-- |
-- Module: Math.NumberTheory.Primes.Testing.Probabilistic
-- Copyright: (c) 2011 Daniel Fischer
-- Licence: MIT
-- Maintainer: Daniel Fischer
-- Stability: Provisional
-- Portability: Non-portable (GHC extensions)
--
-- Probabilistic primality tests, Miller-Rabin and Baillie-PSW.
{-# LANGUAGE CPP, MagicHash, BangPatterns #-}
{-# OPTIONS_HADDOCK hide #-}
module Math.NumberTheory.Primes.Testing.Probabilistic
( isPrime
, millerRabinV
, bailliePSW
, isStrongFermatPP
, isFermatPP
, lucasTest
) where
#include "MachDeps.h"
import Math.NumberTheory.Moduli
import Math.NumberTheory.Utils
import Math.NumberTheory.Powers.Squares
import Data.Bits
import GHC.Base
import GHC.Word
import GHC.Integer.GMP.Internals
-- | @'isPrime' n@ tests whether @n@ is a prime (negative or positive).
-- First, trial division by the primes less than @1200@ is performed.
-- If that hasn't determined primality or compositeness, a Baillie PSW
-- test is performed.
--
-- Since the Baillie PSW test may not be perfect, it is possible that
-- some large composites are wrongly deemed prime, however, no composites
-- passing the test are known and none exist below @2^64@.
isPrime :: Integer -> Bool
isPrime n
| n < 0 = isPrime (-n)
| n < 2 = False
| n < 4 = True
| otherwise = go smallPrimes
where
go (p:ps)
| p*p > n = True
| otherwise = case n `rem` p of
0 -> False
_ -> go ps
go [] = bailliePSW n
-- | A Miller-Rabin like probabilistic primality test with preceding
-- trial division. While the classic Miller-Rabin test uses
-- randomly chosen bases, @'millerRabinV' k n@ uses the @k@
-- smallest primes as bases if trial division has not reached
-- a conclusive result. (Only the primes up to @1200@ are
-- available in this module, so the maximal effective @k@ is @196@.)
millerRabinV :: Int -> Integer -> Bool
millerRabinV k n
| n < 0 = millerRabinV k (-n)
| n < 2 = False
| n < 4 = True
| otherwise = go smallPrimes
where
go (p:ps)
| p*p > n = True
| otherwise = (n `rem` p /= 0) && go ps
go [] = all (isStrongFermatPP n) (take k smallPrimes)
-- | @'isStrongFermatPP' n b@ tests whether @n@ is a strong Fermat
-- probable prime for base @b@, where @n > 2@ and @1 < b < n@.
-- The conditions on the arguments are not checked.
--
-- Apart from primes, also some composite numbers have the tested
-- property, but those are rare. Very rare are composite numbers
-- having the property for many bases, so testing a large prime
-- candidate with several bases can identify composite numbers
-- with high probability. An odd number @n > 3@ is prime if and
-- only if @'isStrongFermatPP' n b@ holds for all @b@ with
-- @2 <= b <= (n-1)/2@, but of course checking all those bases
-- would be less efficient than trial division, so one normally
-- checks only a relatively small number of bases, depending on
-- the desired degree of certainty. The probability that a randomly
-- chosen base doesn't identify a composite number @n@ is less than
-- @1/4@, so five to ten tests give a reasonable level of certainty
-- in general.
--
-- Some notes about the choice of bases: @b@ is a strong Fermat base
-- for @n@ if and only if @n-b@ is, hence one needs only test @b <= (n-1)/2@.
-- If @b@ is a strong Fermat base for @n@, then so is @b^k `mod` n@ for
-- all @k > 1@, hence one needs not test perfect powers, since their
-- base yields a stronger condition. Finally, if @a@ and @b@ are strong
-- Fermat bases for @n@, then @a*b@ is in most cases a strong Fermat
-- base for @n@, it can only fail to be so if @n `mod` 4 == 1@ and
-- the strong Fermat condition is reached at the same step for @a@ as for @b@,
-- so primes are the most powerful bases to test.
isStrongFermatPP :: Integer -> Integer -> Bool
isStrongFermatPP n b = a == 1 || go t a
where
m = n-1
(t,u) = shiftToOddCount m
a = powerModInteger' b u n
go 0 _ = False
go k x = x == m || go (k-1) ((x*x) `rem` n)
-- | @'isFermatPP' n b@ tests whether @n@ is a Fermat probable prime
-- for the base @b@, that is, whether @b^(n-1) `mod` n == 1@.
-- This is a weaker but simpler condition. However, more is lost
-- in strength than is gained in simplicity, so for primality testing,
-- the strong check should be used. The remarks about
-- the choice of bases to test from @'isStrongFermatPP'@ apply
-- with the modification that if @a@ and @b@ are Fermat bases
-- for @n@, then @a*b@ /always/ is a Fermat base for @n@ too.
-- A /Charmichael number/ is a composite number @n@ which is a
-- Fermat probable prime for all bases @b@ coprime to @n@. By the
-- above, only primes @p <= n/2@ not dividing @n@ need to be tested
-- to identify Carmichael numbers (however, testing all those
-- primes would be less efficient than determining Carmichaelness
-- from the prime factorisation; but testing an appropriate number
-- of prime bases is reasonable to find out whether it's worth the
-- effort to undertake the prime factorisation).
isFermatPP :: Integer -> Integer -> Bool
isFermatPP n b = powerModInteger' b (n-1) n == 1
-- | Primality test after Baillie, Pomerance, Selfridge and Wagstaff.
-- The Baillie PSW test consists of a strong Fermat probable primality
-- test followed by a (strong) Lucas primality test. This implementation
-- assumes that the number @n@ to test is odd and larger than @3@.
-- Even and small numbers have to be handled before. Also, before
-- applying this test, trial division by small primes should be performed
-- to identify many composites cheaply (although the Baillie PSW test is
-- rather fast, about the same speed as a strong Fermat test for four or
-- five bases usually, it is, for large numbers, much more costly than
-- trial division by small primes, the primes less than @1000@, say, so
-- eliminating numbers with small prime factors beforehand is more efficient).
--
-- The Baillie PSW test is very reliable, so far no composite numbers
-- passing it are known, and it is known (Gilchrist 2010) that no
-- Baillie PSW pseudoprimes exist below @2^64@. However, a heuristic argument
-- by Pomerance indicates that there are likely infinitely many Baillie PSW
-- pseudoprimes. On the other hand, according to
-- there is
-- reason to believe that there are none with less than several
-- thousand digits, so that for most use cases the test can be
-- considered definitive.
bailliePSW :: Integer -> Bool
bailliePSW n = isStrongFermatPP n 2 && lucasTest n
-- precondition: n odd, > 3 (no small prime factors, typically large)
-- | The Lucas-Selfridge test, including square-check, but without
-- the Fermat test. For package-internal use only.
lucasTest :: Integer -> Bool
lucasTest n
| square || d == 0 = False
| d == 1 = True
| otherwise = uo == 0 || go t vo qo
where
square = isPossibleSquare2 n && r*r == n
r = integerSquareRoot n
d = find True 5
find !pos cd = case jacobi' (n `rem` cd) cd of
0 -> if cd == n then 1 else 0
1 -> find (not pos) (cd+2)
_ -> if pos then cd else (-cd)
q = (1-d) `quot` 4
(t,o) = shiftToOddCount (n+1)
(uo, vo, qo) = testLucas n q o
go 0 _ _ = False
go s vn qn = vn == 0 || go (s-1) ((vn*vn-2*qn) `rem` n) ((qn*qn) `rem` n)
-- n odd positive, n > abs q, index odd
testLucas :: Integer -> Integer -> Integer -> (Integer, Integer, Integer)
testLucas n q (S# i#) = look (WORD_SIZE_IN_BITS - 2)
where
j = I# i#
look k
| testBit j k = go (k-1) 1 1 1 q
| otherwise = look (k-1)
go k un un1 vn qn
| k < 0 = (un, vn, qn)
| testBit j k = go (k-1) u2n1 u2n2 v2n1 q2n1
| otherwise = go (k-1) u2n u2n1 v2n q2n
where
u2n = (un*vn) `rem` n
u2n1 = (un1*vn-qn) `rem` n
u2n2 = ((un1-q*un)*vn-qn) `rem` n
v2n = (vn*vn-2*qn) `rem` n
v2n1 = ((un1 - (2*q)*un)*vn-qn) `rem` n
q2n = (qn*qn) `rem` n
q2n1 = (qn*qn*q) `rem` n
testLucas n q (J# s# ba#) = test (s# -# 1#)
where
test j# = case indexWordArray# ba# j# of
0## -> test (j# -# 1#)
w# -> look (j# -# 1#) (W# w#) (WORD_SIZE_IN_BITS - 1)
look j# w i
| testBit w i = go j# w (i - 1) 1 1 1 q
| otherwise = look j# w (i-1)
go k# w i un un1 vn qn
| i < 0 = if k# <# 0#
then (un,vn,qn)
else go (k# -# 1#) (W# (indexWordArray# ba# k#)) (WORD_SIZE_IN_BITS - 1) un un1 vn qn
| testBit w i = go k# w (i-1) u2n1 u2n2 v2n1 q2n1
| otherwise = go k# w (i-1) u2n u2n1 v2n q2n
where
u2n = (un*vn) `rem` n
u2n1 = (un1*vn-qn) `rem` n
u2n2 = ((un1-q*un)*vn-qn) `rem` n
v2n = (vn*vn-2*qn) `rem` n
v2n1 = ((un1 - (2*q)*un)*vn-qn) `rem` n
q2n = (qn*qn) `rem` n
q2n1 = (qn*qn*q) `rem` n
smallPrimes :: [Integer]
smallPrimes = 2:3:5:prs
where
prs = 7:11:filter isPr (takeWhile (< 1200) . scanl (+) 13 $ cycle [4,2,4,6,2,6,4,2])
isPr n = td n prs
td n (p:ps) = (p*p > n) || (n `rem` p /= 0 && td n ps)
td _ [] = True