爲什麼我的米勒拉賓算法不起作用（Haskell）？

Question

我在Haskell中實現了Miller Rabin測試。我試圖嚴格遵循例如在米勒拉賓測試的維基百科條目中給出的僞代碼。現在我在網上發現，對於某些證人選擇，測試是確定性的，達到某個給定的界限。我對2^64以下的素數感興趣，所以我在這篇文章中找到了足夠的界限。然而，該代碼似乎適用於我測試過的大多數小素數，但對於一些較大的素數則失敗。例如，我嘗試了十位數的素數5915587277，測試返回false。我認爲我的實施是正確的，但希望有人能夠發現我犯了一個錯誤，並誤解了有關MR測試的情況。預先感謝您的幫助。另外，對於凌亂的代碼感到抱歉。

isPrime :: Int -> Bool 
isPrime n = millerRabinTest n (factorizeN (n-1)) 

{- factorizeN finds a number s and odd number d such that n -1 = (2^s)d by 
succesively dividing n by two if it is even. -} 
factorizeN :: Int -> (Int, Int) 
factorizeN n = fN n 0 
    where 
    fN n s | even n = fN (n `div` 2) (s + 1) 
      | otherwise = (n,s) 

{- this is the main function. it takes w values from a set of witnesses 
and checks if n passes the test. If it doesn't, n is not prime, if it does 
for all w, it is probably prime. -} 
millerRabinTest :: Int -> (Int,Int) -> Bool 
millerRabinTest n (d,s) = and [test n (expmod w d n) s | w <- onesToCheck] 

{- this is the test that is used in the millerRabinTest function. it sees if 
w^d = 1 mod n or n-1 mod n, if not it multiplies by two 
and checks again for a total of s-1 times. If it is never true then the number 
is not prime -} 
test :: Int -> Int -> Int -> Bool 
test n w s | w `elem` [1,n-1] = True 
      | otherwise  = or [ (expmod w (2^k) n) `elem` [1,n-1]| k <- [1..s]] 

{- set of witnesses that should make the Miller Rabin test deterministic if 
n < 2^64. -} 
onesToCheck :: [Int] 
onesToCheck = [2,325,9375,28178,450775,9780504,1795265022] 

{- function that calculates a^e mod n. -} 
expmod :: Int -> Int -> Int -> Int 
expmod a e n | e == 1   = a `mod` n 
       | (e `mod` 2) == 0 = (expmod ((a*a) `mod` n) (e `div` 2) n) 
       | otherwise  = (a*(expmod ((a*a) `mod` n) (e `div` 2) n)) `mod` n 

Answer 1

可能是你Int在expmod當你計算a*a四溢。 Int是一個機器大小的整數，不超過64位。您應該使用Integer（即任意精度整數類型）替換程序中Int的一些事件。