[MLton] MLton.IntInf

Wesley W. Terpstra wesley@terpstra.ca
Thu, 28 Apr 2005 16:41:33 +0200


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On Tue, Apr 26, 2005 at 09:30:23PM -0700, Stephen Weeks wrote:
> > Why not also the gcdext, inverse, powm, fdiv_r_2exp, and probab_prime?
> 
> The reasons they are not there are lack of demand, lack of understanding
> of the library on our part, and the fact that adding IntInf primitives
> requires compiler mods.

Yeah, it's not critial for me ATM either.
I just noticed they were missing when comparing with mlgmp for OCaml.

When I find more time for SML/MLton I will be spending it on Unicode, so
don't be expecting me to add these! =) PS. I've got to focus more squarely
on my diss. at the moment; sorry for falling off the planet wrt Unicode.

> IntInf stuff is harder to handle than an ordinary FFI because it deals
> intimately with heap allocation. 

That makes sense.

> If you look at how IntInf.gcd is implemented, you'll see some ...

I see a lot of things. One thing I wonder... do you already optimize the
case of multiplying/adding/etc a big integer by a small one? GMP has special
*_ui methods for this case which are faster.

> The upshot being that we don't have a nice abstraction for adding such
> primitives, and that adding a new one touches quite a few places.

Indeed; probably more trouble than it's worth. Really only cryptographers
will want those methods, and they can always use crypto libraries instead.

> we would be happy for someone in the know to pattern match off of gcd
> to implement the remaining ones and send us a patch.

Maybe some day. =)

> Does mlgmp use the same trick as MLton to represent small integers without
> using an indirection?

No, and that might be the cause of the more extreme speedups I saw (35*) in
other applications.

However, for factoring very large numbers, the work is always with numbers
too big for this to matter... If you look in the attached code, you'll see
the critical section is the loop/try mutually recursive methods. 

> If it's not that, I'd be interested to hear an explanation of the
> difference.

Another possibility is that the FFI for MLton is faster; the OCaml approach
requires a layer of C glue that does conversions to/from OCaml types.

Further work at optimizing the OCaml version has gotten it to be only 1.5*
slower when using mlgmp. W/o mlgmp it's still at 5* despite my efforts.

The OCaml version uses the (faster) powm and probab_prim GMP methods. This
is hardly a fair comparison; I spent zero extra effort on the MLton version
of the program and it uses unoptimized ML code for exp and isPrime.

You know what else I find odd? 

I rebuilt the same code under MacOS X. Supposedly, OCaml has a native code
generator for this platform, so I expected the results to be reversed.

However, factoring one number on my 1.5GHz P4 took 3 minutes with OCaml and
2 minutes with MLton. On the 1.2GHz iBook it took 3 minutes with OCaml and 1
minute with MLton! How weird is that?  Bear in mind, again, that the MLton
version was not hand-optimized like the OCaml one.

For reference, I've attached the two code snippets and a precompiled version
of the mlgmp/libgmp-ocaml built version. To get a good test, you need a
number with two big factors. 903820928350932859017252135432745745642279

-- 
Wesley W. Terpstra

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open LargeInt

(* fun gcd (a, b) = if a = 0 then b else gcd (b mod a, a) *)
val gcd = MLton.IntInf.gcd

val primes = [ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 ]
val cutoff = 53

fun exp (x, 0, n) = 1
  | exp (x, e, n) = 
    let 
      val (q, r) = IntInf.divMod (e, 2)
      val x2 = exp (x*x mod n, q, n) 
    in
      if r = 0 then x2 else x2*x mod n 
    end

fun sort nil = nil
  | sort (p :: r) = 
    let val (s, b) = List.partition (fn x => x < p) r
    in sort s @ p :: sort b end

fun isPrime n = 
  let
    fun chomp (c, e) =
      if c mod 2 = 0 then chomp (c div 2, e+1) else (c, e)
    val (c, e) = chomp (n-1, 0)
    
    fun checkPow (w, e) =
      e > 0 andalso (w+1 = n orelse checkPow (w*w mod n, e-1))
    fun millerTest w = 
      let val v = exp (w, c, n)
      in v = 1 orelse checkPow (v, e) end
  in
    List.foldl (fn (w, a) => a andalso millerTest w) true primes
  end

fun ofindafact n =
  let
    val start = Word.toLargeInt (MLton.Random.rand ()) mod n
    fun f x = (x*x + 2) mod n
    fun loop x y =
      let val g = gcd (n, x-y)
      in if g = 1 orelse g = ~1 then loop (f x) (f (f y)) else g end
  in loop start (f start) end

fun findafact n =
  let
    val start = Word.toLargeInt (MLton.Random.rand ()) mod n
    fun f x = (x*x + 2) mod n
    fun loop l x =
      let
        fun try a y =
          if a = 0 then loop (l+l) y else
          let val g = gcd (n, x-y)
          in if g = 1 orelse g = ~1 then try (a-1) (f y) else g end
      in
        (
          if l > 1024
          then TextIO.output (TextIO.stdErr, "Trying with range " ^ 
                                              toString l ^ " for " ^
                                              toString n ^ "...\n") else (); 
          try l (f x)
        )
      end
  in 
    loop 1 start
  end
  
fun factor n =
  let
    fun suck n l =
      if n < cutoff then l else
      if isPrime n then n :: l else
      let val x = findafact n 
      in suck x (suck (n div x) l) end
      
    fun easy n l =
      case (List.find (fn f => n mod f = 0) primes) of
        NONE => suck n l
        | SOME x => easy (n div x) (x :: l)
    
  in sort (easy n []) end

val n = (valOf o fromString o hd o CommandLine.arguments) ()
val l = factor n

val () = print (toString n ^ ":")
val () = app (fn x => print (" " ^ toString x)) l
val () = print "\n"

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open Gmp

let primes = [ 2; 3; 5; 7; 11; 13; 17; 19; 23; 29; 31; 37; 41; 43; 47 ]
let cutoff = 53

let findafact n =
  let f x = Z.fdiv_r (Z.add_ui (Z.mul x x) 2) n in
  let rec loop l x =
    let rec test a y =
      if a = 0 then loop (l+l) y else
      let g = Z.gcd n (Z.sub x y) in
      if Z.equal_int g 1 || Z.equal_int g (-1) then test (a-1) (f y) else g in
    (
      if l > 1024 then prerr_endline ("Trying with range " ^ string_of_int l ^ 
                                      " for " ^ Z.to_string n ^ "...");
      test l (f x)
    ) in
  loop 1 (Z.fdiv_r (Z.from_int (Random.int 1073741823)) n)

let factor n =
  let rec suck n l =
      if Z.compare_int n cutoff < 0 then l else
      if Z.is_probab_prime n 5 then n :: l else
      let x = findafact n in
      suck x (suck (Z.fdiv_q n x) l) in
      
  let rec easy n l = 
    match List.filter (fun f -> Z.is_zero (Z.fdiv_r_ui n f)) primes with
      | [] -> suck n l
      | p -> easy 
        (List.fold_left (fun n p -> Z.fdiv_q_ui n p) n p)
        (List.map Z.from_int p @ l) in
    
  List.sort Z.compare (easy n [])

let n = Z.from_string Sys.argv.(1)
let l = factor n

let () = print_string (Z.to_string n ^ ":")
let () = List.iter (fun x -> print_string (" " ^ Z.to_string x)) l
let () = print_newline ()

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