If you read the very beginning of this thread, I questioned the randomness of the dice myself. Upon learning that we used mt_rand, I did a lot of research on it and could not find a better PRNG compared to a TRNG which you can not even write a code for, I believe. I referenced a site a long time ago about it. I believe this was the site: Random.Org. There are a lot of questions that can be answered if you want further information about randomness and how it is generated using mt_rand. While I believe the application of the newer random_int as dough suggested is used for other applications, as he just demonstrated for our purposes, the mt_rand is still superior than anything else for our purposes.
Obviously, this is a site that offers both free and paid services but there is valuable information to be had there if you really want to do the research without doing too much work to find it on your own.
Here is a little snippet for anyone who is truly interested on the difference between a PRNG and a TNRG from that site and why mt_rand is a PRNG about as close as you can get to a TNRG at the time I posted it:
Pseudo-Random Number Generators (PRNGs)
As the word ‘pseudo’ suggests, pseudo-random numbers are not random in the way you might expect, at least not if you're used to dice rolls or lottery tickets. Essentially, PRNGs are algorithms that use mathematical formulae or simply precalculated tables to produce sequences of numbers that appear random. A good example of a PRNG is the linear congruential method. A good deal of research has gone into pseudo-random number theory, and modern algorithms for generating pseudo-random numbers are so good that the numbers look exactly like they were really random.
The basic difference between PRNGs and TRNGs is easy to understand if you compare computer-generated random numbers to rolls of a die. Because PRNGs generate random numbers by using mathematical formulae or precalculated lists, using one corresponds to someone rolling a die many times and writing down the results. Whenever you ask for a die roll, you get the next on the list. Effectively, the numbers appear random, but they are really predetermined. TRNGs work by getting a computer to actually roll the die — or, more commonly, use some other physical phenomenon that is easier to connect to a computer than a die is.
PRNGs are efficient, meaning they can produce many numbers in a short time, and deterministic, meaning that a given sequence of numbers can be reproduced at a later date if the starting point in the sequence is known. Efficiency is a nice characteristic if your application needs many numbers, and determinism is handy if you need to replay the same sequence of numbers again at a later stage. PRNGs are typically also periodic, which means that the sequence will eventually repeat itself. While periodicity is hardly ever a desirable characteristic, modern PRNGs have a period that is so long that it can be ignored for most practical purposes.
These characteristics make PRNGs suitable for applications where many numbers are required and where it is useful that the same sequence can be replayed easily. Popular examples of such applications are simulation and modeling applications. PRNGs are not suitable for applications where it is important that the numbers are really unpredictable, such as data encryption and gambling.
It should be noted that even though good PRNG algorithms exist, they aren't always used, and it's easy to get nasty surprises. Take the example of the popular web programming language PHP. If you use PHP for GNU/Linux, chances are you will be perfectly happy with your random numbers. However, if you use PHP for Microsoft Windows, you will probably find that your random numbers aren't quite up to scratch as shown in this visual analysis from 2008. Another example dates back to 2002 when one researcher reported that the PRNG on MacOS was not good enough for scientific simulation of virus infections. The bottom line is that even if a PRNG will serve your application's needs, you still need to be careful about which one you use.
True Random Number Generators (TRNGs)
In comparison with PRNGs, TRNGs extract randomness from physical phenomena and introduce it into a computer. You can imagine this as a die connected to a computer, but typically people use a physical phenomenon that is easier to connect to a computer than a die is. The physical phenomenon can be very simple, like the little variations in somebody's mouse movements or in the amount of time between keystrokes. In practice, however, you have to be careful about which source you choose. For example, it can be tricky to use keystrokes in this fashion, because keystrokes are often buffered by the computer's operating system, meaning that several keystrokes are collected before they are sent to the program waiting for them. To a program waiting for the keystrokes, it will seem as though the keys were pressed almost simultaneously, and there may not be a lot of randomness there after all.
However, there are many other ways to get true randomness into your computer. A really good physical phenomenon to use is a radioactive source. The points in time at which a radioactive source decays are completely unpredictable, and they can quite easily be detected and fed into a computer, avoiding any buffering mechanisms in the operating system. The HotBits service at Fourmilab in Switzerland is an excellent example of a random number generator that uses this technique. Another suitable physical phenomenon is atmospheric noise, which is quite easy to pick up with a normal radio. This is the approach used by RANDOM.ORG. You could also use background noise from an office or laboratory, but you'll have to watch out for patterns. The fan from your computer might contribute to the background noise, and since the fan is a rotating device, chances are the noise it produces won't be as random as atmospheric noise.
Thunderstorm over Denver, USA
Thunderstorms generate atmospheric noise
As long as you are careful, the possibilities are endless. Undoubtedly the visually coolest approach was the lavarand generator, which was built by Silicon Graphics and used snapshots of lava lamps to generate true random numbers. Unfortunately, lavarand is no longer operational, but one of its inventors is carrying on the work (without the lava lamps) at the LavaRnd web site. Yet another approach is the Java EntropyPool, which gathers random bits from a variety of sources including HotBits and RANDOM.ORG, but also from web page hits received by the EntropyPool's own web server.
Regardless of which physical phenomenon is used, the process of generating true random numbers involves identifying little, unpredictable changes in the data. For example, HotBits uses little variations in the delay between occurrences of radioactive decay, and RANDOM.ORG uses little variations in the amplitude of atmospheric noise.
The characteristics of TRNGs are quite different from PRNGs. First, TRNGs are generally rather inefficient compared to PRNGs, taking considerably longer time to produce numbers. They are also nondeterministic, meaning that a given sequence of numbers cannot be reproduced, although the same sequence may of course occur several times by chance. TRNGs have no period.
If you read the very beginning of this thread, I questioned the randomness of the dice myself. Upon learning that we used mt_rand, I did a lot of research on it and could not find a better PRNG compared to a TRNG which you can not even write a code for, I believe. I referenced a site a long time ago about it. I believe this was the site: Random.Org. There are a lot of questions that can be answered if you want further information about randomness and how it is generated using mt_rand. While I believe the application of the newer random_int as dough suggested is used for other applications, as he just demonstrated for our purposes, the mt_rand is still superior than anything else for our purposes.
Obviously, this is a site that offers both free and paid services but there is valuable information to be had there if you really want to do the research without doing too much work to find it on your own.
Here is a little snippet for anyone who is truly interested on the difference between a PRNG and a TNRG from that site and why mt_rand is a PRNG about as close as you can get to a TNRG at the time I posted it:
[spoiler]Pseudo-Random Number Generators (PRNGs)
As the word ‘pseudo’ suggests, pseudo-random numbers are not random in the way you might expect, at least not if you're used to dice rolls or lottery tickets. Essentially, PRNGs are algorithms that use mathematical formulae or simply precalculated tables to produce sequences of numbers that appear random. A good example of a PRNG is the linear congruential method. A good deal of research has gone into pseudo-random number theory, and modern algorithms for generating pseudo-random numbers are so good that the numbers look exactly like they were really random.
The basic difference between PRNGs and TRNGs is easy to understand if you compare computer-generated random numbers to rolls of a die. Because PRNGs generate random numbers by using mathematical formulae or precalculated lists, using one corresponds to someone rolling a die many times and writing down the results. Whenever you ask for a die roll, you get the next on the list. Effectively, the numbers appear random, but they are really predetermined. TRNGs work by getting a computer to actually roll the die — or, more commonly, use some other physical phenomenon that is easier to connect to a computer than a die is.
PRNGs are efficient, meaning they can produce many numbers in a short time, and deterministic, meaning that a given sequence of numbers can be reproduced at a later date if the starting point in the sequence is known. Efficiency is a nice characteristic if your application needs many numbers, and determinism is handy if you need to replay the same sequence of numbers again at a later stage. PRNGs are typically also periodic, which means that the sequence will eventually repeat itself. While periodicity is hardly ever a desirable characteristic, modern PRNGs have a period that is so long that it can be ignored for most practical purposes.
These characteristics make PRNGs suitable for applications where many numbers are required and where it is useful that the same sequence can be replayed easily. Popular examples of such applications are simulation and modeling applications. PRNGs are not suitable for applications where it is important that the numbers are really unpredictable, such as data encryption and gambling.
It should be noted that even though good PRNG algorithms exist, they aren't always used, and it's easy to get nasty surprises. Take the example of the popular web programming language PHP. If you use PHP for GNU/Linux, chances are you will be perfectly happy with your random numbers. However, if you use PHP for Microsoft Windows, you will probably find that your random numbers aren't quite up to scratch as shown in this visual analysis from 2008. Another example dates back to 2002 when one researcher reported that the PRNG on MacOS was not good enough for scientific simulation of virus infections. The bottom line is that even if a PRNG will serve your application's needs, you still need to be careful about which one you use.
True Random Number Generators (TRNGs)
In comparison with PRNGs, TRNGs extract randomness from physical phenomena and introduce it into a computer. You can imagine this as a die connected to a computer, but typically people use a physical phenomenon that is easier to connect to a computer than a die is. The physical phenomenon can be very simple, like the little variations in somebody's mouse movements or in the amount of time between keystrokes. In practice, however, you have to be careful about which source you choose. For example, it can be tricky to use keystrokes in this fashion, because keystrokes are often buffered by the computer's operating system, meaning that several keystrokes are collected before they are sent to the program waiting for them. To a program waiting for the keystrokes, it will seem as though the keys were pressed almost simultaneously, and there may not be a lot of randomness there after all.
However, there are many other ways to get true randomness into your computer. A really good physical phenomenon to use is a radioactive source. The points in time at which a radioactive source decays are completely unpredictable, and they can quite easily be detected and fed into a computer, avoiding any buffering mechanisms in the operating system. The HotBits service at Fourmilab in Switzerland is an excellent example of a random number generator that uses this technique. Another suitable physical phenomenon is atmospheric noise, which is quite easy to pick up with a normal radio. This is the approach used by RANDOM.ORG. You could also use background noise from an office or laboratory, but you'll have to watch out for patterns. The fan from your computer might contribute to the background noise, and since the fan is a rotating device, chances are the noise it produces won't be as random as atmospheric noise.
Thunderstorm over Denver, USA
Thunderstorms generate atmospheric noise
As long as you are careful, the possibilities are endless. Undoubtedly the visually coolest approach was the lavarand generator, which was built by Silicon Graphics and used snapshots of lava lamps to generate true random numbers. Unfortunately, lavarand is no longer operational, but one of its inventors is carrying on the work (without the lava lamps) at the LavaRnd web site. Yet another approach is the Java EntropyPool, which gathers random bits from a variety of sources including HotBits and RANDOM.ORG, but also from web page hits received by the EntropyPool's own web server.
Regardless of which physical phenomenon is used, the process of generating true random numbers involves identifying little, unpredictable changes in the data. For example, HotBits uses little variations in the delay between occurrences of radioactive decay, and RANDOM.ORG uses little variations in the amplitude of atmospheric noise.
The characteristics of TRNGs are quite different from PRNGs. First, TRNGs are generally rather inefficient compared to PRNGs, taking considerably longer time to produce numbers. They are also nondeterministic, meaning that a given sequence of numbers cannot be reproduced, although the same sequence may of course occur several times by chance. TRNGs have no period.[/spoiler]
"I can picture in my mind a world without war, a world without hate. And I can picture us attacking that world, because they'd never expect it."
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