### Re^6: How to generate different random numbers?

by Pragma (Scribe)
 on Sep 08, 2004 at 04:38 UTC Need Help??

an attacker will surely notice...
Again, you are presuming a cryptographic application. No such presumption is warranted based on the root node.

Moreover, in terms of randomness, a random shuffle and just choosing a random number are vastly different.
What part of "indeed, a random shuffle of a known interval isn't the same thing as a set of random numbers... I never said they were the same thing." was unclear?
... the other is chosing a member from an infinite set with replacement. (emphasis mine)

A random number need not necessarily be chosen from an infinite set. Furthermore, an infinite number of randomly chosen numbers from a finite set is equivalent to a random selection from a continuous interval (infinite set). Therefore, a random shuffle (or multiple random shuffles, depending on the desired cardinality) can be transformed algorithmically into a random number suitable for cryptographic use.

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Re^7: How to generate different random numbers?
by thor (Priest) on Sep 08, 2004 at 12:02 UTC
Any time discussions turn to randomness and such, someone will use that discussion for security related purposes at some later date. That having been said, I feel that it's my duty as an upstanding netizen to bring to light the implications of certain things. On we go...
Therefore, a random shuffle (or multiple random shuffles, depending on the desired cardinality) can be transformed algorithmically into a random number
Unless you enumerate all of the permutations of the set, I don't know how this could possibly work. Even if you are, you're reducing the size of your output number drastically.

There are n! ways to permute a set of n items. There are n**n ways to choose n items from a set of n with replacement. The limit as n tends to zero of n!/n**n is zero. What this means is that compared with the cardinality of just choosing n random numbers from 1 to n, generating a random number with from a random shuffle gets progressively worse for larger n. This would imply (well, to me anyways) that if you're looking for a random number, better to do it directly than to first shuffle and then transform that shuffle into one.

A random number need not necessarily be chosen from an infinite set
Point conceded. I mis-spoke.

thor

Feel the white light, the light within
Be your own disciple, fan the sparks of will
For all of us waiting, your kingdom will come

Unless you enumerate all of the permutations of the set, I don't know how this could possibly work.
That's exactly what I'm doing; there is no need to limit yourself to individually selected numbers. If you treat each shuffled set as a random selection from the set of all possible shuffles, then you have a perfectly good random number, equivalent to a randomly selected integer from 1 to N!. If you do M shuffles, you can select random numbers from a set of size N! ^ M. It doesn't take a very large N and/or M to reach the cardinality needed from cryptographic applications.
This works in theory. I think that it would become pretty unwieldy in practice though. Given a permutation of a set (let's say {5 1 3 2 4}), which permutation is that of {1 2 3 4 5}? The fourteenth or the forty-second? Wouldn't you have to pre-compute and store all of the permutations for this to work?

thor

Feel the white light, the light within
Be your own disciple, fan the sparks of will
For all of us waiting, your kingdom will come

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