Hash function in the context of Perfect hash function

In computer science, a hash table is a data structure that implements an associative array, also called a dictionary or simply map; an associative array is an abstract data type that maps keys to values. A hash table uses a hash function to compute an index, also called a hash code, into an array of buckets or slots, from which the desired value can be found. During lookup, the key is hashed and the resulting hash indicates where the corresponding value is stored. A map implemented by a hash table is called a hash map.

Most hash table designs employ an imperfect hash function. Hash collisions, where the hash function generates the same index for more than one key, therefore typically must be accommodated in some way. Common strategies to handle hash collisions include chaining, which stores multiple elements in the same slot using linked lists, and open addressing, which searches for the next available slot according to a probing sequence.

In computer science, a hash collision or hash clash is when two distinct pieces of data in a hash table share the same hash value. The hash value in this case is derived from a hash function which takes a data input and returns a fixed length of bits.

Although hash algorithms, especially cryptographic hash algorithms, have been created with the intent of being collision resistant, they can still sometimes map different data to the same hash (by virtue of the pigeonhole principle). Malicious users can take advantage of this to mimic, access, or alter data.

A cryptographic hash function (CHF) is a hash algorithm (a map of an arbitrary binary string to a binary string with a fixed size of ${\displaystyle n}$ bits) that has special properties desirable for a cryptographic application:

• the probability of a particular ${\displaystyle n}$-bit output result (hash value) for a random input string ("message") is ${\displaystyle 2^{-n}}$ (as for any good hash), so the hash value can be used as a representative of the message;
• finding an input string that matches a given hash value (a pre-image) is infeasible, assuming all input strings are equally likely. The resistance to such search is quantified as security strength: a cryptographic hash with ${\displaystyle n}$ bits of hash value is expected to have a preimage resistance strength of ${\displaystyle n}$ bits, unless the space of possible input values is significantly smaller than ${\displaystyle 2^{n}}$ (a practical example can be found in § Attacks on hashed passwords);
• a second preimage resistance strength, with the same expectations, refers to a similar problem of finding a second message that matches the given hash value when one message is already known;
• finding any pair of different messages that yield the same hash value (a collision) is also infeasible: a cryptographic hash is expected to have a collision resistance strength of ${\displaystyle n/2}$ bits (lower because of the birthday paradox).

Cryptographic hash functions have many information-security applications, notably in digital signatures, message authentication codes (MACs), and other forms of authentication. They can also be used as ordinary hash functions, to index data in hash tables, for fingerprinting, to detect duplicate data or uniquely identify files, and as checksums to detect accidental data corruption. Indeed, in information-security contexts, cryptographic hash values are sometimes called (digital) fingerprints, checksums, (message) digests, or just hash values, even though all these terms stand for more general functions with rather different properties and purposes.