The exploration of these schemes reveals a complex landscape where fungibility and trust play critical roles. One approach to managing large quantities of tokens required for transactions involves mints issuing tokens in various denominations using the BDHKE scheme, which facilitates more efficient transfers by reducing the number of tokens necessary through mechanisms that support token splitting and combining. However, this introduces challenges such as diminished anonymity sets, as each denomination demands a unique minting key, effectively segregating each denomination into its own token. To overcome this, anonymous credential schemes offer enhanced privacy and interoperability among denominations. These schemes differ from BDHKE by allowing tokens to commit to variable values, thus providing a flexible method to represent transaction amounts. Utilizing Pedersen commitments within these frameworks, like WabiSabi, helps conceal individual amounts while ensuring verifiable transactions, thereby achieving fungibility across different denominations.

Despite these advancements, ecash systems inherently possess a "rugpull" capability due to the unilateral control mints have over token or credential issuance, potentially jeopardizing the system's integrity. Mints can create tokens with unbacked values, highlighting a fundamental security risk. This underscores the delicate balance required between privacy, fungibility, and trust within digital currency systems.

Ecash tokens facilitate payments with a high degree of privacy, enabling transactions where only the mint can verify a token's authenticity and its expenditure status. This system allows for anonymous transactions not typically available in traditional digital payment methods. Yet, it faces challenges relating to fraud prevention and transaction verification due to the central role of the mint. Proposed solutions include conducting fraud proofs through arbitration or public challenge-response mechanisms and employing DLEQ proof to validate token creation. However, these measures introduce complexities that may compromise the system's privacy attributes. Additionally, the potential for offline token transfers adds further layers of complexity, potentially affecting fungibility due to additional spending and redemption conditions.

The ecash blind signature scheme, exemplified by Cashu, utilizes cryptographic techniques to preserve user privacy during token creation and redemption. By leveraging elliptic curve Diffie-Hellman points and hash functions, the scheme ensures secure communication and demonstrates possession of valid tokens without revealing private keys. The process involves blinding messages containing public keys, which are then signed by the mint and later unblinded by the user, preventing the mint from tracing redeemed tokens back to their issuance. This system aims to maintain transaction privacy, prevent double-spending, and unauthorized redemptions, highlighting the ongoing efforts to balance privacy benefits against security and verifiability complexities in digital cash systems.