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How mathematicians expose pseudorandom sequences.
Recently, a team of mathematicians presented a new proof that sheds light on the question of whether certain sequences of numbers are random or pseudorandom. Pseudorandom numbers look the same as random numbers, but can actually be obtained using deterministic processes. Research in this area is of great importance for many areas, including modeling, lotteries, casinos, banking and cybersecurity.
In real life, chance surrounds us everywhere. For example, when a dice is rolled, it is believed that the probability of rolling any number is one-sixth. This result is considered random because we don't have enough information to accurately predict the outcome of the roll. However, if we knew all the details—how the hand moves, what forces act on the cube—it would be possible to predict the outcome theoretically. In practice, this is impossible, and that is why such events are perceived as accidental. Mathematicians call such processes pseudorandom: although they appear to be random, detailed knowledge of all the factors makes it clear that they are not.
Similarly, if you ask Google for a random number, it will be generated using a predictable algorithm. This process is not truly random, but since the user does not have access to the details of the generation, the number appears to be random. Such pseudo-random numbers have a wide range of applications, ranging from simulations to sophisticated security systems used in financial and government organizations.
However, the question remains: how do you determine whether a set of numbers or points in space was really created randomly, or whether it is the result of a deterministic process? To do this, mathematicians have developed several randomness tests, one of which is the uniform distribution test. If the points are evenly distributed over the entire area, then the sequence can be considered random. However, this is only a superficial way of analysis. This begs the question: Can there be deterministic sequences that pass this test? If such sequences exist, they are called pseudorandom.
Pseudorandomness is important not only for number theory, but also for practical applications. For example, banks and financial institutions use pseudo-random numbers to ensure the security of transactions. Hackers and other bad actors are constantly trying to unravel the algorithms behind the generation of these numbers, which is driving the development of increasingly sophisticated methods for generating them. For example, Cloudflare uses a process based on video recordings of the movements of lava lamps to generate pseudorandom numbers — the system is called LavaRand.
Returning to the mathematical side of the issue, when analyzing random sequences, it is important to consider not only the uniformity of the distribution of points, but also other parameters, such as the intervals between points and the correlation between them. For example, the gap distribution method evaluates the size of the gaps between points, and the pair correlation method analyzes how likely points are to group or remain disconnected. If the results of these tests match those expected for random data, then such a sequence is considered to have Poisson properties (named after the French mathematician Siméon Poisson).
However, proving that a sequence passes all such tests is much more difficult than demonstrating the presence of a certain pattern. This is what mathematicians Niklas Technau and his colleagues have been doing since the pandemic period. In the course of their work, they searched for sequences that would be able to satisfy even the most stringent requirements for pseudorandom numbers. They turned to the old methods of analysis developed by Hermann Weyl in 1916 to test the uniform distribution.
The breakthrough came when scientists realized that sequences with a small value of the θ parameter exhibit pseudorandom properties better than previously assumed. For example, if θ is less than or equal to 1/3, then the sequences pass the Poisson correlation test, which was an important discovery. This made it possible to find a whole family of sequences that could be proven to meet these strict criteria. The scientists continued their research and were able to show that at even lower values of the θ parameter, these sequences pass an even more rigorous test for so-called triple correlations.
These results not only shed light on the operation of pseudorandom numbers, but also open up new possibilities for their application. For example, sequences that grow slowly, such as sequences like xn = {α(log n)θ}, show all the hallmarks of Poisson spacing and pair correlation, making them extremely useful for use in cybersecurity and cryptography. Moreover, these sequences are so "random" that even with an infinite number of tests, it is impossible to prove that they are not random.
Thus, mathematicians have taken an important step forward in understanding pseudorandom numbers. Despite the complexities in the evidence, the new methods and techniques used in this study could form the basis for further discoveries in this area. In the future, this may lead to even more powerful tools for data analysis, which will find application not only in theoretical mathematics, but also in real-world applications, from security to modeling complex systems.
Source
Recently, a team of mathematicians presented a new proof that sheds light on the question of whether certain sequences of numbers are random or pseudorandom. Pseudorandom numbers look the same as random numbers, but can actually be obtained using deterministic processes. Research in this area is of great importance for many areas, including modeling, lotteries, casinos, banking and cybersecurity.
In real life, chance surrounds us everywhere. For example, when a dice is rolled, it is believed that the probability of rolling any number is one-sixth. This result is considered random because we don't have enough information to accurately predict the outcome of the roll. However, if we knew all the details—how the hand moves, what forces act on the cube—it would be possible to predict the outcome theoretically. In practice, this is impossible, and that is why such events are perceived as accidental. Mathematicians call such processes pseudorandom: although they appear to be random, detailed knowledge of all the factors makes it clear that they are not.
Similarly, if you ask Google for a random number, it will be generated using a predictable algorithm. This process is not truly random, but since the user does not have access to the details of the generation, the number appears to be random. Such pseudo-random numbers have a wide range of applications, ranging from simulations to sophisticated security systems used in financial and government organizations.
However, the question remains: how do you determine whether a set of numbers or points in space was really created randomly, or whether it is the result of a deterministic process? To do this, mathematicians have developed several randomness tests, one of which is the uniform distribution test. If the points are evenly distributed over the entire area, then the sequence can be considered random. However, this is only a superficial way of analysis. This begs the question: Can there be deterministic sequences that pass this test? If such sequences exist, they are called pseudorandom.
Pseudorandomness is important not only for number theory, but also for practical applications. For example, banks and financial institutions use pseudo-random numbers to ensure the security of transactions. Hackers and other bad actors are constantly trying to unravel the algorithms behind the generation of these numbers, which is driving the development of increasingly sophisticated methods for generating them. For example, Cloudflare uses a process based on video recordings of the movements of lava lamps to generate pseudorandom numbers — the system is called LavaRand.
Returning to the mathematical side of the issue, when analyzing random sequences, it is important to consider not only the uniformity of the distribution of points, but also other parameters, such as the intervals between points and the correlation between them. For example, the gap distribution method evaluates the size of the gaps between points, and the pair correlation method analyzes how likely points are to group or remain disconnected. If the results of these tests match those expected for random data, then such a sequence is considered to have Poisson properties (named after the French mathematician Siméon Poisson).
However, proving that a sequence passes all such tests is much more difficult than demonstrating the presence of a certain pattern. This is what mathematicians Niklas Technau and his colleagues have been doing since the pandemic period. In the course of their work, they searched for sequences that would be able to satisfy even the most stringent requirements for pseudorandom numbers. They turned to the old methods of analysis developed by Hermann Weyl in 1916 to test the uniform distribution.
The breakthrough came when scientists realized that sequences with a small value of the θ parameter exhibit pseudorandom properties better than previously assumed. For example, if θ is less than or equal to 1/3, then the sequences pass the Poisson correlation test, which was an important discovery. This made it possible to find a whole family of sequences that could be proven to meet these strict criteria. The scientists continued their research and were able to show that at even lower values of the θ parameter, these sequences pass an even more rigorous test for so-called triple correlations.
These results not only shed light on the operation of pseudorandom numbers, but also open up new possibilities for their application. For example, sequences that grow slowly, such as sequences like xn = {α(log n)θ}, show all the hallmarks of Poisson spacing and pair correlation, making them extremely useful for use in cybersecurity and cryptography. Moreover, these sequences are so "random" that even with an infinite number of tests, it is impossible to prove that they are not random.
Thus, mathematicians have taken an important step forward in understanding pseudorandom numbers. Despite the complexities in the evidence, the new methods and techniques used in this study could form the basis for further discoveries in this area. In the future, this may lead to even more powerful tools for data analysis, which will find application not only in theoretical mathematics, but also in real-world applications, from security to modeling complex systems.
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