Digital signatures are a crucial application of hash functions in cryptography, and ECDSA (Elliptic Curve Digital Signature Algorithm) is a popular digital signature scheme that uses hash functions to ensure the authenticity and integrity of encrypted data.

Here's how it works:

1. Data is hashed using a hash function like SHA-256, producing a fixed-size hash value.

2. The hash value is signed using the sender's private key and the ECDSA algorithm.

3. The resulting digital signature is appended to the encrypted data and transmitted to the recipient.

4. The recipient uses the sender's public key and the ECDSA algorithm to verify the digital signature.

5. If the verification succeeds, the recipient can trust that the data has not been tampered with or altered during transmission, and that it indeed comes from the claimed sender.

Digital signatures provide:

- Authentication: Ensures the data comes from the claimed sender.

- Integrity: Ensures the data has not been modified or tampered with.

- Non-repudiation: Ensures the sender cannot deny having sent the data.

By using hash functions in digital signature schemes like ECDSA, we can ensure the authenticity and integrity of encrypted data, which is critical for various applications, such as secure communication, digital contracts, and cryptocurrency transactions.