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A hash function is a mathematical function that takes in an input (such as a string, a file, or any other data) and produces a fixed-size output called a hash or a message digest. The hash function is designed to take any input and produce a unique, fixed-size output that represents the input data in a condensed form.

The primary purpose of a hash function is to provide a way to verify the integrity of data. For example, if you download a file from the internet, you can use a hash function to compute a hash value for the downloaded file and compare it to the hash value provided by the website. If the two hash values match, you can be reasonably sure that the file has not been tampered with or corrupted during the download.

Hash functions are also commonly used in cryptography for secure message authentication, digital signatures, and password storage. In password storage, for example, a hash function is used to transform a user's password into a fixed-length hash that can be stored in a database instead of the plaintext password. When the user logs in, the system hashes the entered password and compares it to the stored hash to verify the user's identity.