understanding bits and difficulty in a block header

Question

This block: https://blockchain.info/block/00000000000000000025c089d0a7b2bf6241888c4dd90ab7a4c4baa6a2823551

Shows difficulty at 3,007,383,866,429.73, and bits at 392009692.

If I want to see how many zeros need to be in the hash, I believe I can just do:

(log2(3007383866429.73) + 32) / 4) => 18.362911541451258

Which is correct.. But how does bits come from difficulty? How can I calculate the number of zeros from the bits instead of the difficulty?

Answer 1

Contrary to popular belief, Bitcoin's proof of work is not actually based on the number of zeroes. Rather the block hash, when interpreted as a 256 bit integer, must be less than the target value. The target value is what actually determines the difficulty. The target value is represented in a compact form in the nBits field of the block header.

The nBits field of the block header compresses the target from 256 bits into a 32 bits. A description of the format can be found here.

Basically, the nBits field, when represented in Big endian, is split into two parts: the first byte, and the last three bytes. The formula for decompressing the nBits field is as follows: (last three bytes) * 256 ^ ((first byte) - 3). This gives you a 256 bit integer that has the first 3 most significant bytes of the target.

Answer 2

I've created some javascript code as a result of my research in trying to understand how target difficulty works. It seems a good idea to share after finding this post.

First it creates the range of SHA256 powers of 2 thresholds for you to visualize the shear magnitude of these numbers.

Then the bitsToHex function is used to decompress "bits".

Then decompressed "bits" are casted as a BigInt in order to get the integer used for hashing comparison.

Code comments should clarify the rest.

console.log("\nSHA256 in powers of 2\n");

// from easy to hard
for(let i = 256n; i >= 0n; i-=4n) {
  // calc decimal representaion
  const dec = BigInt(2n**i-1n);
  // calc hexadecimal representation
  const hex = BigInt(2n**i-1n).toString(16).padStart(64, "0");
  
  // beautify console
  console.log(`2n**${i}-1n D`.padStart(12, " "), dec.toLocaleString("en-US").padStart(103, "."));
  console.log(`2n**${i}-1n H`.padStart(12, " "), `0x${hex}`.padEnd(116, " "));
}

/**
 * @dev Takes a Bitcoin block header property "bits" and converts it into a 256-bit hex.
 * 
 * The bits property of a Bitcoin block header is a 32-bit integer that 
 * represents the difficulty target for mining that block. 
 * The first three bytes of the bits value are the exponent (in base 256) of
 * the difficulty target, and the last byte is the coefficient.
 * To convert the bits value into a 256-bit hex, we need to extract the 
 * exponent and coefficient from the bits value and calculate the target value. 
 * Then we can convert the target value into a 64-character hex string.
 * 
 */
function bitsToHex(bits) {
  // calc target value from compressed "bits".
  const target = (bits & 0x007fffff) * 2 ** (8 * ((bits >>> 24) - 3));
  // convert to 256bit hex
  return target.toString(16).padStart(64, "0");
}

// console spacer
console.log("\nBitcoin Target Difficulty\n");

// Satoshi's genesis block "bits" A.K.A target hash
// for details see https://bitcoinexplorer.org/block-height/1#JSON
// "bits": "1d00ffff",
const genesisTargetDifficultyHEX = bitsToHex(0x1d00ffff);
const genesisTargetDifficultyDEC = BigInt(`0x${genesisTargetDifficultyHEX}`);
console.log('genesis block target difficulty HEX:', genesisTargetDifficultyHEX);
console.log('genesis block target difficulty DEC:', genesisTargetDifficultyDEC);

// Bitcoins's current block "bits" A.K.A target hash
// for details see https://bitcoinexplorer.org/block-height/776774#JSON
// "bits": "17073039",
const currentTargetDifficultyHEX = bitsToHex(0x17073039);
const currentTargetDifficultyDEC = BigInt(`0x${currentTargetDifficultyHEX}`);
console.log('current block target difficulty HEX:', currentTargetDifficultyHEX);
console.log('current block target difficulty DEC:', currentTargetDifficultyDEC);

...console output:

genesis block target difficulty HEX: 00000000ffff0000000000000000000000000000000000000000000000000000
genesis block target difficulty DEC: 26959535291011309493156476344723991336010898738574164086137773096960n
current block target difficulty HEX: 0000000000000000000730390000000000000000000000000000000000000000
current block target difficulty DEC: 688509036841676372057001313638142781710850853198364672n

As you can see, difficulty has gotten a lot harder since genesis block.