combinat-0.2.4.1: Math/Combinat/Numbers/Primes.hs
-- | Prime numbers and related number theoretical stuff.
module Math.Combinat.Numbers.Primes
( -- * List of prime numbers
primes
, primesSimple
, primesTMWE
-- * Prime factorization
, groupIntegerFactors
, integerFactorsTrialDivision
-- * Integer logarithm
, integerLog2
, ceilingLog2
-- * Integer square root
, isSquare
, integerSquareRoot
, ceilingSquareRoot
, integerSquareRoot'
, integerSquareRootNewton'
-- * Modulo @m@ arithmetic
, powerMod
-- * Prime testing
, millerRabinPrimalityTest
)
where
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-- import Math.Combinat.Numbers
import Data.List ( group , sort )
import Data.Bits
--------------------------------------------------------------------------------
-- List of prime numbers
-- | Infinite list of primes, using the TMWE algorithm.
primes :: [Integer]
primes = primesTMWE
-- | A relatively simple but still quite fast implementation of list of primes.
-- By Will Ness <http://www.haskell.org/pipermail/haskell-cafe/2009-November/068441.html>
primesSimple :: [Integer]
primesSimple = 2 : 3 : sieve 0 primes' 5 where
primes' = tail primesSimple
sieve k (p:ps) x = noDivs k h ++ sieve (k+1) ps (t+2) where
t = p*p
h = [x,x+2..t-2]
noDivs k = filter (\x -> all (\y -> rem x y /= 0) (take k primes'))
-- | List of primes, using tree merge with wheel. Code by Will Ness.
primesTMWE :: [Integer]
primesTMWE = 2:3:5:7: gaps 11 wheel (fold3t $ roll 11 wheel primes') where
primes' = 11: gaps 13 (tail wheel) (fold3t $ roll 11 wheel primes')
fold3t ((x:xs): ~(ys:zs:t))
= x : union xs (union ys zs) `union` fold3t (pairs t)
pairs ((x:xs):ys:t) = (x : union xs ys) : pairs t
wheel = 2:4:2:4:6:2:6:4:2:4:6:6:2:6:4:2:6:4:6:8:4:2:4:2:
4:8:6:4:6:2:4:6:2:6:6:4:2:4:6:2:6:4:2:4:2:10:2:10:wheel
gaps k ws@(w:t) cs@ ~(c:u)
| k==c = gaps (k+w) t u
| True = k : gaps (k+w) t cs
roll k ws@(w:t) ps@ ~(p:u)
| k==p = scanl (\c d->c+p*d) (p*p) ws : roll (k+w) t u
| True = roll (k+w) t ps
minus xxs@(x:xs) yys@(y:ys) = case compare x y of
LT -> x : minus xs yys
EQ -> minus xs ys
GT -> minus xxs ys
minus xs [] = xs
minus [] _ = []
union xxs@(x:xs) yys@(y:ys) = case compare x y of
LT -> x : union xs yys
EQ -> x : union xs ys
GT -> y : union xxs ys
union xs [] = xs
union [] ys =ys
--------------------------------------------------------------------------------
-- Prime factorization
-- | Groups integer factors. Example: from [2,2,2,3,3,5] we produce [(2,3),(3,2),(5,1)]
groupIntegerFactors :: [Integer] -> [(Integer,Int)]
groupIntegerFactors = map f . group . sort where
f xs = (head xs, length xs)
-- | The naive trial division algorithm.
integerFactorsTrialDivision :: Integer -> [Integer]
integerFactorsTrialDivision n
| n<1 = error "integerFactorsTrialDivision: n should be at least 1"
| otherwise = go primes n
where
go _ 1 = []
go rs k = sub ps k where
sub [] k = [k]
sub qqs@(q:qs) k = case mod k q of
0 -> q : go qqs (div k q)
_ -> sub qs k
ps = takeWhile (\p -> p*p <= k) rs
{-
go 1 = []
go k = sub ps k where
sub [] k = [k]
sub (q:qs) k = case mod k q of
0 -> q : go (div k q)
_ -> sub qs k
ps = takeWhile (\p -> p*p <= k) primes
-}
{-
-- brute force testing of factors
ifactorsTest :: (Integer -> [Integer]) -> Integer -> Bool
ifactorsTest alg n = and [ product (alg k) == k | k<-[1..n] ]
-}
--------------------------------------------------------------------------------
-- Integer logarithm
-- | Largest integer @k@ such that @2^k@ is smaller or equal to @n@
integerLog2 :: Integer -> Integer
integerLog2 n = go n where
go 0 = -1
go k = 1 + go (shiftR k 1)
-- | Smallest integer @k@ such that @2^k@ is larger or equal to @n@
ceilingLog2 :: Integer -> Integer
ceilingLog2 0 = 0
ceilingLog2 n = 1 + go (n-1) where
go 0 = -1
go k = 1 + go (shiftR k 1)
--------------------------------------------------------------------------------
-- Integer square root
isSquare :: Integer -> Bool
isSquare n =
if (fromIntegral $ mod n 32) `elem` rs
then snd (integerSquareRoot' n) == 0
else False
where
rs = [0,1,4,9,16,17,25] :: [Int]
-- | Integer square root (largest integer whose square is smaller or equal to the input)
-- using Newton's method, with a faster (for large numbers) inital guess based on bit shifts.
integerSquareRoot :: Integer -> Integer
integerSquareRoot = fst . integerSquareRoot'
-- | Smallest integer whose square is larger or equal to the input
ceilingSquareRoot :: Integer -> Integer
ceilingSquareRoot n = (if r>0 then u+1 else u) where (u,r) = integerSquareRoot' n
-- | We also return the excess residue; that is
--
-- > (a,r) = integerSquareRoot' n
--
-- means that
--
-- > a*a + r = n
-- > a*a <= n < (a+1)*(a+1)
integerSquareRoot' :: Integer -> (Integer,Integer)
integerSquareRoot' n
| n<0 = error "integerSquareRoot: negative input"
| n<2 = (n,0)
| otherwise = go firstGuess
where
k = integerLog2 n
firstGuess = 2^(div (k+2) 2) -- !! note that (div (k+1) 2) is NOT enough !!
go a =
if m < a
then go a'
else (a, r + a*(m-a))
where
(m,r) = divMod n a
a' = div (m + a) 2
-- | Newton's method without an initial guess. For very small numbers (<10^10) it
-- is somewhat faster than the above version.
integerSquareRootNewton' :: Integer -> (Integer,Integer)
integerSquareRootNewton' n
| n<0 = error "integerSquareRootNewton: negative input"
| n<2 = (n,0)
| otherwise = go (div n 2)
where
go a =
if m < a
then go a'
else (a, r + a*(m-a))
where
(m,r) = divMod n a
a' = div (m + a) 2
{-
-- brute force test of integer square root
isqrt_test n1 n2 =
[ k
| k<-[n1..n2]
, let (a,r) = integerSquareRoot' k
, (a*a+r/=k) || (a*a>k) || (a+1)*(a+1)<=k
]
-}
--------------------------------------------------------------------------------
-- Modulo @m@ arithmetic
-- | Efficient powers modulo m.
--
-- > powerMod a k m == (a^k) `mod` m
powerMod :: Integer -> Integer -> Integer -> Integer
powerMod a' k m = {- debug bs $ -} go a bs where
bs = bin k
bin 0 = []
bin x = (x .&. 1 /= 0) : bin (shiftR x 1)
a = mod a' m
go _ [] = 1
go x (b:bs) = -- debug (x,b) $
if b
then mod (x*rest) m
else rest
where
rest = go (mod (x*x) m) bs
--------------------------------------------------------------------------------
-- Prime testing
-- | Miller-Rabin Primality Test (taken from Haskell wiki).
-- We test the primality of the first argument @n@ by using the second argument @a@ as a candidate witness.
-- If it returs @False@, then @n@ is composite. If it returns @True@, then @n@ is either prime or composite.
--
-- A random choice between @2@ and @(n-2)@ is a good choice for @a@.
millerRabinPrimalityTest :: Integer -> Integer -> Bool
millerRabinPrimalityTest n a
| a <= 1 || a >= n-1 =
error $ "millerRabinPrimalityTest: a out of range (" ++ show a ++ " for "++ show n ++ ")"
| n < 2 = False
| even n = False
| b0 == 1 || b0 == n' = True
| otherwise = iter (tail b)
where
n' = n-1
(k,m) = find2km n'
b0 = powMod n a m
b = take (fromIntegral k) $ iterate (squareMod n) b0
iter [] = False
iter (x:xs)
| x == 1 = False
| x == n' = True
| otherwise = iter xs
{-# SPECIALIZE find2km :: Integer -> (Integer,Integer) #-}
find2km :: Integral a => a -> (a,a)
find2km n = f 0 n where
f k m
| r == 1 = (k,m)
| otherwise = f (k+1) q
where (q,r) = quotRem m 2
{-# SPECIALIZE pow' :: (Integer -> Integer -> Integer) -> (Integer -> Integer) -> Integer -> Integer -> Integer #-}
pow' :: (Num a, Integral b) => (a -> a -> a) -> (a -> a) -> a -> b -> a
pow' _ _ _ 0 = 1
pow' mul sq x' n' = f x' n' 1 where
f x n y
| n == 1 = x `mul` y
| r == 0 = f x2 q y
| otherwise = f x2 q (x `mul` y)
where
(q,r) = quotRem n 2
x2 = sq x
{-# SPECIALIZE mulMod :: Integer -> Integer -> Integer -> Integer #-}
mulMod :: Integral a => a -> a -> a -> a
mulMod a b c = (b * c) `mod` a
{-# SPECIALIZE squareMod :: Integer -> Integer -> Integer #-}
squareMod :: Integral a => a -> a -> a
squareMod a b = (b * b) `rem` a
{-# SPECIALIZE powMod :: Integer -> Integer -> Integer -> Integer #-}
powMod :: Integral a => a -> a -> a -> a
powMod m = pow' (mulMod m) (squareMod m)
--------------------------------------------------------------------------------