New degree bounds for polynomial threshold fun tions

Aeùñö¹ìììóööøø Óò×øöù Blockinøøóò× Óó Ae Blockin Blockin Blockinòø È××ùùó¹êêòòóñ Ùò Blockinøøóò×

We des ribe eÆ ient onstru tions for various ryptographi primitives in private-key as well as publi -key ryptography. Our major results are two new onstru tions of pseudorandom fun tions. We prove the pseudo-randomness of one onstru tion under the assumption that fa toring (Blum integers) is hard while the other onstru tion is pseudo-random if the de isional version of the DiÆe-Hellman assumpti...

Äääöòòòò Ùò Blockinøøóò× Óó Ööððúòø Úööööðð×

We onsider a fundamental problem in omputational learning theory: learning an arbitrary Boolean fun tion that depends on an unknown set of k out of n Boolean variables. We give an algorithm for learning su h fun tions from uniform random examples that runs in time roughly (n k ) ! !+1 ; where ! < 2:376 is the matrix multipli ation exponent. We thus obtain the rst polynomial fa tor improvement o...

Interpolation and approximation in Taylor spaces

Barbara Zwi knagl1 and Robert S haba k2 Abstra t: The univariate Taylor formula without remainder allows to reprodu e a fun tion ompletely from ertain derivative values. Thus one an look for Hilbert spa es in whi h the Taylor formula a ts as a reprodu tion formula. It turns out that there are many Hilbert spa es whi h allow this, and they should be alled Taylor spa es. They have ertain reprodu ...

Learning and Lower Bounds for AC with Threshold Gates

In 2002 Jackson et al. [JKS02] asked whether AC circuits augmented with a threshold gate at the output can be efficiently learned from uniform random examples. We answer this question affirmatively by showing that such circuits have fairly strong Fourier concentration; hence the low-degree algorithm of Linial, Mansour and Nisan [LMN93] learns such circuits in sub-exponential time. Under a conje...

On Graph Pebbling , Threshold Fun tions , andSupernormal Posets