248 research outputs found

    Algorithms and lower bounds for de Morgan formulas of low-communication leaf gates

    Get PDF
    The class FORMULA[s]∘GFORMULA[s] \circ \mathcal{G} consists of Boolean functions computable by size-ss de Morgan formulas whose leaves are any Boolean functions from a class G\mathcal{G}. We give lower bounds and (SAT, Learning, and PRG) algorithms for FORMULA[n1.99]∘GFORMULA[n^{1.99}]\circ \mathcal{G}, for classes G\mathcal{G} of functions with low communication complexity. Let R(k)(G)R^{(k)}(\mathcal{G}) be the maximum kk-party NOF randomized communication complexity of G\mathcal{G}. We show: (1) The Generalized Inner Product function GIPnkGIP^k_n cannot be computed in FORMULA[s]∘GFORMULA[s]\circ \mathcal{G} on more than 1/2+Δ1/2+\varepsilon fraction of inputs for s=o ⁣(n2(k⋅4k⋅R(k)(G)⋅log⁥(n/Δ)⋅log⁥(1/Δ))2). s = o \! \left ( \frac{n^2}{ \left(k \cdot 4^k \cdot {R}^{(k)}(\mathcal{G}) \cdot \log (n/\varepsilon) \cdot \log(1/\varepsilon) \right)^{2}} \right). As a corollary, we get an average-case lower bound for GIPnkGIP^k_n against FORMULA[n1.99]∘PTFk−1FORMULA[n^{1.99}]\circ PTF^{k-1}. (2) There is a PRG of seed length n/2+O(s⋅R(2)(G)⋅log⁥(s/Δ)⋅log⁥(1/Δ))n/2 + O\left(\sqrt{s} \cdot R^{(2)}(\mathcal{G}) \cdot\log(s/\varepsilon) \cdot \log (1/\varepsilon) \right) that Δ\varepsilon-fools FORMULA[s]∘GFORMULA[s] \circ \mathcal{G}. For FORMULA[s]∘LTFFORMULA[s] \circ LTF, we get the better seed length O(n1/2⋅s1/4⋅log⁥(n)⋅log⁥(n/Δ))O\left(n^{1/2}\cdot s^{1/4}\cdot \log(n)\cdot \log(n/\varepsilon)\right). This gives the first non-trivial PRG (with seed length o(n)o(n)) for intersections of nn half-spaces in the regime where Δ≀1/n\varepsilon \leq 1/n. (3) There is a randomized 2n−t2^{n-t}-time #\#SAT algorithm for FORMULA[s]∘GFORMULA[s] \circ \mathcal{G}, where t=Ω(ns⋅log⁥2(s)⋅R(2)(G))1/2.t=\Omega\left(\frac{n}{\sqrt{s}\cdot\log^2(s)\cdot R^{(2)}(\mathcal{G})}\right)^{1/2}. In particular, this implies a nontrivial #SAT algorithm for FORMULA[n1.99]∘LTFFORMULA[n^{1.99}]\circ LTF. (4) The Minimum Circuit Size Problem is not in FORMULA[n1.99]∘XORFORMULA[n^{1.99}]\circ XOR. On the algorithmic side, we show that FORMULA[n1.99]∘XORFORMULA[n^{1.99}] \circ XOR can be PAC-learned in time 2O(n/log⁥n)2^{O(n/\log n)}

    The Crypto-democracy and the Trustworthy

    Full text link
    In the current architecture of the Internet, there is a strong asymmetry in terms of power between the entities that gather and process personal data (e.g., major Internet companies, telecom operators, cloud providers, ...) and the individuals from which this personal data is issued. In particular, individuals have no choice but to blindly trust that these entities will respect their privacy and protect their personal data. In this position paper, we address this issue by proposing an utopian crypto-democracy model based on existing scientific achievements from the field of cryptography. More precisely, our main objective is to show that cryptographic primitives, including in particular secure multiparty computation, offer a practical solution to protect privacy while minimizing the trust assumptions. In the crypto-democracy envisioned, individuals do not have to trust a single physical entity with their personal data but rather their data is distributed among several institutions. Together these institutions form a virtual entity called the Trustworthy that is responsible for the storage of this data but which can also compute on it (provided first that all the institutions agree on this). Finally, we also propose a realistic proof-of-concept of the Trustworthy, in which the roles of institutions are played by universities. This proof-of-concept would have an important impact in demonstrating the possibilities offered by the crypto-democracy paradigm.Comment: DPM 201

    Multiparty Karchmer - Wigderson Games and Threshold Circuits

    Get PDF
    We suggest a generalization of Karchmer - Wigderson communication games to the multiparty setting. Our generalization turns out to be tightly connected to circuits consisting of threshold gates. This allows us to obtain new explicit constructions of such circuits for several functions. In particular, we provide an explicit (polynomial-time computable) log-depth monotone formula for Majority function, consisting only of 3-bit majority gates and variables. This resolves a conjecture of Cohen et al. (CRYPTO 2013)

    Secret Sharing and Secure Computing from Monotone Formulae

    Get PDF
    We present a construction of log-depth formulae for various threshold functions based on atomic threshold gates of constant size. From this, we build a new family of linear secret sharing schemes that are multiplicative, scale well as the number of players increases and allows to raise a shared value to the characteristic of the underlying field without interaction. Some of these schemes are in addition strongly multiplicative. Our formulas can also be used to construct multiparty protocols from protocols for a constant number of parties. In particular we implement black-box multiparty computation over non-Abelian groups in a way that is much simpler than previously known and we also show how to get a protocol in this setting that is efficient and actively secure against a constant fraction of corrupted parties, a long standing open problem. Finally, we show a negative result on usage of our scheme for pseudorandom secret sharing as defined by Cramer, DamgÄrd and Ishai

    Making Code Voting Secure against Insider Threats using Unconditionally Secure MIX Schemes and Human PSMT Protocols

    Full text link
    Code voting was introduced by Chaum as a solution for using a possibly infected-by-malware device to cast a vote in an electronic voting application. Chaum's work on code voting assumed voting codes are physically delivered to voters using the mail system, implicitly requiring to trust the mail system. This is not necessarily a valid assumption to make - especially if the mail system cannot be trusted. When conspiring with the recipient of the cast ballots, privacy is broken. It is clear to the public that when it comes to privacy, computers and "secure" communication over the Internet cannot fully be trusted. This emphasizes the importance of using: (1) Unconditional security for secure network communication. (2) Reduce reliance on untrusted computers. In this paper we explore how to remove the mail system trust assumption in code voting. We use PSMT protocols (SCN 2012) where with the help of visual aids, humans can carry out mod  10\mod 10 addition correctly with a 99\% degree of accuracy. We introduce an unconditionally secure MIX based on the combinatorics of set systems. Given that end users of our proposed voting scheme construction are humans we \emph{cannot use} classical Secure Multi Party Computation protocols. Our solutions are for both single and multi-seat elections achieving: \begin{enumerate}[i)] \item An anonymous and perfectly secure communication network secure against a tt-bounded passive adversary used to deliver voting, \item The end step of the protocol can be handled by a human to evade the threat of malware. \end{enumerate} We do not focus on active adversaries
    • 

    corecore