我的RSA算法仅适用于非常短的单词

我的RSA算法适用于像"cat"这样的3个字母的单词,但是一旦单词变长,我会收到一条错误消息:异常:decipher2.hs:(37,1) - (62,19):非详尽的模式函数numToLet

我的程序按照以下步骤工作

用于加密

将一串字母转换为一串数字,其中每个字母都分配给一个特定的数字.

将数字串组合成一个更大的数字.

使用适当的p和q值,计算n并选择一个合适的数字e,它是n的共同素数(因为这是一个内部过程,保持p和q的秘密是不重要的,但可以完成)

执行RSA算法以生成"密码"

用于解密

使用p和q计算总数φ(n)(其中φ(n)=(p-1)*(q-1))

将Euclidean算法写入代码以找出两个数的最大公约数.

将后置替换部分写入代码,使其可交换地形成扩展欧几里得算法.

计算私钥.

包括存储私钥的某种选项,以便每次都不需要执行此功能(为了提高速度).

应该执行RSA加密的解密算法.

结果应该被分成一串数字(与编码过程的第2阶段中它的组合方式相反).

应将数字串转换为字母串(使用加密过程中使用的相同值).

我选择不包括计算关键部分,因为它是一个单独的程序,必须是正确的(因为短字可以被破译).我尝试使用较小的p和q值,但这也不起作用,是否有人知道为什么较长的单词不起作用(即使对于较小的素数)和/或我如何解决它？

我的加密代码是:

import Data.List
letToNum c'' = (let (Just n) = elemIndex c'' ['a'..'z'] in n) + 10

combine = foldl addDigit 0
    where addDigit num i' = (10 ^ (floor $ log (fromIntegral i') / log 10 + 1))*num + i'

firstTwo xs = toInteger (combine (map letToNum xs))

p' = 2^2048 - 1942289
q' = 2^2048 - 2^1056 + 2^736 - 2^320 + 2^128 + 1
n' = p' * q'
e' = 7

newtype ModN = ModN (Integer, Integer) deriving (Eq, Show)
instance Num ModN where
  ModN (x, m) + ModN (y, m') | m == m' = ModN ((x + y) `mod` m, m)
                             | otherwise = undefined
  ModN (x, m) * ModN (y, m') | m == m' = ModN ((x * y) `mod` m, m)
                             | otherwise = undefined
  negate (ModN (x, m)) = ModN ((- x) `mod` m, m)
  abs _ = undefined
  signum _ = undefined
  fromInteger _ = undefined

modPow :: Integer -> Integer -> Integer -> Integer
modPow _ 0 m = 1 `mod` m
modPow a b m = c
  where a' = ModN (a, m)
        ModN (c, _) = a' ^ b

encipherA z' = modPow z' e' n'

encipher xs = encipherA (firstTwo xs)

我的解密代码是:

p' = 2^2048 - 1942289
q' = 2^2048 - 2^1056 + 2^736 - 2^320 + 2^128 + 1
n' = p' * q'
d'=149198411630450358098821815816660626082852035578197682912033354754850558281651065264118115990713936905841443816348466119712510491169399086751890869693052182563708139506244285477194512876340531187775438573122278032339474119913958963667476383477798088213829701243686243438754864105731229873495425397653296705562025639940130672903361904637280085880562426594784029436599468688448179303703337724326069153629191476697420768451884440453280134491448395404958914592441919126201747433884753502442027069825305163272842897505994155682996130741544296475635538035696205346055652365820232813677363525296188331517668300986037331017823989462119054758685224752255850280255541024098528388784743634162996954090230161430468033874779937253936100340449079832503240200984659315256173082971785802328581652375418902448324739292188909509903808503871246982928186837296358844444348158079557757543904495890483033995844854839625810784987612815774450940850973031117459144355531047129541648317845165848539683500066541165782574432790936862217277817101197569391139502130923768985262346549685855947859864917650782062626099395684514986479824104485607000960030713840667632064934158997031779846656288570984548383771118345091911988849410656041321592285731293207858515766711

newtype ModN = ModN (Integer, Integer) deriving (Eq, Show)
instance Num ModN where
  ModN (x, m) + ModN (y, m') | m == m' = ModN ((x + y) `mod` m, m)
                             | otherwise = undefined
  ModN (x, m) * ModN (y, m') | m == m' = ModN ((x * y) `mod` m, m)
                             | otherwise = undefined
  negate (ModN (x, m)) = ModN ((- x) `mod` m, m)
  abs _ = undefined
  signum _ = undefined
  fromInteger _ = undefined

modPow :: Integer -> Integer -> Integer -> Integer
modPow _ 0 m = 1 `mod` m
modPow a b m = c
  where a' = ModN (a, m)
        ModN (c, _) = a' ^ b

decipherA c' = modPow c' d' n'

numToLet "10" = "a"  
numToLet "11" = "b"  
numToLet "12" = "c"
numToLet "13" = "d"
numToLet "14" = "e"
numToLet "15" = "f"
numToLet "16" = "g"
numToLet "17" = "h"
numToLet "18" = "i"
numToLet "19" = "j"
numToLet "20" = "k" 
numToLet "21" = "l"
numToLet "22" = "m"
numToLet "23" = "n"
numToLet "24" = "o"
numToLet "25" = "p"
numToLet "26" = "q"
numToLet "27" = "r"
numToLet "28" = "s"
numToLet "29" = "t"
numToLet "30" = "u"
numToLet "31" = "v"
numToLet "32" = "w"
numToLet "33" = "x"
numToLet "34" = "y"
numToLet "35" = "z"
output xs = putStrLn $ concat (map numToLet xs)

partition :: Int -> [a] -> [[a]]
partition _ [] = []
partition n xs = (take n xs) : (partition n (drop n xs))

decipher j' = map numToLet (partition 2 (show (decipherA j')))

感谢您的帮助,我真的很感激.