Bitcoin Forum

If computers can calculate an address from its private key, then why can't they do the reverse and calculate a private key from an address?

Because the "trapdoor" functions used in public key cryptography are designed with exactly that property.
In your question, the first implicit assumption is already wrong. If a function can be calculated, it does not necessarily follow that its inverse can be calculated with similar effort or can be calculated at all.
For example, the function "abs" returns the absolute value of its argument. Naturally, its inverse cannot be computed, because you don't know whether the original argument was positive or negative.
Other functions have a defined inverse but it's much harder to compute the inverse than computing the original function. Elliptic curve functions as used in bitcoin belong to this class, as well as the operations on large primes that are used in RSA public key crypto (Pretty Good Privacy).
"Much harder" in this case does not mean "requires a faster computer" but "impossible given the natural laws and available ressources of humankind."

Onkel Paul

Not all calculations are reversible. Suppose private keys and address work like this: private keys are long numbers, and addresses are the sum of all digits. Obviously a simplification, but just as an example.

So if you have private key = 134505719542, then you can easily calculate the address: 1+3+4+5+0+5+7+1+9+5+4+2 = 46.

Now I have another address: 43. What's the private key?

Thanks for the response. I like mathematics but never studied too much of it. I would like to understand in more detail both about functions that can't be reversed and the ones which reversion is much harder than the original, where can I read about that?

And if it's possible to create a non-reversible function, then why didn't they do that with bitcoin?

Any number of which the digits' sum is 43?

I don' think your example really explains why a btc private key (actually, as far as I know each address has something like 2^100 private keys), can't be calculated from the address.

His example does explain a lot to be honest , I personally got it ;)
Thanks for the information Kazimir , really helpful

~ Madness

In his example, it is extremely easy to calculate not just one, but many and all the possible private keys of an address, with nearly the same difficulty of the reverse, it surely doesn't explain bitcoin at all.

The examples given here are wrong and misleading.

The problem is not that it is not possible to compute the private key, because there is more than one solution (like in the case of abs() or the sum of numbers). The problem is that it is possible, in theory, to compute it, but the computation is so hard that it is practically infeasible.

Do not confuse asymmetrical cryptography with hashes. A hash is impossible to reverse, because many different inputs can result in the same output. (It's just that there is no easy way of finding all of them - or even any one of them.) A private key can be computed from the public one - but it is very, very hard - hard enough to be practically impossible.

For instance, if I ask you what is the product of 31 and 37, the answer is easy - 1147. But if I ask you what are the prime factors of 1147, that's a hard question. The easiest way to answer it is by trial and error - you try to divide 1147 by every prime number smaller than the integer part of its square root until you find a number that divides it exactly (the first such number is 31). When the numbers involved are very large it becomes practically impossible to answer such questions. (With large numbers there are faster methods than trial division, but they are still too slow to be practical.)

The first few explanations gave me a conceptual idea, this explained everything.  :)

http://arstechnica.com/security/2013/10/a-relatively-easy-to-understand-primer-on-elliptic-curve-cryptography/1/

I hear the young ones like video, so Ill add this: http://media.ccc.de/browse/congress/2014/31c3_-_6369_-_en_-_saal_1_-_201412272145_-_ecchacks_-_djb_-_tanja_lange.html#video

Nope, that does explain Bitcoin. Each address represents around 2^96 possible private keys. It is believed to be computationally infeasible to generate a collision.

Basically they did.  (although no one has been able to mathematically PROVE the
existence of one way functions). 

Hashing is a non reversible function.   Think of like this:  You can turn a cow into
hamburgers, but you cannot turn hamburgers into a cow.

If you watch a video on a pen and paper SHA256 hash, it might become more clear. It's pretty neat.

Stupid people should just be happy with the fact that we let them remain here with us on Earth and they get to benefit from all the smart people who make cool shit.  BitNerd, Don't try to figure out math.  You are annoying the numbers.

I tried to watch the video, and I gotta say, it was extremely IRRITATING to have the math-intensive lecture slides dissapear at the WORST possible timing to only cut to the lecturer in person. What. The. Fuck.

Whoever edited the video needs to be stuffed into a cannon and shot. What is supposed to be an informative learning video is ruined by moronic decision making that killed my concentration.

Trapdoor functions are invertible; the question is how hard they are to invert. There's a long history of crypto functions which were thought to be too hard to solve, but were solved. Sometimes by advances in mathematics, sometimes by sheer computational power, and sometimes by a combination of the two. The ones used for Bitcoin are generally considered strongly resistant to solution, and they're functions which have been examined thoroughly. But SHA-1 and DES were once thought to be highly resistant, too. Now they're routinely solved.

::)
It was a simple example just to understand how difficult it would be determine the key from an address, hence the emphasis on 'suppose', he never implied that it was the concept the bitcoin address generation is based on.

Its a live capture from the conference, the editing was done on the fly and frankly: yes it sucks for many videos.

But all of the 2^96 possible private keys of an address work, don't they? Or only one works?

I don't need to "let you" live in the same planet as I live. You are born with the natural right of private property, no need to worry. And numbers are not alive, don't worry about them either.

Thanks, it's very interesting. Now I'm curious about how are public and private btc keys generated. Do you know a site explaining that?

I understand this.

It's a good example.There are too many combinations

Yes, because a bitcoin address is a 160 bit hash, and since a private key (and its corresponding public key) are 256 bits, there are 2^96 (96 = 256 - 160) public keys which should have the same 160 bit hash.

That has nothing to do with the private key to public key operation.  It has to do with the fact inputting 256 bits into a hash and getting out 160 bits.  The number of possible inputs is higher than the number of possible outputs, so therefore some inputs MUST have the same output as others.

And so yes, any of those 2^96 private keys will work, because the mathematical requirement for sending a bitcoin is only that the signature is valid and that the hash matches, so any key that can produce the hash will satisfy that requirement.

Problem is you have to brute force through 160 bits before you are guaranteed to start getting those collisions...  (And "bits" means it's exponential.  160 bits isn't twice as hard to brute force as 80 bits, it's 2^80 times as hard.)

Also, SHA-256 is used to hash the public key. SHA-256 uses two logical functions.  1. the majority function and 2. the choosing function.

Majority function works how it sounds.  Given 3 bits, it selects the majority.  So;

MAJ(0,0,0) = 0
MAJ(0,0,1) = 0
MAJ(0,1,0) = 0
MAJ(1,0,0) = 0

Thus 4 combinations of inputs yield one and only one output. 

The choosing function works in a similar way.  The first input chooses the second.  If first is positive, choose the second, else choose the third.

CH(0,0,0) = 0
CH(1,0,0) = 0
CH(0,1,0) = 0
CH(1,1,0) = 0

And so on.

Private keys are "generated" as a randomly chosen number between:
0x01 and 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364140

(1 and 115,792,089,237,316,195,423,570,985,008,687,907,852,837,564,279,074,904,382,605,163,141,518,161,494,336)

The programer of a wallet (or address generating software) chooses how they want to generate the random number.  If you like, you can even do it with dice or a deck of cards.

Once you've chosen a random private key, the public key is calculated as described at that arstechnica link above using the secp256k1 curve.

Thank you. Now I wonder how do you convert the big number to the smaller one with upper and lower case letters?

https://en.bitcoin.it/wiki/Technical_background_of_Bitcoin_addresses

(Have you never heard of internet search engines? Most of these answers come up with about 30 seconds of effort on Google, Yahoo, Bing, etc.)

Because you don't have a quantum computer at your disposal, and you never will.

Thanks. So, a private key is nothing more than a random number between 1 and 115,792,089,237,316,195,423,570,985,008,687,907,852,837,564,279,074,904,382,605,163,141,518,161,494,336 converted to base58?

Very interesting. Ok, I will try to google my next questions and will ask here in case I don't find the answer.

Not exactly.

A Bitcoin private key is nothing more than a random number between 1 and 115,792,089,237,316,195,423,570,985,008,687,907,852,837,564,279,074,904,382,605,163,141,518,161,494,336.  You are welcome to convert it into any base you like.  It's just a number.

The Wallet Import Format (WIF) of a Bitcoin private key is a specific representation of that private key that concatenates: a one byte version number, the private key, and a 4 byte checksum.  This value is then converted into base58 using the rules described here:

https://en.bitcoin.it/wiki/Base58Check_encoding

I'd like to thank everyone, especially Kazimir, for laying this all out so clearly. 

I feel a lot better about sites like directory.io and www.Buttcoins.com being out there now.  Even though directory.io was said to be a hoax, the news story about it was focused more on how many address and key pairs there were rather than you can't get the private key from the public one.  Thanks.

Op check out: https://www.khanacademy.org/computing/computer-science/cryptography

Takes less than few hours at most, and its free.

Not every process is reversible.