## The Computational Complexity of Universal Hashing (2002)

Venue: | Theoretical Computer Science |

Citations: | 59 - 2 self |

### BibTeX

@ARTICLE{Mansour02thecomputational,

author = {Yishay Mansour and Noam Nisan and Prasoon Tiwari},

title = {The Computational Complexity of Universal Hashing},

journal = {Theoretical Computer Science},

year = {2002},

volume = {107},

pages = {235--243}

}

### Years of Citing Articles

### OpenURL

### Abstract

Any implementation of Carter-Wegman universal hashing from n-bit strings to m-bit strings requires a time-space tradeoff of TS = Ω(nm). The bound holds in the general boolean branching program model, and thus in essentially any model of computation. As a corollary, computing a+b*c in any field F requires a quadratic time-space tradeoff, and the bound holds for any representation of the elements of the field. Other lower bounds on the...

### Citations

284 |
Constant depth circuits, Fourier transform, and learnability
- Linial, Mansour, et al.
- 1993
(Show Context)
Citation Context ...on a CREW PRAM. 3.sm) of the output bits of h requiressn 2 ) boolean formula size (over the basis consisting of f_; ^; :g). where n = log N is the input size, and m = log M is the output size. Proof: =-=[17]-=- shows that exp(s(f) 1=d ) is a lower bound on the size of depth d circuits computing f . [11] shows that log s(f) is a lower bound on the CREW parallel time. (In fact, they give the bound in terms of... |

192 | Pseudorandom generators for space-bounded computation
- Nisan
- 1992
(Show Context)
Citation Context ...ation of Interactive Proof systems by Arthur-Merlin games; Impagliazzo and Zuckerman [14] used universal hashing for the amplication of the probability of success of randomized algorithms; and Nisan [=-=18]-=- used universal hashing to construct pseudorandom generators for space-bounded computation. Previous research has been directed towardssnding ecient implementation of universal hashing, as well as app... |

187 |
Private coins versus public coins in interactive proof systems
- Goldwasser, Sipser
- 1989
(Show Context)
Citation Context ...purposes as proposed in [9], many new applications have been found. For example, Sipser [20] used universal hashing to obtain simulation of BPP in the polynomial time hierarchy; Goldwasser and Sipser =-=[13-=-] used it for the simulation of Interactive Proof systems by Arthur-Merlin games; Impagliazzo and Zuckerman [14] used universal hashing for the amplication of the probability of success of randomized ... |

186 | How to Recycle Random Bits
- Impagliazzo, Zuckerman
- 1989
(Show Context)
Citation Context ...hing to obtain simulation of BPP in the polynomial time hierarchy; Goldwasser and Sipser [13] used it for the simulation of Interactive Proof systems by Arthur-Merlin games; Impagliazzo and Zuckerman =-=[14-=-] used universal hashing for the amplication of the probability of success of randomized algorithms; and Nisan [18] used universal hashing to construct pseudorandom generators for space-bounded comput... |

146 | A complexity theoretic approach to randomness
- Sipser
- 1983
(Show Context)
Citation Context ... = y 1 ^ h(x 2 ) = y 2 ] = 1=jBj 2 : Apart from the value of such families of hash function for various hashing purposes as proposed in [9], many new applications have been found. For example, Sipser =-=[20]-=- used universal hashing to obtain simulation of BPP in the polynomial time hierarchy; Goldwasser and Sipser [13] used it for the simulation of Interactive Proof systems by Arthur-Merlin games; Impagli... |

114 |
Complexity classes in communication complexity theory (preliminary version
- Babai, Frankl, et al.
- 1986
(Show Context)
Citation Context ...lemma is a stronger variant of a lemma that appears in [18]. It is also interesting to compare this lemma to a result of Lindsey on the distribution of 1's and 1's in submatrices of Hadamard matrices =-=[2-=-, 12]. 8 Lemma 13 Let H = fh : I ! Og be a collection of universal hash functions. Let A I, B O and C H. Then, jProb x2A;h2C [h(x) 2 B] pj s jHj jAjjCj p(1 p) where p = jBj=jOj. Proof: Dene M h;x ... |

110 | A Complexity Theory for VLSI
- Thompson
- 1980
(Show Context)
Citation Context ... This is immediate from the fact that H = fax + bja; b 2 Fg is a universal family of hash functions. 2 11 5 An Area-Time Tradeo for VLSI The model that we are considering is Thompson Grid Model (see [=-=21]-=-) of VLSI Chip Layout. The model assumes that a chip consists of a grid with evenly spaced vertical and horizontal tracks. Circuit nodes are placed on the intersection of vertical and horizontal track... |

86 |
Universal Hash Functions
- Carter, Wegman
- 1979
(Show Context)
Citation Context ... hashing are given as well: Quadratic AT 2 bound for VLSI implementation; tio n) parallel time bound on a CREW PRAM; and exponential size for constant depth circuits. 1 Introduction Carter and Wegman =-=[-=-9] dened the notion of a universal family of hash functions. Laboratory for Computer science, MIT, 545 Tech. Sq., Cambridge, MA 02139. Supported by an IBM graduate fellowship. y Dept. of Computer Sci... |

77 | Probabilistic methods in combinatorics
- Erdos, Spencer
- 1974
(Show Context)
Citation Context ...lemma is a stronger variant of a lemma that appears in [18]. It is also interesting to compare this lemma to a result of Lindsey on the distribution of 1's and 1's in submatrices of Hadamard matrices =-=[2-=-, 12]. 8 Lemma 13 Let H = fh : I ! Og be a collection of universal hash functions. Let A I, B O and C H. Then, jProb x2A;h2C [h(x) 2 B] pj s jHj jAjjCj p(1 p) where p = jBj=jOj. Proof: Dene M h;x ... |

62 | Parallel computation for well-endowed rings and space-bounded probabilistic machines - Borodin, Cook, et al. - 1983 |

47 |
On universal classes of fast high performance hash functions, their time-space tradeo, and their applications
- Siegel
- 1989
(Show Context)
Citation Context ...tion. Previous research has been directed towardssnding ecient implementation of universal hashing, as well as applying universal hashing in varying applications. Perhaps the only exception is Siegel =-=[19-=-], who studied both upper and lower bounds on the tradeos between the storage requirement and the number of random bits versus the computation time of log n-wise independent hash functions, in the alg... |

39 |
The recognition problem for the set of perfect squares
- Cobham
- 1966
(Show Context)
Citation Context ... reasonable model of computation, such as multitape Turing machines or RAMs. In Section 3, we recall the denition of the model, and the notions of time and space in this model. With the exception of [=-=10-=-],srst time-space tradeos were obtained on structured models of compuation. Tompa [23] contains a discussion of these results. In the case of non-structured models, Cobham [10] shows a timespace trade... |

25 |
The entropic limitations on VLSI computations
- Yao
- 1981
(Show Context)
Citation Context ... x 2 f1 : : : Ng and h 2 H outputs h(x) requires a area-time tradeo of AT 2 =sn 2 ), where n = log N . Again, this theorem implies some of the known AT 2 lower bounds, and in particular Yao's result [=-=24] giv-=-ing a quadratic AT 2 lower bound for the computation of a + b c, in any representation. We then make the observation that universal hash functions must have high \average sensitivity", a fact wh... |

19 |
Upper and lower bounds for parallel random access machines without simultaneous writes
- Cook, Dwork, et al.
- 1986
(Show Context)
Citation Context ...sis consisting of f_; ^; :g). where n = log N is the input size, and m = log M is the output size. Proof: [17] shows that exp(s(f) 1=d ) is a lower bound on the size of depth d circuits computing f . =-=[11]-=- shows that log s(f) is a lower bound on the CREW parallel time. (In fact, they give the bound in terms of the stronger worst case sensitivity.) Krapchenko's lower bound technique [15] shows that s(f)... |

11 |
A time-space tradeo for sorting on a general sequential model of computation
- Borodin, Cook
- 1982
(Show Context)
Citation Context ...t any implementation of a universal family of hash functions must incur. Our main result is a tight, quadratic, time-space tradeo on the general model of boolean branching programs. Borodin and Cook [=-=-=-6] dened and justied the boolean branching program as a general model of computation. It imposes no structure on the computation, allows any pattern of access to the input bits, and also allows the ou... |

10 |
Trade-offs between communication and space
- Lam, Tiwari, et al.
- 1992
(Show Context)
Citation Context ... communication-space tradeo problem was posed in Tiwari [22]. First results in this direction were obtained on the algebraic straight-line model of communicating processors by Lam, Tiwari, and Tompa [=-=1-=-6]. They proved a tight communication-space tradeo for convolution and matrix multiplication. Recently, Beame, Tompa, and Yan [4] have used our main lemma in order to prove a tradeo between communicat... |

8 |
The complexity of functions
- Boppana, Sipser
- 1990
(Show Context)
Citation Context ...(In fact, they give the bound in terms of the stronger worst case sensitivity.) Krapchenko's lower bound technique [15] shows that s(f) 2 is a lower bound on the boolean formula complexity. (See also =-=[5]-=-). 2 7 Open problems We would like to conjecture about the complexity of universal hash function in the circuit and Turing machine models. We conjecture that any implementation of universal hashing ha... |

5 |
A time-space tradeo# for sorting on non-oblivious machines
- Borodin, Fischer, et al.
- 1981
(Show Context)
Citation Context ...d space S, can be transformed to a branching program whose graph is acyclic (no directed cycles) and leveled (the underline graph is leveled) , and whose time is T and space is at most S + log T (see =-=[8-=-]). For this reason, we restrict our discussion to acyclic and leveled branching programs. 4 A Time-Space Tradeo for Hash Functions For the lower bound we use a version of the proof technique introduc... |

5 |
Time-space tradeo s for computing functions, using connectivity properties of their circuits
- Tompa
- 1980
(Show Context)
Citation Context ...n 3, we recall the denition of the model, and the notions of time and space in this model. With the exception of [10],srst time-space tradeos were obtained on structured models of compuation. Tompa [2=-=3-=-] contains a discussion of these results. In the case of non-structured models, Cobham [10] shows a timespace tradeo for any computational device having one head-read only input tape. The boolean bran... |

3 |
A Method of Determining Lower Bounds for the Complexity of Π-Schemes,” Aametki 10
- Krapchenko
- 1971
(Show Context)
Citation Context ...s computing f . [11] shows that log s(f) is a lower bound on the CREW parallel time. (In fact, they give the bound in terms of the stronger worst case sensitivity.) Krapchenko's lower bound technique =-=[15]-=- shows that s(f) 2 is a lower bound on the boolean formula complexity. (See also [5]). 2 7 Open problems We would like to conjecture about the complexity of universal hash function in the circuit and ... |

2 |
Time-space tradeos for matrix multiplication and the discrete Fourier transform on any general sequential random-access computer
- Yesha
- 1984
(Show Context)
Citation Context ...ut bits. Thesrst non-trivial time-space tradeos in this model were 2 given by Borodin and Cook [6] who proved that sorting requires a nearly quadratic tradeo. Subsequently, Abrahamson [1] and Yesha [25] proved tradeos for several algebraic problems, and Beame [3] proved a tight tradeo for sorting and related problems. In contrast to earlier results that provide tradeos for specic problems, we p... |

1 |
Time-space tradeos for branching programs constructed with those for straight line programs
- Abrahamson
- 1986
(Show Context)
Citation Context ...essing the input bits. Thesrst non-trivial time-space tradeos in this model were 2 given by Borodin and Cook [6] who proved that sorting requires a nearly quadratic tradeo. Subsequently, Abrahamson [1] and Yesha [25] proved tradeos for several algebraic problems, and Beame [3] proved a tight tradeo for sorting and related problems. In contrast to earlier results that provide tradeos for specic... |

1 |
A general sequential time space tradeo for unique elements
- Beame
- 1989
(Show Context)
Citation Context ...were 2 given by Borodin and Cook [6] who proved that sorting requires a nearly quadratic tradeo. Subsequently, Abrahamson [1] and Yesha [25] proved tradeos for several algebraic problems, and Beame [3] proved a tight tradeo for sorting and related problems. In contrast to earlier results that provide tradeos for specic problems, we prove tradeos for a whole family of problems. Theorem 2 Let H ... |

1 |
Communication-space tradeos in the boolean model. manuscript
- Beame, Tompa, et al.
(Show Context)
Citation Context ...aight-line model of communicating processors by Lam, Tiwari, and Tompa [16]. They proved a tight communication-space tradeo for convolution and matrix multiplication. Recently, Beame, Tompa, and Yan [=-=4-=-] have used our main lemma in order to prove a tradeo between communication and space on a general model of communicating processors. The rest of this paper is organized as following. Section 2 gives ... |

1 |
The communication complexity of distributed computing and a parallel algorithm for polynomial roots
- Tiwari
- 1986
(Show Context)
Citation Context ... n = log N is the input size, and m = log M is the output size. Another interesting resource tradeo is that between communication and space. A communication-space tradeo problem was posed in Tiwari [2=-=2-=-]. First results in this direction were obtained on the algebraic straight-line model of communicating processors by Lam, Tiwari, and Tompa [16]. They proved a tight communication-space tradeo for con... |