Traditionally, signature grinding has been used to create PoW-locked output scripts in Bitcoin, exploiting the variable size of DER-encoded ECDSA signatures. This method, however, is limited by its coarse adjustment capabilities — difficulty can only be multiplied or divided by 256 due to the reliance on the byte size of signatures.

A new method has been proposed to circumvent these limitations by utilizing a fixed set of twelve signatures, a known short nonce, and a specific selection of private keys. This approach does not rely on operations within the finite field or elliptic curve calculations but instead on simple comparisons within the 256-bit integer space, making it potentially more efficient and flexible. The concept leverages the mathematical structure inherent in the size requirements for short signatures, which dictates that the signature's s-value must fall within a predefined range to be considered "short."

By employing two private keys and requiring that both signatures for them meet the shortness criteria against a single transaction hash, this method ensures that the hash falls within the intersection of two intervals, each defined by one of the private keys. Adjusting the difficulty of this PoW scheme involves manipulating the difference between the private keys' corresponding x-values, allowing for an arbitrary degree of difficulty adjustment far beyond the binary scale imposed by signature grinding.

To implement this process practically, miners would need to produce signatures under all possible sighash flags, facilitated by using pairs of private keys designed to generate overlapping or non-overlapping intervals within which the transaction hash must fall. The overall difficulty of generating a qualifying transaction hash can thus be finely tuned, providing a way to create PoW-locked outputs that can adjust more smoothly to changing network conditions.

An exemplary Node.JS application demonstrates the feasibility of creating and mining PoW-locked outputs with arbitrary difficulty levels, offering a step-by-step guide through the necessary calculations and considerations for setting up such a system. This includes deriving private key pairs from chosen x-values, constructing the locking script to enforce the signature size constraints, and crafting transactions that satisfy the defined conditions.

In summary, the introduction of arbitrary difficulty adjustments for PoW-locked Bitcoin outputs via short DER-encoded ECDSA signatures represents a significant advancement in the flexibility and precision of such mechanisms. By moving away from the limitations of signature grinding and towards a system based on mathematical intervals and comparisons, developers can offer a more scalable and adaptable solution for securing Bitcoin transactions against unauthorized modifications.