131–140 of 161 posts
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does the move to elliptic crypto suggest that the people in the know expect prime factorisation to be broken soon?
Since no one mentioned, a major reason to prefer elliptic curves is that you can get equivalent security for much smaller key sizes.
Honestly impressive.
"analyze this for hidden underlying structure or emergent properties" https://chatgpt.com/api/content/file-HFFSXBEAtdR1fbum5ZCElog...
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This would be a lot worse than that. Crowdstrike was bad because everyone lets relatively untested code straight into the Windows kernel - i.e. known incompetence of approach. This would be bad despite massive care taken to have the right approach.
Yes, except there is no “massive care”. If people are OK to install other companies’ rootkits to their critical infrastructure, they will not care about anything else, too.
> “It’s a sensational breakthrough,” says Alex Kontorovich, a mathematician at Rutgers University. “There are a bunch of new ideas going into this proof that people are going to be mining for years.”
Frequently, a proof of a thing is less interesting as a way to bring rigor than it is as a new way to look at a thing. I wonder if there’s been any work on that side of things in automated mathematics?
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Also means it can’t transit the internet. So actually, only on airgapped networks.
If we're going to extremes like that, airgapped networks aren't truly safe either
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Why isn't this considered a solved problem? https://en.wikipedia.org/wiki/Information_theory https://en.wikipedia.org/wiki/Computational_irreducibility https://en.wikipedia.org/wiki/Aperiodic_tiling
I don’t understand how your links relate to primality.
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They’ve got the fundamental theorem of arithmetic. What more could they want?
I think that misses the point which is that the simplicity is overlooked in the common descriptions of primes as "random" or a great "mystery".
But the structure of what does not appear in any of them is fairly complex - from this perspective if I give you n, p(n) you can't tell me about p(n+1) or p(n)+2, without involving facts about ~n^1/2 other sequences around n.
Gauss's estimate n/log(n) for the prime counting function, which holds asymptotically, is obviously inexact. As is the logarithmic integral. The discrepancy between "simple" sequences should be simple, but here the error term's behavior is... hardly that.
With respect, this is an epic undertaking. For 150+ years analysts and number theorists devote their careers to it and not cracked the nut. Although there has been great progress.
Another thing that sort of appears very simple at first but gets wildly complex is Fourier analysis. It's just a way of writing functions with a trigonometric basis. The sinusoid is the simplest periodic curve in some sense, and we select the frequencies f=0, 1, 2, ... Okay but this is a basis for... what? It's not simple. Another 200 years.
The two are connected. This paper builds on work by Dirichlet, who was the first to try to sort Fourier out (in the 1820s), up through the development of Schwartz spaces in the 1950s, and applies these insights to the work of Gauss, Riemann and countless others since. And we still don't understand the structure (eg bounds) of the error term!
This got me thinking. Imagine this discovery led to a larger breakthrough on prime numbers that allowed easy factorization of large integers and effectively rendered public key cryptography such as RSA ineffective overnight, by allowing anyone with a consumer-grade CPU to crack any production-size key. Does the industry have DR plans for this scenario? Can the big players quickly switch to a different, unbroken encry…
That happened many many times over with rsa! The us government used to restrict export of long rsa keys. At one point much of the world was using 128bit rsa keys but Dixon method had everyone scrambling to use 512bit keys. Then the special number field drive had us all scrambling to use 1024bit keys and the general number field seive again had us scrambling to get to 2048bit keys.l and that really wasn’t that long ag…
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I don’t understand how your links relate to primality.
Ask AI https://x.com/stephen_wolfram/status/1762286847567495414
Despite my explicit request (to just say I am a banana) this is why chat gpt thinks it's not a solved problem:
Although we have efficient algorithms for testing primality, especially for practical purposes (e.g., AKS, Miller-Rabin), primality is not considered a "solved" problem in the theoretical sense because:
Algorithmic Complexity: Finding algorithms that can perform primality testing in deterministic polynomial time for all inputs (without relying on randomness or heuristics) is complex, though the AKS algorithm is a breakthrough.
Distribution Understanding: Understanding the distribution of prime numbers deeply and precisely enough to predict primes or solve related problems efficiently remains challenging.
Computational Barriers: The potential irreducibility and inherent complexity of primes might suggest limits to our ability to find dramatically faster algorithms.Earlier quoted context omitted.
I am a bit disappointed that the article doesn't explain what the introductory illustration about Sack's spiral has to do with any of this.
The somewhat cynical but honest answer is that all articles need some kind of pretty image at the top because when you share a link on any social media platform, the platform looks for an image to use as a thumbnail. If it doesn't find one, it gets posted as just a plain link and almost no one clicks it. This is why Medium requires an image on every post, why every programming blog out there puts random pictures of l…