computing with very weak random sources we give an efficient algorithm to extract randomness from a very weak random source using a small additional number t of truly random bits our work extends that of nisan and zuckerman j comput system sci 52 1996 pp 43 52 in that t remains small even if the entropy rate is well below constant a key application of this is in running randomized algorithms using such a very weak source of randomness for any fixed gamma 0 we show how to simulate rp algorithms in time n o log n using the output of a ds with min entropy r gamma such a weak random source is asked once for r bits it outputs an r bit string according to any probability distribution that places probability at most 2 r gamma on each string if gamma 1 2 our simulation also works for bpp for gamma 1 1 k 1 our simulation takes time n o logk n log k is the logarithm iterated k times we also give a polynomial time bpp simulation using chor goldreich sources of min entropy r omega 1 which is optimal we present applications to time space tradeoffs expander constructions and to the hardness of approximation of independent interest is our randomness efficient leftover hash lemma a key tool for extracting randomness from weak random sources

on serre s conjecture for 2 dimensional mod p representations of gal q q we prove the existence in many cases of minimally ramified p adic lifts of 2 dimensional continuous odd absolutely irreducible mod p representations of the absolute galois group of q it is predicted by serre s conjecture that such representations arise from newforms of optimal level and weight

robust fuzzy extractors and authenticated key agreement from close secrets consider two parties holding samples from correlated distributions w and w prime respectively where these samples are within distance t of each other in some metric space the parties wish to agree on a close to uniformly distributed secret key r by sending a single message over an insecure channel controlled by an all powerful adversary who may read and modify anything sent over the channel we consider both the keyless case where the parties share no additional secret information and the keyed case where the parties share a long term secret ssr sk ssr ext that they can use to generate a sequence of session keys r j using multiple pairs w j w prime j the former has applications to e g biometric authentication while the latter arises in e g the bounded storage model with errors we show solutions that improve upon previous work in several respects the best prior solution for the keyless case with no errors i e t 0 requires the min entropy of w to exceed 2n 3 where n is the bit length of w our solution applies whenever the min entropy of w exceeds the minimal threshold n 2 and yields a longer key

construction of some families of 2 dimensional crystalline representations we construct explicitly some analytic families of etale modules which give rise to analytic families of 2 dimensional crystalline representations as an application of our constructions we verify some conjectures of breuil on the reduction modulo p of those representations and extend some results of deligne edixhoven fontaine and serre on the representations arising from modular forms

reductions of galois representations of slope 1 abstract we compute the reductions of irreducible crystalline two dimensional representations of g q p of slope 1 for primes p 5 and all weights we describe the semisimplification of the reductions completely in particular we show that the reduction is often reducible we also investigate whether the extension obtained is peu or tres ramifiee in the relevant reducible non semisimple cases the proof uses the compatibility between the p adic and mod p local langlands correspondences and involves a detailed study of the reductions of both the standard and non standard lattices in certain p adic banach spaces

extracting kolmogorov complexity with applications to dimension zero one laws we apply results on extracting randomness from independent sources to extract kolmogorov complexity for any a e 0 given a string x with k x a x we show how to use a constant number of advice bits to efficiently compute another string y y w x with k y 1 e y this result holds for both unbounded and space bounded kolmogorov complexity we use the extraction procedure for space bounded complexity to establish zero one laws for the strong dimensions of complexity classes within espace the unbounded extraction procedure yields a zero one law for the constructive strong dimensions of turing degrees

unbalanced expanders and randomness extractors from parvaresh vardy codes we give an improved explicit construction of highly unbalanced bipartite expander graphs with expansion arbitrarily close to the degree which is polylogarithmic in the number of vertices both the degree and the number of right hand vertices are polynomially close to optimal whereas the previous constructions of ta shma et al 2007 required at least one of these to be quasipolynomial in the optimal our expanders have a short and self contained description and analysis based on the ideas underlying the recent list decodable error correcting codes of parvaresh and vardy 2005 our expanders can be interpreted as near optimal randomness condensers that reduce the task of extracting randomness from sources of arbitrary min entropy rate to extracting randomness from sources of min entropy rate arbitrarily close to 1 which is a much easier task using this connection we obtain a new self contained construction of randomness extractors that is optimal up to constant factors while being much simpler than the previous construction of lu et al 2003 and improving upon it when the error parameter is small e g 1 poly n

better short seed quantum proof extractors we construct a strong extractor against quantum storage that works for every min entropy k has logarithmic seed length and outputs w k bits provided that the quantum adversary has at most bk qubits of memory for any b 12 the construction works by first condensing the source with minimal entropy loss and then applying an extractor that works well against quantum adversaries when the source is close to uniform we also obtain an improved construction of a strong quantum proof extractor in the high min entropy regime specifically we construct an extractor that uses a logarithmic seed length and extracts w n bits from any source over 0 1 n provided that the min entropy of the source conditioned on the quantum adversary s state is at least 1 b n for any b 12

serre s modularity conjecture ii we provide proofs of theorems 4 1 and 5 1 of khare and wintenberger invent math doi 10 1007 s00222 009 0205 7 2009

multiplicites modulaires et representations de rm gl sb 2 mathbf z sb p et de rm gal overline mathbf q sb p mathbf q sb p en ell p appendice par guy henniart sur l unicite des types pour rm gl sb 2

simple extractors for all min entropies and a new pseudorandom generator a randomness extractor is an algorithm that given a sample from a distribution with sufficiently high min entropy and a short random seed produces an output that is statistically indistinguishable from uniform min entropy is a measure of the amount of randomness in a distribution we present a simple self contained extractor construction that produces good extractors for all min entropies our construction is algebraic and builds on a new polynomial based approach introduced by ta shma et al 2001b using our improvements we obtain for example an extractor with output length m k log n o 1 and seed length 1 p log n for an arbitrary 0 1 where n is the input length and k is the min entropy of the input distribution a pseudorandom generator is an algorithm that given a short random seed produces a long output that is computationally indistinguishable from uniform our technique also gives a new way to construct pseudorandom generators from functions that require large circuits our pseudorandom generator construction is not based on the nisan wigderson generator nisan and wigderson 1994 and turns worst case hardness directly into pseudorandomness the parameters of our generator match those in impagliazzo and wigderson 1997 and sudan et al 2001 and in particular are strong enough to obtain a new proof that p bpp if e requires exponential size circuits our construction also gives the following improvements over previous work we construct an optimal hitting set generator that stretches o log n random bits into s 1 pseudorandom bits when given a function on log n bits that requires circuits of size s this yields a quantitatively optimal hardness versus randomness tradeoff for both rp and bpp and solves an open problem raised in impagliazzo et al 1999 we give the first construction of pseudorandom generators that fool nondeterministic circuits when given a function that requires large nondeterministic circuits this technique also give a quantitatively optimal hardness versus randomness tradeoff for am and the first hardness amplification result for nondeterministic circuits

extraction of optimally unbiased bits from a biased source we explore the problem of transforming n independent and identically biased 1 1 valued random variables x sub 1 x sub n into a single 1 1 random variable f x sub 1 x sub n so that this result is as unbiased as possible in general no function f produces a completely unbiased result we perform the first study of the relationship between the bias b of these x sub i and the rate at which f x sub 1 x sub n can converge to an unbiased 1 1 random variable as n spl rarr spl infin a 1 1 random variable has bias b if e x sub i b fixing a bias b we explore the rate at which the output bias e f x sub 1 x sub n can tend to zero for a function f 1 1 spl rarr 1 1 this is accomplished by classifying the behavior of the natural normalized quantity spl xi b spl delta inf sub f lim sub n spl rarr spl infin n spl radic e f x sub 1 x sub n this infimum taken over all such f we show that for rational b spl xi b 1 s where 1 b 2 r s r and s relatively prime developing the theory of uniform distribution of sequences to suit our problem we then explore the case where b is irrational we prove a new metrical theorem concerning multidimensional diophantine approximation type from which we show that for lebesgue almost all biases b spl xi b 0 finally we show that algebraic biases exhibit curious boundary behavior falling into two classes class 1 those algebraics b for which spl xi b 0 and furthermore c sub 1 spl les spl xi b spl les c sub 2 where c sub 1 and c sub 2 are positive constants depending only on b s algebraic characteristics class 2 those algebraics b for which there exist n 0 and f 1 1 sup n spl rarr 1 1 so that e f x sub 1 x sub n 0 notice that this classification excludes the possibility that n spl radic e f x sub 1 x sub n limits to zero for algebraics for rational and algebraic biases we also study the computational problem by restricting f to be a polynomial time computable function finally we discuss natural extensions where output distributions other than the uniform distribution on 1 1 are sought

the fontaine mazur conjecture for gl _2 introduction 641 1 breuil mezard conjecture and the p adic local langlands 644 1 1 the breuil mezard conjecture 644 1 2 review of colmez s functor 647 1 3 hilbert samuel multiplicities 650 1 4 representations and pseudo representations 653 1 5 gl2 qp representations mod p 657 1 6 local patching and multiplicities 661 1 7 from pseudo representations to representations 669 2 modularity via the breuil mezard conjecture 675 2 1 quaternionic forms 675 2 2 global patching and multiplicities 677 2 3 the breuil mezard conjecture 686 addendum to ki 1 687 references 689

a fast randomized algorithm for overdetermined linear least squares regression we introduce a randomized algorithm for overdetermined linear least squares regression given an arbitrary full rank m n matrix a with m n any m 1 vector b and any positive real number e the procedure computes an n 1 vector x such that x minimizes the euclidean norm ax b to relative precision e the algorithm typically requires log n log 1 e mn n 3 floating point operations this cost is less than the mn 2 required by the classical schemes based on qr decompositions or bidiagonalization we present several numerical examples illustrating the performance of the algorithm

leftover hashing against quantum side information the leftover hash lemma states that the output of a two universal hash function applied to an input with sufficiently high entropy is almost uniformly random in its standard formulation the lemma refers to a notion of randomness that is usually implicitly defined with respect to classical side information here a strictly more general version of the leftover hash lemma that is valid even if side information is represented by the state of a quantum system is shown our result applies to almost two universal families of hash functions the generalized leftover hash lemma has applications in cryptography e g for key agreement in the presence of an adversary who is not restricted to classical information processing

langlands base change for gl 2 let f be a totally real galois number eld we prove the existence of base change relative to the extension f q for every holomorphic newform of weight at least 2 and odd level under simple local assumptions on the eld f