How secure is 256 bit security?

How hard is it to find a 256-bit hash just by guessing and checking?
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Several people have commented about how 2^256 would be the maximum number of attempts, not the average. This depends on the thing being attempted. If it's guessing a private key, you are correct, but for something like guessing which input to a hash function gives the desired output (as in bitcoin mining, for example), which is the kind of thing I had in mind here, 2^256 would indeed be the average number of attempts needed, at least for a true cryptographic hash function. Think of rolling a die until you get a 6, how many rolls do you need to make, on average?

Music by Vince Rubinetti:

Thanks to these viewers for their contributions to translations
Dutch: @bvdeijzen
Hebrew: Omer Tuchfeld
Italian: retr00h

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3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe, and click the bell to receive notifications (if you're into that).

Various social media stuffs:

[TaranVH]

the funny thing is, even if you had 4 billion galaxies of computers working on that problem, even if someone found the solution, because light has a speed limit, and the universe is expanding, you wouldn't even be able to communicate the solution to everyone.

[decycle]

The funny thing is, even though 256 bit is really secure, a number of people are just dumb enough to just tell attackers their password.

[ExplosiveLizard]

* 507 Billion years later

"I'm in"

[charikakashyap]

Imagine a lucky bastard getting it right in the first guess.

[nyx]

Turns out it's preeeettty secure...

[vieuetcon]

Theres always some kind of simplicity behind complexity. In computers, sometimes, the "security" looks like a big iron door, until you realize you can pass beside that door, coz theres no walls beside that door..

[hpetty]

Everyone of this man's videos is like a weird mixture of extremely informative yet peaceful and therapeutic LOL. Another amazing video!

[VRC-tk]

In reality:
hashed "qwerty" and "password"
Boom!
Hit!

[Superphilipp]

"next, try to imagine 4 billion copies of the milky way" ... okay, I'm out.

[FourthDerivative]

"...so you're telling me there's a chance!"

[SSM_]

Bonus fun fact: If you actually take 4, 000, 000, 000 to the 8th power, what you'd get is closer to 2^255 than 2^256 (specifically, about 2^(255.18), or 1.13 * 2^255). Approximating 2^32 (4, 294, 967, 296) as just "4 billion" ends up losing more overall value here than you might expect.

[TunaBear]

4:33 Now the channel surpassed 2^21 subscribers
And close to reach 2^22 subscribers
Congratulations
I always loved your videos, the quality of the explanation of the topics is simply unmatched, I swear I learned more from this and other educational channels than from school/college on the last 4 years
Keep going, the world need more channels like this.

[bluebrown]

Edit: One thing I wish I had said explicitly is that even though a perfect and idealized cryptographic hash function would behave like a random function, in reality, there is some element of predictability to functions like those in the SHA-2 family. So even though SHA256 has a 256-bit output, it's actual level of security is lower than 256 bits.

[quaternaryyy]

Worth mentioning: Some cryptocurrencies (I think Litecoin does this) use a different hash function called scrypt instead of SHA-256. scrypt is designed to be impossible to create application-specific integrated circuits for, because it requires large amounts of RAM and computing power, unlike something like SHA-256 which is essentially a bunch of logic gates.

Neat!

[awesomeSquirel]

For all those confused about why you need 2^256 guesses and not 2^255. Here is an answer from 3Blue1Brown from a reply:

To the point about it really requiring half as many guesses, (i.e. 2^255 not 2^256): This is true for something like hacking a digital signature, where you are methodically going through all possibilities. But if a cryptographic hash function truly behaves like a random function, guessing and checking a nonce with a hash will not look like going through all possible hashes one-by-one, it's more like rolling a die over and over until you hit a 6, in which the expected number of rolls needed is 6, not 3. While running this GigaGalactic supercomputer, many of the guesses will actually collide, so it is not a methodical search through all possible hashes.

However, as you point out, the actual security on SHA256 is indeed lower than 256-bits. It turns out not to quite behave like a nice random function. But this discussion just centered on an idealized cryptographic hash function.

[amconners]

"Next, try to imagine four billion copies of the Milky Way."

No. My brain will break if I try to imagine that.

[itschilled]

Small note: it would likely not take you 2^256 guesses to get it correct, as that is every single possible combination of guesses. Rather, it would take an average of 2^255, which is the halfway point between 0 and 2^256 (as any power of 2 is twice as many as the previous power). Minor detail but helps with the general understanding.

[BossManTee]

But can all these computers combined able to run crysis?

[ucihyyu]

3 blue 1 brown always comes In clutch with the visualizations. This reminds of combinatorial explosion, and how often it comes up in real world problems, it would be interesting to see you make a video going into depth on this topic.

[Shrooblord]

It's so fun to hear Grant say "wow, we recently passed 262k subscribers" and I look over to the sub count today and there's 6.21 MILLION of us. <3