In mathematics and computing, universal hashing (in a randomized algorithm or data structure) refers to selecting a hash function at random from a family of hash functions with a certain mathematical property (see definition below). This guarantees a low number of collisions in expectation, even if the data is chosen by an adversary. Many universal families are known (for hashing integers, vectors, strings), and their evaluation is often very efficient. Universal hashing has numerous uses in computer science, for example in implementations of hash tables, randomized algorithms, and cryptography.
Introduction
Assume we want to map keys from some universe U {\displaystyle U} into m {\displaystyle m} bins (labelled [ m ] = { 0 , … , m − 1 } {\displaystyle [m]=\{0,\dots ,m-1\}} ). The algorithm will have to handle some data set S ⊆ U {\displaystyle S\subseteq U} of | S | = n {\displaystyle |S|=n} keys, which is not known in advance. Usually, the goal of hashing is to obtain a low number of collisions (keys from S {\displaystyle S} that land in the same bin). A deterministic hash function cannot offer any guarantee in an adversarial setting if | U | > m ⋅ n {\displaystyle |U|>m\cdot n} , since the adversary may choose S {\displaystyle S} to be precisely the preimage of a bin. This means that all data keys land in the same bin, making hashing useless. Furthermore, a deterministic hash function does not allow for rehashing: sometimes the input data turns out to be bad for the hash function (e.g. there are too many collisions), so one would like to change the hash function. The solution to these problems is to pick a function randomly from a family of hash functions. A family of functions H = { h : U → [ m ] } {\displaystyle H=\{h:U\to [m]\}} is called a universal family if, ∀ x , y ∈ U , x ≠ y : | { h ∈ H : h ( x ) = h ( y ) } | ≤ | H | m {\displaystyle \forall x,y\in U,~x\neq y:~~|\{h\in H:h(x)=h(y)\}|\leq {\frac {|H|}{m}}} . In other words, any two different keys of the universe collide with probability at most 1 / m {\displaystyle 1/m} when the hash function h {\displaystyle h} is drawn uniformly at random from H {\displaystyle H} . This is exactly the probability of collision we would expect if the hash function assigned truly random hash codes to every key. Sometimes, the definition is relaxed by a constant factor, only requiring collision probability O ( 1 / m ) {\displaystyle O(1/m)} rather than ≤ 1 / m {\displaystyle \leq 1/m} . This concept was introduced by Carter and Wegman in 1977, and has found numerous applications in computer science (see, for example). If we have an upper bound of ϵ < 1 {\displaystyle \epsilon <1} on the collision probability, we say that we have ϵ {\displaystyle \epsilon } -almost universality. So for example, a universal family has 1 / m {\displaystyle 1/m} -almost universality. Many, but not all, universal families have the following stronger uniform difference property:
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