Describe the specific role of hashing functions in cryptographic systems, and discuss their importance beyond simple data integrity checks.

Hashing functions play a crucial and multifaceted role in cryptographic systems, extending far beyond simple data integrity checks. At their core, hashing functions are mathematical algorithms that take an input of any size (a message, a file, data, etc.) and produce a fixed-size output, known as a hash or message digest. These functions are designed to be one-way, meaning that it's computationally infeasible to reverse the process and derive the original input from the hash. Hashing functions are also designed to be deterministic, meaning that the same input will always produce the same output, and they are designed to have a ‘avalanche effect’ – a small change in the input will drastically change the output hash. This makes them extremely useful in various aspects of cryptography.

One of the primary roles of hashing functions, as stated in the question, is to ensure data integrity. When data is hashed, the resulting hash acts as a unique fingerprint. If the data is altered even slightly during transmission or storage, the hash will change dramatically. By comparing the original hash with a hash of the received data, one can verify whether the data has been tampered with. For example, when downloading a file, a checksum hash is often provided along with the file. This hash is produced using a hashing algorithm such as SHA-256. Once the file is downloaded, you can hash it yourself and compare it to the provided hash. If they match, you can be sure that the file has been downloaded without any alterations and hasn’t been corrupted or modified in transit. This check ensures that the file you have is exactly the same as the file that was originally provided and helps prevent various types of malicious injection attacks that aim to deliver modified files.

However, hashing functions have many uses beyond simple integrity checks. They are also vital for password storage. Instead of storing passwords in plaintext, which is a major security risk, systems store hashes of the passwords. When a user tries to log in, the system hashes the provided password and compares it to the stored hash. If the hashes match, the user is authenticated without ever exposing the original password. This means that if a database containing password hashes is breached, the attackers will not know the users’ actual passwords, although attackers may still use the stored hashes to attempt a ‘rainbow table’ attack. However, a combination with techniques like salting prevents common rainbow table attacks and other attacks on hashed passwords.

Another important application of hashing is in digital signatures. In digital signatures, the message to be signed is first hashed, and then the hash is encrypted using the sender's private key. The recipient can then decrypt this encrypted hash using the sender’s public key and also calculate the hash of the original message. If the decrypted hash matches the calculated hash, it confirms that the message came from the sender and that it has not been altered. This process ensures both authentication and integrity of the data in a digital signature system.

Furthermore, hashing is used in message authentication codes (MACs), which provide a way to verify that a message comes from the expected source and that it hasn’t been tampered with. A MAC combines the message with a secret key and then hashes the result to produce a tag that's appended to the message. Only parties knowing the secret key can recreate the MAC to verify the message's origin and integrity. This ensures message origin validation and data integrity.

Hashing is also used in cryptographic protocols and various algorithms. It is the fundamental part of many cryptographic schemes. For example, blockchain technologies use cryptographic hashing to create tamper-proof ledgers by creating hash chains. In this process, each block includes the hash of the preceding block, creating a chain of hashes where every block is inextricably linked to all its predecessors. This process ensures that data within the blockchain cannot be modified without invalidating all the subsequent blocks which creates a system that is resistant to tampering.

Hashing functions are therefore a cornerstone of modern cryptographic systems. They are crucial not only for simple checks like data integrity, but also for a range of functionalities, from secure password storage and digital signatures to authentication and various other cryptographic protocols. Their ability to provide a one-way, unique, fixed-size output, combined with their sensitivity to even small changes in input, makes them an indispensable tool for ensuring data security and integrity.