def miller_rabin_test(candidate, iterations, randfunc=None):
    """Perform a Miller-Rabin primality test on an integer.

    The test is specified in Section C.3.1 of `FIPS PUB 186-4`__.

    :Parameters:
      candidate : integer
        The number to test for primality.
      iterations : integer
        The maximum number of iterations to perform before
        declaring a candidate a probable prime.
      randfunc : callable
        An RNG function where bases are taken from.

    :Returns:
      ``Primality.COMPOSITE`` or ``Primality.PROBABLY_PRIME``.

    .. __: http://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.186-4.pdf
    """

    if not isinstance(candidate, Integer):
        candidate = Integer(candidate)

    if candidate in (1, 2, 3, 5):
        return PROBABLY_PRIME

    if candidate.is_even():
        return COMPOSITE

    one = Integer(1)
    minus_one = Integer(candidate - 1)

    if randfunc is None:
        randfunc = Random.new().read

    # Step 1 and 2
    m = Integer(minus_one)
    a = 0
    while m.is_even():
        m >>= 1
        a += 1

    # Skip step 3

    # Step 4
    for i in iter_range(iterations):

        # Step 4.1-2
        base = 1
        while base in (one, minus_one):
            base = Integer.random_range(min_inclusive=2,
                                        max_inclusive=candidate - 2)
            assert (2 <= base <= candidate - 2)

        # Step 4.3-4.4
        z = pow(base, m, candidate)
        if z in (one, minus_one):
            continue

        # Step 4.5
        for j in iter_range(1, a):
            z = pow(z, 2, candidate)
            if z == minus_one:
                break
            if z == one:
                return COMPOSITE
        else:
            return COMPOSITE

    # Step 5
    return PROBABLY_PRIME
def lucas_test(candidate):
    """Perform a Lucas primality test on an integer.

    The test is specified in Section C.3.3 of `FIPS PUB 186-4`__.

    :Parameters:
      candidate : integer
        The number to test for primality.

    :Returns:
      ``Primality.COMPOSITE`` or ``Primality.PROBABLY_PRIME``.

    .. __: http://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.186-4.pdf
    """

    if not isinstance(candidate, Integer):
        candidate = Integer(candidate)

    # Step 1
    if candidate in (1, 2, 3, 5):
        return PROBABLY_PRIME
    if candidate.is_even() or candidate.is_perfect_square():
        return COMPOSITE

    # Step 2
    def alternate():
        value = 5
        while True:
            yield value
            if value > 0:
                value += 2
            else:
                value -= 2
            value = -value

    for D in alternate():
        if candidate in (D, -D):
            continue
        js = Integer.jacobi_symbol(D, candidate)
        if js == 0:
            return COMPOSITE
        if js == -1:
            break
    # Found D. P=1 and Q=(1-D)/4 (note that Q is guaranteed to be an integer)

    # Step 3
    # This is \delta(n) = n - jacobi(D/n)
    K = candidate + 1
    # Step 4
    r = K.size_in_bits() - 1
    # Step 5
    # U_1=1 and V_1=P
    U_i = Integer(1)
    V_i = Integer(1)
    U_temp = Integer(0)
    V_temp = Integer(0)
    # Step 6
    for i in iter_range(r - 1, -1, -1):
        # Square
        # U_temp = U_i * V_i % candidate
        U_temp.set(U_i)
        U_temp *= V_i
        U_temp %= candidate
        # V_temp = (((V_i ** 2 + (U_i ** 2 * D)) * K) >> 1) % candidate
        V_temp.set(U_i)
        V_temp *= U_i
        V_temp *= D
        V_temp.multiply_accumulate(V_i, V_i)
        if V_temp.is_odd():
            V_temp += candidate
        V_temp >>= 1
        V_temp %= candidate
        # Multiply
        if K.get_bit(i):
            # U_i = (((U_temp + V_temp) * K) >> 1) % candidate
            U_i.set(U_temp)
            U_i += V_temp
            if U_i.is_odd():
                U_i += candidate
            U_i >>= 1
            U_i %= candidate
            # V_i = (((V_temp + U_temp * D) * K) >> 1) % candidate
            V_i.set(V_temp)
            V_i.multiply_accumulate(U_temp, D)
            if V_i.is_odd():
                V_i += candidate
            V_i >>= 1
            V_i %= candidate
        else:
            U_i.set(U_temp)
            V_i.set(V_temp)
    # Step 7
    if U_i == 0:
        return PROBABLY_PRIME
    return COMPOSITE