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    Learning, cryptography, and the average case.

    机译:学习,密码学和一般情况。

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    摘要

    This thesis explores problems in computational learning theory from an average-case perspective. Through this perspective we obtain a variety of new results for learning theory and cryptography.;Several major open questions in computational learning theory revolve around the problem of efficiently learning polynomial-size DNF formulas, which dates back to Valiant's introduction of the PAC learning model [Valiant, 1984]. We apply an average-case analysis to make progress on this problem in two ways. (1) We prove that Mansour's conjecture is true for random DNF. In 1994, Y. Mansour conjectured that for every DNF formula on n variables with t terms there exists a polynomial p with tO(log(1/epsilon)) non-zero coefficients such that Ex∈0,1 n [(p(x) - f( x))2] ≤ epsilon. We make the first progress on this conjecture and show that it is true for several natural subclasses of DNF formulas including randomly chosen DNF formulas and read-k DNF formulas. Our result yields the first polynomial-time query algorithm for agnostically learning these subclasses of DNF formulas with respect to the uniform distribution on {0, 1}n (for any constant error parameter and constant k). Applying recent work on sandwiching polynomials, our results imply that t -O(log 1/epsilon)-biased distributions fool the above subclasses of DNF formulas. This gives pseudorandom generators for these subclasses with shorter seed length than all previous work. (2) We give an efficient algortihm that learns random monotone DNF. The problem of efficiently learning the monotone subclass of polynomial-size DNF formulas from random examples was also posed in [Valiant, 1984]. This notoriously difficult question is still open, despite much study and the fact that known impediments to learning the non-monotone class (cf. [Blum et al., 1994; Blum, 2003a]) do not exist for monotone DNF formulas. We give the first algorithm that learns randomly chosen monotone DNF formulas of arbitrary polynomial size, improving results which efficiently learn n2-epsilon-size random monotone DNF formulas [Jackson and Servedio, 2005b]. Our main structural result is that most monotone DNF formulas reveal their term structure in their constant-degree Fourier coefficients.;In this thesis, we also see that connections between learning and cryptography are naturally made through average-case analysis. First, by applying techniques from average-case complexity, we demonstrate new ways of using cryptographic assumptions to prove limitations on learning. As counterpoint, we also exploit the average-case connection in the service of cryptography. Below is a more detailed description of these contributions. (1) We show that monotone polynomial-sized circuits are hard to learn if one-way functions exist. We establish the first cryptographic hardness results for learning polynomial-size classes of monotone circuits, giving a computational analogue of the information-theoretic hardness results of [Blum et al., 1998]. Some of our results show the cryptographic hardness of learning polynomial-size monotone circuits to accuracy only slightly greater than 1/2 + 1/ n ; this is close to the optimal accuracy bound, by positive results of Blum, et al. Our main tool is a complexity-theoretic approach to hardness amplification via noise sensitivity of monotone functions that was pioneered by O'Donnell [O'Donnell, 2004a]. (2) Learning an overcomplete basis: analysis of lattice-based signatures with perturbations. Lattice-based cryptographic constructions are desirable not only because they provide security based on worst-case hardness assumptions, but also because they can be extremely efficient and practical. We propose a general technique for recovering parts of the secret key in lattice-based signature schemes that follow the Goldreich-Goldwasser-Halevi (GGH) and NTRUSign design with perturbations. Our technique is based on solving a learning problem in the average-case. To solve the average-case problem, we propose a special-purpose optimization algorithm based on higher-order cumulants of the signature distribution, and give theoretical and experimental evidence of its efficacy. Our results suggest (but do not conclusively prove) that NTRUSign is vulnerable to a polynomial-time attack.
    机译:本文从平均案例的角度探讨了计算学习理论中的问题。通过这种观点,我们获得了学习理论和密码学的各种新结果。;计算学习理论中的几个主要开放性问题都围绕着有效学习多项式大小的DNF公式的问题,这可以追溯到Valiant引入PAC学习模型[ Valiant,1984年]。我们应用平均案例分析以两种方式在此问题上取得进展。 (1)我们证明曼苏尔猜想对于随机DNF是正确的。 1994年,Y。Mansour推测,对于n个带有t项的变量的每个DNF公式,都存在一个具有tO(log(1 / epsilon))非零系数的多项式p,使得Ex∈0,1n [(p(x )-f(x))2]≤epsilon。我们对此猜想取得了第一个进展,并证明了DNF公式的几个自然子类都是正确的,包括随机选择的DNF公式和read-k DNF公式。我们的结果产生了第一个多项式时间查询算法,用于相对于{0,1} n(对于任何恒定误差参数和常数k)的均匀分布,无知地学习DNF公式的这些子类。将最新工作应用于三明治多项式,我们的结果表明,t -O(log 1 / epsilon)偏向的分布欺骗了DNF公式的上述子类。这为这些子类提供了伪随机数生成器,其种子长度比以前的所有作品都短。 (2)我们给出了学习随机单调DNF的有效算法。 [Valiant,1984]也提出了从随机示例中有效学习多项式大小DNF公式的单调子类的问题。尽管进行了大量研究,并且对于单调DNF公式不存在学习非单调类的已知障碍(参见[Blum等,1994; Blum,2003a]),但这个臭名昭著的难题仍然悬而未决。我们给出了第一个学习任意多项式大小的随机选择单调DNF公式的算法,改进了有效学习n2-ε大小的随机单调DNF公式的结果[Jackson and Servedio,2005b]。我们的主要结构结果是,大多数单调DNF公式在其恒定度傅立叶系数中揭示了其项结构。在本论文中,我们还看到,学习和密码学之间的联系自然是通过平均案例分析得出的。首先,通过应用来自平均情况复杂度的技术,我们演示了使用密码假设来证明学习限制的新方法。作为对策,我们还利用加密服务中的平均情况连接。以下是这些贡献的更详细说明。 (1)我们证明,如果存在单向函数,则很难学习单调多项式大小的电路。我们建立了用于学习单调电路的多项式大小类的第一个密码学硬度结果,给出了[Blum et al。,1998]信息理论硬度结果的计算类似物。我们的一些结果表明,学习多项式大小的单调电路的密码硬度只有略大于1/2 + 1 / n的精度;根据Blum等人的积极结果,这接近最佳精度范围。我们的主要工具是由O'Donnell率先提出的通过单调函数的噪声敏感性来提高硬度的复杂度理论方法[O'Donnell,2004a]。 (2)学习过完备的基?。悍治龃腥哦幕诟竦那┟??;诟竦募用芙峁怪粤钊似谕?,不仅因为它们基于最坏情况的硬度假设提供了安全性,而且还因为它们非常高效和实用。我们提出了一种通用技术,用于恢复基于格的签名方案中的部分秘密密钥,该方案遵循具有扰动的Goldreich-Goldwasser-Halevi(GGH)和NTRUSign设计。我们的技术基于解决平均情况下的学习问题。为了解决平均情况下的问题,我们提出了一种基于签名分布的高阶累积量的专用优化算法,并给出了其有效性的理论和实验证据。我们的结果表明(但不能最终证明)NTRUSign容易受到多项式时间攻击。

    著录项

    • 作者

      Wan, Andrew.;

    • 作者单位

      Columbia University.;

    • 授予单位 Columbia University.;
    • 学科 Applied Mathematics.;Computer Science.
    • 学位 Ph.D.
    • 年度 2010
    • 页码 134 p.
    • 总页数 134
    • 原文格式 PDF
    • 正文语种 eng
    • 中图分类 ;
    • 原文服务方 国家工程技术数字图书馆
    • 关键词

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