What is the current computing speed of performing an elliptic curve multiplication? Hi I was just looking for some information on how long it would take to crack a private key in bitcoin using a brute force approach and I couldn't find a very good answer for how long it takes to check whether one specific key (or every key) would work.

So essentially what I am asking is how long would the elliptic curve multiplication process take to check whether a single private key would work for a given public key (on average), thanks :)
The only way to determine the private key associated with a given public key is by a brute force method, spanning the entire 2^256 key space. More specifically, any number from 0x1 to 0xFFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE BAAEDCE6 AF48A03B BFD25E8C D0364140 (order of G) is a valid private key.

According to an article on wikipedia on brute force cryptography, it would take 50 supercomputers that could check 10^18 keys per second about 3x10^51 years to exhaust the 256 bit key space. However, this number fails to account for theoretical limits of energy consumption and computation from Landauer's principle. I decided to put in some work to show the theoretical limits.


  Landauer's Principle


Laundauer's principle basically provides us with the lower limit of energy consumption needed to flip a bit as a function of temperature (it is much more complex, but we can leave that aside for now.) That lower limit is given by the formula: kT * ln(2) (where k is the Boltzmann constant (1.38*10^-23) and T is temperature in Kelvin scale.) So, for a 2^256 key space, that energy is 2^256*kT*ln(2). The energy that I'm calculating below, is only to generate keys in the 2^256 key space, and does not involve the energy to generate public keys from them and check the result. 

Since I'm in Mumbai, I will take the liberty to use temperature of 25 degree celsius or 298 Kelvin. Using the above formula, E = 2^256*kT*ln(2), we get energy as 3.3x10^56. According to this article, the global electricity consumption was 21,700 TWh in 2017. That's equivalent to 7.8*10^19 Joules. So, even assuming that world's entire electricity supply is diverted to generation of the keys in 256-bit key space, we would still need electricity that is 5*10^36 times more than that is generated today.