**Definition**: A cryptographic hash function is a mathematical algorithm that takes an input (or message) and produces a fixed-size string of characters, which is typically a hexadecimal number. The output, known as the hash value or digest, is unique to the input data and appears random. Cryptographic hash functions are designed to be fast to compute and irreversible, meaning it should be infeasible to derive the original input data from the hash value.

**Key Properties of Cryptographic Hash Functions**:

**Deterministic**: For the same input, a cryptographic hash function always produces the same hash value.**Fixed Output Size**: The hash function generates hash values of a fixed and predefined length, regardless of the input size.**Fast Computation**: Hash functions are designed to be computationally efficient, making them suitable for use in various applications.**Preimage Resistance**: It should be computationally infeasible to reverse the hash function and find the original input data from the hash value.**Collision Resistance**: It should be extremely unlikely for two different inputs to produce the same hash value (collision) when hashed using the same cryptographic hash function.**Avalanche Effect**: A small change in the input data should result in a significantly different hash value.**Non-Reversibility**: Given a hash value, it should be computationally infeasible to determine a valid input that produces that hash value.**Pseudorandomness**: The hash value should appear random, even when the input data has patterns.

**Common Cryptographic Hash Functions**:

**SHA-256 (Secure Hash Algorithm 256-bit)**: A widely used cryptographic hash function, commonly associated with Bitcoin and blockchain technology.**SHA-3**: The latest member of the Secure Hash Algorithm family, designed to provide better security and resistance to cryptographic attacks.**MD5 (Message Digest Algorithm 5)**: An older hash function known for its vulnerabilities and susceptibility to collisions. It is no longer considered secure for critical applications.**SHA-1 (Secure Hash Algorithm 1)**: Another older hash function, deprecated due to vulnerabilities.

**Applications**:

**Data Integrity**: Hash functions are used to verify data integrity. A sender can compute the hash value of a file and send both the file and its hash value. The recipient can verify the file’s integrity by recomputing the hash and comparing it to the received hash value.**Password Storage**: Hash functions are used to securely store user passwords. Instead of storing plaintext passwords, systems store the hash of the password. During login, the entered password is hashed and compared to the stored hash.**Digital Signatures**: Hash functions are used in digital signatures to create a unique representation of a document or message. The hash value is signed by the sender’s private key to prove document authenticity.**Blockchain Technology**: Cryptographic hash functions are used extensively in blockchain networks to link blocks and secure the blockchain’s immutability.**Data Deduplication**: Hashing is used in data storage systems to identify and eliminate duplicate data blocks, saving storage space.

**Security Considerations**: Cryptographic hash functions should meet stringent security requirements to resist attacks, such as collision attacks and preimage attacks. The choice of a hash function depends on the specific security needs of an application, with stronger functions recommended for critical applications.

Cryptographic hash functions are a fundamental tool in computer security and cryptography, providing data integrity, authentication, and a wide range of applications where data integrity and uniqueness are essential.