def is_prime(n):
  if n < 2:
    return False
  for i in xrange(2, int(n**0.5+1)):
    if n%i == 0:
      return False
  return True

(a^b) % p = ((a % p)^b) % p
(a * b) % p = (a % p * b % p) % p

def xn_mod_p(x, n, p):
  if n == 0:
    return 1
  res = xn_mod_p((x*x)%p, n>>1, p)
  if n&1 != 0:
    res = (res*x)%p
  return res

def xn_mod_p2(x, n, p):
  res = 1
  n_bin = bin(n)[2:]
  for i in range(0, len(n_bin)):
    res = res**2 % p
    if n_bin[i] == '1':
      res = res * x % p
  return res

def fermat_test_prime(n):
  if n == 1:
    return False
  if n == 2:
    return True
  res = xn_mod_p(2, n-1, n)
  return res == 1

def miller_rabin_witness(a, p):
  if p == 1:
    return False
  if p == 2:
    return True
  #p-1 = u*2^t 求解 u, t
  n = p - 1
  t = int(math.floor(math.log(n, 2)))
  u = 1
  while t > 0:
    u = n / 2**t
    if n % 2**t == 0 and u % 2 == 1:
      break
    t = t - 1
  b1 = b2 = xn_mod_p2(a, u, p)
  for i in range(1, t + 1):
    b2 = b1**2 % p
    if b2 == 1 and b1 != 1 and b1 != (p - 1):
      return False
    b1 = b2
  if b1 != 1:
    return False
  return True
def prime_test_miller_rabin(p, k):
  while k > 0:
    a = randint(1, p - 1)
    if not miller_rabin_witness(a, p):
      return False
    k = k - 1
  return True