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Breakthrough a step toward revealing hidden structure of prime numbers

science.org

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Re: Breakthrough a step toward revealing hidden structure of prime numbers

#31

Post by jiggawatts »

JackSlateur wrote: post #6
EMIRELADERO wrote: post #2

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…

The world has moved away from RSA etc to elliptic curves. Not everybody did, through. RSA is no longer the standard, and has not been for many years.

You can’t imagine how many places I’ve seen “Elliptic curve certificates are not supported” as recently as this year.

It’s the IPv6 of the security world.

Re: Breakthrough a step toward revealing hidden structure of prime numbers

#33

Post by seanhunter »

codeduck wrote: post #32

[flagged]

    That this subject [imaginary numbers] has hitherto been surrounded by mysterious obscurity, is to be attributed largely to an ill adapted notation. If, for example, +1, -1, and the square root of -1 had been called direct, inverse and lateral units, instead of positive, negative and imaginary (or even impossible), such an obscurity would have been out of the question.
- Gauss

Re: Breakthrough a step toward revealing hidden structure of prime numbers

#35

Post by keepamovin »

EMIRELADERO wrote: post #2

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…

I guess one perspective is finding fast factoring is considered super rare. The story is so many smart people have looked at it, it's probably impossible...for now. But that story may be its own weakness.

Anyway, the risk, while just as real as something like the grid being downed by a massive solar storm with multiyear recovery period from stone age due to transformer manufacturing delays and no stockpiles, just seems too minuscule/theoretical to spend much time on - from that point of view.

Regarding any plan, I don't know if it's so easy to just switch to ECC, because actual asymmetric encryption with ECC depends on shared secret, which (if you're assuming an unsecured exchange channel due to RSA being broken), is more vulnerable to MITM than RSA. I don't think it's an easy swap out.

All that aside, another point of view is RSA is probably already broken, the break is just secret to the codebreaking agencies. It would be very desirable for them to keep their breakthrough secret. That might even involve trying to find ways to suppress any "sudden number theory breakthroughs" hahaha! :)

Re: Breakthrough a step toward revealing hidden structure of prime numbers

#37

Post by odyssey7 »

lmpdev wrote: post #26
keepamovin wrote: post #25

People always think the structure of primes is complex, but it's not really, it's just a recursive structure of the magnitude gaps not landed on by multiples of previous gaps. It doesn't make it easier to "predict" without tracking all prior gaps, but it's not essentially a complex structure. Kind of funny that like such a simple structure is so elusive. Sorta like how the 3n + 1 sequence gives rise to such complexit…

Ah yes nothing simpler than providing the foundational theory to one of the most rigorous and intellectually intimidating areas of mathematics - number theory /s

They’ve got the fundamental theorem of arithmetic. What more could they want?

Re: Breakthrough a step toward revealing hidden structure of prime numbers

#38

Post by remus »

testaccount135 wrote: post #28

"they pulled some unorthodox moves to finally break Ingham’s bound" Why is taking methods from other fields an unorthodox move? I come from an engineering background an there it is the common case. The usage of harmonic analysis is a staple in many fields (audio, waves, electrical analysis, statistics) and of course the algorithms are pure math under the hood. If I want to find a reaccuring structure in an underlying…

Unorthodox is maybe a bit strong, but what they're saying is that it's a novel application of an existing technique from another field. Fairly often you will see big breakthroughs like this in maths, where someone has the insight to see that there are parallels between two seemingly unconnected areas of maths, and you can use ideas from one area to give you insight in to the other area.

The tricky bit is that these connections between areas are not usually obvious, so to see the similarity can require a considerable step in understanding.

Re: Breakthrough a step toward revealing hidden structure of prime numbers

#39

Post by keepamovin »

lmpdev wrote: post #26
keepamovin wrote: post #25

People always think the structure of primes is complex, but it's not really, it's just a recursive structure of the magnitude gaps not landed on by multiples of previous gaps. It doesn't make it easier to "predict" without tracking all prior gaps, but it's not essentially a complex structure. Kind of funny that like such a simple structure is so elusive. Sorta like how the 3n + 1 sequence gives rise to such complexit…

Ah yes nothing simpler than providing the foundational theory to one of the most rigorous and intellectually intimidating areas of mathematics - number theory /s

Nah, it's not that complex. Did you ever take introductory number theory? Anyway, I think you missed the point that the description is really simple, and overlooked. Hahaha! :)

Re: Breakthrough a step toward revealing hidden structure of prime numbers

#40

Post by keepamovin »

odyssey7 wrote: post #37
lmpdev wrote: post #26

Earlier quoted context omitted.

Ah yes nothing simpler than providing the foundational theory to one of the most rigorous and intellectually intimidating areas of mathematics - number theory /s

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".
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