Faster fwdNTT

Resolves some timing discrepancy between the FwdNTT and InverseNTT in the NTT benchmarks. In particular, the FwdNTT benchmark was calling an expensive conversion from NTL::ZZX to NTL::zz_pX, which took about 50us on the helib_fft_forward/hexl_F4_params benchmark.

This PR adds a FFT function using NTL::zz_pX, which avoids this overhead and is called by the pre-existing FFT conversion routines

Co-authored-by: @faberga 
Co-authored-by: @hamishun

Author: fboemer

benchmarks/fft_bench.cpp

@@ -34,14 +34,13 @@ static void helib_fft_forward(benchmark::State& state, Meta& meta)
   long q = prime_generator.next();
   helib::Cmodulus cmod(zms, q, 0);
 
-  NTL::ZZX poly;
-  poly.SetLength(N);
-  for (long i = 0; i < N; ++i)
-    poly[i] = i;
+  NTL::zz_pX poly(N, 1);
 
-  NTL::vec_long transformed;
-  for (auto _ : state)
+  NTL::vec_long transformed(NTL::INIT_SIZE, N);
+
+  for (auto _ : state) {
     cmod.FFT(transformed, poly);
+  }
 }
 
 static void helib_fft_inverse(benchmark::State& state, Meta& meta)

include/helib/CModulus.h

@@ -9,6 +9,20 @@
  * See the License for the specific language governing permissions and
  * limitations under the License. See accompanying LICENSE file.
  */
+
+/* Intel HEXL integration.
+ * Copyright (C) 2021 Intel Corporation
+ * Licensed under the Apache License, Version 2.0 (the "License");
+ * you may not use this file except in compliance with the License.
+ * You may obtain a copy of the License at
+ *  http://www.apache.org/licenses/LICENSE-2.0
+ * Unless required by applicable law or agreed to in writing, software
+ * distributed under the License is distributed on an "AS IS" BASIS,
+ * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+ * See the License for the specific language governing permissions and
+ * limitations under the License.
+ */
+
 #ifndef HELIB_CMODULUS_H
 #define HELIB_CMODULUS_H
 /**
@@ -123,6 +137,9 @@ class Cmodulus
   void FFT(NTL::vec_long& y, const NTL::ZZX& x) const;
   // y = FFT(x)
   void FFT(NTL::vec_long& y, const zzX& x) const;
+  // y = FFT(x)
+  void FFT(NTL::vec_long& y, NTL::zz_pX& x) const;
+
 
   // expects zp context to be set externally
   // x = FFT^{-1}(y)

src/CModulus.cpp

@@ -370,7 +370,7 @@ void Cmodulus::FFT_aux(NTL::vec_long& y, NTL::zz_pX& tmp) const
     y.SetLength(phim);
     long* yp = y.elts();
 
-    NTL::zz_p* tmp_p = tmp.rep.elts();
+    const NTL::zz_p* tmp_p = tmp.rep.elts();
 
 #ifdef USE_INTEL_HEXL
 
@@ -456,7 +456,7 @@ void Cmodulus::FFT(NTL::vec_long& y, const NTL::ZZX& x) const
     convert(tmp, x); // convert input to zpx format
   }
 
-  FFT_aux(y, tmp);
+  FFT(y, tmp);
 }
 
 void Cmodulus::FFT(NTL::vec_long& y, const zzX& x) const
@@ -471,8 +471,16 @@ void Cmodulus::FFT(NTL::vec_long& y, const zzX& x) const
     HELIB_NTIMER_START(FFT_remainder);
     convert(tmp, x); // convert input to zpx format
   }
+  FFT(y, tmp);
+}
 
-  FFT_aux(y, tmp);
+void Cmodulus::FFT(NTL::vec_long& y, NTL::zz_pX& x) const
+{
+  HELIB_TIMER_START;
+  NTL::zz_pBak bak;
+  bak.save();
+  context.restore();
+  FFT_aux(y, x);
 }
 
 void Cmodulus::iFFT(NTL::zz_pX& x, const NTL::vec_long& y) const