1,887 research outputs found
Quantum Algorithms for Boolean Equation Solving and Quantum Algebraic Attack on Cryptosystems
Decision of whether a Boolean equation system has a solution is an NPC
problem and finding a solution is NP hard. In this paper, we present a quantum
algorithm to decide whether a Boolean equation system FS has a solution and
compute one if FS does have solutions with any given success probability. The
runtime complexity of the algorithm is polynomial in the size of FS and the
condition number of FS. As a consequence, we give a polynomial-time quantum
algorithm for solving Boolean equation systems if their condition numbers are
small, say polynomial in the size of FS. We apply our quantum algorithm for
solving Boolean equations to the cryptanalysis of several important
cryptosystems: the stream cipher Trivum, the block cipher AES, the hash
function SHA-3/Keccak, and the multivariate public key cryptosystems, and show
that they are secure under quantum algebraic attack only if the condition
numbers of the corresponding equation systems are large. This leads to a new
criterion for designing cryptosystems that can against the attack of quantum
computers: their corresponding equation systems must have large condition
numbers
Attribute-Efficient PAC Learning of Low-Degree Polynomial Threshold Functions with Nasty Noise
The concept class of low-degree polynomial threshold functions (PTFs) plays a
fundamental role in machine learning. In this paper, we study PAC learning of
-sparse degree- PTFs on , where any such concept depends
only on out of attributes of the input. Our main contribution is a new
algorithm that runs in time and under the Gaussian
marginal distribution, PAC learns the class up to error rate with
samples even when an fraction of them are corrupted by the nasty noise of
Bshouty et al. (2002), possibly the strongest corruption model. Prior to this
work, attribute-efficient robust algorithms are established only for the
special case of sparse homogeneous halfspaces. Our key ingredients are: 1) a
structural result that translates the attribute sparsity to a sparsity pattern
of the Chow vector under the basis of Hermite polynomials, and 2) a novel
attribute-efficient robust Chow vector estimation algorithm which uses
exclusively a restricted Frobenius norm to either certify a good approximation
or to validate a sparsity-induced degree- polynomial as a filter to detect
corrupted samples.Comment: ICML 202
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