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- #Random - Number Generator
- Random Number Generator
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Random Number Generator Tool version 1. Just click the green Download button above to start. Until now the program was downloaded times.
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#Random - Number Generator
Rated 4. To rate and review, sign in. Sign in. Showing out of 1 reviews. We need to generate perfect, uniform entropy from a limited sets of these values around 21 values in perfect conditions, or down to 7 values in the worst conditions. And the distribution of these values will be highly biased — in our distribution the sample zero is happening most often with a frequency of 8. How much entropy can we hope to extract, theoretically?
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In this sample, zero happens 8. The simple calculation of -log2 8. If we have to work with worse camera conditions i. The good news is that our 12 megapixel camera has lots and lots of data points produced every second! However, even after encoding, our bits are still heavily biased, obviously with zero bits being far more prevalent since our most common value is zero.
How can we go from a zero-biased bitstream to a bitstream with uniform distribution? To solve this challenge, we are going to use the granddaddy of all randomness extractors — the famous Von Neumann extractor. John Von Neumann figured out how you can get perfect entropy even if your source is a highly biased coin: a coin that lands on one side far more often then another. Yet despite having a coin that is so defective, we can still extract perfectly uniform entropy from it!
Obviously, a toss of 0,0 is far more likely than a toss of 1,1 since zero is far more probable to happen. But what will be the probability of a 0,1 toss compared with a 1,0 toss? We just changed the order of variables, but the result is exactly the same. Here is a classical Von Neumann extractor of biased random bits:.
The Von Neumann extractor works amazingly well and has the added benefit that we do not even need to know how biased our incoming bits are. However, we are throwing away much of our raw data all these 0,0 and 1,1 pairs. Can we extract more entropy? We will explain Peres method since it is a bit simpler to explain and implement.
But what about longer sequences? If we use straight Von Neumann method we will get nothing from these: both 11 and 00 will be tossed away.
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But let's calculate probabilities: will be 20x20x80x80 , while will be 80x80x20x20 - the same numbers again. There is clearly something very similar to what we have already seen. We have two sequences with the same probabilities. This is a case of recursion: we are using the same algorithm just on numbers doubled up instead of originals.
Here is the Von Neumann extractor with simple recursion:.
Since all 1,1 and 0,0 sequences become just single bits, the Von Neumann classic algorithm by definition will extract more uniform bits if the initial data contained any 1,1,0,0 or 0,0,1,1. And since this algorithm is recursive we can keep running it as long as something collects in our temp buffer and extract every last bit of entropy from longer sequences. Finally, the Peres algorithm has one more abstraction that allows us to extract even more entropy from a Von Neumann source by using the position of sequences that produce bits in initial stream, which is implemented in our codebase.
Now we have a powerful extractor that can deal with an input of biased bits, and we are almost ready to extract some entropy. But there is one more step we need to do first. What might look like a perfect random entropy coming from, say, quantum physics, can be in fact completely deterministically created by a software generator and fully controlled by an attacker selecting an initial seed.
What is going on? Khan Academy has an excellent video explaining step by step how to use the chi-squared test — check it out to learn more. The chi-square test is fairly simple and consists of the following:.
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If any of your expectations are too far out from reality, then the squared difference will grow very fast! The bigger it becomes, the more wrong you are. We want to produce random bytes, that can take values from 0 to We want the distribution of our bytes to be uniform , so that the probability of every byte is exactly the same. Observing every byte is an event, so we have events total. That is our expected value for each event.
That number is our chi-squared statistic. What do we do with it? The most interesting part is the second parameter P , a probability that we can calculate from The only printed table we could find tantalizingly cuts off at — just 5 short from what we need! However, we can use a chi-squared calculator to get the exact values we need.
What this table tells us is the probability that our observation fits our hypothesis — the hypothesis we used to formulate all expected values. That is not bad actually — if you run the chi-square test 10 times on 10 perfectly random samples, it is likely that you will get one sample around If you run hundreds of random samples, one might have a chi-square value of , i. If you willing to test a thousand samples, one might have a chi-squared value of 0.
As you can see, the chi-square test, like all other randomness tests, do NOT tell us anything about the nature of our randomness!