78 lines
2.6 KiB
Haskell
78 lines
2.6 KiB
Haskell
-- |
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-- Module : Crypto.Number.Basic
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-- License : BSD-style
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-- Maintainer : Vincent Hanquez <vincent@snarc.org>
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-- Stability : experimental
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-- Portability : Good
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{-# LANGUAGE BangPatterns #-}
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module Crypto.Number.Basic
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( sqrti
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, gcde
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, areEven
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, log2
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) where
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import Crypto.Number.Compat
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-- | sqrti returns two integer (l,b) so that l <= sqrt i <= b
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-- the implementation is quite naive, use an approximation for the first number
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-- and use a dichotomy algorithm to compute the bound relatively efficiently.
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sqrti :: Integer -> (Integer, Integer)
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sqrti i
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| i < 0 = error "cannot compute negative square root"
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| i == 0 = (0,0)
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| i == 1 = (1,1)
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| i == 2 = (1,2)
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| otherwise = loop x0
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where
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nbdigits = length $ show i
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x0n = (if even nbdigits then nbdigits - 2 else nbdigits - 1) `div` 2
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x0 = if even nbdigits then 2 * 10 ^ x0n else 6 * 10 ^ x0n
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loop x = case compare (sq x) i of
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LT -> iterUp x
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EQ -> (x, x)
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GT -> iterDown x
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iterUp lb = if sq ub >= i then iter lb ub else iterUp ub
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where ub = lb * 2
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iterDown ub = if sq lb >= i then iterDown lb else iter lb ub
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where lb = ub `div` 2
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iter lb ub
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| lb == ub = (lb, ub)
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| lb+1 == ub = (lb, ub)
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| otherwise =
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let d = (ub - lb) `div` 2 in
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if sq (lb + d) >= i
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then iter lb (ub-d)
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else iter (lb+d) ub
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sq a = a * a
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-- | get the extended GCD of two integer using integer divMod
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--
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-- gcde 'a' 'b' find (x,y,gcd(a,b)) where ax + by = d
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--
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gcde :: Integer -> Integer -> (Integer, Integer, Integer)
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gcde a b = onGmpUnsupported (gmpGcde a b) $
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if d < 0 then (-x,-y,-d) else (x,y,d)
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where
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(d, x, y) = f (a,1,0) (b,0,1)
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f t (0, _, _) = t
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f (a', sa, ta) t@(b', sb, tb) =
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let (q, r) = a' `divMod` b' in
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f t (r, sa - (q * sb), ta - (q * tb))
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-- | check if a list of integer are all even
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areEven :: [Integer] -> Bool
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areEven = and . map even
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-- | Compute the binary logarithm of a integer
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log2 :: Integer -> Int
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log2 n = onGmpUnsupported (gmpLog2 n) $ imLog 2 n
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where
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-- http://www.haskell.org/pipermail/haskell-cafe/2008-February/039465.html
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imLog b x = if x < b then 0 else (x `div` b^l) `doDiv` l
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where
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l = 2 * imLog (b * b) x
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doDiv x' l' = if x' < b then l' else (x' `div` b) `doDiv` (l' + 1)
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{-# INLINE log2 #-}
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