## The communication complexity of addition (2011)

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### BibTeX

@MISC{Viola11thecommunication,

author = {Emanuele Viola},

title = {The communication complexity of addition},

year = {2011}

}

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### Abstract

Suppose each of k ≤ no(1) players holds an n-bit number xi in its hand. The players wish to determine if ∑ i≤k xi = s. We give a public-coin protocol with error 1% and communication O(k lg k). The communication bound is independent of n, and for k ≥ 3 improves on the O(k lg n) bound by Nisan (Bolyai Soc. Math. Studies; 1993). Our protocol also applies to addition modulo m. In this case we give a matching (public-coin) Ω(k lg k) lower bound for various m. We also obtain some lower bounds over the integers, including Ω(k lg lg k) for protocols that are one-way, like ours. We give a protocol to determine if ∑ xi> s with error 1 % and communication O(k lg k) lg n. For k ≥ 3 this improves on Nisan’s O(k lg 2 n) bound. A similar improvement holds for computing degree-(k − 1) polynomial-threshold functions in the number-on-forehead model. We give a (public-coin, 2-player, tight) Ω(lg n) lower bound to determine if x1> x2. This improves on the Ω ( √ lg n) bound by Smirnov (1988).

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Citation Context ...for public randomness (as in Def. 3). However, in [Smi88] Smirnov proves a round-communication tradeoff for 2-player Sum-Greater, according to Miltersen, Nisan, Safra, and Wigderson who reprove it in =-=[MNSW98]-=-. This tradeoff implies an Ω( √ lg n) lower bound. In this paper we obtain a tight Ω(lg n) lower bound. Theorem 6. The communication complexity of 2-player Sum-Greater is Ω(lg n). Together with our pr... |

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Citation Context ...ng in time nO(lg n) instead of nO(1) . To handle more threshold gates we still show that with high probability the restricted circuit is computable by a low-communication protocol, using an idea from =-=[Vio07]-=-. We note that the result [Bei94] of Beigel shows that O(1) symmetric gates can be reduced to 1 with only a polynomial blow-up. However no equivalent of Beigel’s result holds even for two threshold ga... |

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Citation Context ...(1) . To handle more threshold gates we still show that with high probability the restricted circuit is computable by a low-communication protocol, using an idea from [Vio07]. We note that the result =-=[Bei94]-=- of Beigel shows that O(1) symmetric gates can be reduced to 1 with only a polynomial blow-up. However no equivalent of Beigel’s result holds even for two threshold gates, as was shown by Gopalan and ... |

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Citation Context ...k)(d lg n + lg 1/ɛ). Nisan uses the upper bound for d-PTF to obtain three lower bounds for functions with high (d + 1)-player number-on-forehead communication, such as Generalized Inner Product (GIP) =-=[BNS92]-=-. He proves that for any d, to compute GIP: any circuit of d-PTF requires Ω(n/ lg 2 n) gates, any tree of d-PTF requires depth Ω(n/ lg 2 n), and any majority of d-PTF requires 2 Ω(n/ lg n) gates. Usin... |

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Citation Context ..., 1} n → {0, 1} in space O(lg n), or even NL, with security 2nΩ(1) . We use the TC 0 candidates by Naor and Reingold, later with Rosen, in [NR99, NRR02]. For concreteness we focus on the candidate in =-=[NRR02]-=- which is based on the hardness of factoring. See [NRR02, §4.1,5] for background. For a bound t(n), the t(n)-factoring assumption is the assumption that any algorithm running in time t(n) has success ... |

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Citation Context ...) with error 1/4. This protocol has error ≤ 1/2 + 1/4 < 1, and the result follows by repeating it O(lg 1/ɛ) times. Mul-Greater: By a result of Baker and Wüstholz on logarithmic forms, Theorem 2.15 in =-=[BW07]-=-, we have that the quantity | ∑ i≤k lg xi − lg s| is either 0 or at least β := 1/2 nO(k) in absolute value. Player i computes privately approximations x ′ i and s ′ to lg xi and lg s respectively, to ... |

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Citation Context ...his, previous communication lower bounds suffice.) In the case of a single majority gate, this improves on the quasipolynomial-time distinguisher by Razborov and Rudich [RR97], based on [ABFR94]; cf. =-=[KL01]-=-. Theorem 10. Let F be a distribution on functions from {0, 1} n to {0, 1} such that each function in the support is computable by a circuit of size n d , depth d, with d threshold (or arbitrary symme... |

10 |
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Citation Context ...k ) ⎤ ⎦ . x 1 1 ,x1 2 ,...,x1 k ∈{0,1}ℓ ɛ1,...,ɛk∈{0,1} The next lemma shows that if we can compute f well on average using little communication, then Rk(f) is large. (Note the contrapositive is used =-=[VW08]-=-.) Lemma 38 (Corollary 3.11 in [VW08]). Let f : ({0, 1} n ) k → {−1, 1} be a function. Let Π : ({0, 1} n ) k → {−1, 1} be a function computable by a k-player number-on-forehead protocol using c bits o... |

9 |
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Citation Context ...h a larger, constant error, then one is in the setting of performing binary search with noisy comparisons. For a recent account of the latter, and the many solutions available, we refer the reader to =-=[BOH08]-=-. Nisan and Safra use the cute random-walk-with-backtrack algorithm by Feige, Raghavan, Peleg, and Upfal [FRPU94]. The algorithm [FRPU94] shows that such a search can be accomplished with O(lg n) comp... |

9 |
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Citation Context ...ater, and polynomialthreshold functions. 3.1 Sum-Equal We define the hash function originating in the work [DHKP97, §2.2] by Dietzfelbinger, Hagerup, Katajainen, and Penttonen. Definition 12 (§2.2 in =-=[DHKP97]-=-). The hash function h : {0, 1} n → {0, 1} r has a seed of n − 1 bits and is defined as follows. Interpret the seed as an odd integer a ∈ {0, 1} n . Then ha(x) := (a · x mod 2 n ) ≫ n − r. Here (a · x... |

7 |
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Citation Context ...mnjaˇsčiĭ [Nep70], this function is computable by poly(n)-size AC0 circuits. What remains to do is to increase the input length from n ′ to n while maintaining security. Here we use the idea of Levin =-=[Lev87]-=- and combine f ′ s with a hash function ht : {0, 1} n → {0, 1} n′ with seed t to obtain the pseudorandom function fs,t : {0, 1} n → {0, 1} defined as fs,t(x) := f ′ s(ht(x)). A hashing property that i... |

5 |
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Citation Context ...ounds. In the special case of private randomness, it is known that 2-player SumGreater requires communication Ω(lg n).[KN97] This bound was not known for public randomness (as in Def. 3). However, in =-=[Smi88]-=- Smirnov proves a round-communication tradeoff for 2-player Sum-Greater, according to Miltersen, Nisan, Safra, and Wigderson who reprove it in [MNSW98]. This tradeoff implies an Ω( √ lg n) lower bound... |

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Citation Context ...hows that O(1) symmetric gates can be reduced to 1 with only a polynomial blow-up. However no equivalent of Beigel’s result holds even for two threshold gates, as was shown by Gopalan and Servedio in =-=[GS10]-=-. 3 Upper bounds In this section we prove our upper bounds for Sum-Equal, Sum-Greater, and polynomialthreshold functions. 3.1 Sum-Equal We define the hash function originating in the work [DHKP97, §2.... |

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Citation Context ...lg n) lower bound may also be obtained by combining a result in learning theory by Forster et al. [FSSS03] with a result by Linial and Shraibman [LS09]. After our work, Braverman and Weinstein obtain =-=[BW12]-=- yet another proof. Together with our previous upper bound (Theorem 4), we obtain that for every k = O(1), k-player Sum-Greater has communication complexity Θ(lg n). We then move to lower bounds for S... |

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Citation Context ...heorem 5, we show that truth-tables of poly(n)-size constant-depth AC0 circuits augmented with O(1) threshold gates cannot support cryptographic pseudorandom functions. Introduced in the seminal work =-=[GGM86]-=- of Goldreich, Goldwasser, and Micali, a pseudorandom function is a random function F : {0, 1} n → {0, 1} such that for every c and sufficiently large n, any oracle algorithm M running time ≤ nc has a... |

1 |
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Citation Context ...(lg n). Troy Lee informed us in January 2013 that an Ω(lg n) lower bound may also be obtained by combining a result in learning theory by Forster et al. [FSSS03] with a result by Linial and Shraibman =-=[LS09]-=-. After our work, Braverman and Weinstein obtain [BW12] yet another proof. Together with our previous upper bound (Theorem 4), we obtain that for every k = O(1), k-player Sum-Greater has communication... |

1 |
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Citation Context ...can be solved with error, say, 1% and communication O(1).[KN97, Example 3.13] Remarkably, the communication is independent of n. In the more challenging case k ≥ 3, Nisan gives in his beautiful paper =-=[Nis93]-=- a randomized protocol with O(k lg(n + lg k)) communication. Note now the communication depends (logarithmically) on n. In his protocol, Player i communicates xi modulo a small, randomly-chosen prime.... |

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