These Numbers Look Random but Aren’t, Mathematicians Prove

A new mathematical proof helps show whether a sequence of numbers is “pseudorandom”

published : 24 March 2024

In the real world, probability is a tough thing to characterize. If I roll a die, what does it mean to say that it has a one-sixth chance of coming up 5? We say that the outcome is random because we lack the information needed to predict which side will land facing up. But it’s not truly random. If we know all the details about how I move my hand and what forces act on the die as I toss it, we might be able to predict the outcome of the roll. Practically speaking, however, we usually lack that prediction power. Mathematicians call this situation “pseudorandom”: although it looks and seems random to us, we know that, in truth, if we were to have all the information we wanted, the die roll would not be random. Similarly, if I ask Google for a randomly generated number, it needs a process to spit one out, yet Google doesn’t have access to purely theoretical random mathematical models. So it uses a nonrandom process that results in a number that looks randomly generated to us because we don’t know what that process was—or at least we don’t know what some of the inputs were. The number produced by Google is also pseudorandom. Pseudorandom numbers pop up in all sorts of areas: modeling, casinos and lotteries need pseudorandom numbers as inputs. Moreover, banks and financial agencies use such numbers for security and fraud detection. Because hackers have a lot of interest in cracking these pseudorandom codes, people have developed sophisticated ways of generating these numbers. For example, many “random” number generators rely on a physical process. The cybersecurity and Internet services company Cloudflare has a system called LavaRand, for instance, that uses lava lamps to produce pseudorandom numbers. Now suppose we have a collection of numbers or, more generally, points in space. How could you tell if the set of points came from a truly random process or if the set was produced by a deterministic process? This question is central to numerous fields of mathematics, and mathematicians have a variety of tests to detect randomness. We wonder, however, “Just how good are these tests? Are there sets of points that aren’t randomly generated but can satisfy the tests?” If so, we call them pseudorandom. Pseudorandom properties and searches for them have all sorts of applications, from understanding prime numbers to something called quantum chaos (my pick for the area of mathematical physics that would make the best band name). During the pandemic, I began working on a problem related to pseudorandomness: the search for random behavior where there is no randomness. The work started with an e-mail I sent to Niclas Technau, a researcher who was at Tel Aviv University at the time. By mid-2022, nearly three years later, despite never having met face-to-face, the two of us had written three papers together and had found some of the only examples of sequences that we could prove passed extremely strong pseudorandom tests. Detecting Randomness How can we detect randomness? Perhaps the crudest way is something called uniformdistribution. Consider two boxes with dots that are apparently sprinkled within them—one where the dots fill the whole square and another where half of the square is empty.