Difficulty

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Difficulty is a value used to show how hard is it to find a hash that will be lower than target defined by system.

Principles of operation

In Bitcoin network there's global difficulty set for all blocks. For block to be considered legitimate it has to have hash value lower than set target.

Difficulty is changed every 2016 blocks. This is calculated using following formula:

difficulty = difficulty_1_target / current_target

where target is a 256-bit number.

Difficulty_1_target can take various values. Traditionally it's a hash function first 32 bits of which are equal to 0 while all the rest are 1 (it is also called pdiff or pool difficulty). Bitcoin protocol provides target as a type with floating point and limited accuracy. Different Bitcoin clients often determine difficulty based on this data.

Storing difficulty in blocks

Every block contains a packed version (called "Bits") of hexadecimal target.

Using following formula target can be obtained from any block. For example if a target packed in a block appears as 0x1b0404cb its hexadecimal version will look as following:

0x0404cb * 2**(8*(0x1b - 3)) = 0x00000000000404CB000000000000000000000000000000000000000000000000

Maximum value for this field is 0x7fffff while minimum is 0x008000.

Maximum possible target (with difficulty equal to 1) is defined as 0x1d00ffff which appears as following in hexadecimal numeration:

0x00ffff * 2**(8*(0x1d - 3)) = 0x00000000FFFF0000000000000000000000000000000000000000000000000000

Next is an easy way of difficulty calculation. It uses an altered version of Taylor series to logarithm and relies on logs to transform difficulty calculation.

#include <iostream>
#include <cmath>
 
inline float fast_log(float val)
{
   int * const exp_ptr = reinterpret_cast <int *>(&val);
   int x = *exp_ptr;
   const int log_2 = ((x >> 23) & 255) - 128;
   x &= ~(255 << 23);
   x += 127 << 23;
   *exp_ptr = x;
 
   val = ((-1.0f/3) * val + 2) * val - 2.0f/3;
   return ((val + log_2) * 0.69314718f);
}
 
float difficulty(unsigned int bits)
{
    static double max_body = fast_log(0x00ffff), scaland = fast_log(256);
    return exp(max_body - fast_log(bits & 0x00ffffff) + scaland * (0x1d - ((bits & 0xff000000) >> 24)));
}
 
int main()
{
    std::cout << difficulty(0x1b0404cb) << std::endl;
    return 0;
}

Maximum, current and minimum difficulty

Current difficulty can be found out by using Bitcoin command line 'getDifficulty'.

Due to target function not having minimum value maximum difficulty can be calculated only approximately as following: maximum_target / 1 (as 0 would lead the equation to being infinitely big) which is an inconcievable number (~2 to the 224).

Minimum difficulty is just equal to 1.

Difficulty changes

Difficulty is changed every 2016 blocks based on the time it took to discover 2016 previous blocks. If a block is found every 10 minutes (as it was intended initially for even emission) finding 2016 blocks will take exactly 2 weeks. If previous 2016 blocks were found in more than two weeks the difficulty will be lowered, and if they were mined faster then that it will be raised. The more (or less) time was spent on finding the previous 2016 blocks the more will difficulty be lowered (raised).

To mine a block hash has to be lower than targer (proof-of-work). Hash is a random number between 0 and 2*256-1.

Shift for difficulty of 1: 0xffff * 2**208
For difficulty D: (0xffff * 2**208)/D
Amount of hash-functions needed to be solved for a block to be found at difficulty D: D * 2**256 / (0xffff * 2**208)
Or simply: D * 2**48 / 0xffff

Difficulty is set as if we found previous 2016 blocks at the speed of one block per 10 minutes

According to this we were calculating (D * 2**48 / 0xffff) hashes every 600 seconds.
Our network's hashrate for previous 2016 blocks was: D * 2**32 / 0xffff / 600.
Without significant accuracy loss we can simplify it down to: D * 2**32 / 600.
At difficulty of 1 that is roughly 7 Mhash/s.

Average time of finding a single block can be calculated using this formula: time = difficulty * 2**32 / hashrate where 'difficulty' is the current difficulty level of Bitcoin network and 'hashrate' is the amount of hashes a miner finds per second.

See also

References