## Check against boundary using integer division

Ethereum Yellow Paper in section 11.5 defines PoW difficulty check as

n \leq \frac{2^{256}}{H_d}

where n is Ethash final hash and H_d is current block difficulty.

The \frac{2^{256}}{H_d} is called *boundary*. In practical implementations integer division is used to validate the check.

n \leq \left\lfloor\frac{2^{256}}{H_d}\right\rfloor

This is correct because of the following fact (proof):

x \in \mathbb{R}, n \in \mathbb{Z}: n \leq x \iff n \leq \left\lfloor x \right\rfloor

## Check against difficulty using multiplication

We can avoid using big integer division (which is slow and complex to implement) by transforming the orignal check formula into:

nH_d \leq 2^{256}

### Benefits

- Integer division has been replaced with multiplication. The 256 x 256 → 512 multiplication is straight forward to implement.
- Degenerated values of difficulty (0 and 1) do not require special handling.

### Implementation

This has been implemented in ethash 0.8.0.

## Side notes

- The check nH_d \leq 2^{256} can be further decomposed into nH_d \lt 2^{256} \lor nH_d = 2^{256} where the first part is 256-bit multiplication overflow check and the second part is very unlikely (or even impossible considering the difficulty update formula).
- The difficulty values on Ethereum Mainnet safely stay within 64-bit boundaries. Therefore, optimized path can be used for such values for both integer division and multiplication.