Journal of the ACM最新文献

{"title":"Query lower bounds for log-concave sampling","authors":"Sinho Chewi, Jaume de Dios Pont, Jerry Li, Chen Lu, Shyam Narayanan","doi":"10.1145/3673651","DOIUrl":"https://doi.org/10.1145/3673651","url":null,"abstract":"<p>Log-concave sampling has witnessed remarkable algorithmic advances in recent years, but the corresponding problem of proving <i>lower bounds</i> for this task has remained elusive, with lower bounds previously known only in dimension one. In this work, we establish the following query lower bounds: (1) sampling from strongly log-concave and log-smooth distributions in dimension <i>d</i> ≥ 2 requires <i>Ω</i>(log <i>κ</i>) queries, which is sharp in any constant dimension, and (2) sampling from Gaussians in dimension <i>d</i> (hence also from general log-concave and log-smooth distributions in dimension <i>d</i>) requires (widetilde{Omega }(min (sqrt kappa log d, d)) ) queries, which is nearly sharp for the class of Gaussians. Here <i>κ</i> denotes the condition number of the target distribution. Our proofs rely upon (1) a multiscale construction inspired by work on the Kakeya conjecture in geometric measure theory, and (2) a novel reduction that demonstrates that block Krylov algorithms are optimal for this problem, as well as connections to lower bound techniques based on Wishart matrices developed in the matrix-vector query literature.</p>","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":"39 1","pages":""},"PeriodicalIF":2.5,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141505503","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Transaction Fee Mechanism Design","authors":"Tim Roughgarden","doi":"10.1145/3674143","DOIUrl":"https://doi.org/10.1145/3674143","url":null,"abstract":"<p>Demand for blockchains such as Bitcoin and Ethereum is far larger than supply, necessitating a mechanism that selects a subset of transactions to include “on-chain” from the pool of all pending transactions. This paper investigates the problem of designing a blockchain transaction fee mechanism through the lens of mechanism design. We introduce two new forms of incentive-compatibility that capture some of the idiosyncrasies of the blockchain setting, one (MMIC) that protects against deviations by profit-maximizing miners and one (OCA-proofness) that protects against off-chain collusion between miners and users. </p><p>This study is immediately applicable to major change (made on August 5, 2021) to Ethereum’s transaction fee mechanism, based on a proposal called “EIP-1559.” Originally, Ethereum’s transaction fee mechanism was a first-price (pay-as-bid) auction. EIP-1559 suggested making several tightly coupled changes, including the introduction of variable-size blocks, a history-dependent reserve price, and the burning of a significant portion of the transaction fees. We prove that this new mechanism earns an impressive report card: it satisfies the MMIC and OCA-proofness conditions, and is also dominant-strategy incentive compatible (DSIC) except when there is a sudden demand spike. We also introduce an alternative design, the “tipless mechanism,” which offers an incomparable slate of incentive-compatibility guarantees—it is MMIC and DSIC, and OCA-proof unless in the midst of a demand spike.</p>","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":"9 1","pages":""},"PeriodicalIF":2.5,"publicationDate":"2024-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141505504","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Sparse Higher Order Čech Filtrations","authors":"Mickaël Buchet, Bianca B Dornelas, Michael Kerber","doi":"10.1145/3666085","DOIUrl":"https://doi.org/10.1145/3666085","url":null,"abstract":"<p>For a finite set of balls of radius <i>r</i>, the <i>k</i>-fold cover is the space covered by at least <i>k</i> balls. Fixing the ball centers and varying the radius, we obtain a nested sequence of spaces that is called the <i>k</i>-fold filtration of the centers. For <i>k</i> = 1, the construction is the union-of-balls filtration that is popular in topological data analysis. For larger <i>k</i>, it yields a cleaner shape reconstruction in the presence of outliers. We contribute a sparsification algorithm to approximate the topology of the <i>k</i>-fold filtration. Our method is a combination and adaptation of several techniques from the well-studied case <i>k</i> = 1, resulting in a sparsification of linear size that can be computed in expected near-linear time with respect to the number of input points. Our method also extends to the multicover bifiltration, composed of the <i>k</i>-fold filtrations for several values of <i>k</i>, with the same size and complexity bounds.</p>","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":"2 1","pages":""},"PeriodicalIF":2.5,"publicationDate":"2024-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141170701","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Killing a Vortex","authors":"Dimitrios Thilikos, Sebastian Wiederrecht","doi":"10.1145/3664648","DOIUrl":"https://doi.org/10.1145/3664648","url":null,"abstract":"<p>The Graph Minors Structure Theorem of Robertson and Seymour asserts that, for every graph <i>H</i>, every <i>H</i>-minor-free graph can be obtained by clique-sums of “almost embeddable” graphs. Here a graph is “almost embeddable” if it can be obtained from a graph of bounded Euler-genus by pasting graphs of bounded pathwidth in an “orderly fashion” into a bounded number of faces, called the <i>vortices</i>, and then adding a bounded number of additional vertices, called <i>apices</i>, with arbitrary neighborhoods. Our main result is a full classification of all graphs <i>H</i> for which the use of vortices in the theorem above can be avoided. To this end we identify a (parametric) graph (mathscr{S}_t) and prove that all (mathscr{S}_t)-minor-free graphs can be obtained by clique-sums of graphs embeddable in a surface of bounded Euler-genus after deleting a bounded number of vertices. We show that this result is tight in the sense that the appearance of vortices cannot be avoided for <i>H</i>-minor-free graphs, whenever <i>H</i> is not a minor of (mathscr{S}_t) for some (tin mathbb {N}. ) Using our new structure theorem, we design an algorithm that, given an (mathscr{S}_t)-minor-free graph <i>G</i>, computes the generating function of all perfect matchings of <i>G</i> in polynomial time. Our results, combined with known complexity results, imply a complete characterization of minor-closed graph classes where the number of perfect matchings is polynomially computable: They are exactly those graph classes that do not contain every (mathscr{S}_t) as a minor. This provides a <i>sharp</i> complexity dichotomy for the problem of counting perfect matchings in minor-closed classes.</p>","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":"24 1","pages":""},"PeriodicalIF":2.5,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140940924","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Separations in Proof Complexity and TFNP","authors":"Mika Göös, Alexandros Hollender, Siddhartha Jain, Gilbert Maystre, William Pires, Robert Robere, Ran Tao","doi":"10.1145/3663758","DOIUrl":"https://doi.org/10.1145/3663758","url":null,"abstract":"<p>It is well-known that Resolution proofs can be efficiently simulated by Sherali–Adams (SA) proofs. We show, however, that any such simulation needs to exploit huge coefficients: Resolution cannot be efficiently simulated by SA when the coefficients are written in unary. We also show that <i>Reversible Resolution</i> (a variant of MaxSAT Resolution) cannot be efficiently simulated by Nullstellensatz (NS). </p><p>These results have consequences for total ({text{upshape sffamily NP}} ) search problems. First, we characterise the classes ({text{upshape sffamily PPADS}} ), ({text{upshape sffamily PPAD}} ), ({text{upshape sffamily SOPL}} ) by unary-SA, unary-NS, and Reversible Resolution, respectively. Second, we show that, relative to an oracle, ({text{upshape sffamily PLS}} notsubseteq {text{upshape sffamily PPP}} ), ({text{upshape sffamily SOPL}} notsubseteq {text{upshape sffamily PPA}} ), and ({text{upshape sffamily EOPL}} notsubseteq {text{upshape sffamily UEOPL}} ). In particular, together with prior work, this gives a complete picture of the black-box relationships between all classical ({text{upshape sffamily TFNP}} ) classes introduced in the 1990s.</p>","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":"20 1","pages":""},"PeriodicalIF":2.5,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140940739","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Smoothed Analysis of Information Spreading in Dynamic Networks","authors":"Michael Dinitz, Jeremy Fineman, Seth Gilbert, Calvin Newport","doi":"10.1145/3661831","DOIUrl":"https://doi.org/10.1145/3661831","url":null,"abstract":"<p>The best known solutions for <i>k</i>-message broadcast in dynamic networks of size <i>n</i> require <i>Ω</i>(<i>nk</i>) rounds. In this paper, we see if these bounds can be improved by smoothed analysis. To do so, we study perhaps the most natural randomized algorithm for disseminating tokens in this setting: at every time step, choose a token to broadcast randomly from the set of tokens you know. We show that with even a small amount of smoothing (i.e., one random edge added per round), this natural strategy solves <i>k</i>-message broadcast in (tilde{O}(n+k^3) ) rounds, with high probability, beating the best known bounds for (k=o(sqrt {n}) ) and matching the <i>Ω</i>(<i>n</i> + <i>k</i>) lower bound for static networks for <i>k</i> = <i>O</i>(<i>n</i>^1/3) (ignoring logarithmic factors). In fact, the main result we show is even stronger and more general: given ℓ-smoothing (i.e., ℓ random edges added per round), this simple strategy terminates in <i>O</i>(<i>kn</i>^2/3log ^1/3(<i>n</i>)ℓ^{− 1/3}) rounds. We then prove this analysis close to tight with an almost-matching lower bound. To better understand the impact of smoothing on information spreading, we next turn our attention to static networks, proving a tight bound of (tilde{O}(ksqrt {n}) ) rounds to solve <i>k</i>-message broadcast, which is better than what our strategy can achieve in the dynamic setting. This confirms the intuition that although smoothed analysis reduces the difficulties induced by changing graph structures, it does not eliminate them altogether. Finally, we apply tools developed to support our smoothed analysis to prove an optimal result for <i>k</i>-message broadcast in so-called well-mixed networks in the absence of smoothing. By comparing this result to an existing lower bound for well-mixed networks, we establish a formal separation between oblivious and strongly adaptive adversaries with respect to well-mixed token spreading, partially resolving an open question on the impact of adversary strength on the <i>k</i>-message broadcast problem.</p>","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":"48 1","pages":""},"PeriodicalIF":2.5,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140830493","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Verifiable Quantum Advantage without Structure","authors":"Takashi Yamakawa, Mark Zhandry","doi":"10.1145/3658665","DOIUrl":"https://doi.org/10.1145/3658665","url":null,"abstract":"<p>We show the following hold, unconditionally unless otherwise stated, relative to a random oracle: <p><table border=\"0\" list-type=\"bullet\" width=\"95%\"><tr><td valign=\"top\"><p>•</p></td><td colspan=\"5\" valign=\"top\"><p>There are NP <i>search</i> problems solvable by quantum polynomial-time machines but not classical probabilistic polynomial-time machines.</p></td></tr><tr><td valign=\"top\"><p>•</p></td><td colspan=\"5\" valign=\"top\"><p>There exist functions that are one-way, and even collision resistant, against classical adversaries but are easily inverted quantumly. Similar counterexamples exist for digital signatures and CPA-secure public key encryption (the latter requiring the assumption of a classically CPA-secure encryption scheme). Interestingly, the counterexample does not necessarily extend to the case of other cryptographic objects such as PRGs.</p></td></tr><tr><td valign=\"top\"><p>•</p></td><td colspan=\"5\" valign=\"top\"><p>There are unconditional publicly verifiable proofs of quantumness with the minimal rounds of interaction: for uniform adversaries, the proofs are non-interactive, whereas for non-uniform adversaries the proofs are two message public coin.</p></td></tr><tr><td valign=\"top\"><p>•</p></td><td colspan=\"5\" valign=\"top\"><p>Our results do not appear to contradict the Aaronson-Ambanis conjecture. Assuming this conjecture, there exist publicly verifiable certifiable randomness, again with the minimal rounds of interaction.</p></td></tr></table></p>\u0000By replacing the random oracle with a concrete cryptographic hash function such as SHA2, we obtain plausible Minicrypt instantiations of the above results. Previous analogous results all required substantial structure, either in terms of highly structured oracles and/or algebraic assumptions in Cryptomania and beyond.</p>","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":"95 1","pages":""},"PeriodicalIF":2.5,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140636980","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Bitcoin Backbone Protocol: Analysis and Applications","authors":"Juan A. Garay, Aggelos Kiayias, Nikos Leonardos","doi":"10.1145/3653445","DOIUrl":"https://doi.org/10.1145/3653445","url":null,"abstract":"<p>Bitcoin is the first and most popular decentralized cryptocurrency to date. In this work, we extract and analyze the core of the Bitcoin protocol, which we term the Bitcoin <i>backbone</i>, and prove three of its fundamental properties which we call <i>Common Prefix</i>, <i>Chain Quality</i> and <i>Chain Growth</i> in the static setting where the number of players remains fixed. Our proofs hinge on appropriate and novel assumptions on the “hashing power” of the protocol participants and their interplay with the protocol parameters and the time needed for reliable message passing between honest parties in terms of computational steps. A takeaway from our analysis is that, all else being equal, the protocol’s provable tolerance in terms of the number of adversarial parties (or, equivalently, their “hashing power” in our model) decreases as the duration of a message passing round increases. </p><p>Next, we propose and analyze applications that can be built “on top” of the backbone protocol, specifically focusing on Byzantine agreement (BA) and on the notion of a public transaction ledger. Regarding BA, we observe that a proposal due to Nakamoto falls short of solving it, and present a simple alternative which works assuming that the adversary’s hashing power is bounded by 1/3. The public transaction ledger captures the essence of Bitcoin’s operation as a cryptocurrency, in the sense that it guarantees the liveness and persistence of committed transactions. Based on this notion we describe and analyze the Bitcoin system as well as a more elaborate BA protocol and we prove them secure assuming the adversary’s hashing power is strictly less than 1/2. Instrumental to this latter result is a technique we call <i>2-for-1 proof-of-work</i>\u0000(PoW) that has proven to be useful in the design of other PoW-based protocols.</p>","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":"19 1","pages":""},"PeriodicalIF":2.5,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140609941","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Smoothed Analysis with Adaptive Adversaries","authors":"Nika Haghtalab, Tim Roughgarden, Abhishek Shetty","doi":"10.1145/3656638","DOIUrl":"https://doi.org/10.1145/3656638","url":null,"abstract":"&lt;p&gt;We prove novel algorithmic guarantees for several online problems in the smoothed analysis model. In this model, at each time step an adversary chooses an input distribution with density function bounded above pointwise by (tfrac{1}{sigma } ) times that of the uniform distribution; nature then samples an input from this distribution. Here, &lt;i&gt;σ&lt;/i&gt; is a parameter that interpolates between the extremes of worst-case and average case analysis. Crucially, our results hold for &lt;i&gt;adaptive&lt;/i&gt; adversaries that can base their choice of an input distribution on the decisions of the algorithm and the realizations of the inputs in the previous time steps. An adaptive adversary can nontrivially correlate inputs at different time steps with each other and with the algorithm’s current state; this appears to rule out the standard proof approaches in smoothed analysis. &lt;/p&gt;&lt;p&gt;This paper presents a general technique for proving smoothed algorithmic guarantees against adaptive adversaries, in effect reducing the setting of an adaptive adversary to the much simpler case of an oblivious adversary (i.e., an adversary that commits in advance to the entire sequence of input distributions). We apply this technique to prove strong smoothed guarantees for three different problems: &lt;p&gt;&lt;table border=\"0\" list-type=\"ordered\" width=\"95%\"&gt;&lt;tr&gt;&lt;td valign=\"top\"&gt;&lt;p&gt;(1)&lt;/p&gt;&lt;/td&gt;&lt;td colspan=\"5\" valign=\"top\"&gt;&lt;p&gt;Online learning: We consider the online prediction problem, where instances are generated from an adaptive sequence of &lt;i&gt;σ&lt;/i&gt;-smooth distributions and the hypothesis class has VC dimension &lt;i&gt;d&lt;/i&gt;. We bound the regret by (tilde{O}big (sqrt {T dln (1/sigma)} + dln (T/sigma) big) ) and provide a near-matching lower bound. Our result shows that under smoothed analysis, learnability against adaptive adversaries is characterized by the finiteness of the VC dimension. This is as opposed to the worst-case analysis, where online learnability is characterized by Littlestone dimension (which is infinite even in the extremely restricted case of one-dimensional threshold functions). Our results fully answer an open question of Rakhlin et al. [64]. &lt;/p&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td valign=\"top\"&gt;&lt;p&gt;(2)&lt;/p&gt;&lt;/td&gt;&lt;td colspan=\"5\" valign=\"top\"&gt;&lt;p&gt;Online discrepancy minimization: We consider the setting of the online Komlós problem, where the input is generated from an adaptive sequence of &lt;i&gt;σ&lt;/i&gt;-smooth and isotropic distributions on the ℓ&lt;sub&gt;2&lt;/sub&gt; unit ball. We bound the ℓ&lt;sub&gt;∞&lt;/sub&gt; norm of the discrepancy vector by (tilde{O}big (ln ^2big (frac{nT}{sigma }big) big) ). This is as opposed to the worst-case analysis, where the tight discrepancy bound is (Theta (sqrt {T/n}) ). We show such polylog(&lt;i&gt;nT&lt;/i&gt;/&lt;i&gt;σ&lt;/i&gt;) discrepancy guarantees are not achievable for non-isotropic &lt;i&gt;σ&lt;/i&gt;-smooth distributions. &lt;/p&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td valign=\"top\"&gt;&lt;p&gt;(3)&lt;/p&gt;&lt;/td&gt;&lt;td colspan=\"5\" valign=\"top\"&gt;&lt;p&gt;Dispersion in online optimization: We consider online optimization with piecewise Lipschitz functions where fun","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":"72 1","pages":""},"PeriodicalIF":2.5,"publicationDate":"2024-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140600098","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Fast Multivariate Multipoint Evaluation Over All Finite Fields","authors":"Vishwas Bhargava, Sumanta Ghosh, Zeyu Guo, Mrinal Kumar, Chris Umans","doi":"10.1145/3652025","DOIUrl":"https://doi.org/10.1145/3652025","url":null,"abstract":"<p>Multivariate multipoint evaluation is the problem of evaluating a multivariate polynomial, given as a coefficient vector, simultaneously at multiple evaluation points. In this work, we show that there exists a deterministic algorithm for multivariate multipoint evaluation over any finite field (mathbb {F} ) that outputs the evaluations of an <i>m</i>-variate polynomial of degree less than <i>d</i> in each variable at <i>N</i> points in time <span>[ (d^m+N)^{1+o(1)}cdot {rm poly}(m,d,log |mathbb {F}|) ]</span>\u0000for all (min mathbb {N} ) and all sufficiently large (din mathbb {N} ). </p><p>A previous work of Kedlaya and Umans (FOCS 2008, SICOMP 2011) achieved the same time complexity when the number of variables <i>m</i> is at most <i>d</i>^<i>o</i>(1) and had left the problem of removing this condition as an open problem. A recent work of Bhargava, Ghosh, Kumar and Mohapatra (STOC 2022) answered this question when the underlying field is not <i>too</i> large and has characteristic less than <i>d</i>^<i>o</i>(1). In this work, we remove this constraint on the number of variables over all finite fields, thereby answering the question of Kedlaya and Umans over all finite fields. </p><p>Our algorithm relies on a non-trivial combination of ideas from three seemingly different previously known algorithms for multivariate multipoint evaluation, namely the algorithms of Kedlaya and Umans, that of Björklund, Kaski and Williams (IPEC 2017, Algorithmica 2019), and that of Bhargava, Ghosh, Kumar and Mohapatra, together with a result of Bombieri and Vinogradov from analytic number theory about the distribution of primes in an arithmetic progression. </p><p>We also present a second algorithm for multivariate multipoint evaluation that is completely elementary and in particular, avoids the use of the Bombieri–Vinogradov Theorem. However, it requires a mild assumption that the field size is bounded by an exponential tower in <i>d</i> of bounded <i>height</i>. More specifically, our second algorithm solves the multivariate multipoint evaluation problem over a finite field (mathbb {F} ) in time <span>[ (d^m+N)^{1+o(1)}cdot {rm poly}(m,d,log |mathbb {F}|) ]</span>\u0000for all (min mathbb {N} ) and all sufficiently large (din mathbb {N} ), provided that the size of the finite field (mathbb {F} ) is at most (exp(exp(exp(⋅⋅⋅(exp(<i>d</i>))))), where the height of this tower of exponentials is fixed.</p>","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":"8 1","pages":""},"PeriodicalIF":2.5,"publicationDate":"2024-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140203269","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}