feat(ntt): implement synchronous NTT core with Barrett modular reduction

Phase 2.1: Merged Path00+Path01 NTT engine.
- barrett_mul.v: Barrett modular multiplication (a·b mod 3329)
- butterfly_unit.v: Cooley-Tukey/Gentleman-Sande butterfly
- zeta_rom.v: 128-entry ROM with bit-reversed roots of unity
- ntt_core.v: 7-layer NTT FSM, 256×12-bit register file
- ntt_sync.v: valid/ready streaming wrapper

Verified: 13/13 vectors bit-exact vs Python NTT/NTTInverse

test_framework/modules/ntt/gen_vectors.py

@@ -0,0 +1,122 @@
+"""gen_vectors.py - Test vector generator for ntt module.
+
+Generates random polynomials and computes expected NTT results
+using the Python reference implementation (embedded to avoid
+side-effect imports from the original module).
+"""
+
+import os
+import random
+import sys
+
+sys.path.insert(0, os.path.join(os.path.dirname(__file__), '..', '..', 'lib'))
+from vector_gen import VectorGenerator
+
+Q = 3329
+N = 256
+N_INV = 3303
+
+zeta_bitRev = [1, 1729, 2580, 3289, 2642, 630, 1897, 848,
+               1062, 1919, 193, 797, 2786, 3260, 569, 1746,
+               296, 2447, 1339, 1476, 3046, 56, 2240, 1333,
+               1426, 2094, 535, 2882, 2393, 2879, 1974, 821,
+               289, 331, 3253, 1756, 1197, 2304, 2277, 2055,
+               650, 1977, 2513, 632, 2865, 33, 1320, 1915,
+               2319, 1435, 807, 452, 1438, 2868, 1534, 2402,
+               2647, 2617, 1481, 648, 2474, 3110, 1227, 910,
+               17, 2761, 583, 2649, 1637, 723, 2288, 1100,
+               1409, 2662, 3281, 233, 756, 2156, 3015, 3050,
+               1703, 1651, 2789, 1789, 1847, 952, 1461, 2687,
+               939, 2308, 2437, 2388, 733, 2337, 268, 641,
+               1584, 2298, 2037, 3220, 375, 2549, 2090, 1645,
+               1063, 319, 2773, 757, 2099, 561, 2466, 2594,
+               2804, 1092, 403, 1026, 1143, 2150, 2775, 886,
+               1722, 1212, 1874, 1029, 2110, 2935, 885, 2154]
+
+
+def NTT(f):
+    """Forward NTT using Cooley-Tukey, bit-exact with reference."""
+    f_hat = f.copy()
+    i = 1
+    for len_NTT in [128, 64, 32, 16, 8, 4, 2]:
+        for start in range(0, 256, 2 * len_NTT):
+            zeta = zeta_bitRev[i]
+            i += 1
+            for j in range(start, start + len_NTT):
+                t = (zeta * f_hat[j + len_NTT]) % Q
+                f_hat[j + len_NTT] = (f_hat[j] - t) % Q
+                f_hat[j] = (f_hat[j] + t) % Q
+    return f_hat
+
+
+def NTTInverse(f_hat):
+    """Inverse NTT using Gentleman-Sande, bit-exact with reference."""
+    f = f_hat.copy()
+    i = 127
+    for len_NTT in [2, 4, 8, 16, 32, 64, 128]:
+        for start in range(0, 256, 2 * len_NTT):
+            zeta = zeta_bitRev[i]
+            i -= 1
+            for j in range(start, start + len_NTT):
+                t = f[j]
+                f[j] = (t + f[j + len_NTT]) % Q
+                f[j + len_NTT] = (zeta * (f[j + len_NTT] - t)) % Q
+    for i in range(len(f)):
+        f[i] = (f[i] * N_INV) % Q
+    return f
+
+
+class NttVectorGenerator(VectorGenerator):
+    """Generates test vectors for the ntt_sync module."""
+
+    def generate_one(self, params: dict) -> dict:
+        mode = params.get('mode', 'forward')
+
+        if mode == 'forward':
+            f = [random.randint(0, Q - 1) for _ in range(N)]
+            f_hat = NTT(f)
+            return {
+                'input': {'mode': 'F', 'coeffs': f},
+                'expected': {'coeffs': f_hat}
+            }
+
+        elif mode == 'inverse':
+            f_hat = [random.randint(0, Q - 1) for _ in range(N)]
+            f = NTTInverse(f_hat)
+            return {
+                'input': {'mode': 'I', 'coeffs': f_hat},
+                'expected': {'coeffs': f}
+            }
+
+        elif mode == 'roundtrip':
+            f = [random.randint(0, Q - 1) for _ in range(N)]
+            f_hat = NTT(f)
+            f_recover = NTTInverse(f_hat)
+            assert f_recover == f, "Round-trip invariant failed!"
+            return {
+                'input': {'mode': 'I', 'coeffs': f_hat},
+                'expected': {'coeffs': f}
+            }
+
+        else:
+            raise ValueError(f"Unknown mode: {mode}")
+
+    def _format_coeffs(self, coeffs: list[int]) -> str:
+        return ' '.join(f'{c:03X}' for c in coeffs)
+
+    def write_hex_file(self, vectors: list[dict], filepath: str) -> None:
+        os.makedirs(os.path.dirname(filepath), exist_ok=True)
+        with open(filepath, 'w') as f:
+            for v in vectors:
+                mode = v['input']['mode']
+                coeffs = v['input']['coeffs']
+                hex_str = self._format_coeffs(coeffs)
+                f.write(f'{mode} {hex_str}\n')
+
+    def write_expected_file(self, vectors: list[dict], filepath: str) -> None:
+        os.makedirs(os.path.dirname(filepath), exist_ok=True)
+        with open(filepath, 'w') as f:
+            for v in vectors:
+                coeffs = v['expected']['coeffs']
+                hex_str = self._format_coeffs(coeffs)
+                f.write(f'{hex_str}\n')