Random strings of character and some stats

[Jean-Fran?ois Lacrampe]

Hello,

I've got two random number/statistics questions I'd like you to
review. My first question is not directly related to PHP, but will be
implemented in PHP, as explained in my second question, so let's go:

I want to generate 10000 strings of x characters, with one chance (or
less) on a million that you can guess them by just randomly typing
them. So I need to know what is the value of x.

I wrote the following equation :

36^x/10000 = 1000000
<=> 36^x = 10000 * 1000000
<=> 36^x = 1010
<=> x = ln(1010)/ln(36)
<=> x = 23.025850929940 456840179914546 844/3.5835189384561 100016249547167 614
<=> x = 6.4254860446923 437997173954827 712

So, a 7-characters string would be good enough.

So my first question is: is my reasoning OK? Knowing my math
abilities, I doubt it very much! ;)

The second question I have is related to PHP's rand() function. I've
read many times that rand() is not random enough, especially when
generating long lists of this kind. Would you use something that's
more powerful than rand(), are there stronger random functions, within
PEAR for instance, or anything?

Thanks,
JFLac

[Andy Hassall]

Re: Random strings of character and some stats

On 2 May 2005 13:55:18 -0700, jflacrampe@gmai l.com (Jean-Fran?ois Lacrampe)
wrote:
[color=blue]
>I've got two random number/statistics questions I'd like you to
>review. My first question is not directly related to PHP, but will be
>implemented in PHP, as explained in my second question, so let's go:
>
>I want to generate 10000 strings of x characters, with one chance (or
>less) on a million that you can guess them by just randomly typing
>them. So I need to know what is the value of x.[/color]

I'm not 100% clear on the "them" in the sentence; are you saying you want less
than 1/1000000 chance of guessing ONE of the 10000 strings, or 1/1000000 chance
of guessing the ENTIRE SET of 10000 strings?
[color=blue]
>I wrote the following equation :
>
> 36^x/10000 = 1000000[/color]

Depends on the interpretation above. Not sure I get how the 10000 is involved
here though.
[color=blue]
><=> 36^x = 10000 * 1000000
><=> 36^x = 1010[/color]

10000 * 1000000 = 1010 ? Is that supposed to be 10^10 ?
[color=blue]
><=> x = ln(1010)/ln(36)
><=> x = 23.025850929940 456840179914546 844/3.5835189384561 100016249547167 614[/color]

Apparently so :-)
[color=blue]
><=> x = 6.4254860446923 437997173954827 712
>
>So, a 7-characters string would be good enough.[/color]

If you want at worst 1/1000000 chance of guessing any string, isn't the number
of strings irrelevant if they're random?

i.e. it's just

36^x > 1000000
=> x > ln(1000000)/ln(36)
=> x > 3.855

So minimum number of chars = 4.

(36^3 = 46656, 36^4 = 1679616)


The odds of guessing ALL the strings surely head well out of the 1 in 1000000
range for 10000 strings very quickly...
[color=blue]
>So my first question is: is my reasoning OK? Knowing my math
>abilities, I doubt it very much! ;)
>
>The second question I have is related to PHP's rand() function. I've
>read many times that rand() is not random enough, especially when
>generating long lists of this kind. Would you use something that's
>more powerful than rand(), are there stronger random functions, within
>PEAR for instance, or anything?[/color]

mt_rand() uses the Mersenne Twister pseudorandom algorithm, which is typically
better (and as a bonus it's faster too).

If you want to get really serious you'll need to base it on some sort of truly
physical phenomenon, e.g. with RNG hardware, which is often based on random
thermal fluctuations.

--
Andy Hassall / <andy@andyh.co. uk> / <http://www.andyh.co.uk >
<http://www.andyhsoftwa re.co.uk/space> Space: disk usage analysis tool

[Chung Leong]

Re: Random strings of character and some stats


Andy Hassall wrote:[color=blue]
> If you want at worst 1/1000000 chance of guessing any string, isn't[/color]
the[color=blue]
> number of strings irrelevant if they're random?[/color]

No, because each additional string means an additional try. Think of it
this way: A person comes with a number of letters randomly and the
computer tries 10000 times to guess it.

[Jean-François Lacrampe]

Re: Random strings of character and some stats

Andy Hassall wrote:[color=blue]
> I'm not 100% clear on the "them" in the sentence; are you saying you[/color]
want less[color=blue]
> than 1/1000000 chance of guessing ONE of the 10000 strings, or[/color]
1/1000000 chance[color=blue]
> of guessing the ENTIRE SET of 10000 strings?[/color]

I meant the odds of guessing any of the 10000 strings, of course! :-)
The odds of guessing the entire set must be really, really low!
[color=blue]
> 10000 * 1000000 = 1010 ? Is that supposed to be 10^10 ?[/color]

Well, I wrote the equation in another editor who was sooo happy to show
me it was able to display the 10^10 graphically. Too bad it forgot to
copy/paste it back to me with the circumflex.
[color=blue]
> If you want at worst 1/1000000 chance of guessing any string, isn't[/color]
the number[color=blue]
> of strings irrelevant if they're random?[/color]

Well, keeping in mind my very 'intuitive' and weak knowledge of math,
I'd guess that the more strings you put in the list, the more chances
you have to guess one (any) of them. If for instance I had a list long
enough to contain all the possible combinations, the odds would be 1/1,
right? If you divide the list by two, the odds are 1/2. And so on.
So the number of items in the list seems to matter: that's how I came
with the 10000 * 1000000 thing (by doing lots of intermediate and
stupid steps on a sheet of paper).

I'm not sure at all that I put the 10000 where I should have in the
equation, though, hence my initial question.

Now, I'm talking about things I don't understand (math) in a language
that isn't my native language and I reckon that I'm a bit awkward at
explaining my thoughts. :-)
[color=blue]
> mt_rand() uses the Mersenne Twister pseudorandom algorithm, which is[/color]
typically[color=blue]
> better (and as a bonus it's faster too).
>
> If you want to get really serious you'll need to base it on some[/color]
sort of truly[color=blue]
> physical phenomenon, e.g. with RNG hardware, which is often based on[/color]
random[color=blue]
> thermal fluctuations.[/color]

I could also use a webcam on a lava lamp and produce my results using
the webcam info, but I guess I don't need that randomness. I just
wanted to know what was my best bet with what PHP can give me, with
minimal hassle. ;-)

Thanks for your answers,
JFLac

[NC]

Re: Random strings of character and some stats

Jean-Francois Lacrampe wrote:[color=blue]
>
> I want to generate 10000 strings of x characters, with one chance (or
> less) on a million that you can guess them by just randomly typing
> them. So I need to know what is the value of x.[/color]

OK, one in a million chance of successfully guessing 10,000 strings
equals 0.9986 chance of successfully guessing a single string:

0.9986 ^ 10000 = 8.23412E-07 ~ 1E-06 (one in a million)

In other words, even if you are virtually certain to get a single
string right, it's still virtually impossible to get 10,000 of them
right. So a one-character string will suffice. In fact, even a
one-bit value (0 or 1) would be an overkill. :)
[color=blue]
> The second question I have is related to PHP's rand() function. I've
> read many times that rand() is not random enough, especially when
> generating long lists of this kind. Would you use something that's
> more powerful than rand(), are there stronger random functions,[/color]
within[color=blue]
> PEAR for instance, or anything?[/color]

Check out mt_rand():

http://www.php.net/mt_rand

Cheers,
NC

[Jean-François Lacrampe]

Re: Random strings of character and some stats

NC wrote:[color=blue]
> Jean-Francois Lacrampe wrote:[color=green]
> >
> > I want to generate 10000 strings of x characters, with one chance[/color][/color]
(or[color=blue][color=green]
> > less) on a million that you can guess them by just randomly typing
> > them. So I need to know what is the value of x.[/color]
>
> OK, one in a million chance of successfully guessing 10,000 strings
> equals 0.9986 chance of successfully guessing a single string:
>
> 0.9986 ^ 10000 = 8.23412E-07 ~ 1E-06 (one in a million)[/color]

As I said in another branch of the thread, I wasn't clear enough: I
meant 'one chance (or less) on a million that you can guess _any_ of
them'.

Anyway... Here's the code I wrote to generate my 10000 strings, just in
case it's useful to somebody browsing the archives, someday. The
random function is pretty much the same as the one you see on every php
tutorial, but it uses mt_rand() instead of rand() as many of you have
advised me. and I wrote a (very inefficient) dupe checker.
Optimizations and ideas are welcome, but that's just for the fun of it:
I'll generate these strings just once, so it doesn't matter if it takes
one full minute, it will only be ran once. :-)

<?php
set_time_limit (600); // We need this because of the time-consuming
// in_array()
$values = array();
function random_string() {
$allowed_chars = "0123456789AZER TYUIOPQSDFGHJKL MWXCVBN";
mt_srand((doubl e)microtime()*1 000000);
$string = 'A'; // I put a control char at the start of my
// password just in case I want to generate
// a second, third,... series in the future.
for($i = 0 ;$i <= 6; $i++) {
$position = mt_rand()%36;
$temp = substr($allowed _chars, $position, 1);
$string .= $temp;
}
return $string;
}

// The odds are low, but it's possible that the same string
// is generated twice. So I check if each new string found
// isn't in the previous string found in the array, but it
// slows down the script, which is not a problem in my
// case, but I welcome optimizations ideas.

// If a dupe is detected, I just decrement $i, which forces
// the loop to loop once more for this value of $i.

for ($i = 1 ; $i <= 10000; $i++) {
if (!in_array($str ing,$values)) {
$values[$i] = random_string() ;
} else {
$i--;
}
}
echo '<pre>';
print_r ($values);
echo '</pre>';
?>

JFLac

[Andy Hassall]

Re: Random strings of character and some stats

On 3 May 2005 03:44:19 -0700, "Jean-François Lacrampe" <jflacrampe@gma il.com>
wrote:
[color=blue]
>Well, keeping in mind my very 'intuitive' and weak knowledge of math,
>I'd guess that the more strings you put in the list, the more chances
>you have to guess one (any) of them. If for instance I had a list long
>enough to contain all the possible combinations, the odds would be 1/1,
>right? If you divide the list by two, the odds are 1/2. And so on.
>So the number of items in the list seems to matter: that's how I came
>with the 10000 * 1000000 thing (by doing lots of intermediate and
>stupid steps on a sheet of paper).[/color]

Ah, yes of course. OK, I agree with your maths, looks right.

--
Andy Hassall / <andy@andyh.co. uk> / <http://www.andyh.co.uk >
<http://www.andyhsoftwa re.co.uk/space> Space: disk usage analysis tool