Add FFT & Roots of Unity in Sage

fft.sage

@@ -0,0 +1,42 @@
+# Primitive Root of Unity
+def get_primitive_root_of_unity(F, n):
+    # using the method described by Thomas Pornin in
+    # https://crypto.stackexchange.com/a/63616
+    q = F.order()
+    for k in range(q):
+        if k==0:
+            continue
+        g = F(k)
+        #  g = F.random_element()
+        if g==0:
+            continue
+        w = g ^ ((q-1)/n)
+        if w^(n/2) != 1:
+            return g, w
+
+# Roots of Unity
+def get_nth_roots_of_unity(n, primitive_w):
+    w = [0]*n
+    for i in range(n):
+        w[i] = primitive_w^i
+    return w
+
+
+# fft (Fast Fourier Transform) returns:
+# - nth roots of unity
+# - Vandermonde matrix for the nth roots of unity
+# - Inverse Vandermonde matrix
+def fft(F, n):
+    g, primitive_w = get_primitive_root_of_unity(F, n)
+
+    w = get_nth_roots_of_unity(n, primitive_w)
+
+    ft = matrix(F, n)
+    for j in range(n):
+        row = []
+        for k in range(n):
+            row.append(primitive_w^(j*k))
+        ft.set_row(j, row)
+    ft_inv = ft^-1
+    return w, ft, ft_inv
+