> This Agreement does not have an end date and may only be amended by mutual consent in accordance with Article XIII of the 1951 Defense Agreement.
> In case Greenland exercises its right to self-determination to become independent, the Government of the Kingdom of Denmark and the Government of Greenland shall together ensure that the independent Greenlandic state will (i) agree to remain in NATO, including by applying to become a member if necessary, and (ii) affirmatively assume, as from the date of independence, all of the rights and obligations of the Kingdom of Denmark specified in this Agreement, including any implementation agreements or arrangements between the Parties, as well as the Defense Arrangements.
With this, Denmark has effectively ceded sovereignty over Greenland. In time, they are likely to lose it entirely.
The weakening Empire is squeezing its satrapies for all they are worth. But don’t expect genuine resistance and elite resentment any time soon, there’s a century of conditioning and control to undo. Hard power, soft power, covert power.
At this point, it does not appear that (reasonably strong and modern) symmetric cryptography (ciphers, hash functions, etc.) is realistically threatened by quantum computing: https://words.filippo.io/128-bits/
Thanks for the great link! This has been vexing me, as intuitively it seems like if quantum computers can test all factors they should be able to test all keys.
But the article helps. Basically Grover’s is not as potent as Shorr’s. And it seems like everyone is convinced there is no dramatically better quantum algorithm than Grover’s?
No. Not quite.
In fact, that blog post ignores something important from the very papers it cites.
Grover's assumes the function is a black box that you cannot look inside and that your only way of finding a certain result is through repeated invocation.
Under this assumption, Grover's is optimal in the number of invocations of the function required to find the result.
However, this assumption may be quite wrong for AES and friends. It may be the structure allows for non brute force attacks that are totally impractical classically but not subject to Grover's optimality limitation quantumly.
The only thing you are guaranteed here is that if you cannot take advantage of structure at all then Grover's is the best you can do.
Given that we have pretty much always found a way to take some advantage of structure, I would bet we will do so here.
That may or may not make it viable to break at all, I just wouldn't bet that it must be treated like a black box forever.
Yes. Structure comes in many forms.
In this case, Shor is turning the function into a strongly periodic algebraic structure.
I'll explain it without going too far into why any of this is true, which is much more complicated to prove. This will let me use relatively simple math.
Let's say you want to factor N. Pick some number that is coprime to N, which we'll call a, and consider f(x) = a^x (mod N).
Since it's a modular function, it repeats at some point. Shor calculates the period of this function (r), rather than seeing which of the 2^n numbers is "the answer".
Once you know the period of this function, there is a high chance that the factors fall out of gcd(a^(r/2) - 1, N) and gcd(a^(r/2)+1, N).
The point here is not to explain Shor's as much as to point out it is finding a strong amount of structure to take advantage of, quantumly.
This is actually the same way the oracle separation of BQP and the entire polynomial hiearchy works[1] - It depends on forrelation, which is a problem where quantum computers can extract a global property of the function without needing to learn all the individual values, by taking advantage of structure.
Which is why i go to "The idea that there is literally no structure that can be taken advantage of in AES strikes me as a bad bet".
There are already reduced round quantum attacks on AES as well. Again, more to the point, the idea that symmetric key ciphers and cryptographic hashes in general are safe because grover's is slower than shor's is not a thing i would bet on at all. Even if AES ends up relatively safe, that tells you basically nothing about the other practically-used ciphers and functions since there are a lot of different construction mechanisms being used.
[1] People still seem to believe there are no functions which quantum computing models have been been proven to be faster at than classical computing models. This is false. Forrelation is the canonical example - and shows that BQP can perform things exponentially faster than you can classically even given access to an infinite polynomial hierarchy.
It is the current physical actualization of these computing models that have the "is it really faster than classical computers" issue, not the theory ;)
(IE it is a perfect example of "in theory there is no difference between theory and practice, and in practice, there is")
Thanks so much for the patient and accessible explanation. So much more useful and grounded than the naive “quantum is faster because it can consider all possible solutions at once” canard.
Especially when the whole point of such lies is to kill the story before it gets much public interest. A device the sunny state of Israel uses all the time and to great effect.
If only the Taliban had received half a trillion of military and financial support. If only China had shipped them thousands of FPV drones per day and given them access to low-latency satcom and a trillion-dollar ISR ecosystem.
> how do you win them over
Putin would have loved to. But, finally, even he has realized that that brotherly ship has sailed for a while. Luckily for him there’s an alternative: he just has to make sure Ukraine doesn’t come back from this in any threatening fashion any time soon. An effort that’s well underway and intensifying.
His other option would be what the West’s darling is doing in the Middle East, genocide. But Putin is not a monster.
As things stand, without fresh funds, Ukraine can keep the war going for about another year. I wouldn’t expect Zelensky to concede this round one day too soon.
The problematic thing here is that there remains an abundance of cowardly and delusional elites in positions of influence and power. 4.5 years in. Willing to sell out their countrymen for a mirage of normalcy that will not come. Not to the 99.9%, anyway. We wouldn‘t be here if Putin was a fraction of the despot he’s presented to be in the West.
The firing of Fedorov should tell you something about how well this latest gambit/escalation is working. And about the chasm between PR and reality. Syrskyi knew where this is going. And he knew what the AFU are in dire need of: men, not robots.
> This Agreement does not have an end date and may only be amended by mutual consent in accordance with Article XIII of the 1951 Defense Agreement.
> In case Greenland exercises its right to self-determination to become independent, the Government of the Kingdom of Denmark and the Government of Greenland shall together ensure that the independent Greenlandic state will (i) agree to remain in NATO, including by applying to become a member if necessary, and (ii) affirmatively assume, as from the date of independence, all of the rights and obligations of the Kingdom of Denmark specified in this Agreement, including any implementation agreements or arrangements between the Parties, as well as the Defense Arrangements.
With this, Denmark has effectively ceded sovereignty over Greenland. In time, they are likely to lose it entirely.
The weakening Empire is squeezing its satrapies for all they are worth. But don’t expect genuine resistance and elite resentment any time soon, there’s a century of conditioning and control to undo. Hard power, soft power, covert power.
This isn’t about Trump, he’s a convenient vessel.
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