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
+

fft_test.sage

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+load("fft.sage")
+
+#####
+# Roots of Unity test:
+q = 17
+F = GF(q)
+n = 4
+g, primitive_w = get_primitive_root_of_unity(F, n)
+print("generator:", g)
+print("primitive_w:", primitive_w)
+
+w = get_nth_roots_of_unity(n, primitive_w)
+print(f"{n}th roots of unity: {w}")
+assert w == [1, 13, 16, 4]
+
+
+#####
+# FFT test:
+
+def isprime(num):
+    for n in range(2,int(num^1/2)+1):
+        if num%n==0:
+            return False
+    return True
+
+# list valid values for q
+for i in range(20):
+    if isprime(8*i+1):
+        print("q =", 8*i+1)
+
+q = 41
+F = GF(q)
+n = 4
+# q needs to be a prime, s.t. q-1 is divisible by n
+assert (q-1)%n==0
+print("q =", q, "n = ", n)
+
+# ft: Vandermonde matrix for the nth roots of unity
+w, ft, ft_inv = fft(F, n)
+print("nth roots of unity:", w)
+print("Vandermonde matrix:")
+print(ft)
+
+a = vector([3,4,5,9])
+print("a:", a)
+
+# interpolate f_a(x)
+fa_coef = ft_inv * a
+print("fa_coef:", fa_coef)
+
+P.<x> = PolynomialRing(F)
+fa = P(list(fa_coef))
+print("f_a(x):", fa)
+
+# check that evaluating fa(x) at the roots of unity returns the expected values of a
+for i in range(len(a)):
+    assert fa(w[i]) == a[i]
+
+