Programming Praxis

2 Responses to “Reversible Random Number Generator”

1. Globules said

   November 29, 2015 at 7:26 AM

   A Haskell solution.

   import System.Random
   import Text.Printf
   
   -- Extended Euclidean algorithm.  Given non-negative a and b, return x, y and g
   -- such that ax + by = g, where g = gcd(a,b).  Note that x or y may be negative.
   gcdExt :: Integral a => a -> a -> (a, a, a)
   gcdExt a 0 = (1, 0, a)
   gcdExt a b = let (q, r) = a `quotRem` b
                    (s, t, g) = gcdExt b r
                in (t, s - q*t, g)
   
   -- Given a and m, return Just x such that ax = 1 mod m.  If there is no such x 
   -- return Nothing.
   modInv :: Integral a => a -> a -> Maybe a
   modInv a m = let (i, _, g) = gcdExt a m
                in if g == 1 then Just (pos i) else Nothing
     where pos x = if x < 0 then x + m else x
   
   
   type LCG a = a -> a
   
   -- Given a, c and m return Just (fwd, rev), where fwd is a forward linear
   -- congruential generator and rev is its corresponding reverse generator.
   -- Return Nothing if a and m are not coprime.
   lcgPair :: Integral a => a -> a -> a -> Maybe (LCG a, LCG a)
   lcgPair a c m = case a `modInv` m of
     Nothing -> Nothing
     Just a' -> let fwd prev = (a*prev + c) `mod` m
                    rev next = a'*(next - c) `mod` m
                in Just (fwd, rev)
   
   -- Test the LCG pairs by generating one sequence of numbers using the forward
   -- generator, then a second sequence using the reverse generator, starting
   -- with the last number produced by the forward generator.  Compare the first
   -- sequence with the reverse of the second one, indicating success if they are
   -- the same, or failure if they are not.
   testLcg :: String -> Integer -> Integer -> Integer -> Integer -> IO ()
   testLcg name seed a c m = case lcgPair a c m of
     Nothing -> printf "%s: %d and %d are not coprime.\n" name a m
     Just (fwd, rev) -> let fs = take 10000 $ iterate fwd seed
                            rs = take 10000 $ iterate rev (last fs)
                        in if fs == reverse rs
                           then printf "%s: success!\n" name
                           else printf "%s: failure!\n" name
   
   main :: IO ()
   main = do
     -- Ensure the seed is less than the smallest (valid) modulus below.
     seed <- randomRIO (0, 2^24-1)
   
     -- Test a few LCGs listed on the Linear congruential generator page
     -- (https://en.wikipedia.org/wiki/Linear_congruential_generator).
   
     testLcg "mmix"        seed 6364136223846793005 1442695040888963407 (2^64)
     testLcg "vms"         seed               69069                   1 (2^32)
     testLcg "vb"          seed          1140671485            12820163 (2^24)
     testLcg "minstd_rand" seed               48271                   0 (2^31-1)
     testLcg "bogus"       seed                1234                   0 123456
   

   $ ./revrand 
   mmix: success!
   vms: success!
   vb: success!
   minstd_rand: success!
   bogus: 1234 and 123456 are not coprime.
   
2. Jussi Piitulainen said

   December 12, 2015 at 9:33 AM

   This implements the Julia (v0.4.1) iterator protocol (methods for start, done, next). I believe each version of the generator (Gen, ForwardIter, ReverseIter) is actually stored as a single 64-bit value as long as the compiler can infer its type. So the whole thing could even be held in a register. A call to next returns both the generated value and the next state. There are no stateful objects in this implementation.

   module ReversibleGenerator
   
   export Gen, ForwardIter, ReverseIter
   import Base: start, done, next
   
   immutable Gen  # 64-bit state
       n::UInt64
   end
   
   immutable ForwardIter
       gen::Gen
   end
   
   immutable ReverseIter
       gen::Gen
   end
   
   # Linear congruential generator parameters
   # of "POSIX [jm]rand48" from Wikipedia,
   # reverse formula from Praxis.
   # m = 2^48, c = 11, return bits 47..15 (a 32-bit integer)
   a = 0x5deece66d
   b = 0xffffdfe05bcb1365 # = invmod(a, m)
   
   start(i::ForwardIter) = i.gen
   done(::ForwardIter, ::Gen) = false
   next(::ForwardIter, x::Gen) = let
       (UInt((x.n >>> 15) & 0xFFFFFFFF), # 32-bit value as system UInt
        Gen(mod(a * x.n + 11, 2^48)))
   end
   
   start(i::ReverseIter) = i.gen
   done(::ReverseIter, ::Gen) = false
   next(::ReverseIter, x::Gen) = let
       (UInt((x.n >>> 15) & 0xFFFFFFFF), # 32-bit value as system UInt
        Gen(mod(b * (x.n - 11), 2^48)))
   end
   
   end
   

   Julia for-loops use the iterator protocol implicitly, but then one gets access only to the values. To reverse the process in the middle requires access to the state, hence the explicit calls to start and next (but never done).

   using ReversibleGenerator
   
   xs = Array(UInt, 0) # empty vector of UInt (system size)
   
   # i serves as a type for start/done/next
   # s is an actual generator/state
   
   i = ForwardIter(Gen(31415926))
   s = start(i)
   for k in 1:3
       x, s = next(i, s)
       push!(xs, x)
   end
   
   i = ReverseIter(s)
   for k in 1:4
       x, s = next(i, s)
       push!(xs, x)
   end
   
   # prints a palindromic random array (in 0-padded hex :)
   println(xs)