Programming Praxis - RSA Cryptography

In today’s Programming Praxis exercise, our task is to implement a key generator and encryption/decription functions for RSA cryptography. Let’s get started, shall we?

A few imports:

import Data.Bits
import Data.List
import System.Random
import Data.Numbers.Primes

Like the Scheme solution, we’ll be recycling a few functions from previous solutions.

expm :: Integer -> Integer -> Integer -> Integer
expm b e m = foldl' (\r (b', _) -> mod (r * b') m) 1 .
             filter (flip testBit 0 . snd) .
             zip (iterate (flip mod m . (^ 2)) b) $
             takeWhile (> 0) $ iterate (`shiftR` 1) e

euclid :: Integral a => a -> a -> a
euclid x y = f 1 0 x 0 1 y where
    f a b g u v w | w == 0    = mod a y
                  | otherwise = f u v w (a - q*u) (b - q*v) (g - q*w)
                                where q = div g w

inv :: Integral a => a -> a -> a
inv x m | gcd x m == 1 = euclid x m
        | otherwise    = error "divisor must be coprime to modulus"

The keygen returns the modulus and the decryption key.

keygen :: Integer -> Integer -> IO (Integer, Integer)
keygen bits key = do p <- gen (div bits 2)
                     q <- gen (div bits 2)
                     let d = inv key ((p - 1) * (q - 1))
                     return (p * q, d) where
    gen k = fmap (until valid succ) $ randomRIO (2 ^ (k - 1), 2 ^ k)
    valid v = gcd key (v - 1) == 1 && mod v 4 == 3 && isPrime v

For encrypting and decrypting we could just use expm directly, but we flip the last two arguments to correspond to the Scheme solution.

crypt :: Integer -> Integer -> Integer -> Integer
crypt = flip . expm

Some tests to see if everything’s working correctly:

main :: IO ()
main = do let e = 65537
          (m, d) <- keygen 32 e
          print $ crypt (crypt 42 m e) m d == 42
          print $ crypt 42 1893932189 e == 1118138102

Everything seems to be working fine.

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Tags: bonsai, code, cryptography, Haskell, kata, praxis, programming, rsa