Hash function

A hash function that maps names to integers from 0 to 15.. There is a collision between keys "John Smith" and "Sandra Dee".

A hash function is any algorithm or subroutine that maps large data sets, called keys, to smaller data sets. For example, a single integer can serve as an index to an array (cf. associative array). The values returned by a hash function are called hash values, hash codes, hash sums, checksums or simply hashes.

A hash function that assigns unique indices to strings, even if inconsistent between runs, is still a perfectly valid hash.

Descriptions

Hash functions are mostly used to accelerate table lookup or data comparison tasks such as finding items in a database, detecting duplicated or similar records in a large file, finding similar stretches in DNA sequences, and so on.

A hash function should be referentially transparent, i.e. if called twice on input that is "equal" (e.g. strings that consist of the same sequence of characters), it should give the same result. This is a contract in many programming languages that allow the user to override equality and hash functions for an object, that if two objects are equal their hash codes must be the same. This is important in order for it to be possible to find an element in a hash table quickly since two of the same element would both hash to the same slot.

Some hash functions may map two or more keys to the same hash value, causing a collision. Such hash functions try to map the keys to the hash values as evenly as possible because, as Hash Tables fill up, collisions become more frequent. Thus single digit hash values are frequently restricted to 80% of the size of the Table. Depending on the algorithm used, other properties may be required as well, such as double Hashing and Linear Probing. Although the idea was conceived in the 1950s, the design of good hash functions is still a topic of active research.

Hash functions are related to (and often confused with) checksums, check digits, fingerprints, randomization functions, error correcting codes, and cryptographic hash functions. Although these concepts overlap to some extent, each has its own uses and requirements and is designed and optimized differently. The HashKeeper database maintained by the American National Drug Intelligence Center, for instance, is more aptly described as a catalog of file fingerprints than of hash values.

Hash tables

Hash functions are primarily used in hash tables, to quickly locate a data record (for example, a dictionary definition) given its search key (the headword). Specifically, the hash function is used to map the search key to the hash. The index gives the place where the corresponding record should be stored. Hash tables, in turn, are used to implement associative arrays and dynamic sets.

In general, a hashing function may map several different keys to the same index. Therefore, each slot of a hash table is associated with (implicitly or explicitly) a set of records, rather than a single record. For this reason, each of a hash table is often called a bucket, and hash values are also called bucket indices.

Thus, the hash function only hints at the record's location—it tells where one should start looking for it. Still, in a half-full table, a good hash function will typically narrow the search down to only one or two entries.

Caches

Hash functions are also used to build caches for large data sets stored in slow media. A cache is generally simpler than a hashed search table, since any collision can be resolved by discarding or writing back the older of the two colliding items.

Bloom filters

Hash functions are an essential ingredient of the Bloom filter, a compact data structure that provides an enclosing approximation to a set of them.

Finding duplicate records

Main article: Hash table

When storing records in a large unsorted file, one may use a hash function to map each record to an index into a table T, and collect in each bucket T[i] a list of the numbers of all records with the same hash value i. Once the table is complete, any two duplicate records will end up in the same bucket. The duplicates can then be found by scanning every bucket T[i] which contains two or more members, fetching those records, and comparing them. With a table of appropriate size, this method is likely to be much faster than any alternative approach (such as sorting the file and comparing all consecutive pairs).

Finding similar records

Main article: Locality sensitive hashing

Hash functions can also be used to locate table records whose key is similar, but not identical, to a given key; or pairs of records in a large file which have similar keys. For that purpose, one needs a hash function that maps similar keys to hash values that differ by at most m, where m is a small integer (say, 1 or 2). If one builds a table T of all record numbers, using such a hash function, then similar records will end up in the same bucket, or in nearby buckets. Then one need only check the records in each bucket T[i] against those in buckets T[i+k] where k ranges between -m and m.

This class includes the so-called acoustic fingerprint algorithms, that are used to locate similar-sounding entries in large collection of audio files. For this application, the hash function must be as insensitive as possible to data capture or transmission errors, and to "trivial" changes such as timing and volume changes, compression, etc.

Finding similar substrings

The same techniques can be used to find equal or similar stretches in a large collection of strings, such as a document repository or a genomic database. In this case, the input strings are broken into many small pieces, and a hash function is used to detect potentially equal pieces, as above.

The Rabin-Karp algorithm is a relatively fast string searching algorithm that works in O(n) time on average. It is based on the use of hashing to compare strings.

Geometric hashing

This principle is widely used in computer graphics, computational geometry and many other disciplines, to solve many proximity problems in the plane or in three-dimensional space, such as finding closest pairs in a set of points, similar shapes in a list of shapes, similar images in an image database, and so on. In these applications, the set of all inputs is some sort of metric space, and the hashing function can be interpreted as a partition of that space into a grid of cells. The table is often an array with two or more indices (called a grid file, grid index, bucket grid, and similar names), and the hash function returns an index tuple. This special case of hashing is known as geometric hashing or the grid method. Geometric hashing is also used in telecommunications (usually under the name vector quantization) to encode and compress multi-dimensional signals.

Properties

Good hash functions, in the original sense of the term, are usually required to satisfy certain properties listed below. Note that different requirements apply to the other related concepts (cryptographic hash functions, checksums, etc.).

Low cost

The cost of computing a hash function must be small enough to make a hashing-based solution more efficient than alternative approaches. For instance, a self-balancing binary tree can locate an item in a sorted table of n items with O(log n) key comparisons. Therefore, a hash table solution will be more efficient than a self-balancing binary tree if the number of items is large and the hash function produces few collisions and less efficient if the number of items is small and the hash function is complex.

Determinism

A hash procedure must be deterministic—meaning that for a given input value it must always generate the same hash value. In other words, it must be a function of the hashed data, in the mathematical sense of the term. This requirement excludes hash functions that depend on external variable parameters, such as pseudo-random number generators or the time of day. It also excludes functions that depend on the memory address of the object being hashed, because that address may change during execution (as may happen on systems that use certain methods of garbage collection), although sometimes rehashing of the item is possible).

Uniformity

A good hash function should map the expected inputs as evenly as possible over its output range. That is, every hash value in the output range should be generated with roughly the same probability. The reason for this last requirement is that the cost of hashing-based methods goes up sharply as the number of collisions—pairs of inputs that are mapped to the same hash value—increases. Basically, if some hash values are more likely to occur than others, a larger fraction of the lookup operations will have to search through a larger set of colliding table entries.

Note that this criterion only requires the value to be uniformly distributed, not random in any sense. A good randomizing function is (barring computational efficiency concerns) generally a good choice as a hash function, but the converse need not be true.

Hash tables often contain only a small subset of the valid inputs. For instance, a club membership list may contain only a hundred or so member names, out of the very large set of all possible names. In these cases, the uniformity criterion should hold for almost all typical subsets of entries that may be found in the table, not just for the global set of all possible entries.

In other words, if a typical set of m records is hashed to n table slots, the probability of a bucket receiving many more than m/n records should be vanishingly small. In particular, if m is less than n, very few buckets should have more than one or two records. (In an ideal "perfect hash function", no bucket should have more than one record; but a small number of collisions is virtually inevitable, even if n is much larger than m -- see the birthday paradox).

When testing a hash function, the uniformity of the distribution of hash values can be evaluated by the chi-squared test.

Variable range

In many applications, the range of hash values may be different for each run of the program, or may change along the same run (for instance, when a hash table needs to be expanded). In those situations, one needs a hash function which takes two parameters—the input data z, and the number n of allowed hash values.

A common solution is to compute a fixed hash function with a very large range (say, 0 to 2³²−1), divide the result by n, and use the division's remainder. If n is itself a power of 2, this can be done by bit masking and bit shifting. When this approach is used, the hash function must be chosen so that the result has fairly uniform distribution between 0 and n−1, for any n that may occur in the application. Depending on the function, the remainder may be uniform only for certain n, e.g. odd or prime numbers.

It is possible to relax the restriction of the table size being a power of 2 and not having to perform any modulo, remainder or division operation -as these operation are considered computational costly in some contexts. For example, when n is significantly less than 2^b begin with a pseudo random number generator (PRNG) function P(key), uniform on the interval [0, 2^b−1]. Consider the ratio q = 2^b / n. Now the hash function can be seen as the value of P(key) / q. Rearranging the calculation and replacing the 2^b-division by bit shifting right (>>) b times you end up with hash function n * P(key) >> b.

Variable range with minimal movement (dynamic hash function)

When the hash function is used to store values in a hash table that outlives the run of the program, and the hash table needs to be expanded or shrunk, the hash table is referred to as a dynamic hash table.

A hash function that will relocate the minimum number of records when the table is resized is desirable. What is needed is a hash function H(z,n) – where z is the key being hashed and n is the number of allowed hash values – such that H(z,n+1) = H(z,n) with probability close to n/(n+1).

Linear hashing and spiral storage are examples of dynamic hash functions that execute in constant time but relax the property of uniformity to achieve the minimal movement property.

Extendible hashing uses a dynamic hash function that requires space proportional to n to compute the hash function, and it becomes a function of the previous keys that have been inserted.

Several algorithms that preserve the uniformity property but require time proportional to n to compute the value of H(z,n) have been invented.

Data normalization

In some applications, the input data may contain features that are irrelevant for comparison purposes. For example, when looking up a personal name, it may be desirable to ignore the distinction between upper and lower case letters. For such data, one must use a hash function that is compatible with the data equivalence criterion being used: that is, any two inputs that are considered equivalent must yield the same hash value. This can be accomplished by normalizing the input before hashing it, as by upper-casing all letters.

Continuity

A hash function that is used to search for similar (as opposed to equivalent) data must be as continuous as possible; two inputs that differ by a little should be mapped to equal or nearly equal hash values.

Note that continuity is usually considered a fatal flaw for checksums, cryptographic hash functions, and other related concepts. Continuity is desirable for hash functions only in some applications, such as hash tables that use linear search.

Hash function algorithms

For most types of hashing functions the choice of the function depends strongly on the nature of the input data, and their probability distribution in the intended application.

Trivial hash function

If the datum to be hashed is small enough, one can use the datum itself (reinterpreted as an integer in binary notation) as the hashed value. The cost of computing this "trivial" (identity) hash function is effectively zero. This hash function is perfect, as it maps each input to a distinct hash value.

The meaning of "small enough" depends on the size of the type that is used as the hashed value. For example, in Java, the hash code is a 32-bit integer. Thus the 32-bit integer Integer and 32-bit floating-point Float objects can simply use the value directly; whereas the 64-bit integer Long and 64-bit floating-point Double cannot use this method.

Other types of data can also use this perfect hashing scheme. For example, when mapping character strings between upper and lower case, one can use the binary encoding of each character, interpreted as an integer, to index a table that gives the alternative form of that character ("A" for "a", "8" for "8", etc.). If each character is stored in 8 bits (as in ASCII or ISO Latin 1), the table has only 2⁸ = 256 entries; in the case of Unicode characters, the table would have 17×2¹⁶ = 1114112 entries.

The same technique can be used to map two-letter country codes like "us" or "za" to country names (26²=676 table entries), 5-digit zip codes like 13083 to city names (100000 entries), etc. Invalid data values (such as the country code "xx" or the zip code 00000) may be left undefined in the table, or mapped to some appropriate "null" value.

Perfect hashing

Main article: Perfect hash function

A perfect hash function for the four names shown

A hash function that is injective—that is, maps each valid input to a different hash value—is said to be perfect. With such a function one can directly locate the desired entry in a hash table, without any additional searching.

Minimal perfect hashing

A minimal perfect hash function for the four names shown

A perfect hash function for n keys is said to be minimal if its range consists of n consecutive integers, usually from 0 to n−1. Besides providing single-step lookup, a minimal perfect hash function also yields a compact hash table, without any vacant slots. Minimal perfect hash functions are much harder to find than perfect ones with a wider range.

Hashing uniformly distributed data

If the inputs are bounded-length strings (such as telephone numbers, car license plates, invoice numbers, etc.), and each input may independently occur with uniform probability, then a hash function need only map roughly the same number of inputs to each hash value. For instance, suppose that each input is an integer z in the range 0 to N−1, and the output must be an integer h in the range 0 to n−1, where N is much larger than n. Then the hash function could be h = z mod n (the remainder of z divided by n), or h = (z × n) ÷ N (the value z scaled down by n/N and truncated to an integer), or many other formulas.

Warning: h = z mod n was used in many of the original random number generators, but was found to have a number of issues. One of which is that as n approaches N, this function becomes less and less uniform.

Hashing data with other distributions

These simple formulas will not do if the input values are not equally likely, or are not independent. For instance, most patrons of a supermarket will live in the same geographic area, so their telephone numbers are likely to begin with the same 3 to 4 digits. In that case, if n is 10000 or so, the division formula (z × n) ÷ N, which depends mainly on the leading digits, will generate a lot of collisions; whereas the remainder formula z mod n, which is quite sensitive to the trailing digits, may still yield a fairly even distribution.

Hashing variable-length data

When the data values are long (or variable-length) character strings—such as personal names, web page addresses, or mail messages—their distribution is usually very uneven, with complicated dependencies. For example, text in any natural language has highly non-uniform distributions of characters, and character pairs, very characteristic of the language. For such data, it is prudent to use a hash function that depends on all characters of the string—and depends on each character in a different way.

In cryptographic hash functions, a Merkle–Damgård construction is usually used. In general, the scheme for hashing such data is to break the input into a sequence of small units (bits, bytes, words, etc.) and combine all the units b[1], b[2], ..., b[m] sequentially, as follows

S ← S0;                             // Initialize the state.
for k in 1, 2, ..., m do             // Scan the input data units:
  S ← F(S, b[k]);                   //   Combine data unit k into the state.
return G(S, n)                      // Extract the hash value from the state.

This schema is also used in many text checksum and fingerprint algorithms. The state variable S may be a 32- or 64-bit unsigned integer; in that case, S0 can be 0, and G(S,n) can be just S mod n. The best choice of F is a complex issue and depends on the nature of the data. If the units b[k] are single bits, then F(S,b) could be, for instance

 if highbit(S) = 0 then
   return 2 * S + b
 else
   return (2 * S + b) ^ P

Here highbit(S) denotes the most significant bit of S; the '*' operator denotes unsigned integer multiplication with lost overflow; '^' is the bitwise exclusive or operation applied to words; and P is a suitable fixed word.

Special-purpose hash functions

In many cases, one can design a special-purpose (heuristic) hash function that yields many fewer collisions than a good general-purpose hash function. For example, suppose that the input data are file names such as FILE0000.CHK, FILE0001.CHK, FILE0002.CHK, etc., with mostly sequential numbers. For such data, a function that extracts the numeric part k of the file name and returns k mod n would be nearly optimal. Needless to say, a function that is exceptionally good for a specific kind of data may have dismal performance on data with different distribution.

Rolling hash

Main article: rolling hash

In some applications, such as substring search, one must compute a hash function h for every k-character substring of a given n-character string t; where k is a fixed integer, and n is k. The straightforward solution, which is to extract every such substring s of t and compute h(s) separately, requires a number of operations proportional to k·n. However, with the proper choice of h, one can use the technique of rolling hash to compute all those hashes with an effort proportional to k+n.

Universal hashing

A universal hashing scheme is a randomized algorithm that selects a hashing function h among a family of such functions, in such a way that the probability of a collision of any two distinct keys is 1/n, where n is the number of distinct hash values desired—independently of the two keys. Universal hashing ensures (in a probabilistic sense) that the hash function application will behave as well as if it were using a random function, for any distribution of the input data. It will however have more collisions than perfect hashing, and may require more operations than a special-purpose hash function.

Hashing with checksum functions

One can adapt certain checksum or fingerprinting algorithms for use as hash functions. Some of those algorithms will map arbitrary long string data z, with any typical real-world distribution—no matter how non-uniform and dependent—to a 32-bit or 64-bit string, from which one can extract a hash value in 0 through n−1.

This method may produce a sufficiently uniform distribution of hash values, as long as the hash range size n is small compared to the range of the checksum or fingerprint function. However, some checksums fare poorly in the avalanche test, which may be a concern in some applications. In particular, the popular CRC32 checksum provides only 16 bits (the higher half of the result) that are usable for hashing. Moreover, each bit of the input has a deterministic effect on each bit of the CRC32, that is one can tell without looking at the rest of the input, which bits of the output will flip if the input bit is flipped; so care must be taken to use all 32 bits when computing the hash from the checksum.

Hashing with cryptographic hash functions

Some cryptographic hash functions, such as SHA-1, have even stronger uniformity guarantees than checksums or fingerprints, and thus can provide very good general-purpose hashing functions.

In ordinary applications, this advantage may be too small to offset their much higher cost. However, this method can provide uniformly distributed hashes even when the keys are chosen by a malicious agent. This feature may help protect services against denial of service attacks.

Origins of the term

The term "hash" comes by way of analogy with its non-technical meaning, to "chop and mix". Indeed, typical hash functions, like the mod operation, "chop" the input domain into many sub-domains that get "mixed" into the output range to improve the uniformity of the key distribution.

Donald Knuth notes that Hans Peter Luhn of IBM appears to have been the first to use the concept, in a memo dated January 1953, and that Robert Morris used the term in a survey paper in CACM which elevated the term from technical jargon to formal terminology.

List of hash functions

Main article: List of hash functions

• Bernstein hash
• Fowler-Noll-Vo hash function (32, 64, 128, 256, 512, or 1024 bits)
• Jenkins hash function (32 bits)
• Pearson hashing (8 bits)
• Zobrist hashing

See also

• Bloom filter
• Coalesced hashing
• Cuckoo hashing
• Cryptographic hash function
• Distributed hash table
• Geometric hashing
• Hash table
• HMAC
• Identicon
• Linear hash
• List of hash functions
• Locality sensitive hashing
• Perfect hash function
• Rabin-Karp string search algorithm
• Rolling hash
• Transposition table
• Universal hashing

External links

• General purpose hash function algorithms (C/C++/Pascal/Java/Python/Ruby)
• Hash Functions and Block Ciphers by Bob Jenkins
• The Goulburn Hashing Function (PDF) by Mayur Patel
• MIT's Introduction to Algorithms: Hashing 1 MIT OCW lecture Video
• MIT's Introduction to Algorithms: Hashing 2 MIT OCW lecture Video
• Hash Fuction Construction for Textual and Geometrical Data Retrieval Latest Trends on Computers, Vol.2, pp.483-489, CSCC conference, Corfu, 2010