Hacking at Quantum Speed with Shor's Algorithm | Infinite Series 

but somehow, that's still useful in certain cases

Similar however i came here while going on a tangent while i was studding for my cryptography midterm

That is pretty cool. Kudos to you!

"Oh, Ok. Thank you, Dr. Graham [happy face emoticon]" To be continued...

Came here to say the same thing at 3:39

But it's not a PRIME factor

1 isn't prime

have you yet submitted this new prime? heard you can make a lot of money for each of those

it didnt click for me until she explained about waves constructively interfering or destructively interfering.

it was a parker square of a computer

the only thing that could beat it is if it were turing complete

Ethan Rojek I was about to wrote that too.

lol XD

PlayTheMind That thing is unhackable.

Google would say it's KINDS BLUE , amirite ?

Schrodinger's truth, a.k.a. politics.

God you are sooo indecent

Luke Amendolara my love of math (aside from trig) is continous

and after that, check out KhanAcademy.org

When they toss in a decimal 😶

smash or pass SMASH

In mathematics unity is the multiplicative identity a.k.a. 1. So she's saying the complex numbers that can be raised to a whole number power to become 1. Examples: (-1)² *=* 1² *= 1* (-1/2 + i*sqrt(3)/2)³ *=* (-1/2 - i*sqrt(3)/2)³ *=* (9/8 - 1/8) *= 1* (i)⁴ *=* (-i)⁴ *=* (-1)⁴ *=* 1⁴ *= 1*

this was actually really helpful, thanks!

I never thought of it that way till now. Thank you for opening our minds to a mathematical pun!

I'd love to see a band with that name :)

The quantum states in this algorithm are the periods r, as you are trying to find the period r such that a^r mod N = 1. In the example from the video, 2^3 mod 7 = 1, so r = 3. This period is then used to derive the prime factors q and q of N using the formula in step 4. For a period of 3 (as in that example), a^x mod N = 1 every time x is a multiple of 3 (or the period r in the general case). In the example, a^x mod N = [2, 4, *1*, 2, 4, *1*, 2, 4, *1*, ...] for x = [1, 2, *3*, 4, 5, *6*, 7, 8, *9*, ...]. That is, a^x mod N repeats at a length equal to the period and ends with a value of 1. The arrows in the explanation are complex numbers (or 2D vectors) around a unit circle. For example, the arrows pointing east have coordinates (1,0). Lets call these arrows f. Now, scale all the arrows down by a^x mod N (i.e. f' = f / a^x mod N), then sum the resulting arrows up to some x value (e.g. a), what happens to that sum? Notice that if a^x mod N = 1 the magnitude of the arrow at x does not change, but if it is something different the magnitude gets smaller. Thus, the arrows at the multiples of the period stay the same and the arrows at other positions get smaller, so the arrows relating to the period have more of an effect on the result than those that don't. Next, for each "clock", look at the direction the arrows are facing whenever a^x mod N = 1 (i.e. at period r). The clock with r points has these arrows all pointing east. This means that summing over these values will result in an arrow (number) that gets bigger the more numbers you apply to it. The clocks without r points result in arrows pointing in different directions at intervals of the period r. They also have the property that the arrows will return to the starting point. This means that the sum of these arrows will wonder around 0 and a small value. The key here is that this fourier transform is a function of the period r (the thing we are interested in finding), so can be applied to all the quantum states (which are the different values of r). This then makes the correct value of r more likely to be found when you then read the answer from the quantum computer.

Thanks Gerben for the general intuition, and msclrhd for the details about the amplification. Imo the video didn't make very clear, that we're scaling the unit vectors and adding them. From the video it seemed like we were trying one period at a time, rather than all at the same time and amplifying the right result. Or maybe it's just that the details of the amplification are missing, which, at least for me, made it seem like we're missing or skipping something.

trying answers (if check takes polynomial time) until you find the correct one is the very definition NP.

Scott Brown I think you misunderstand how it works. Shor's algorithm is only probabilistic in the sense that you might not get an even number for r (the period). The algorithm actually deterministically solves for a multiple of the period m*r, from which the period can easily be retrieved. All the randomness associated with quantum mechanics is actually utilized, not a hindrance.

Also, this really doesn't have much to do with P vs NP, besides the fact that this is an NP problem that happens to also be QP. If you could prove factorization is an NP-hard problem, then it would be very connected to P vs NP

As far as I know, Shor's algorithm only has a high probability of returning the correct value for the period, which then is checked classically. This would make the algorithm at least partially probabalistic, which in the worst-case is the same as truly probabalistic right? This would make it functionally, but not truly, deterministic I think.

True, I should probably say amortized O(n^k) time, since we would probably ignore the pathological/absolute worst case times.

Mitch Kovacs Yes please!

Ordinal arithmetic would also be cool since we already talked about them.

:P

Yes, if I recall correctly, there are types of encryption that are currently resistant to quantum computing and some are in use. That said, quantum encryption puts it into another level. There is this thing called quantum encryption. The basic idea is that if you transmit your message in qubits in just the right way, you can create a message that will destroy itself after first try decoding it.

WeepingAngel256 There are dozens of alternative classical crypto-systems for which there are no quantum solutions, and even if there are, if quantum computers take off, then one-way functions will be possible, meaning a provably secure crypto-system could be developed. Mathematicians are always a century ahead of the game.

Well in worst case scenario, public cryptography is abolished. We can still use encryption securely, but you'd have to keep a different password/key for every secure site you visit. The problem gets only worse for the web site, it has to keep a unique key for every user. The biggest problem would be to negotiate these keys over the internet. One way to do it, is to have some trusted authority by both parties.

Yes, search term is post-quantum cryptography.

Ditto. I have to view this several more times.

ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-spUNpyF58BY.html You can watch 3Blue1Brown's video about the Fourier transform for more info about that interference thing.

10:00 starts to look a bit like phasor analysis, right?

Yes. "Shor's algorithm" is actually a pair of algorithms with similar properties. One factors numbers; the other computes discrete logarithms, and can be made to compute discrete logarithms over elliptic curves, thus breaking elliptic curve crypto.

It's not my area but I've read from reliable sources that elliptic curve cryptography is even more susceptible to quantum attacks.

Those are hashing algorithms, not encryption protocols

What's "reality" ?