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Generation of random values

This module contains functions which let us generate random values.

module Random
  (
    randomBytes, randomBytesR
  , randomResidue
  , randomPrimeR
  , randomWECPoint
  , randomGF
  ) where

import Bytes ( Bytes, convBytes )
import EllipticCurve ( WECParameters(..), WECPoint, mkWECPoint )
import GCM ( GF )
import Math ( modSqrt )

import qualified Data.ByteString as B
import qualified Control.Monad.Random as R

import Math.NumberTheory.Primes.Testing ( isPrime )

Random bytes

randomBytes produces a sequence of the given number of random Bytes.

randomBytes :: R.MonadRandom m => Int -> m Bytes
randomBytes len = B.pack <$> R.replicateM len R.getRandom

randomBytesR produces a random number of random Bytes. Its argument is the lower and upper bound on the number of Bytes to produce.

randomBytesR :: R.MonadRandom m => (Int,Int) -> m Bytes
randomBytesR bounds = R.getRandomR bounds >>= randomBytes

Random integers

randomResidue produces a random element of the finite field of the given size. For usability as a key, we exclude 0, 1, and p-1.

randomResidue :: R.MonadRandom m => Integer -> m Integer
randomResidue p = R.getRandomR (2,p-2)

randomPrimeR produces a prime number within the given bounds. It generates a random integer, then tests its primality using isPrime from the arithmoi package.

randomPrimeR :: R.MonadRandom m => (Integer,Integer) -> m Integer
randomPrimeR bounds = do
  n <- R.getRandomR bounds
  if isPrime n
    then pure n
    else randomPrimeR bounds

Random elliptic curve points

randomWECPoint produces a random point on the elliptic curve with the given Weierstrass-form parameters.

randomWECPoint :: R.MonadRandom m => WECParameters -> m WECPoint
randomWECPoint params@WECParameters{wecA=a, wecB=b, wecP=p} = do

x can potentially take any value in the prime field.

  x <- randomResidue p

We can then compute y^2 from the elliptic curve equation.

  let x2 = (x*x) `rem` p
      y2 = ((x2 + a)*x + b) `rem` p

Unfortunately, this value might not have a square root. If not, then we have to pick another x and try again.

  case modSqrt p y2 of
    Nothing -> randomWECPoint params

If there is a square root y, the point could be either (x,y) or (x,-y); we randomly determine which and return.

    Just y  -> do
      neg <- R.getRandom
      pure $ mkWECPoint params x $ if neg then (-y) else y

Random Galois field points

randomGF creates a random element of GF(2^128); this is just sixteen random bytes, converted into GF.

randomGF :: R.MonadRandom m => m GF
randomGF = convBytes <$> randomBytes 16