Add fast polynomial multiplication using FFT

fft.sage

@@ -40,3 +40,23 @@ def fft(F, n):
     ft_inv = ft^-1
     return w, ft, ft_inv
 
+
+# Fast polynomial multiplicaton using FFT
+def poly_mul(fa, fb, F, n):
+    w, ft, ft_inv = fft(F, n)
+
+    # compute evaluation points from polynomials fa & fb at the roots of unity
+    a_evals = []
+    b_evals = []
+    for i in range(n):
+        a_evals.append(fa(w[i]))
+        b_evals.append(fb(w[i]))
+
+    # multiply elements in a_evals by b_evals
+    c_evals = map(operator.mul, a_evals, b_evals)
+    c_evals = vector(c_evals)
+
+    # using FFT, convert the c_evals into fc(x)
+    fc_coef = c_evals*ft_inv
+    fc2=P(fc_coef.list())
+    return fc2, c_evals

fft_test.sage

@@ -57,3 +57,23 @@ for i in range(len(a)):
     assert fa(w[i]) == a[i]
 
 
+
+# Fast polynomial multiplicaton using FFT
+print("\n---------")
+print("---Fast polynomial multiplication using FFT")
+
+n = 8
+# q needs to be a prime, s.t. q-1 is divisible by n
+assert (q-1)%n==0
+print("q =", q, "n = ", n)
+
+fa=P([1,2,3,4])
+fb=P([1,2,3,4])
+fc_expected = fa*fb
+print("fc expected result:", fc_expected) # expected result
+print("fc expected coef", fc_expected.coefficients())
+
+fc, c_evals = poly_mul(fa, fb, F, n)
+print("c_evals=(a_evals*b_evals)=", c_evals)
+print("fc:", fc)
+assert fc_expected == fc