module Math.NumberTheory.Moduli
(
jacobi
, invertMod
, powerMod
, powerModInteger
, jacobi'
, powerMod'
, powerModInteger'
) where
#include "MachDeps.h"
import Data.Word
import Data.Bits
import Data.Array.Unboxed
import Data.Array.Base (unsafeAt)
import Math.NumberTheory.GCD (extendedGCD)
import Math.NumberTheory.Utils (shiftToOddCount)
invertMod :: Integer -> Integer -> Maybe Integer
invertMod k 0 = if k == 1 || k == (1) then Just k else Nothing
invertMod k m = case extendedGCD k' m' of
(1, u, _) -> Just (if u < 0 then m' + u else u)
_ -> Nothing
where
m' = abs m
k' | k >= m' || k < 0 = k `mod` m'
| otherwise = k
jacobi :: (Integral a, Bits a) => a -> a -> Int
jacobi a b
| b < 0 = error "Math.NumberTheory.Moduli.jacobi: negative denominator"
| evenI b = error "Math.NumberTheory.Moduli.jacobi: even denominator"
| b == 1 = 1
| a == 0 = 0
| a == 1 = 1
| otherwise = jacobi' a b
jacobi' :: (Integral a, Bits a) => a -> a -> Int
jacobi' a b
| a == 0 = 0
| a == 1 = 1
| a < 0 = let n | rem4 b == 1 = 1
| otherwise = 1
(z,o) = shiftToOddCount (abs $ toInteger a)
s | evenI z || unsafeAt jac2 (rem8 b) == 1 = n
| otherwise = (n)
in s*jacobi' (fromInteger o) b
| a >= b = case a `rem` b of
0 -> 0
r -> jacPS 1 r b
| evenI a = case shiftToOddCount a of
(z,o) -> let r = 2 (rem4 o .&. rem4 b)
s | evenI z || unsafeAt jac2 (rem8 b) == 1 = r
| otherwise = (r)
in jacOL s b o
| otherwise = case rem4 a .&. rem4 b of
3 -> jacOL (1) b a
_ -> jacOL 1 b a
jacPS :: (Integral a, Bits a) => Int -> a -> a -> Int
jacPS !j a b
| evenI a = case shiftToOddCount a of
(z,o) | evenI z || unsafeAt jac2 (rem8 b) == 1 ->
jacOL (if rem4 o .&. rem4 b == 3 then (j) else j) b o
| otherwise ->
jacOL (if rem4 o .&. rem4 b == 3 then j else (j)) b o
| otherwise = jacOL (if rem4 a .&. rem4 b == 3 then (j) else j) b a
jacOL :: (Integral a, Bits a) => Int -> a -> a -> Int
jacOL !j a b
| b == 1 = j
| otherwise = case a `rem` b of
0 -> 0
r -> jacPS j r b
powerMod :: (Integral a, Bits a) => Integer -> a -> Integer -> Integer
powerMod base expo md
| md == 0 = base ^ expo
| md' == 1 = 0
| expo == 0 = 1
| bse' == 1 = 1
| expo < 0 = case invertMod bse' md' of
Just i -> powerMod' i (negate expo) md'
Nothing -> error "Math.NumberTheory.Moduli.powerMod: Base isn't invertible with respect to modulus"
| bse' == 0 = 0
| otherwise = powerMod' bse' expo md'
where
md' = abs md
bse' = if base < 0 || md' <= base then base `mod` md' else base
powerMod' :: (Integral a, Bits a) => Integer -> a -> Integer -> Integer
powerMod' base expo md = go expo 1 base
where
go 1 !a !s = (a*s) `rem` md
go e a s
| testBit e 0 = go (e `shiftR` 1) a ((s*s) `rem` md)
| otherwise = go (e `shiftR` 1) ((a*s) `rem` md) ((s*s) `rem` md)
powerModInteger :: Integer -> Integer -> Integer -> Integer
powerModInteger base ex mdl
| mdl == 0 = base ^ ex
| mdl' == 1 = 0
| ex == 0 = 1
| ex < 0 = case invertMod bse' mdl' of
Just i -> powerModInteger' i (negate ex) mdl'
Nothing -> error "Math.NumberTheory.Moduli.powerMod: Base isn't invertible with respect to modulus"
| bse' == 0 = 0
| bse' == 1 = 1
| otherwise = powerModInteger' bse' ex mdl'
where
mdl' = abs mdl
bse' = if base < 0 || mdl' <= base then base `mod` mdl' else base
powerModInteger' :: Integer -> Integer -> Integer -> Integer
powerModInteger' base expo md = go e1 w1 1 base
where
w1 = fromInteger expo
e1 = expo `shiftR` 64
#if WORD_SIZE_IN_BITS == 32
go :: Integer -> Word64 -> Integer -> Integer -> Integer
go 0 !w !a !s = end w a s
go e w a s = inner1 0 a s
where
wl :: Word
!wl = fromIntegral w
wh :: Word
!wh = fromIntegral (w `shiftR` 32)
inner1 32 !au !sq = inner2 0 au sq
inner1 i au sq
| testBit wl i = inner1 (i+1) ((au*sq) `rem` md) ((sq*sq) `rem` md)
| otherwise = inner1 (i+1) au ((sq*sq) `rem` md)
inner2 32 !au !sq = go (e `shiftR` 64) (fromInteger e) au sq
inner2 i au sq
| testBit wh i = inner2 (i+1) ((au*sq) `rem` md) ((sq*sq) `rem` md)
| otherwise = inner2 (i+1) au ((sq*sq) `rem` md)
end w !a !s
| wh == 0 = fin wl a s
| otherwise = innerE 0 a s
where
wl :: Word
!wl = fromIntegral w
wh :: Word
!wh = fromIntegral (w `shiftR` 32)
innerE 32 !au !sq = fin wh au sq
innerE i au sq
| testBit wl i = innerE (i+1) ((au*sq) `rem` md) ((sq*sq) `rem` md)
| otherwise = innerE (i+1) au ((sq*sq) `rem` md)
fin :: Word -> Integer -> Integer -> Integer
fin 1 !a !s = (a*s) `rem` md
fin w a s
| testBit w 0 = fin (w `shiftR` 1) ((a*s) `rem` md) ((s*s) `rem` md)
| otherwise = fin (w `shiftR` 1) a ((s*s) `rem` md)
#else
go :: Integer -> Word -> Integer -> Integer -> Integer
go 0 !w !a !s = end w a s
go e w a s = inner 0 a s
where
inner 64 !au !sq = go (e `shiftR` 64) (fromInteger e) au sq
inner i au sq
| testBit w i = inner (i+1) ((au*sq) `rem` md) ((sq*sq) `rem` md)
| otherwise = inner (i+1) au ((sq*sq) `rem` md)
end 1 !a !s = (a*s) `rem` md
end w a s
| testBit w 0 = end (w `shiftR` 1) ((a*s) `rem` md) ((s*s) `rem` md)
| otherwise = end (w `shiftR` 1) a ((s*s) `rem` md)
#endif
evenI :: (Integral a, Bits a) => a -> Bool
evenI n = fromIntegral n .&. 1 == (0 :: Int)
rem4 :: (Integral a, Bits a) => a -> Int
rem4 n = fromIntegral n .&. 3
rem8 :: (Integral a, Bits a) => a -> Int
rem8 n = fromIntegral n .&. 7
jac2 :: UArray Int Int
jac2 = array (0,7) [(0,0),(1,1),(2,0),(3,1),(4,0),(5,1),(6,0),(7,1)]