct0i = Poly::<F>::new(ct0is[z].clone());
-            let ct0i_at_gamma = ct0i.eval(gamma);
-            let ct0i_at_gamma_assigned = ctx_gate.load_witness(ct0i_at_gamma);
-            ct0is_at_gamma_assigned.push(ct0i_at_gamma_assigned);
-
+        for z in 0..ct0is_assigned.len() {
             let qi_constant = Constant(F::from_str_vartime(QIS[z]).unwrap());
             qi_constants.push(qi_constant);
 
@@ -193,39 +196,40 @@ impl<F: ScalarField> RlcCircuitInstructions<F> for BfvSkEncryptionCircuit {
             k0i_constants.push(k0i_constant);
         }
 
-        // cyclo poly is equal to x^N + 1
-        let cyclo_at_gamma = gamma.pow_vartime([N as u64]) + F::from(1);
-        let cyclo_at_gamma_assigned = ctx_gate.load_witness(cyclo_at_gamma);
-
-        // RANGE CHECK
         s_assigned.range_check(ctx_gate, range, S_BOUND);
         e_assigned.range_check(ctx_gate, range, E_BOUND);
         k1_assigned.range_check(ctx_gate, range, K1_BOUND);
 
-        for z in 0..ct0is.len() {
+        for z in 0..ct0is_assigned.len() {
             r2is_assigned[z].range_check(ctx_gate, range, R2_BOUNDS[z]);
             r1is_assigned[z].range_check(ctx_gate, range, R1_BOUNDS[z]);
         }
 
         // EVALUATION AT GAMMA CONSTRAINT
 
+        let bits_used = usize::BITS as usize - N.leading_zeros() as usize;
+        rlc.load_rlc_cache((ctx_gate, ctx_rlc), gate, bits_used);
+        let cyclo_at_gamma_assigned = rlc.rlc_pow_fixed(ctx_gate, gate, N);
+        let cyclo_at_gamma_assigned =
+            gate.add(ctx_gate, cyclo_at_gamma_assigned, Constant(F::from(1)));
+
         let s_at_gamma = s_assigned.enforce_eval_at_gamma(ctx_rlc, rlc);
         let e_at_gamma = e_assigned.enforce_eval_at_gamma(ctx_rlc, rlc);
         let k1_at_gamma = k1_assigned.enforce_eval_at_gamma(ctx_rlc, rlc);
 
-        let gate = range.gate();
-
         // For each `i` Prove that LHS(gamma) = RHS(gamma)
         // LHS = ct0i(gamma)
         // RHS = ai(gamma) * s(gamma) + e(gamma) + k1(gamma) * k0i + r1i(gamma) * qi + r2i(gamma) * cyclo(gamma)
-        for z in 0..ct0is.len() {
+        for z in 0..ct0is_assigned.len() {
             let r1i_at_gamma = r1is_assigned[z].enforce_eval_at_gamma(ctx_rlc, rlc);
             let r2i_at_gamma = r2is_assigned[z].enforce_eval_at_gamma(ctx_rlc, rlc);
+            let ai_at_gamma = ais_assigned[z].enforce_eval_at_gamma(ctx_rlc, rlc);
+            let ct0i_at_gamma = ct0is_assigned[z].enforce_eval_at_gamma(ctx_rlc, rlc);
 
             // CORRECT ENCRYPTION CONSTRAINT
 
             // rhs = ai(gamma) * s(gamma) + e(gamma)
-            let rhs = gate.mul_add(ctx_gate, ais_at_gamma_assigned[z], s_at_gamma, e_at_gamma);
+            let rhs = gate.mul_add(ctx_gate, ai_at_gamma, s_at_gamma, e_at_gamma);
 
             // rhs = rhs + k1(gamma) * k0i
             let rhs = gate.mul_add(ctx_gate, k1_at_gamma, k0i_constants[z], rhs);
@@ -235,13 +239,26 @@ impl<F: ScalarField> RlcCircuitInstructions<F> for BfvSkEncryptionCircuit {
 
             // rhs = rhs + r2i(gamma) * cyclo(gamma)
             let rhs = gate.mul_add(ctx_gate, r2i_at_gamma, cyclo_at_gamma_assigned, rhs);
-            let lhs = ct0is_at_gamma_assigned[z];
+            let lhs = ct0i_at_gamma;
 
             // LHS(gamma) = RHS(gamma)
             let res = gate.is_equal(ctx_gate, lhs, rhs);
             gate.assert_is_const(ctx_gate, &res, &F::from(1));
         }
     }
+
+    fn instances(&self) -> Vec<Vec<F>> {
+        let mut instance = vec![];
+        for ai in self.ais.iter() {
+            let ai_poly = Poly::<F>::new(ai.clone());
+            instance.push(ai_poly.coefficients[0]);
+        }
+        for ct0i in self.ct0is.iter() {
+            let ct0i_poly = Poly::<F>::new(ct0i.clone());
+            instance.push(ct0i_poly.coefficients[0]);
+        }
+        vec![instance]
+    }
 }
 
 #[cfg(test)]
@@ -252,7 +269,10 @@ mod test {
         constants::sk_enc_constants_4096_2x55_65537::R1_BOUNDS,
         sk_encryption_circuit::BfvSkEncryptionCircuit,
     };
-    use axiom_eth::rlc::{circuit::builder::RlcCircuitBuilder, utils::executor::RlcExecutor};
+    use axiom_eth::rlc::{
+        circuit::{builder::RlcCircuitBuilder, instructions::RlcCircuitInstructions},
+        utils::executor::RlcExecutor,
+    };
     use halo2_base::{
         gates::circuit::CircuitBuilderStage,
         halo2_proofs::{
@@ -262,7 +282,7 @@ mod test {
         },
         utils::{
             fs::gen_srs,
-            testing::{check_proof, gen_proof},
+            testing::{check_proof_with_instances, gen_proof_with_instances},
         },
     };
     use std::{fs::File, io::Read};
@@ -290,13 +310,15 @@ mod test {
                 .use_params(rlc_circuit_params.clone());
         mock_builder.base.set_lookup_bits(8);
 
+        let instances = sk_enc_circuit.instances();
+
         let rlc_circuit = RlcExecutor::new(mock_builder, sk_enc_circuit);
 
         // 3. Run the mock prover. The circuit should be satisfied
         MockProver::run(
             rlc_circuit_params.base.k.try_into().unwrap(),
             &rlc_circuit,
-            vec![],
+            instances,
         )
         .unwrap()
         .assert_satisfied();
@@ -321,6 +343,7 @@ mod test {
         let mut key_gen_builder =
             RlcCircuitBuilder::from_stage(CircuitBuilderStage::Keygen, 0).use_k(k);
         key_gen_builder.base.set_lookup_bits(k - 1); // lookup bits set to `k-1` as suggested [here](https://docs.axiom.xyz/protocol/zero-knowledge-proofs/getting-started-with-halo2#technical-detail-how-to-choose-lookup_bits)
+        key_gen_builder.base.set_instance_columns(1);
 
         let rlc_circuit = RlcExecutor::new(key_gen_builder, empty_sk_enc_circuit.clone());
 
@@ -334,10 +357,9 @@ mod test {
         drop(rlc_circuit);
 
         // 5. Generate the proof, here we pass the actual inputs
-        let mut proof_gen_builder: RlcCircuitBuilder<Fr> =
+        let proof_gen_builder: RlcCircuitBuilder<Fr> =
             RlcCircuitBuilder::from_stage(CircuitBuilderStage::Prover, 0)
                 .use_params(rlc_circuit_params);
-        proof_gen_builder.base.set_lookup_bits(k - 1);
 
         let file_path = "src/data/sk_enc_4096_2x55_65537.json";
         let mut file = File::open(file_path).unwrap();
@@ -345,6 +367,8 @@ mod test {
         file.read_to_string(&mut data).unwrap();
         let sk_enc_circuit = serde_json::from_str::<BfvSkEncryptionCircuit>(&data).unwrap();
 
+        let instances = sk_enc_circuit.instances();
+
         let rlc_circuit = RlcExecutor::new(proof_gen_builder, sk_enc_circuit.clone());
 
         rlc_circuit
@@ -352,10 +376,10 @@ mod test {
             .builder
             .borrow_mut()
             .set_break_points(break_points);
-        let proof = gen_proof(&kzg_params, &pk, rlc_circuit);
+        let proof = gen_proof_with_instances(&kzg_params, &pk, rlc_circuit, &[&instances[0]]);
 
         // 6. Verify the proof
-        check_proof(&kzg_params, pk.get_vk(), &proof, true);
+        check_proof_with_instances(&kzg_params, pk.get_vk(), &proof, &[&instances[0]], true);
     }
 
     #[test]
@@ -378,13 +402,15 @@ mod test {
                 .use_params(rlc_circuit_params.clone());
         mock_builder.base.set_lookup_bits(8); // Set the lookup bits to 8
 
+        let instances = sk_enc_circuit.instances();
+
         let rlc_circuit = RlcExecutor::new(mock_builder, sk_enc_circuit);
 
         // 4. Run the mock prover
         let invalid_mock_prover = MockProver::run(
             rlc_circuit_params.base.k.try_into().unwrap(),
             &rlc_circuit,
-            vec![],
+            instances,
         )
         .unwrap();
 
@@ -395,19 +421,19 @@ mod test {
             Err(vec![
                 VerifyFailure::Permutation {
                     column: (Any::advice_in(SecondPhase), 1).into(),
-                    location: FailureLocation::OutsideRegion { row: 393180 }
+                    location: FailureLocation::OutsideRegion { row: 393175 }
                 },
                 VerifyFailure::Permutation {
                     column: (Any::advice_in(SecondPhase), 1).into(),
-                    location: FailureLocation::OutsideRegion { row: 393190 }
+                    location: FailureLocation::OutsideRegion { row: 393185 }
                 },
                 VerifyFailure::Permutation {
                     column: (Any::advice_in(SecondPhase), 1).into(),
-                    location: FailureLocation::OutsideRegion { row: 905111 }
+                    location: FailureLocation::OutsideRegion { row: 905158 }
                 },
                 VerifyFailure::Permutation {
                     column: (Any::advice_in(SecondPhase), 1).into(),
-                    location: FailureLocation::OutsideRegion { row: 905131 }
+                    location: FailureLocation::OutsideRegion { row: 905178 }
                 },
             ])
         );
@@ -433,13 +459,14 @@ mod test {
                 .use_params(rlc_circuit_params.clone());
         mock_builder.base.set_lookup_bits(8); // Set the lookup bits to 8
 
+        let instances = sk_enc_circuit.instances();
         let rlc_circuit = RlcExecutor::new(mock_builder, sk_enc_circuit);
 
         // 4. Run the mock prover
         let invalid_mock_prover = MockProver::run(
             rlc_circuit_params.base.k.try_into().unwrap(),
             &rlc_circuit,
-            vec![],
+            instances,
         )
         .unwrap();
 
@@ -457,15 +484,15 @@ mod test {
                 },
                 VerifyFailure::Permutation {
                     column: (Any::advice_in(SecondPhase), 1).into(),
-                    location: FailureLocation::OutsideRegion { row: 871319 }
+                    location: FailureLocation::OutsideRegion { row: 871366 }
                 },
                 VerifyFailure::Permutation {
                     column: (Any::advice_in(SecondPhase), 1).into(),
-                    location: FailureLocation::OutsideRegion { row: 871339 }
+                    location: FailureLocation::OutsideRegion { row: 871386 }
                 },
                 VerifyFailure::Permutation {
                     column: (Any::advice_in(SecondPhase), 1).into(),
-                    location: FailureLocation::OutsideRegion { row: 871347 }
+                    location: FailureLocation::OutsideRegion { row: 871394 }
                 },
             ])
         );
@@ -514,6 +541,7 @@ mod test {
             let mut key_gen_builder =
                 RlcCircuitBuilder::from_stage(CircuitBuilderStage::Keygen, 0).use_k(config.k);
             key_gen_builder.base.set_lookup_bits(config.k - 1); // lookup bits set to `k-1` as suggested [here](https://docs.axiom.xyz/protocol/zero-knowledge-proofs/getting-started-with-halo2#technical-detail-how-to-choose-lookup_bits)
+            key_gen_builder.base.set_instance_columns(1);
 
             let rlc_circuit = RlcExecutor::new(key_gen_builder, empty_sk_enc_circuit.clone());
 
@@ -531,10 +559,9 @@ mod test {
             drop(rlc_circuit);
 
             // 4. Generate the proof, here we pass the actual inputs
-            let mut proof_gen_builder: RlcCircuitBuilder<Fr> =
+            let proof_gen_builder: RlcCircuitBuilder<Fr> =
                 RlcCircuitBuilder::from_stage(CircuitBuilderStage::Prover, 0)
                     .use_params(rlc_circuit_params);
-            proof_gen_builder.base.set_lookup_bits(config.k - 1);
 
             let file_path = format!("{}.json", file_path);
             let mut file = File::open(file_path).unwrap();
@@ -549,13 +576,23 @@ mod test {
                 .builder
                 .borrow_mut()
                 .set_break_points(break_points);
+
+            let instances = sk_enc_circuit.instances();
+
             let timer = std::time::Instant::now();
-            let proof = gen_proof(&config.kzg_params, &pk, rlc_circuit);
+            let proof =
+                gen_proof_with_instances(&config.kzg_params, &pk, rlc_circuit, &[&instances[0]]);
             let proof_gen_time = timer.elapsed();
 
             // 5. Verify the proof
             let timer = std::time::Instant::now();
-            check_proof(&config.kzg_params, pk.get_vk(), &proof, true);
+            check_proof_with_instances(
+                &config.kzg_params,
+                pk.get_vk(),
+                &proof,
+                &[&instances[0]],
+                true,
+            );
             let proof_verification_time = timer.elapsed();
 
             table.add_row(row![