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
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.is_even() or candidate.is_perfect_square(): return COMPOSITE # Step 2 def alternate(): sgn = 1 value = 5 for x in xrange(10): yield sgn * value sgn, value = -sgn, value + 2 for D in alternate(): js = Integer.jacobi_symbol(D, candidate) if js == 0: return COMPOSITE if js == -1: break else: return COMPOSITE # 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 xrange(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