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|
--Copyright 2001, 2002, 2003 David J. Sankel
--
--This file is part of rsa-haskell.
--rsa-haskell is free software; you can redistribute it and/or modify
--it under the terms of the GNU General Public License as published by
--the Free Software Foundation; either version 2 of the License, or
--(at your option) any later version.
--
--rsa-haskell is distributed in the hope that it will be useful,
--but WITHOUT ANY WARRANTY; without even the implied warranty of
--MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
--GNU General Public License for more details.
--
--You should have received a copy of the GNU General Public License
--along with rsa-haskell; if not, write to the Free Software
--Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
module Codec.Encryption.RSA.NumberTheory(
inverse, extEuclGcd, simplePrimalityTest, getPrime, pg, isPrime,
rabinMillerPrimalityTest, expmod, factor, testInverse, primes, (/|),
randomOctet
) where
import System.Random(getStdRandom,randomR)
--The following line is required for ghc optomized implementation
-- (see comments beginning with GHC):
-- import Bits(setBit)
import Data.List(elemIndex)
import Data.Maybe(fromJust)
import Data.Char(chr,ord)
import Data.Bits(xor)
--Precondition: the integer is >= 0
randomOctet :: Int -> IO( String )
randomOctet n
| n < 0 = error "randomOctet argument doesn't meet preconditions"
| otherwise = (sequence $ take n $ repeat $ getStdRandom (randomR( 0,255) ))
>>= (return . (map chr) )
--Returns a list [r_1,r_2,r_3,r_4, . . ., r_n ] where
-- a = p_1^r_1 * p_2^r_2 * p_3^r_3 * . . . * p_n^r_n
factor :: Integer -> [Int]
factor = factor_1
--An implimentation of factor
factor_1 :: Integer -> [Int]
factor_1 a = reverse . dropWhile (== 0) . reverse
. map (\x -> largestPower x a) . takeWhile (<= a ) $ primes
--Another implimentation of factor
factor_2 :: Integer -> [Integer]
factor_2 a =
let
p = map (fromIntegral) . reverse . dropWhile (== 0)
. reverse . map (\x -> largestPower x a)
. takeWhile (<= a `div` 2) $ primes
in
if (length p == 0)
then (take ((fromIntegral . fromJust $ elemIndex a primes)-1) (repeat 0))
++ [1]
else p
--Find the inverse of x (mod n)
inverse :: Integer -> Integer -> Integer
inverse x n = (fst (extEuclGcd x n)) `mod` n
testInverse :: Integer ->Integer -> Bool
testInverse a b = ((inverse a b)*a) `mod` b == 1
--Extended Eucildean algorithm
--Returns (x,y) where gcd(a,b) = xa + yb
extEuclGcd :: Integer -> Integer -> (Integer,Integer)
extEuclGcd a b = extEuclGcd_iter a b (1,0) (0,1)
extEuclGcd_iter :: Integer -> Integer
-> (Integer,Integer) -> (Integer,Integer) -> (Integer,Integer)
extEuclGcd_iter a b (c1,c2) (d1,d2)
| (a > b) && (r1 == 0) = (d1,d2)
| (a > b) && (r1 /= 0) = extEuclGcd_iter
(a - (q1*b)) b (c1 - (q1*d1), c2 - (q1*d2)) (d1,d2)
| (a <= b) && (r2 == 0) = (c1,c2)
| (a <= b) && (r2 /= 0) = extEuclGcd_iter
a (b - (q2*a)) (c1,c2) ( d1 - (q2*c1), d2- (q2*c2))
where
q1 = a `div` b
q2 = b `div` a
r1 = a `mod` b
r2 = b `mod` a
-- This will return a random Integer of n bits. The highest order bit
-- will always be 1.
-- GHC optomized implementation
-- getNumber :: Int -> IO Integer
-- getNumber n = do
-- i <- getStdRandom ( randomR (0, a-1 ) )
-- return (setBit i (n-1))
-- where
-- a = (2^n) ::Integer
--This is the portable version
getNumber :: Int -> IO Integer
getNumber n = do
i <- getStdRandom ( randomR (0, a-1 ) )
return (i+(2^(n-1)))
where
a = (2^(n-1)) ::Integer
--Returns a probable prime number of nBits bits
-- GHC optomized implementation
-- getPrime :: Int -> IO Integer
-- getPrime nBits = do
-- r <- getNumber nBits
-- let p = (setBit r 0) --Make it odd for speed
-- pIsPrime <- isPrime p
-- if( pIsPrime )
-- then return p
-- else getPrime nBits
--This is the portable version
getPrime :: Int -> IO Integer
getPrime nBits = do
r <- getNumber nBits
let p = if( 2 /| r ) then r else r+1
pIsPrime <- isPrime p
if( pIsPrime )
then return p
else getPrime nBits
--Prime Generate:
--Generates a prime p | minimum <= p <= maximum and gcd p e == 1
pg :: Integer -> Integer -> Integer -> IO(Integer)
pg minimum maximum e = do
p <- getStdRandom( randomR( minimum, maximum ) )
pIsPrime <- isPrime p
if( pIsPrime && (gcd p e) == 1 )
then return p
else pg minimum maximum e
isPrime :: Integer -> IO Bool
isPrime a
| (a <= 1) = return False
| (a <= 2000) = return (simplePrimalityTest a)
| otherwise = if (simplePrimalityTest a)
then do --Do this 5 times for saftey
test <- mapM rabinMillerPrimalityTest $ take 5 $ repeat a
return (and test)
else return False
simplePrimalityTest :: Integer -> Bool
simplePrimalityTest a = foldr (&&) True (map (/| a)(takeWhile (<it) primes))
where it = min 2000 a
--returns greatest z where x^z | y
largestPower :: Integer -> Integer -> Int
largestPower x y = fromJust . elemIndex False
. map (\b -> (y `mod` x^b) == 0) $ [1..]
rabinMillerPrimalityTest :: Integer -> IO Bool
rabinMillerPrimalityTest p = rabinMillerPrimalityTest_iter_1 p b m
where
b = fromIntegral $ largestPower 2 (p-1)
m = (p-1) `div` (2^b)
--The ?prime? Number -> The amount of iterations -> b -> m
rabinMillerPrimalityTest_iter_1 :: Integer -> Integer -> Integer -> IO Bool
rabinMillerPrimalityTest_iter_1 p b m =
do
a <- getStdRandom ( randomR (0, 2000 ) )
return (rabinMillerPrimalityTest_iter_2 p b 0 (expmod a m p))
rabinMillerPrimalityTest_iter_2 :: Integer -> Integer -> Integer -> Integer
-> Bool
rabinMillerPrimalityTest_iter_2 p b j z
| (z == 1) || (z == p-1) = True
| (j > 0) && (z == 1) = False
| (j+1 < b) && (z /= p-1) =
(rabinMillerPrimalityTest_iter_2 p b (j+1) ((z^2) `mod` p ))
| z == p - 1 = True
| (j+1 == b) && (z /= p-1) = False
--a^x (mod m)
expmod :: Integer -> Integer -> Integer -> Integer
expmod a x m | x == 0 = 1
| x == 1 = a `mod` m
| even x = let p = (expmod a (x `div` 2) m) `mod` m
in (p^2) `mod` m
| otherwise = (a * expmod a (x-1) m) `mod` m
--Largest x where x^2 < i
intSqrt :: Integer -> Integer
intSqrt i = floor (sqrt (fromIntegral i ) )
--The doesn't divide function
(/|) :: Integer -> Integer -> Bool
a /| b = b `mod` a /= 0
--List of primes
primes :: [Integer]
primes = 2:[x | x <- [3,5..], foldr (&&) True
( map ( /| x ) (takeWhile (<=(intSqrt x)) primes ) ) ]
|
