K-wise independent hash functions pdf

We seek a hash function that is both easy to compute and uniformly distributes the keys. A standard solution to this problem is accomplished by using a hash function h. Prf and pairwise independent hash function cryptography. A small approximately minwise independent family of hash functions piotr indyk1 departmentofcomputerscience,stanforduniversity,stanford,california94305 email. Key independent uniform segmentation of arbitrary input using a hash function paul dorfman, independent consultant don henderson, henderson consulting services, llc. It is not hard to see that this hash function is indeed a 2wise independent function. That is, for any i,jand values x,y, it holds that pr xi x. Pairwise independent hash functions 1 hash functions the goal of hash functions is to map elements from a large domain to a small one. It would be desirable to store and query a fewer number of bits in order to compute a kwise independent hash function. Such a result may say that if the algorithm behaves well under any kwise independent distribution then it would behave essentially as well also under any almost kwise independent distribution, provided that the parameter governing this measure of closeness is small enough. Keyindependent uniform segmentation of arbitrary input. We also defined pairwise independence for hash functions hs. Typically, to obtain the required guarantees, we would need not just one function, but a family of functions, where we would use randomness to sample a hash function from this family.

It is common to think of h as a hash function, because it is a randomlike function. In computer science, a family of hash functions is said to be kindependent or kuniversal if. Family of functions with properties similar to kwise. Kwise hash functions are important because they allow for e cient construction of hash families. We will prove this by induction of s k, the size of the subset t. Family of functions with properties similar to kwise independent hash functions. Advanced data structures spring mit opencourseware.

Theorem 7 for any m 1, there is a hash function h h. Recursive ngram hashing is pairwise independent, at best. Approximation, randomization and combinatorial optimization. Our construction of pairwise independent hash functions from section1. Lecture 1 kwise independence ubc computer science. 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. Lecture 5 1 overview 2 pairwise independent hash functions. Nick harvey university of british columbia 1 kwise independence a set of events e 1e. Definition 2 pairwise independent family of hash functions a family of hash. The hash function allows us to map a universe uof u keys to a slot in a table of size m. Fully independent hash functions generally require large space requirements.

Also, observe that if we want a kwise independent construction for independent permutations. Although pairwise independence is already sufficient for our application today, kwise independent hash functions are very important objects in. In computer science, a family of hash functions is said to be kindependent or k universal if. It is enough to show that for any subset of size kof x, the variables are linearly independent when written as linear functions of r 1. Design strategies for minimal perfect hash functions. The use of cryptographic hash functions like sha1, has been suggested 11. We will assume that u 2k, by possible increasing the universe size, and identify u 0,1k. The connection is to hash functions is more compelling if we consider the case f. How to prove pairwise independence of a family of hash. A small approximately minwise independent family of hash functions. New hash functions and their use in authentication and set equality pdf. A similar result for kwise independent hash functions was obtained by austrin and h astad ah11.

Thorup, mikkel 2010, on the kindependence required by linear probing and minwise independence pdf, automata, languages and programming. Request pdf dkminwise independent family of hash functions in this paper we introduce a general framework that exponentially improves the space, degree of independence, and time needed by. The original technique for constructing k independent hash functions, given by carter and wegman, was to select a large prime number p, choose k random numbers modulo p, and use these numbers as the coefficients of a polynomial of degree k. Moreover, the idea of pairwise independence can be generalized.

The proof that this construction is kwise independent follows. Although pairwise independence is already su cient for our application today, kwise independent hash functions are very important objects in computer science, and thus have found a lot of applications elsewhere. Finally, as a usage example, we show how to apply those hash functions to the estimation of similarity and rarity over data streams. One such hash function presented by thorup and zhang has query time as a function of k4. Instead of using a defined hash function, for which an adversary can always find a bad set of keys.

Almost kwise independence versus kwise independence. For alternative, we can use the \universal hash functions or kwise independent hash functions, which can save randomness while having the same running time for hashing algorithms. Typically, to obtain the required guarantees, we would need not just one function, but a family of functions, where we would use randomness to sample a hash function from this. First, we extend the notion of a minwise independent family of hash functions by defining a dkminwise independent family of hash functions. We say that a distribution over 0,1n is, k wise independent if its restriction to every k coordinates results in a distribution that is close to the uniform distribution. A natural question regarding, k wise independent distributions is how close they are to some k. Already carter and wegman, in their seminal paper on universal hashing, raised the question of extending their analysis to linear probing. The following facts about kwise independent hash families are. One such hash function presented by thorup and zhang has query time as a function of k 4.

This guarantees a low number of collisions in expectation. Our result implies a somewhat surprising consequence for search algorithms which work given any kwise independent distribution over permutations, which allows to. I need to use a hash function which belongs to a family of kwise independent hash functions. A family of hash functions is kwise independent or k independent if the hash values of any k distinct elements are independent. Hash, displace, and compress djamal belazzougui1, fabiano c. H as uniform or a pairwise independent hash function when the family in question can be inferred from the context. The reason why it works is exactly the interpolation theorem for finite fields. I want to prove pairwise independence of a family of hash functions, but i dont know where to start. If we have an array that can hold m keyvalue pairs, then we need a function that can transform any given key into an index into that array. Universal kwise independent classes of hash functions are recommended along with their construction mechanisms 16. M is a prime and m iui so how do i show that the family is pairwise independent. A small approximately minwise independent family of hash. In this section we will give three examples of generating kwise independent random variables with simple families of hash functions.

As a consequence, pairwise independent hash families 2. Universal hashing and kwise independent random variables via integer arithmetic without primes. X j y prx i xpr x j y hash functions with similar properties are called kwise independent and pairwise indepen dent hash functions, respectively. There are constructions that are almost kwise independent, but we leave the discussion of approximation notions to later in the course. A natural candidate is a pairwise independent hash family, for we are simply seeking to minimize collisions, and collisions are pairwise events, so the statistics will be the same. Almost kwise independence versus kwise independence noga alon. The power of simple tabulation hashing journal of the acm. T is a randomized function that provides the guarantee that, for any kdistinct. Cs621 theory gems lecture 17 massachusetts institute of. Finally, as a usage example, we show how to apply those hash functions to the. Definition 1 hash function a hash function is a \random looking function mapping values from a domain d to its range r the solution to the dictionary problem using hashing is to store the set s d in an.