----------------------------------------------------------------------------- -- | -- Module : Data.SBV.BitVectors.Polynomials -- Copyright : (c) Levent Erkok -- License : BSD3 -- Maintainer : erkokl@gmail.com -- Stability : experimental -- Portability : portable -- -- Implementation of polynomial arithmetic ----------------------------------------------------------------------------- {-# LANGUAGE TypeSynonymInstances #-} {-# LANGUAGE FlexibleContexts #-} {-# LANGUAGE PatternGuards #-} module Data.SBV.BitVectors.Polynomial (Polynomial(..)) where import Data.Bits import Data.List(genericTake) import Data.Maybe(fromJust) import Data.Word import Data.SBV.BitVectors.Data import Data.SBV.BitVectors.Model import Data.SBV.BitVectors.Splittable import Data.SBV.Utils.Boolean -- | Implements polynomial addition, multiplication, division, and modulus operations -- over GF(2^n). NB. Similar to 'bvQuotRem', division by @0@ is interpreted as follows: -- -- @x `pDivMod` 0 = (0, x)@ -- -- for all @x@ (including @0@) -- -- Minimal complete definiton: 'pMult', 'pDivMod', 'showPoly' class Bits a => Polynomial a where -- | Given bit-positions to be set, create a polynomial -- For instance -- -- @polynomial [0, 1, 3] :: SWord8@ -- -- will evaluate to @11@, since it sets the bits @0@, @1@, and @3@. Mathematicans would write this polynomial -- as @x^3 + x + 1@. And in fact, 'showPoly' will show it like that. polynomial :: [Int] -> a -- | Add two polynomials in GF(2^n) pAdd :: a -> a -> a -- | Multiply two polynomials in GF(2^n), and reduce it by the irreducible specified by -- the polynomial as specified by coefficients of the third argument. Note that the third -- argument is specifically left in this form as it is usally in GF(2^(n+1)), which is not available in our -- formalism. (That is, we would need SWord9 for SWord8 multiplication, etc.) Also note that we do not -- support symbolic irreducibles, which is a minor shortcoming. (Most GF's will come with fixed irreducibles, -- so this should not be a problem in practice.) -- -- Passing [] for the third argument will multiply the polynomials and then ignore the higher bits that won't -- fit into the resulting size. pMult :: (a, a, [Int]) -> a -- | Divide two polynomials in GF(2^n), see above note for division by 0 pDiv :: a -> a -> a -- | Compute modulus of two polynomials in GF(2^n), see above note for modulus by 0 pMod :: a -> a -> a -- | Division and modulus packed together pDivMod :: a -> a -> (a, a) -- | Display a polynomial like a mathematician would (over the monomial @x@) showPoly :: a -> String -- defaults.. Minumum complete definition: pMult, pDivMod, showPoly polynomial = foldr (flip setBit) 0 pAdd = xor pDiv x y = fst (pDivMod x y) pMod x y = snd (pDivMod x y) instance Polynomial Word8 where {showPoly = sp; pMult = lift polyMult; pDivMod = liftC polyDivMod} instance Polynomial Word16 where {showPoly = sp; pMult = lift polyMult; pDivMod = liftC polyDivMod} instance Polynomial Word32 where {showPoly = sp; pMult = lift polyMult; pDivMod = liftC polyDivMod} instance Polynomial Word64 where {showPoly = sp; pMult = lift polyMult; pDivMod = liftC polyDivMod} instance Polynomial SWord8 where {showPoly = liftS sp; pMult = polyMult; pDivMod = polyDivMod} instance Polynomial SWord16 where {showPoly = liftS sp; pMult = polyMult; pDivMod = polyDivMod} instance Polynomial SWord32 where {showPoly = liftS sp; pMult = polyMult; pDivMod = polyDivMod} instance Polynomial SWord64 where {showPoly = liftS sp; pMult = polyMult; pDivMod = polyDivMod} lift :: SymWord a => ((SBV a, SBV a, [Int]) -> SBV a) -> (a, a, [Int]) -> a lift f (x, y, z) = fromJust $ unliteral $ f (literal x, literal y, z) liftC :: SymWord a => (SBV a -> SBV a -> (SBV a, SBV a)) -> a -> a -> (a, a) liftC f x y = let (a, b) = f (literal x) (literal y) in (fromJust (unliteral a), fromJust (unliteral b)) liftS :: SymWord a => (a -> String) -> SBV a -> String liftS f s | Just x <- unliteral s = f x | True = show s -- | Pretty print as a polynomial sp :: Bits a => a -> String sp a | null cs = "0" ++ t | True = foldr (\x y -> sh x ++ " + " ++ y) (sh (last cs)) (init cs) ++ t where t = " :: GF(2^" ++ show n ++ ")" n = bitSize a is = [n-1, n-2 .. 0] cs = map fst $ filter snd $ zip is (map (testBit a) is) sh 0 = "1" sh 1 = "x" sh i = "x^" ++ show i -- | Add two polynomials addPoly :: [SBool] -> [SBool] -> [SBool] addPoly xs [] = xs addPoly [] ys = ys addPoly (x:xs) (y:ys) = x <+> y : addPoly xs ys ites :: SBool -> [SBool] -> [SBool] -> [SBool] ites s xs ys | Just t <- unliteral s = if t then xs else ys | True = go xs ys where go [] [] = [] go [] (b:bs) = ite s false b : go [] bs go (a:as) [] = ite s a false : go as [] go (a:as) (b:bs) = ite s a b : go as bs -- | Multiply two polynomials and reduce by the third (concrete) irreducible, given by its coefficients. -- See the remarks for the 'pMult' function for this design choice polyMult :: (Bits a, SymWord a, FromBits (SBV a)) => (SBV a, SBV a, [Int]) -> SBV a polyMult (x, y, red) = fromBitsLE $ genericTake sz $ r ++ repeat false where (_, r) = mdp ms rs ms = genericTake (2*sz) $ mul (blastLE x) (blastLE y) [] ++ repeat false rs = genericTake (2*sz) $ [if i `elem` red then true else false | i <- [0 .. foldr max 0 red] ] ++ repeat false sz = sizeOf x mul _ [] ps = ps mul as (b:bs) ps = mul (false:as) bs (ites b (as `addPoly` ps) ps) polyDivMod :: (Bits a, SymWord a, FromBits (SBV a)) => SBV a -> SBV a -> (SBV a, SBV a) polyDivMod x y = ite (y .== 0) (0, x) (adjust d, adjust r) where adjust xs = fromBitsLE $ genericTake sz $ xs ++ repeat false sz = sizeOf x (d, r) = mdp (blastLE x) (blastLE y) -- conservative over-approximation of the degree degree :: [SBool] -> Int degree xs = walk (length xs - 1) $ reverse xs where walk n [] = n walk n (b:bs) | Just t <- unliteral b = if t then n else walk (n-1) bs | True = n -- over-estimate mdp :: [SBool] -> [SBool] -> ([SBool], [SBool]) mdp xs ys = go (length ys - 1) (reverse ys) where degTop = degree xs go _ [] = error "SBV.Polynomial.mdp: Impossible happened; exhausted ys before hitting 0" go n (b:bs) | n == 0 = (reverse qs, rs) | True = let (rqs, rrs) = go (n-1) bs in (ites b (reverse qs) rqs, ites b rs rrs) where degQuot = degTop - n ys' = replicate degQuot false ++ ys (qs, rs) = divx (degQuot+1) degTop xs ys' -- return the element at index i; if not enough elements, return false -- N.B. equivalent to '(xs ++ repeat false) !! i', but more efficient idx :: [SBool] -> Int -> SBool idx [] _ = false idx (x:_) 0 = x idx (_:xs) i = idx xs (i-1) divx :: Int -> Int -> [SBool] -> [SBool] -> ([SBool], [SBool]) divx n _ xs _ | n <= 0 = ([], xs) divx n i xs ys' = (q:qs, rs) where q = xs `idx` i xs' = ites q (xs `addPoly` ys') xs (qs, rs) = divx (n-1) (i-1) xs' (tail ys')