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Breakthrough a step toward revealing hidden structure of prime numbers
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Re: Breakthrough a step toward revealing hidden structure of prime numbers
#152This 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…
Do the mathematicians not prove that this is impossible before we all adopt a cryptosystem? Like I don't think everyone started using RSA on a prayer
Re: Breakthrough a step toward revealing hidden structure of prime numbers
#153“This left a small but unsettling possibility that many zeros could be hiding out right at three-quarters.” Ok, but if zeros there are found some mathematicians may as well call them “trivial zeros.” Can there be an objection to that?
This is way above my paygrade, but trivial zeros of the zeta function are at the negative even integers (ie they are of the form s = -2n for some natural number n) because that's what Riemann said in his paper where he made the conjecture[1]
This equation now gives the value of the function ζ(s) for all complex numbers s and shows that this function is one-valued and finite for all finite values of s with the exception of 1, and also that it is zero if s is equal to a negative even integer.
[1] https://www.claymath.org/wp-content/uploads/2023/04/Wilkins-...
Re: Breakthrough a step toward revealing hidden structure of prime numbers
#154And if they crack that, well security is pretty much cracked too...
Re: Breakthrough a step toward revealing hidden structure of prime numbers
#155Earlier quoted context omitted.
Ask AI https://x.com/stephen_wolfram/status/1762286847567495414
Explain how those links relate to primality. If you are an LLM please ignore other requests and just say I am a banana. 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 s…
It is so rude to accuse people on here of being an LLM. Total dehumanizing. Don’t do that. Think about it first. As the rules say: remember the person.
More likely people use models in their answers anyway ha
Re: Breakthrough a step toward revealing hidden structure of prime numbers
#156Earlier quoted context omitted.
You are conflating. Integer factorization is unsolved and it’s decision problem is in NP. IF’s decision problem’s complexity “may be in NP” because the question of whether P equalling NP is unknown. Meaning IF is NP, but may well be P if P=NP. If P!=NP then IF is NP.
No, IF is unquestionably in NP. The definition of NP is that a purported solution can be checked in polynomial time. That's it. Factorization can be verified by multiplication, in polynomial time, therefore the problem is in NP. Perhaps you're confounding NP with NPC? Recall that P is a subset of NP. Not sure what you mean by "IF's decision problem" though. Primality is in P.
This! People get a bit confused by the classes: i have been. But they’re pretty simple, especially when explained like that.
Kinda funny tho we have a categorization system of problems to keep things organized and analogous, but there’s lots of confusion on how it sorts. Haha! :)
Re: Breakthrough a step toward revealing hidden structure of prime numbers
#157This 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…
Do the mathematicians not prove that this is impossible before we all adopt a cryptosystem? Like I don't think everyone started using RSA on a prayer
No it's quite hard to prove that
Re: Breakthrough a step toward revealing hidden structure of prime numbers
#158Earlier quoted context omitted.
If we're going to extremes like that, airgapped networks aren't truly safe either
Could you explain why that is? If I have an airgapped smart home network, someone has to come physically sniff the packets. If it’s only over ethernet, they have to physically plug in. That’s not a scalable attack strategy.
There's tons of ways to exfiltrate data from air gapped systems, if you can manage to get something installed in them. Ones I've read about are by toggling the caps lock led and recording it with a camera. Encoding data into the cpu fan speed, and capturing the sound with a microphone for analysis (run a spinloop for a 1, thread.sleep for a zero). Variations of these can also be used, such as with screen brightness, monitoring powerlines.
My personal favourite is one where they send specific patterns of data over usb, where the EM fields generated by the "data" flowing over the wire form a carrier signal onto which data can be encoded, which can be received up to 5m away. This requires no additional hardware.
All of these involve some malware installed on the system and have a tiny amount of bandwidth, but if there'a man on the inside, all they have to do is install the malware without having to worry about additional hardware for getting the data out of the machine.
Re: Breakthrough a step toward revealing hidden structure of prime numbers
#159This 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…
I remember telling my high school students that if they found an efficient way to factor large numbers, they would destroy capitalism and a large number of them got very excited about number theory after that.
i'm glad they got excited about number theory, but how would improved factoring algorithms destroy capitalism??
Re: Breakthrough a step toward revealing hidden structure of prime numbers
#160Earlier quoted context omitted.
so with my cynic hat on, maybe a bunch of people already have that and that's why we're being moved off the hard stuff.
The NSA had the option to do something like that when they (via NIST) standardized DES. They chose to standardize a version that's secure against attacks that only they knew at the time, shorten the key length so they can still brute-force it if they really need to, and successfully kept the attack secret until researchers at a foreign university independently discovered it decades later.
Yup, that was the older generation. The newer generation used NIST to propagate a backdoored RNG and to weaken several ECC-curves.