Hash function - Knowledge

869:: whenever a single input bit is complemented, each of the output bits changes with a 50% probability. The reason for this property is that selected subsets of the keyspace may have low variability. For the output to be uniformly distributed, a low amount of variability, even one bit, should translate into a high amount of variability (i.e. distribution over the tablespace) in the output. Each bit should change with a probability of 50% because, if some bits are reluctant to change, then the keys become clustered around those values. If the bits want to change too readily, then the mapping is approaching a fixed XOR function of a single bit. Standard tests for this property have been described in the literature. The relevance of the criterion to a multiplicative hash function is assessed here. 2517:

initialized at the start of the program. The random numbers could be any length, but 64 bits was natural due to the 64 squares on the board. A position was transcribed by cycling through the pieces in a position, indexing the corresponding random numbers (vacant spaces were not included in the calculation) and XORing them together (the starting value could be 0 (the identity value for XOR) or a random seed). The resulting value was reduced by modulo, folding, or some other operation to produce a hash table index. The original Zobrist hash was stored in the table as the representation of the position.

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addition is also a plausible alternative. The final operation would be a modulo, mask, or other function to reduce the word value to an index the size of the table. The weakness of this procedure is that information may cluster in the upper or lower bits of the bytes; this clustering will remain in the hashed result and cause more collisions than a proper randomizing hash. ASCII byte codes, for example, have an upper bit of 0, and printable strings do not use the first 32 byte codes, so the information (95 bytecodes) is clustered in the remaining bits in an unobvious manner.

143: 897:) by a constant can be inverted to become a multiplication by the word-size multiplicative-inverse of that constant. This can be done by the programmer, or by the compiler. Division can also be reduced directly into a series of shift-subtracts and shift-adds, though minimizing the number of such operations required is a daunting problem; the number of assembly instructions resulting may be more than a dozen and swamp the pipeline. If the architecture has 482: 46: 878:

indexable by the key-value would be very large and very sparse, but very fast. A hash function takes a finite amount of time to map a potentially large keyspace to a feasible amount of storage space searchable in a bounded amount of time regardless of the number of keys. In most applications, the hash function should be computable with minimum latency and secondarily in a minimum number of instructions.

1545:) and can be 10 times slower than multiplication. A second drawback is that it will not break up clustered keys. For example, the keys 123000, 456000, 789000, etc. modulo 1000 all map to the same address. This technique works well in practice because many key sets are sufficiently random already, and the probability that a key set will be cyclical by a large prime number is small. 2732:

example, a 128-bit word will hash only a 26-character alphabetic string (ignoring case) with a radix of 29; a printable ASCII string is limited to 9 characters using radix 97 and a 64-bit word. However, alphabetic keys are usually of modest length, because keys must be stored in the hash table. Numeric character strings are usually not a problem; 64 bits can count up to

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the item is added to the table there. If the hash code indexes a full slot, then some kind of collision resolution is required: the new item may be omitted (not added to the table), or replace the old item, or be added to the table in some other location by a specified procedure. That procedure depends on the structure of the hash table. In

1502:, so the hash code is taken as the middle 4 digits of the 17-digit number (ignoring the high digit) 8750. The mid-squares method produces a reasonable hash code if there is not a lot of leading or trailing zeros in the key. This is a variant of multiplicative hashing, but not as good because an arbitrary key is not a good multiplier. 4020:

implementation of the service and present solutions for avoiding single points of failure and guaranteeing a service with reasonable and stable delay. Guardtime AS has been operating a KSI Infrastructure for 5 years. We summarize how the KSI Infrastructure is built, and the lessons learned during the operational period of the service.

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operations (e.g. multiplication by constant and bit-shifting). The final word, which may have unoccupied byte positions, is filled with zeros or a specified randomizing value before being folded into the hash. The accumulated hash code is reduced by a final modulo or other operation to yield an index into the table.

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other pathologies in the key set. Such strategies may be effective as a custom hash function if the structure of the keys is such that either the middle, ends, or other fields are zero or some other invariant constant that does not differentiate the keys; then the invariant parts of the keys can be ignored.

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In a hash table, a hash function takes a key as an input, which is associated with a datum or record and used to identify it to the data storage and retrieval application. The keys may be fixed-length, like an integer, or variable-length, like a name. In some cases, the key is the datum itself. The

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Worst case results for a hash function can be assessed two ways: theoretical and practical. The theoretical worst case is the probability that all keys map to a single slot. The practical worst case is the expected longest probe sequence (hash function + collision resolution method). This analysis

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The paradigmatic example of folding by characters is to add up the integer values of all the characters in the string. A better idea is to multiply the hash total by a constant, typically a sizable prime number, before adding in the next character, ignoring overflow. Using exclusive-or instead of

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A hash function can be designed to exploit existing entropy in the keys. If the keys have leading or trailing zeros, or particular fields that are unused, always zero or some other constant, or generally vary little, then masking out only the volatile bits and hashing on those will provide a better

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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

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Later, the method was extended to hashing integers by representing each byte in each of 4 possible positions in the word by a unique 32-bit random number. Thus, a table of 2×4 random numbers is constructed. A 32-bit hashed integer is transcribed by successively indexing the table with the value of

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Multiplicative hashing is susceptible to a "common mistake" that leads to poor diffusion—higher-value input bits do not affect lower-value output bits. A transmutation on the input which shifts the span of retained top bits down and XORs or ADDs them to the key before the multiplication step

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In data storage and retrieval applications, the use of a hash function is a trade-off between search time and data storage space. If search time were unbounded, then a very compact unordered linear list would be the best medium; if storage space were unbounded, then a randomly accessible structure

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function of the number of keys to be mapped versus the number of table slots that they are mapped into. Finding a perfect hash function over more than a very small set of keys is usually computationally infeasible; the resulting function is likely to be more computationally complex than a standard

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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

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is usually a prime number large enough to hold the number of different characters in the character set of potential keys. Radix conversion hashing of strings minimizes the number of collisions. Available data sizes may restrict the maximum length of string that can be hashed with this method. For

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characters of a string along with the length, or form a word-size hash from the middle 4 characters of a string. This saves iterating over the (potentially long) string, but hash functions that do not hash on all characters of a string can readily become linear due to redundancies, clustering, or

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to store and retrieve data items or data records. The hash function translates the key associated with each datum or record into a hash code, which is used to index the hash table. When an item is to be added to the table, the hash code may index an empty slot (also called a bucket), in which case

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If the keys are uniformly or sufficiently uniformly distributed over the key space, so that the key values are essentially random, then they may be considered to be already "hashed". In this case, any number of any bits in the key may be extracted and collated as an index into the hash table. For

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It is often desirable that the output of a hash function have fixed size (but see below). If, for example, the output is constrained to 32-bit integer values, then the hash values can be used to index into an array. Such hashing is commonly used to accelerate data searches. Producing fixed-length

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Hash functions and their associated hash tables are used in data storage and retrieval applications to access data in a small and nearly constant time per retrieval. They require an amount of storage space only fractionally greater than the total space required for the data or records themselves.

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Zobrist hashing was originally introduced as a means of compactly representing chess positions in computer game-playing programs. A unique random number was assigned to represent each type of piece (six each for black and white) on each space of the board. Thus a table of 64×12 such numbers is

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key sets, and poorly designed hash functions can result in access times approaching linear in the number of items in the table. Hash functions can be designed to give the best worst-case performance, good performance under high table loading factors, and in special cases, perfect (collisionless)

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Modern microprocessors will allow for much faster processing if 8-bit character strings are not hashed by processing one character at a time, but by interpreting the string as an array of 32-bit or 64-bit integers and hashing/accumulating these "wide word" integer values by means of arithmetic

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is the table size, and using a parity-preserving bitwise operation such as ADD or XOR to combine the sections, followed by a mask or shifts to trim off any excess bits at the high or low end. For example, for a table size of 15 bits and a 64-bit key value of 0x0123456789ABCDEF, there are five

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If keys are being hashed repeatedly, and the hash function is costly, then computing time can be saved by precomputing the hash codes and storing them with the keys. Matching hash codes almost certainly means that the keys are identical. This technique is used for the transposition table in

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Keyless Signatures Infrastructure (KSI) is a globally distributed system for providing time-stamping and server-supported digital signature services. Global per-second hash trees are created and their root hash values published. We discuss some service quality issues that arise in practical

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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

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output from variable-length input can be accomplished by breaking the input data into chunks of specific size. Hash functions used for data searches use some arithmetic expression that iteratively processes chunks of the input (such as the characters in a string) to produce the hash value.

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Hashing is a computationally- and storage-space-efficient form of data access that avoids the non-constant access time of ordered and unordered lists and structured trees, and the often-exponential storage requirements of direct access of state spaces of large or variable-length keys.

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Computational complexity varies with the number of instructions required and latency of individual instructions, with the simplest being the bitwise methods (folding), followed by the multiplicative methods, and the most complex (slowest) are the division-based methods.

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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

3754: 332:, each slot is the head of a linked list or chain, and items that collide at the slot are added to the chain. Chains may be kept in random order and searched linearly, or in serial order, or as a self-ordering list by frequency to speed up access. In 305:

mapping of keys into hash codes. Implementation is based on parity-preserving bit operations (XOR and ADD), multiply, or divide. A necessary adjunct to the hash function is a collision-resolution method that employs an auxiliary data structure like

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is a 7-bit character encoding, although it is often stored in 8-bit bytes with the highest-order bit always clear (zero). Therefore, for plain ASCII, the bytes have only 2 = 128 valid values, and the character translation table has only this many

2513:, is a method for constructing universal families of hash functions by combining table lookup with XOR operations. This algorithm has proven to be very fast and of high quality for hashing purposes (especially hashing of integer-number keys). 591:—pairs of inputs that are mapped to the same hash value—increases. If some hash values are more likely to occur than others, then a larger fraction of the lookup operations will have to search through a larger set of colliding table entries. 2521:

each byte of the plain text integer and XORing the loaded values together (again, the starting value can be the identity value or a random seed). The natural extension to 64-bit integers is by use of a table of 2×8 64-bit random numbers.

1084:) is still a valid hash function when used within a single run, but if the values are persisted (for example, written to disk), they can no longer be treated as valid hash values, since in the next run the random value might differ. 348:

until an open slot is located or the entire table is probed (overflow). Searching for the item follows the same procedure until the item is located, an open slot is found, or the entire table has been searched (item not in table).

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offers a natural analogy with its non-technical meaning (to chop up or make a mess out of something), given how hash functions scramble their input data to derive their output. In his research for the precise origin of the term,

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pHash is an open source software library released under the GPLv3 license that implements several perceptual hashing algorithms, and provides a C-like API to use those functions in your own programs. pHash itself is written in

1384:, 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 832: 2331:

of consecutive keys with respect to any block of bits in the key. Consecutive keys within the high bits or low bits of the key (or some other field) are relatively common. The multipliers for various word lengths are:

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Because collisions should be infrequent, and cause a marginal delay but are otherwise harmless, it is usually preferable to choose a faster hash function over one that needs more computation but saves a few collisions.

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or the time of day. It also excludes functions that depend on the memory address of the object being hashed, because the address may change during execution (as may happen on systems that use certain methods of

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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.

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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.

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There are several common algorithms for hashing integers. The method giving the best distribution is data-dependent. One of the simplest and most common methods in practice is the modulo division method.

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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.

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and possibly faster hash function. Selected divisors or multipliers in the division and multiplicative schemes may make more uniform hash functions if the keys are cyclic or have other redundancies.

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appears to have been the first to use the concept of a hash function in a memo dated January 1953, the term itself did not appear in published literature until the late 1960s, in Herbert Hellerman's

242:. Although the concepts overlap to some extent, each one has its own uses and requirements and is designed and optimized differently. The hash function differs from these concepts mainly in terms of 660:

In special cases when the keys are known in advance and the key set is static, a hash function can be found that achieves absolute (or collisionless) uniformity. Such a hash function is said to be

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Use of hash functions relies on statistical properties of key and function interaction: worst-case behavior is intolerably bad but rare, and average-case behavior can be nearly optimal (minimal

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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

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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

2922:, which are designed to have significantly different hashes for even minor differences. Fuzzy hashing has been used to identify malware and has potential for other applications, like 2612:

back into the low byte of the cumulative quantity. The result is a word-size hash code to which a modulo or other reducing operation can be applied to produce the final hash index.

2565:, 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. 3010:

worries about adversarial attack on real time systems, Gonnet has shown that the probability of such a case is "ridiculously small". His representation was that the probability of

1541:. This gives good results over a large number of key sets. A significant drawback of division hashing is that division is microprogrammed on most modern architectures (including 1012:

A hash function that allows only certain table sizes or strings only up to a certain length, or cannot accept a seed (i.e. allow double hashing) is less useful than one that does.

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Buldas, Ahto; Kroonmaa, Andres; Laanoja, Risto (2013). "Keyless Signatures' Infrastructure: How to Build Global Distributed Hash-Trees". In Riis, Nielson H.; Gollmann, D. (eds.).

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adds the feature that hash functions make use of a randomized seed that is generated once when the Python process starts in addition to the input to be hashed. The Python hash (

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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.

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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.

687:. This test is a goodness-of-fit measure: it is the actual distribution of items in buckets versus the expected (or uniform) distribution of items. The formula is 1028:: Minor input changes result in a random-looking output alteration, known as the diffusion property. Thus, hash functions are valuable for key derivation functions. 166:

of arbitrary size to fixed-size values, though there are some hash functions that support variable-length output. The values returned by a hash function are called

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Password storage: The password's hash value does not expose any password details, emphasizing the importance of securely storing hashed passwords on the server.

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of the data to be hashed, in the mathematical sense of the term. This requirement excludes hash functions that depend on external variable parameters, such as

1034:(MACs): Through the integration of a confidential key with the input data, hash functions can generate MACs ensuring the genuineness of the data, such as in 1340:

If the data to be hashed is small enough, then one can use the data itself (reinterpreted as an integer) as the hashed value. The cost of computing this

2879:(in all fairness, the worst case here is gravely pathological: both the text string and substring are composed of a repeated single character, such as 1466:

A mid-squares hash code is produced by squaring the input and extracting an appropriate number of middle digits or bits. For example, if the input is

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to not be a power of 2 and still not have to perform any remainder or division operation, as these computations are sometimes costly. For example, let

2608:. This hash function offsets the bytes 4 bits before adding them together. When the quantity wraps, the high 4 bits are shifted out and if non-zero, 671:

hash function and provides only a marginal advantage over a function with good statistical properties that yields a minimum number of collisions. See

2725:. It can be used directly as the hash code, or a hash function applied to it to map the potentially large value to the hash table size. The value of 1271:

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

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A ratio within one confidence interval (such as 0.95 to 1.05) is indicative that the hash function evaluated has an expected uniform distribution.

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of a small change in input value creating a drastic change in output value. Perceptual hash functions are widely used in finding cases of online

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Hash functions can have some technical properties that make it more likely that they will have a uniform distribution when applied. One is the

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will get large, or both, for the scheme to be computationally feasible. Therefore, it is more suited to hardware or microcode implementation.

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character string representing a decimal number is converted to a numeric quantity for computing, a variable-length string can be converted as

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A good hash function satisfies two basic properties: it should be very fast to compute, and it should minimize duplication of output values (

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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

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Algebraic coding is a variant of the division method of hashing which uses division by a polynomial modulo 2 instead of an integer to map

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Today, especially with the advent of 64-bit word sizes, much more efficient variable-length string hashing by word chunks is available.

2487:. The last two values given above are rounded (up and down, respectively) by more than 1/2 of a least-significant bit to achieve this. 3495:

Wagner, Urs; Lugrin, Thomas (2023), Mulder, Valentin; Mermoud, Alain; Lenders, Vincent; Tellenbach, Bernhard (eds.), "Hash Functions",

4170: 2222:; it should be large, and its binary representation a random mix of 1s and 0s. An important practical special case occurs when 641:, then very few buckets should have more than one or two records. A small number of collisions is virtually inevitable, even if 2293:

this translates into a single integer multiplication and right-shift, making it one of the fastest hash functions to compute.

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is done by default in low-level programming languages and integer division by a power of 2 is simply a right-shift, so, in

1153:. When this approach is used, the hash function must be chosen so that the result has fairly uniform distribution between 1015:

A hash function is applicable in a variety of situations. Particularly within cryptography, notable applications include:

518: 82: 2804:. The straightforward solution, which is to extract such a substring at every character position in the text and compute 1908: 27: 2982:

because of the ability to have a correlation between hashes so similar data can be found (for instance with a differing

3565: 3418: 2553:, or mail messages—their distribution is usually very uneven, with complicated dependencies. For example, text in any 1170:

that may occur in the application. Depending on the function, the remainder may be uniform only for certain values of

3831: 3723: 3676: 565: 525: 129: 89: 1022:: Identical hash values for different files imply equality, providing a reliable means to detect file modifications. 587:. The reason for this last requirement is that the cost of hashing-based methods goes up sharply as the number of 3159: 247: 4152: 3434:

Castro, Julio Cesar Hernandez; et al. (3 February 2005). "The strict avalanche criterion randomness test".

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The meaning of "small enough" depends on the size of the type that is used as the hashed value. For example, in

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among a family of such functions, in such a way that the probability of a collision of any two distinct keys is

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This is useful in cases where keys are devised by a malicious agent, for example in pursuit of a DOS attack.

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in the 1970s, was originally designed for hashing identifiers into compiler symbol tables as given in the

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When testing a hash function, the uniformity of the distribution of hash values can be evaluated by the

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This article is about a computer programming construct. For other meanings of "hash" and "hashing", see

4436: 4387: 4261: 4251: 4231: 2826:, one can use the technique of rolling hash to compute all those hashes with an effort proportional to 2757: 866: 301: 3905: 3614: 1458:

sections consisting of 0x4DEF, 0x1357, 0x159E, 0x091A, and 0x0. Adding yields 0x7FFE, a 15-bit value.

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collisions than perfect hashing and may require more operations than a special-purpose hash function.

4266: 3529: 2624: 3941: 1060:—for a given input value, it must always generate the same hash value. In other words, it must be a 4156: 4146: 3706: 3108: 2528:, meaning that every 3-tuple of keys is equally likely to be mapped to any 3-tuple of hash values. 2276: 465: 297: 282: 263:

output is a hash code used to index a hash table holding the data or records, or pointers to them.

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for their effectiveness, reducing access time to nearly constant. High table loading factors,

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The multiplier should be odd, so the least significant bit of the output is invertible modulo

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considers uniform hashing, that is, any key will map to any particular slot with probability

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Scramble the bits of the key so that the resulting values are uniformly distributed over the

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game-playing programs, which stores a 64-bit hashed representation of the board position.

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unsigned hash(unsigned K) { K ^= K >> (w-m); return (a*K) >> (w-m); }

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modulo 2. If follows that the corresponding hash function will map keys with fewer than

3356: 336:, the table is probed starting from the occupied slot in a specified manner, usually by 4410: 4312: 4307: 4221: 3886: 3729: 3455: 3407: 2943: 2937: 2496: 1410: 1272: 666:. There is no algorithmic way of constructing such a function—searching for one is a 4372: 4216: 4113: 4010: 4000: 3960: 3876: 3827: 3816:

Proceedings of the Eighth ACM Conference on Data and Application Security and Privacy

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A standard technique is to use a modulo function on the key, by selecting a divisor

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A common solution is to compute a fixed hash function with a very large range (say,

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Division-based implementations can be of particular concern because the division is

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This kind of function has some nice theoretical properties, one of which is called

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least significant bits and use the result as an index into a hash table of size

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Other types of data can also use this hashing scheme. For example, when mapping

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may be found in the table, not just for the global set of all possible entries.

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that is uniform on the interval . A hash function uniform on the interval is

827:{\displaystyle {\frac {\sum _{j=0}^{m-1}(b_{j})(b_{j}+1)/2}{(n/2m)(n+2m-1)}},} 142: 4425: 4297: 4136: 4014: 2911: 2905: 1076:

The determinism is in the context of the reuse of the function. For example,

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Unique permutation hashing has a guaranteed best worst-case insertion time.

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Map the key values into ones less than or equal to the size of the table.

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hashing is a form of multiplicative hashing in which the multiplier is

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A folding hash code is produced by dividing the input into sections of

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table slots, then the probability of a bucket receiving many more than

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objects can simply use the value directly, whereas the 64-bit integer

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are used in cybersecurity to secure sensitive data such as passwords.

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Hash Function Construction for Textual and Geometrical Data Retrieval

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Menezes, Alfred J.; van Oorschot, Paul C.; Vanstone, Scott A (1996).

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like "us" or "za" to country names (26 = 676 table entries), 5-digit

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A hash function that maps names to integers from 0 to 15. There is a

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Expected Length of the Longest Probe Sequence in Hash Code Searching

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Pagani, Fabio; Dell'Amico, Matteo; Balzarotti, Davide (2018-03-13).

481: 186:. The values are usually used to index a fixed-size table called a 45: 4034:"pHash.org: Home of pHash, the open source perceptual hash library" 3443: 1414: 20: 2918:, but not exactly the same, as other data. This is in contrast to 2891:="AAA"). The hash function used for the algorithm is usually the 4347: 1773:

or fewer non-zero coefficients, then keys which share fewer than

1393: 1275:

uses a dynamic hash function that requires space proportional to

1081: 407:. The table is often an array with two or more indices (called a 26:"Hash code" redirects here. For the programming competition, see 449:

in a set of points, similar shapes in a list of shapes, similar

3398: 2639: 2282:

unsigned hash(unsigned K) { return (a*K) >> (w-m); }

395:. In these applications, the set of all inputs is some sort of 239: 4110:

The Art of Computer Programming, Vol. 3, Sorting and Searching

4066:

The Art of Computer Programming, Vol. 3, Sorting and Searching

3854:

Sarantinos, Nikolaos; Benzaïd, Chafika; Arabiat, Omar (2016).

3806: 3345:

The Art of Computer Programming, Vol. 3, Sorting and Searching

1305:

A hash function with minimal movement is especially useful in

266:

A hash function may be considered to perform three functions:

3763: 3143: 2682: 1357:, the hash code is a 32-bit integer. Thus the 32-bit integer 422: 421:, and similar names), and the hash function returns an index 19:"hashlink" redirects here. For the Haxe virtual machine, see 2810:

separately, requires a number of operations proportional to

1188:

Variable range with minimal movement (dynamic hash function)

904:, then the multiply-by-inverse is likely a better approach. 309:, or systematic probing of the table to find an empty slot. 4276: 3980: 3478:"Fibonacci Hashing: The Optimization that the World Forgot" 3433: 3404: 1035: 944:. We can replace the division by a (possibly faster) right 4025: 3250:

Knuth conveniently leaves the proof of this to the reader.

1047:: Message hashes are signed rather than the whole message. 3949:

2013 Fourth Cybercrime and Trustworthy Computing Workshop

3071: 2550: 1542: 1346:

hash function is effectively zero. This hash function is

1073:), although sometimes rehashing of the item is possible. 192:. Use of a hash function to index a hash table is called 4033: 3853: 3078:, even though it was already widespread jargon by then. 1673:. The remainder using polynomial arithmetic modulo 2 is 270:

Convert variable-length keys into fixed-length (usually

214:

Hash functions are related to (and often confused with)

2914:, also known as similarity hashing, is a technique for 631:

records should be vanishingly small. In particular, if

3634:

A New Hashing Method with Application for Game Playing

2966:

of the multimedia are similar. This is in contrast to

2297:

corrects for this. The resulting function looks like:

1392:), the table has only 2 = 256 entries; in the case of 3986: 3940:

Oliver, Jonathan; Cheng, Chun; Chen, Yanggui (2013).

3694:"Performance in Practice of String Hashing Functions" 3581:

Dolev, Shlomi; Lahiani, Limor; Haviv, Yinnon (2013).

3551: 3549: 3499:, Cham: Springer Nature Switzerland, pp. 21–24, 3497:

Trends in Data Protection and Encryption Technologies

3162: 2573:

Simplistic hash functions may add the first and last

1911: 695: 3897: 1514:

which is a prime number close to the table size, so

3471: 3469: 3283:

Hash_RC6 — Variable length Hash algorithm using RC6

3280:Aggarwal, Kirti; Verma, Harsh K. (March 19, 2015). 2545:When the data values are long (or variable-length) 1968:{\displaystyle P(x)=\prod _{j\in S}(x-\alpha ^{j})} 1433:example, a simple hash function might mask off the 399:, and the hashing function can be interpreted as a 70:. Unsourced material may be challenged and removed. 4192: 3933: 3546: 3406: 3234: 1967: 1350:, as it maps each input to a distinct hash value. 826: 365:Hash functions are an essential ingredient of the 3580: 2782:-character string by advancing a window of width 2163:Standard multiplicative hashing uses the formula 4423: 3691: 3466: 3003:, a characteristic of universal hash functions. 2719:as the characters of the input string of length 2214:is an appropriately chosen value that should be 2106:is any nonzero polynomial modulo 2 with at most 1990:are computed in this field. Then the degree of 1223:is the number of allowed hash values) such that 296:). Hash functions rely on generating favorable 3939: 2842:is the number of occurrences of the substring. 2540: 3698:Database Systems for Advanced Applications '97 3651: 4178: 4100: 4098: 3279: 2271:. This is special because arithmetic modulo 2150: 1326: 893:on nearly all chip architectures. Division ( 594:This criterion only requires the value to be 3818:. New York, NY, USA: ACM. pp. 354–365. 3530:"3. Data model — Python 3.6.1 documentation" 3494: 3335: 3333: 3331: 3329: 3229: 3169: 2845:The most familiar algorithm of this type is 2505:Tabulation hashing, more generally known as 2026:is a root, it follows that the coefficients 433:, and many other disciplines, to solve many 323:Hash functions are used in conjunction with 3665:Compilers: Principles, Techniques and Tools 3382:"Understanding CPU caching and performance" 3235:{\displaystyle S=\{1,2,3,4,5,6,8,10,12,9\}} 2633: 2531: 150:between keys "John Smith" and "Sandra Dee". 16:Mapping arbitrary data to fixed-size values 4185: 4171: 4095: 3769: 3692:Ramakrishna, M. V.; Zobel, Justin (1997). 1868:be the smallest set of integers such that 1396:characters, the table would have 17 × 2 = 678: 3917: 3912:. 3, Supplement (September 2006): 91–97. 3705: 3598: 3556:Sedgewick, Robert (2002). "14. Hashing". 3555: 3504: 3326: 2158: 1335: 566:Learn how and when to remove this message 130:Learn how and when to remove this message 3903: 3475: 2926:and detecting multiple versions of code. 2625:Universal hashing § Hashing strings 2557:has highly non-uniform distributions of 1427: 141: 3627: 3436:Mathematics and Computers in Simulation 2849:with best and average case performance 460:Hash tables are also used to implement 387:A special case of hashing is known as 4424: 4082: 3379: 2736:, or 19 decimal digits with radix 10. 2618: 1409:The same technique can be used to map 357:Hash functions are also used to build 4166: 4104: 4087:(Technical report). Ontario, Canada: 4060: 4031: 3339: 2820:. However, with the proper choice of 2279:, for example, this function becomes 2244:. In this case, this formula becomes 1312: 4032:Klinger, Evan; Starkweather, David. 3615:"CS 3110 Lecture 21: Hash functions" 2788:characters along the string, where 2583: 2303: 1777:bits are guaranteed to not collide. 609:In other words, if a typical set of 504:adding citations to reliable sources 475: 425:. This principle is widely used in 68:adding citations to reliable sources 39: 3942:"TLSH -- A Locality Sensitive Hash" 3617:. Section "Multiplicative hashing". 2950:that produces a snippet, hash, or 2681:. This is simply a polynomial in a 1795:(the last of which is a divisor of 1548: 1505: 352: 28:Hash Code (programming competition) 13: 3772:"NIST Special Publication 800-168" 3076:Digital Computer System Principles 2930: 2760:, one can compute a hash function 2568: 2490: 2136:bits in common to unique indices. 1633:can be regarded as the polynomial 14: 4448: 4130: 3861:2016 IEEE Trustcom/BigDataSE/ISPA 3752: 3700:. DASFAA 1997. pp. 215–224. 3156:For example, for n=15, k=4, t=6, 3022:keys mapping to a single slot is 2958:. A perceptual hash is a type of 2638:Analogous to the way an ASCII or 2592:The classic approach, dubbed the 2202:, which produces a hash value in 2139:The usual outcome is that either 1483:, then squaring the key produces 1096: 855:is the number of items in bucket 3476:Sharupke, Malte (16 June 2018). 3409:Handbook of Applied Cryptography 2936:This section is an excerpt from 2904:This section is an excerpt from 1861:. The derivation is as follows: 1087: 1051: 1007: 480: 376:that is used to test whether an 248:non-cryptographic hash functions 44: 4137:Calculate hash of a given value 4076: 4054: 3989:Secure IT Systems. NordSec 2013 3746: 3685: 3645: 3621: 3607: 3574: 3522: 3347:. Reading, MA., United States: 3253: 3244: 3150: 3136: 2739: 2319:is the machine word length and 1066:pseudo-random number generators 965: 491:needs additional citations for 55:needs additional citations for 3770:Breitinger, Frank (May 2014). 3560:(3 ed.). Addison Wesley. 3488: 3427: 3373: 3305: 3273: 3127: 2916:detecting data that is similar 2756:In some applications, such as 1979:and where the coefficients of 1962: 1943: 1921: 1915: 1799:) and is constructed from the 1461: 844:is the number of buckets, and 815: 794: 791: 774: 761: 742: 739: 726: 312: 1: 4143:The Goulburn Hashing Function 3893:. 10.1109/TrustCom.2016.0274. 3809:"Beyond Precision and Recall" 3313:"NIST Glossary — hash digest" 3266: 2898: 2752:Linear congruential generator 2066:, so they are all 0 or 1. If 984:that selects a hash function 925:pseudorandom number generator 872: 578: 471: 403:of that space into a grid of 3997:10.1007/978-3-642-41488-6_21 3587:Theoretical Computer Science 3583:"Unique permutation hashing" 3452:10.1016/j.matcom.2004.09.001 2920:cryptographic hash functions 2541:Hashing variable-length data 1217:is the key being hashed and 1032:Message Authentication Codes 907:We can allow the table size 252:cryptographic hash functions 7: 3506:10.1007/978-3-031-33386-6_5 3292:10.1109/ICACEA.2015.7164747 3081: 2990: 2110:nonzero coefficients, then 919:be significantly less than 257: 10: 4453: 3919:10.1016/j.diin.2006.06.015 3873:10.1109/TrustCom.2016.0274 3716:10.1142/9789812819536_0023 3380:Stokes, Jon (2002-07-08). 3052: 2935: 2903: 2749: 2743: 2691:that takes the components 2622: 2494: 2151:Unique permutation hashing 1444: 1417:like 13083 to city names ( 1369:and 64-bit floating-point 1361:and 32-bit floating-point 1327:Hashing integer data types 1250:with probability close to 969: 867:strict avalanche criterion 316: 276:parity-preserving operator 198:scatter-storage addressing 32: 25: 18: 4401: 4285: 4204: 3600:10.1016/j.tcs.2012.12.047 2549:—such as personal names, 1810:gives an example: taking 1131:, and use the division's 1056:A hash procedure must be 298:probability distributions 3904:Kornblum, Jesse (2006). 3756:A Handbook of Algorithms 3120: 3109:Low-discrepancy sequence 2962:, which is analogous if 2948:fingerprinting algorithm 2794:is a fixed integer, and 2634:Radix conversion hashing 2532:Customized hash function 1561:bits. In this approach, 1476:and the hash table size 1411:two-letter country codes 1287:to compute the value of 1125:), divide the result by 1114:of allowed hash values. 162:that can be used to map 4406:List of data structures 3951:. IEEE. pp. 7–13. 3824:10.1145/3176258.3176306 3787:10.6028/NIST.SP.800-168 3094:Nearest neighbor search 2960:locality-sensitive hash 1769:is constructed to have 1307:distributed hash tables 840:is the number of keys, 679:Testing and measurement 673:universal hash function 443:three-dimensional space 232:randomization functions 4089:University of Waterloo 3867:. pp. 1782–1787. 3236: 3099:Distributed hash table 3089:List of hash functions 2976:copyright infringement 2970:, which relies on the 2159:Multiplicative hashing 1969: 1568:, and we postulate an 1336:Identity hash function 1145:, this can be done by 828: 725: 431:computational geometry 246:. Hash tables may use 236:error-correcting codes 151: 3910:Digital Investigation 3259:Unisys large systems. 3237: 2968:cryptographic hashing 2596:based on the work of 2121:is not a multiple of 1970: 1572:th-degree polynomial 1428:Trivial hash function 1141:is itself a power of 829: 699: 615:records is hashed to 596:uniformly distributed 145: 35:Hash (disambiguation) 4303:Breadth-first search 3160: 3039:is the load factor, 2954:of various forms of 2924:data loss prevention 2885:="AAAAAAAAAAA", and 2526:3-tuple independence 2236:are powers of 2 and 1977:α ∈ GF(2) 1909: 1382:upper and lower case 1302:have been invented. 982:randomized algorithm 693: 647:is much larger than 500:improve this article 369:, a space-efficient 334:open address hashing 64:improve this article 4393:Topological sorting 4323:Dynamic programming 4083:Gonnet, G. (1978). 3357:1973acp..book.....K 3114:Transposition table 2619:Word length folding 2598:Peter J. Weinberger 2143:will get large, or 1164:, for any value of 4411:List of algorithms 4318:Divide and conquer 4313:Depth-first search 4308:Brute-force search 4222:Binary search tree 3957:10.1109/ctc.2013.9 3629:Zobrist, Albert L. 3558:Algorithms in Java 3232: 3066:notes that, while 2944:Perceptual hashing 2938:Perceptual hashing 2551:web page addresses 2497:Tabulation hashing 1965: 1942: 1595:+ ⋯ + ζ 1313:Data normalization 1273:Extendible hashing 1236: + 1) = 1071:garbage collection 1020:Integrity checking 824: 462:associative arrays 445:, such as finding 435:proximity problems 152: 4437:Search algorithms 4419: 4418: 4217:Associative array 4119:978-0-201-89685-5 4006:978-3-642-41487-9 3966:978-1-4799-3076-0 3882:978-1-5090-3205-1 3779:NIST Publications 3516:978-3-031-33386-6 3366:978-0-201-03803-3 2980:digital forensics 2893:Rabin fingerprint 2689: > 1 2584:Character folding 2547:character strings 2304:Fibonacci hashing 1927: 1783:is a function of 1378:character strings 1108:, and the number 978:universal hashing 972:Universal hashing 899:hardware multiply 819: 576: 575: 568: 550: 427:computer graphics 389:geometric hashing 380:is a member of a 342:quadratic probing 228:lossy compression 140: 139: 132: 114: 4444: 4388:String-searching 4187: 4180: 4173: 4164: 4163: 4149:) by Mayur Patel 4124: 4123: 4106:Knuth, Donald E. 4102: 4093: 4092: 4080: 4074: 4073: 4062:Knuth, Donald E. 4058: 4052: 4051: 4045: 4044: 4029: 4023: 4022: 3984: 3978: 3977: 3975: 3973: 3946: 3937: 3931: 3930: 3928: 3926: 3921: 3901: 3895: 3894: 3866: 3851: 3845: 3844: 3842: 3840: 3813: 3804: 3798: 3797: 3795: 3793: 3776: 3767: 3761: 3760: 3750: 3744: 3743: 3741: 3740: 3709: 3689: 3683: 3682: 3649: 3643: 3641: 3639: 3625: 3619: 3618: 3611: 3605: 3604: 3602: 3578: 3572: 3571: 3553: 3544: 3543: 3541: 3540: 3526: 3520: 3519: 3508: 3492: 3486: 3485: 3473: 3464: 3463: 3431: 3425: 3424: 3412: 3402: 3396: 3395: 3393: 3392: 3377: 3371: 3370: 3341:Knuth, Donald E. 3337: 3324: 3323: 3321: 3319: 3309: 3303: 3302: 3300: 3298: 3277: 3260: 3257: 3251: 3248: 3242: 3241: 3239: 3238: 3233: 3154: 3148: 3140: 3134: 3131: 3048: 3038: 3032: 3021: 3015: 3002: 2972:avalanche effect 2946:is the use of a 2890: 2884: 2878: 2863: 2841: 2835: 2825: 2819: 2809: 2803: 2793: 2787: 2781: 2771: 2765: 2758:substring search 2735: 2730: 2724: 2718: 2690: 2680: 2578: 2555:natural language 2486: 2476: 2475: 2472: 2469: 2466: 2463: 2460: 2454: 2450: 2449: 2446: 2443: 2427: 2426: 2423: 2420: 2417: 2411: 2407: 2406: 2403: 2387: 2386: 2383: 2380: 2374: 2370: 2369: 2353: 2352: 2346: 2322: 2318: 2314: 2292: 2288: 2274: 2270: 2269: 2261: 2239: 2235: 2228: 2221: 2216:relatively prime 2213: 2209: 2201: 2200: 2180: 2146: 2142: 2135: 2131: 2120: 2109: 2105: 2065: 2057: 2056: 2042: 2031: 2025: 2021: 2010: 2006: 2004: 1989: 1978: 1974: 1972: 1971: 1966: 1961: 1960: 1941: 1901: 1878: 1867: 1860: 1825: 1805: 1798: 1794: 1790: 1786: 1782: 1776: 1772: 1768: 1757: 1720: 1672: 1632: 1599: 1571: 1567: 1560: 1556: 1549:Algebraic coding 1540: 1532: 1513: 1506:Division hashing 1501: 1500: 1497: 1494: 1491: 1488: 1482: 1481: 1475: 1474: 1471: 1456: 1452: 1440: 1436: 1423: 1422: 1405: 1404: 1401: 1372: 1368: 1364: 1360: 1301: 1286: 1280: 1260: 1249: 1222: 1216: 1210: 1175: 1169: 1163: 1156: 1144: 1140: 1130: 1124: 1120: 1113: 1107: 1002: 996: 989: 957: 943: 933: 922: 918: 912: 902:functional units 858: 854: 843: 839: 833: 831: 830: 825: 820: 818: 784: 772: 768: 754: 753: 738: 737: 724: 713: 697: 685:chi-squared test 655:birthday problem 652: 646: 640: 630: 620: 614: 571: 564: 560: 557: 551: 549: 508: 484: 476: 353:Specialized uses 278:like ADD or XOR, 135: 128: 124: 121: 115: 113: 72: 48: 40: 4452: 4451: 4447: 4446: 4445: 4443: 4442: 4441: 4422: 4421: 4420: 4415: 4397: 4328:Graph traversal 4281: 4205:Data structures 4200: 4194:Data structures 4191: 4133: 4128: 4127: 4120: 4103: 4096: 4081: 4077: 4068:. Reading, MA: 4059: 4055: 4042: 4040: 4030: 4026: 4007: 3985: 3981: 3971: 3969: 3967: 3944: 3938: 3934: 3924: 3922: 3902: 3898: 3883: 3864: 3852: 3848: 3838: 3836: 3834: 3811: 3805: 3801: 3791: 3789: 3774: 3768: 3764: 3751: 3747: 3738: 3736: 3726: 3690: 3686: 3679: 3671:. p. 435. 3667:. Reading, MA: 3650: 3646: 3637: 3626: 3622: 3613: 3612: 3608: 3579: 3575: 3568: 3554: 3547: 3538: 3536: 3534:docs.python.org 3528: 3527: 3523: 3517: 3493: 3489: 3474: 3467: 3432: 3428: 3421: 3403: 3399: 3390: 3388: 3378: 3374: 3367: 3338: 3327: 3317: 3315: 3311: 3310: 3306: 3296: 3294: 3278: 3274: 3269: 3264: 3263: 3258: 3254: 3249: 3245: 3161: 3158: 3157: 3155: 3151: 3141: 3137: 3132: 3128: 3123: 3118: 3084: 3068:Hans Peter Luhn 3055: 3040: 3034: 3023: 3017: 3011: 2997: 2993: 2988: 2987: 2941: 2933: 2931:Perceptual hash 2928: 2927: 2909: 2901: 2886: 2880: 2865: 2864:and worst case 2850: 2837: 2827: 2821: 2811: 2805: 2795: 2789: 2783: 2777: 2767: 2761: 2754: 2748: 2742: 2733: 2726: 2720: 2716: 2706: 2699: 2692: 2685: 2679: 2669: 2662: 2652: 2643: 2636: 2627: 2621: 2586: 2574: 2571: 2569:Middle and ends 2563:character pairs 2543: 2534: 2507:Zobrist hashing 2503: 2501:Zobrist hashing 2495:Main articles: 2493: 2491:Zobrist hashing 2484: 2479: 2473: 2470: 2467: 2464: 2461: 2458: 2456: 2453: 2447: 2444: 2441: 2439: 2438: 2430: 2424: 2421: 2418: 2415: 2413: 2410: 2404: 2401: 2399: 2398: 2390: 2384: 2381: 2378: 2376: 2373: 2367: 2365: 2364: 2356: 2350: 2348: 2345: 2341: 2320: 2316: 2312: 2306: 2301: 2290: 2286: 2283: 2272: 2267: 2259: 2253: 2245: 2240:is the machine 2237: 2230: 2223: 2219: 2211: 2203: 2198: 2178: 2172: 2164: 2161: 2153: 2144: 2140: 2133: 2122: 2111: 2107: 2104: 2094: 2084: 2067: 2064: 2055: 2050: 2049: 2048: 2044: 2033: 2027: 2023: 2012: 2008: 2000: 1991: 1980: 1976: 1956: 1952: 1931: 1910: 1907: 1906: 1880: 1869: 1865: 1827: 1811: 1803: 1796: 1792: 1788: 1784: 1780: 1774: 1770: 1759: 1756: 1752: 1746: 1739: 1722: 1719: 1709: 1699: 1674: 1671: 1661: 1651: 1634: 1631: 1627: 1621: 1614: 1601: 1598: 1591: 1573: 1569: 1562: 1558: 1554: 1551: 1534: 1515: 1511: 1508: 1498: 1495: 1492: 1489: 1486: 1484: 1479: 1477: 1472: 1469: 1467: 1464: 1454: 1450: 1447: 1438: 1434: 1430: 1420: 1418: 1402: 1399: 1397: 1370: 1366: 1362: 1358: 1338: 1329: 1315: 1288: 1282: 1276: 1259: + 1) 1251: 1224: 1218: 1212: 1197: 1190: 1171: 1165: 1158: 1154: 1142: 1136: 1126: 1123:2 − 1 1122: 1118: 1109: 1103: 1099: 1090: 1054: 1010: 998: 991: 985: 974: 968: 953:(key) >> 949: 935: 928: 920: 914: 908: 891:microprogrammed 875: 856: 853: 845: 841: 837: 780: 773: 764: 749: 745: 733: 729: 714: 703: 698: 696: 694: 691: 690: 681: 648: 642: 632: 622: 616: 610: 581: 572: 561: 555: 552: 515:"Hash function" 509: 507: 497: 485: 474: 355: 330:chained hashing 321: 315: 260: 136: 125: 119: 116: 79:"Hash function" 73: 71: 61: 49: 38: 31: 24: 17: 12: 11: 5: 4450: 4440: 4439: 4434: 4432:Hash functions 4417: 4416: 4414: 4413: 4408: 4402: 4399: 4398: 4396: 4395: 4390: 4385: 4380: 4375: 4370: 4365: 4360: 4355: 4350: 4345: 4340: 4335: 4330: 4325: 4320: 4315: 4310: 4305: 4300: 4295: 4289: 4287: 4283: 4282: 4280: 4279: 4274: 4269: 4264: 4259: 4254: 4249: 4244: 4239: 4234: 4229: 4224: 4219: 4214: 4208: 4206: 4202: 4201: 4190: 4189: 4182: 4175: 4167: 4161: 4160: 4150: 4140: 4132: 4131:External links 4129: 4126: 4125: 4118: 4094: 4091:. CS-RR-78-46. 4075: 4072:. p. 540. 4070:Addison-Wesley 4053: 4024: 4005: 3979: 3965: 3932: 3896: 3881: 3846: 3832: 3799: 3762: 3745: 3724: 3707:10.1.1.18.7520 3684: 3677: 3669:Addison-Wesley 3644: 3631:(April 1970), 3620: 3606: 3573: 3567:978-0201361209 3566: 3545: 3521: 3515: 3487: 3482:Probably Dance 3465: 3426: 3420:978-0849385230 3419: 3397: 3372: 3365: 3349:Addison-Wesley 3325: 3304: 3271: 3270: 3268: 3265: 3262: 3261: 3252: 3243: 3231: 3228: 3225: 3222: 3219: 3216: 3213: 3210: 3207: 3204: 3201: 3198: 3195: 3192: 3189: 3186: 3183: 3180: 3177: 3174: 3171: 3168: 3165: 3149: 3135: 3125: 3124: 3122: 3119: 3117: 3116: 3111: 3106: 3101: 3096: 3091: 3085: 3083: 3080: 3054: 3051: 2992: 2989: 2978:as well as in 2942: 2934: 2932: 2929: 2910: 2902: 2900: 2897: 2744:Main article: 2741: 2738: 2711: 2704: 2697: 2677: 2667: 2663:a + ⋯ + 2657: 2647: 2635: 2632: 2620: 2617: 2585: 2582: 2570: 2567: 2542: 2539: 2533: 2530: 2511:Albert Zobrist 2492: 2489: 2481: 2480: 2477: 2451: 2431: 2428: 2408: 2391: 2388: 2371: 2357: 2354: 2343: 2305: 2302: 2299: 2285:and for fixed 2281: 2249: 2168: 2160: 2157: 2152: 2149: 2102: 2092: 2079: 2062: 2051: 1964: 1959: 1955: 1951: 1948: 1945: 1940: 1937: 1934: 1930: 1926: 1923: 1920: 1917: 1914: 1754: 1750: 1744: 1734: 1717: 1707: 1694: 1669: 1659: 1646: 1629: 1625: 1619: 1609: 1596: 1586: 1550: 1547: 1507: 1504: 1463: 1460: 1446: 1443: 1429: 1426: 1386:extended ASCII 1337: 1334: 1328: 1325: 1314: 1311: 1269:spiral hashing 1265:Linear hashing 1189: 1186: 1162: − 1 1098: 1097:Variable range 1095: 1089: 1086: 1053: 1050: 1049: 1048: 1042: 1039: 1029: 1026:Key derivation 1023: 1009: 1006: 970:Main article: 967: 964: 874: 871: 849: 823: 817: 814: 811: 808: 805: 802: 799: 796: 793: 790: 787: 783: 779: 776: 771: 767: 763: 760: 757: 752: 748: 744: 741: 736: 732: 728: 723: 720: 717: 712: 709: 706: 702: 680: 677: 580: 577: 574: 573: 488: 486: 479: 473: 470: 455:image database 374:data structure 354: 351: 346:double hashing 338:linear probing 317:Main article: 314: 311: 290: 289: 286: 279: 259: 256: 244:data integrity 138: 137: 52: 50: 43: 15: 9: 6: 4: 3: 2: 4449: 4438: 4435: 4433: 4430: 4429: 4427: 4412: 4409: 4407: 4404: 4403: 4400: 4394: 4391: 4389: 4386: 4384: 4381: 4379: 4376: 4374: 4371: 4369: 4366: 4364: 4361: 4359: 4356: 4354: 4351: 4349: 4346: 4344: 4343:Hash function 4341: 4339: 4336: 4334: 4331: 4329: 4326: 4324: 4321: 4319: 4316: 4314: 4311: 4309: 4306: 4304: 4301: 4299: 4298:Binary search 4296: 4294: 4291: 4290: 4288: 4284: 4278: 4275: 4273: 4270: 4268: 4265: 4263: 4260: 4258: 4255: 4253: 4250: 4248: 4245: 4243: 4240: 4238: 4235: 4233: 4230: 4228: 4225: 4223: 4220: 4218: 4215: 4213: 4210: 4209: 4207: 4203: 4199: 4195: 4188: 4183: 4181: 4176: 4174: 4169: 4168: 4165: 4158: 4154: 4151: 4148: 4144: 4141: 4138: 4135: 4134: 4121: 4115: 4111: 4107: 4101: 4099: 4090: 4086: 4079: 4071: 4067: 4063: 4057: 4050: 4039: 4035: 4028: 4021: 4016: 4012: 4008: 4002: 3998: 3994: 3990: 3983: 3968: 3962: 3958: 3954: 3950: 3943: 3936: 3920: 3915: 3911: 3907: 3900: 3892: 3888: 3884: 3878: 3874: 3870: 3863: 3862: 3857: 3850: 3835: 3833:9781450356329 3829: 3825: 3821: 3817: 3810: 3803: 3788: 3784: 3780: 3773: 3766: 3759:. N.B. Singh. 3758: 3757: 3753:Singh, N. B. 3749: 3735: 3731: 3727: 3725:981-02-3107-5 3721: 3717: 3713: 3708: 3703: 3699: 3695: 3688: 3680: 3678:0-201-10088-6 3674: 3670: 3666: 3662: 3661:Ullman, J. D. 3658: 3654: 3648: 3636: 3635: 3630: 3624: 3616: 3610: 3601: 3596: 3592: 3588: 3584: 3577: 3569: 3563: 3559: 3552: 3550: 3535: 3531: 3525: 3518: 3512: 3507: 3502: 3498: 3491: 3483: 3479: 3472: 3470: 3461: 3457: 3453: 3449: 3445: 3441: 3437: 3430: 3422: 3416: 3413:. CRC Press. 3411: 3410: 3401: 3387: 3383: 3376: 3368: 3362: 3358: 3354: 3350: 3346: 3342: 3336: 3334: 3332: 3330: 3314: 3308: 3293: 3289: 3285: 3284: 3276: 3272: 3256: 3247: 3226: 3223: 3220: 3217: 3214: 3211: 3208: 3205: 3202: 3199: 3196: 3193: 3190: 3187: 3184: 3181: 3178: 3175: 3172: 3166: 3163: 3153: 3145: 3139: 3130: 3126: 3115: 3112: 3110: 3107: 3105: 3102: 3100: 3097: 3095: 3092: 3090: 3087: 3086: 3079: 3077: 3073: 3069: 3065: 3060: 3050: 3047: 3043: 3037: 3030: 3027: 3020: 3014: 3009: 3004: 3001: 2985: 2981: 2977: 2973: 2969: 2965: 2961: 2957: 2953: 2949: 2945: 2939: 2925: 2921: 2917: 2913: 2912:Fuzzy hashing 2907: 2906:Fuzzy hashing 2896: 2894: 2889: 2883: 2876: 2872: 2868: 2861: 2857: 2853: 2848: 2843: 2840: 2834: 2831: + 2830: 2824: 2818: 2814: 2808: 2802: 2798: 2792: 2786: 2780: 2775: 2770: 2764: 2759: 2753: 2747: 2737: 2729: 2723: 2714: 2710: 2703: 2696: 2688: 2684: 2676: 2672: 2666: 2660: 2656: 2650: 2646: 2641: 2631: 2626: 2616: 2613: 2611: 2607: 2606:"Dragon Book" 2603: 2599: 2595: 2590: 2581: 2577: 2566: 2564: 2560: 2556: 2552: 2548: 2538: 2529: 2527: 2522: 2518: 2514: 2512: 2508: 2502: 2498: 2488: 2436: 2432: 2396: 2392: 2362: 2358: 2339: 2335: 2334: 2333: 2330: 2326: 2323:(phi) is the 2310: 2298: 2294: 2280: 2278: 2265: 2257: 2252: 2248: 2243: 2233: 2226: 2217: 2210:. The value 2207: 2204:{0, …, 2196: 2192: 2188: 2184: 2176: 2171: 2167: 2156: 2148: 2137: 2129: 2125: 2118: 2114: 2101: 2097: 2091: 2087: 2082: 2078: 2074: 2070: 2061: 2054: 2047: 2040: 2036: 2030: 2019: 2015: 2011:is a root of 2003: 1998: 1994: 1987: 1983: 1957: 1953: 1949: 1946: 1938: 1935: 1932: 1928: 1924: 1918: 1912: 1903: 1900: 1896: 1892: 1888: 1884: 1877: 1873: 1870:{1,2,…, 1862: 1858: 1854: 1850: 1846: 1842: 1838: 1834: 1830: 1824:) = (15,10,7) 1823: 1819: 1815: 1809: 1802: 1778: 1766: 1762: 1749: 1743: 1737: 1733: 1729: 1725: 1716: 1712: 1706: 1702: 1697: 1693: 1689: 1685: 1681: 1677: 1668: 1664: 1658: 1654: 1649: 1645: 1641: 1637: 1624: 1618: 1612: 1608: 1604: 1594: 1589: 1584: 1580: 1576: 1565: 1546: 1544: 1538: 1530: 1526: 1522: 1518: 1503: 1459: 1442: 1425: 1416: 1412: 1407: 1395: 1391: 1387: 1383: 1379: 1374: 1356: 1351: 1349: 1345: 1344: 1333: 1324: 1321: 1310: 1308: 1303: 1299: 1295: 1291: 1285: 1279: 1274: 1270: 1266: 1262: 1258: 1254: 1247: 1243: 1239: 1235: 1231: 1227: 1221: 1215: 1208: 1204: 1200: 1194: 1185: 1183: 1182:prime numbers 1179: 1174: 1168: 1161: 1152: 1148: 1139: 1134: 1129: 1115: 1112: 1106: 1094: 1088:Defined range 1085: 1083: 1079: 1074: 1072: 1067: 1063: 1059: 1058:deterministic 1052:Deterministic 1046: 1043: 1040: 1037: 1033: 1030: 1027: 1024: 1021: 1018: 1017: 1016: 1013: 1008:Applicability 1005: 1001: 995: 988: 983: 979: 973: 963: 959: 956: 952: 947: 941: 938: 931: 926: 923:. Consider a 917: 911: 905: 903: 900: 896: 892: 887: 883: 879: 870: 868: 863: 860: 852: 848: 834: 821: 812: 809: 806: 803: 800: 797: 788: 785: 781: 777: 769: 765: 758: 755: 750: 746: 734: 730: 721: 718: 715: 710: 707: 704: 700: 688: 686: 676: 674: 669: 665: 664: 658: 656: 651: 645: 639: 635: 629: 625: 619: 613: 607: 603: 601: 597: 592: 590: 586: 570: 567: 559: 548: 545: 541: 538: 534: 531: 527: 524: 520: 517: – 516: 512: 511:Find sources: 505: 501: 495: 494: 489:This section 487: 483: 478: 477: 469: 467: 463: 458: 457:, and so on. 456: 452: 448: 447:closest pairs 444: 440: 436: 432: 428: 424: 420: 416: 412: 411: 406: 402: 398: 394: 390: 385: 383: 379: 375: 372: 371:probabilistic 368: 363: 360: 350: 347: 343: 339: 335: 331: 326: 320: 310: 308: 303: 299: 295: 287: 284: 280: 277: 273: 269: 268: 267: 264: 255: 253: 249: 245: 241: 237: 233: 229: 225: 221: 217: 212: 210: 205: 201: 199: 195: 191: 190: 185: 181: 177: 173: 169: 165: 161: 157: 156:hash function 149: 144: 134: 131: 123: 112: 109: 105: 102: 98: 95: 91: 88: 84: 81: – 80: 76: 75:Find sources: 69: 65: 59: 58: 53:This article 51: 47: 42: 41: 36: 29: 22: 4368:Root-finding 4342: 4293:Backtracking 4257:Segment tree 4227:Fenwick tree 4139:by Timo Denk 4109: 4084: 4078: 4065: 4056: 4047: 4041:. Retrieved 4037: 4027: 4018: 3988: 3982: 3972:December 12, 3970:. Retrieved 3948: 3935: 3923:. Retrieved 3909: 3899: 3860: 3849: 3839:December 12, 3837:. Retrieved 3815: 3802: 3790:. Retrieved 3778: 3765: 3755: 3748: 3737:. Retrieved 3697: 3687: 3664: 3647: 3633: 3623: 3609: 3590: 3586: 3576: 3557: 3537:. Retrieved 3533: 3524: 3496: 3490: 3481: 3439: 3435: 3429: 3408: 3400: 3389:. Retrieved 3386:Ars Technica 3385: 3375: 3344: 3316:. Retrieved 3307: 3295:. Retrieved 3282: 3275: 3255: 3246: 3152: 3138: 3129: 3075: 3064:Donald Knuth 3058: 3056: 3045: 3041: 3035: 3028: 3025: 3018: 3012: 3005: 2999: 2994: 2887: 2881: 2874: 2870: 2866: 2859: 2855: 2851: 2844: 2838: 2832: 2828: 2822: 2816: 2812: 2806: 2800: 2796: 2790: 2784: 2778: 2768: 2762: 2755: 2746:Rolling hash 2740:Rolling hash 2727: 2721: 2712: 2708: 2701: 2694: 2686: 2674: 2670: 2664: 2658: 2654: 2648: 2644: 2637: 2628: 2614: 2591: 2587: 2575: 2572: 2544: 2535: 2525: 2523: 2519: 2515: 2506: 2504: 2482: 2434: 2394: 2360: 2337: 2325:golden ratio 2307: 2295: 2284: 2263: 2255: 2250: 2246: 2231: 2224: 2205: 2194: 2190: 2186: 2182: 2174: 2169: 2165: 2162: 2154: 2138: 2127: 2123: 2116: 2112: 2099: 2095: 2089: 2088:+ ⋯ + 2085: 2080: 2076: 2072: 2068: 2059: 2052: 2045: 2038: 2034: 2028: 2017: 2013: 2001: 1996: 1992: 1985: 1981: 1904: 1898: 1894: 1890: 1886: 1882: 1875: 1871: 1863: 1856: 1852: 1848: 1844: 1840: 1836: 1832: 1828: 1821: 1817: 1813: 1801:finite field 1779: 1764: 1760: 1747: 1741: 1735: 1731: 1727: 1723: 1714: 1710: 1704: 1700: 1695: 1691: 1687: 1683: 1679: 1675: 1666: 1662: 1656: 1655:+ ⋯ + 1652: 1647: 1643: 1639: 1635: 1622: 1616: 1610: 1606: 1602: 1592: 1587: 1582: 1578: 1574: 1563: 1552: 1536: 1528: 1524: 1520: 1516: 1509: 1465: 1453:bits, where 1448: 1431: 1408: 1375: 1352: 1341: 1339: 1330: 1316: 1304: 1297: 1293: 1289: 1283: 1277: 1263: 1256: 1252: 1245: 1241: 1237: 1233: 1229: 1225: 1219: 1213: 1206: 1202: 1198: 1195: 1191: 1172: 1166: 1159: 1151:bit shifting 1137: 1127: 1116: 1110: 1104: 1100: 1091: 1075: 1055: 1014: 1011: 999: 993: 986: 980:scheme is a 977: 975: 966:Universality 960: 954: 950: 939: 936: 929: 915: 909: 906: 888: 884: 880: 876: 864: 861: 850: 846: 835: 689: 682: 661: 659: 649: 643: 637: 633: 627: 623: 617: 611: 608: 604: 599: 595: 593: 588: 582: 562: 556:October 2017 553: 543: 536: 529: 522: 510: 498:Please help 493:verification 490: 466:dynamic sets 459: 418: 414: 408: 404: 397:metric space 392: 386: 367:Bloom filter 364: 356: 333: 329: 322: 307:linked lists 302:pathological 291: 272:machine-word 265: 261: 224:fingerprints 220:check digits 213: 206: 202: 197: 193: 187: 183: 182:, or simply 179: 176:hash digests 175: 171: 167: 155: 153: 126: 117: 107: 100: 93: 86: 74: 62:Please help 57:verification 54: 4247:Linked list 3792:January 11, 3297:January 24, 2952:fingerprint 2776:of a given 2772:-character 1797:2 − 1 1462:Mid-squares 1390:ISO Latin 1 1320:equivalence 1147:bit masking 585:probability 419:bucket grid 393:grid method 325:hash tables 313:Hash tables 168:hash values 4426:Categories 4383:Sweep line 4358:Randomized 4286:Algorithms 4237:Hash table 4198:algorithms 4043:2018-07-05 3739:2021-12-06 3539:2017-03-24 3391:2022-02-06 3318:January 1, 3267:References 3024:α / ( 2956:multimedia 2899:Fuzzy hash 2847:Rabin-Karp 2766:for every 2750:See also: 2623:See also: 2559:characters 2313:2 / ϕ 2266:mod 2) / 2 2208:− 1} 1999:) = | 1889:) ∈ 1874:} ⊆ 1703:+ ⋯ 1539:− 1} 1523:) ≡ 1045:Signatures 873:Efficiency 589:collisions 579:Uniformity 526:newspapers 472:Properties 415:grid index 319:Hash table 294:collisions 189:hash table 172:hash codes 90:newspapers 4378:Streaming 4363:Recursion 4038:pHash.org 4015:0302-9743 3702:CiteSeerX 3657:Sethi, R. 3593:: 59–65. 3104:Identicon 3057:The term 2984:watermark 2774:substring 2602:Bell Labs 2309:Fibonacci 2242:word size 2022:whenever 1954:α 1950:− 1936:∈ 1929:∏ 1600:. A key 1415:ZIP codes 1406:entries. 1318:the data 1133:remainder 946:bit shift 942:(key) / 2 927:function 810:− 719:− 701:∑ 668:factorial 653:—see the 410:grid file 401:partition 216:checksums 209:collision 148:collision 120:July 2010 4108:(2000). 4064:(1975). 3925:June 30, 3891:32568938 3663:(1986). 3460:18086276 3444:Elsevier 3343:(1973). 3147:entries. 3082:See also 3033:, where 2991:Analysis 2964:features 2661:−2 2651:−1 2594:PJW hash 2315:, where 2083:−1 2043:satisfy 2007:. Since 1897:∈ 1738:−1 1698:−1 1650:−1 1613:−1 1590:−1 1585:+ ζ 1557:bits to 1380:between 1373:cannot. 1343:identity 1062:function 997:, where 283:keyspace 258:Overview 250:, while 160:function 21:HashLink 4373:Sorting 4348:Minimax 3734:8250194 3653:Aho, A. 3446:: 1–7. 3353:Bibcode 3053:History 1905:Define 1893:∀ 1826:yields 1740:… 1721:. Then 1615:… 1445:Folding 1394:Unicode 1359:Integer 1348:perfect 1211:(where 1176:, e.g. 1082:SipHash 663:perfect 540:scholar 437:in the 391:or the 378:element 240:ciphers 194:hashing 180:digests 158:is any 104:scholar 4353:Online 4338:Greedy 4267:String 4116: 4013: 4003: 3963: 3889: 3879: 3830: 3732: 3722: 3704: 3675: 3564: 3513: 3458: 3417: 3363: 3142:Plain 3036:α 3006:While 2836:where 2640:EBCDIC 2561:, and 2509:after 2329:blocks 2321:ϕ 2024:α 2009:α 2005:| 1975:where 1791:, and 1682:) mod 1371:Double 1078:Python 895:modulo 836:where 600:random 598:, not 542: 535: 528: 521: 513: 453:in an 451:images 441:or in 359:caches 238:, and 184:hashes 106: 99: 92: 85: 77: 4262:Stack 4252:Queue 4232:Graph 4212:Array 3945:(PDF) 3887:S2CID 3865:(PDF) 3812:(PDF) 3775:(PDF) 3730:S2CID 3638:(PDF) 3456:S2CID 3442:(1). 3144:ASCII 3121:Notes 3008:Knuth 2799:> 2707:,..., 2683:radix 2610:xored 2433:64: 2393:48: 2359:32: 2336:16: 2189:) / ( 1808:Knuth 1804:GF(2) 1758:. If 1730:) = ( 1527:(mod 1363:Float 1135:. 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