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|
-- |
-- Module: Math.NumberTheory.Moduli.Equations
-- Copyright: (c) 2018 Andrew Lelechenko
-- Licence: MIT
-- Maintainer: Andrew Lelechenko <andrew.lelechenko@gmail.com>
--
-- Polynomial modular equations.
--
{-# LANGUAGE MagicHash #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE UnboxedSums #-}
{-# LANGUAGE ViewPatterns #-}
{-# OPTIONS_GHC -Wno-incomplete-uni-patterns #-}
module Math.NumberTheory.Moduli.Equations
( solveLinear
, solveQuadratic
) where
import Data.Constraint
import Data.Maybe
import Data.Mod
import GHC.Num.Integer
import GHC.TypeNats (KnownNat, natVal)
import Math.NumberTheory.Moduli.Chinese
import Math.NumberTheory.Moduli.Singleton
import Math.NumberTheory.Moduli.Sqrt
import Math.NumberTheory.Primes
import Math.NumberTheory.Utils (recipMod)
-------------------------------------------------------------------------------
-- Linear equations
-- | Find all solutions of ax + b ≡ 0 (mod m).
--
-- >>> :set -XDataKinds
-- >>> solveLinear (6 :: Mod 10) 4 -- solving 6x + 4 ≡ 0 (mod 10)
-- [(1 `modulo` 10),(6 `modulo` 10)]
solveLinear
:: KnownNat m
=> Mod m -- ^ a
-> Mod m -- ^ b
-> [Mod m] -- ^ list of x
solveLinear a b = map fromInteger $ solveLinear' (toInteger (natVal a)) (toInteger (unMod a)) (toInteger (unMod b))
solveLinear' :: Integer -> Integer -> Integer -> [Integer]
solveLinear' m a b = case solveLinearCoprime m' (a `quot` d) (b `quot` d) of
Nothing -> []
Just x -> map (\i -> x + m' * i) [0 .. d - 1]
where
d = m `gcd` a `gcd` b
m' = m `quot` d
solveLinearCoprime :: Integer -> Integer -> Integer -> Maybe Integer
solveLinearCoprime 1 _ _ = Just 0
solveLinearCoprime m a b = (\a1 -> negate b * a1 `mod` m) <$> recipMod a m
-------------------------------------------------------------------------------
-- Quadratic equations
-- | Find all solutions of ax² + bx + c ≡ 0 (mod m).
--
-- >>> :set -XDataKinds
-- >>> solveQuadratic sfactors (1 :: Mod 32) 0 (-17) -- solving x² - 17 ≡ 0 (mod 32)
-- [(9 `modulo` 32),(25 `modulo` 32),(7 `modulo` 32),(23 `modulo` 32)]
solveQuadratic
:: SFactors Integer m
-> Mod m -- ^ a
-> Mod m -- ^ b
-> Mod m -- ^ c
-> [Mod m] -- ^ list of x
solveQuadratic sm a b c = case proofFromSFactors sm of
Sub Dict ->
map fromInteger
$ fst
$ combine
$ map (\(p, n) -> (solveQuadraticPrimePower a' b' c' p n, unPrime p ^ n))
$ unSFactors sm
where
a' = toInteger $ unMod a
b' = toInteger $ unMod b
c' = toInteger $ unMod c
combine :: [([Integer], Integer)] -> ([Integer], Integer)
combine = foldl
(\(xs, xm) (ys, ym) -> ([ fst $ fromJust $ chinese (x, xm) (y, ym) | x <- xs, y <- ys ], xm * ym))
([0], 1)
solveQuadraticPrimePower
:: Integer
-> Integer
-> Integer
-> Prime Integer
-> Word
-> [Integer]
solveQuadraticPrimePower a b c p = go
where
go :: Word -> [Integer]
go 0 = [0]
go 1 = solveQuadraticPrime a b c p
go k = concatMap (liftRoot k) (go (k - 1))
-- Hensel lifting
-- https://en.wikipedia.org/wiki/Hensel%27s_lemma#Hensel_lifting
liftRoot :: Word -> Integer -> [Integer]
liftRoot k r = case recipMod (2 * a * r + b) pk of
Nothing -> case fr of
0 -> map (\i -> r + pk `quot` p' * i) [0 .. p' - 1]
_ -> []
Just invDeriv -> [(r - fr * invDeriv) `mod` pk]
where
pk = p' ^ k
fr = (a * r * r + b * r + c) `mod` pk
p' :: Integer
p' = unPrime p
solveQuadraticPrime
:: Integer
-> Integer
-> Integer
-> Prime Integer
-> [Integer]
solveQuadraticPrime a b c (unPrime -> (2 :: Integer))
= case (even c, even (a + b)) of
(True, True) -> [0, 1]
(True, _) -> [0]
(_, False) -> [1]
_ -> []
solveQuadraticPrime a b c p
| a `rem` p' == 0
= solveLinear' p' b c
| otherwise
= map (\n -> let (# t | #) = integerRecipMod# (2 * a) (fromInteger p') in (n - b) * toInteger t `mod` p')
$ sqrtsModPrime (b * b - 4 * a * c) p
where
p' :: Integer
p' = unPrime p
|
