classical dirichlet forms on topological vector spaces the construction of the associated diffusion process given a minimal classical dirichlet form onl 2 e we construct the associated diffusion process heree is a locally convex topological vector space and is a not necessarily quasi invariant probability measure one the construction is carried out under certain assumptions one and which can be easily verified in many examples in particular we explicitly apply our results to time zero and space time quantum fields with or with out cut off

density and tails of unimodal convolution semigroups abstract we give sharp bounds for the isotropic unimodal probability convolution semigroups when their levy khintchine exponent has matuszewska indices strictly between 0 and 2

classical dirichlet forms on topological vector spaces closability and a cameron martin formula abstract we prove a sufficient condition for the closability of classical dirichlet forms on l 2 e which is also necessary if all components of the dirichlet form are closable here e is a locally convex topological vector space and a not necessarily quasi invariant probability measure on e the same condition is shown to imply the existence of a closed extension whose domain can be described explicitly in the special case where is quasi invariant with respect to certain vectors in e our result generalises previous theorems on closability in addition we prove a cameron martin type formula for a large class of measures if e is finite dimensional our characterisation of closability is the analogue of the corresponding one dimensional result applications to quantum fields and the connection with the well studied case of abstract wiener spaces are discussed

uniform random number generation from markov chains non asymptotic and asymptotic analyses in this paper we derive non asymptotic achievability and converse bounds on the random number generation with without side information our bounds are efficiently computable in the sense that the computational complexity does not depend on the block length we also characterize the asymptotic behaviors of the large deviation regime and the moderate deviation regime by using our bounds which implies that our bounds are asymptotically tight in those regimes we also show the second order rates of those problems and derive single letter forms of the variances characterizing the second order rates furthermore we address the relative entropy rate and the modified mutual information rate for these problems

asymptotic behavior for the heat equation in nonhomogeneous media with critical density abstract we study the long time behavior of solutions to the heat equation in nonhomogeneous media with critical singular density x 2 t u u in r n 0 in dimensions n 3 the asymptotic behavior proves to have some interesting and quite striking properties we show that there are two completely different asymptotic profiles depending on whether the initial data u 0 vanishes at x 0 or not moreover in the former the results are true only for radially symmetric solutions and we provide counterexamples to convergence to symmetric profiles in the general case

pseudorandom generators from regular one way functions we revisit the problem of basing pseudorandom generators on regular one way functions and present the following constructions for any known regular one way function on n bit inputs that is known to be e hard to invert we give a neat and tighter proof for the folklore construction of pseudorandom generator of seed length n by making a single call to the underlying one way function for any unknown regular one way function with known e hardness we give a new construction with seed length n and o n log 1 e calls here the number of calls is also optimal by matching the lower bounds of holenstein and sinha 2012 6 both constructions require the knowledge about e but the dependency can be removed while keeping nearly the same parameters in the latter case we get a construction of pseudo random generator from any unknown regular one way function using seed length o n and o n log n calls where o omits a factor that can be made arbitrarily close to constant e g log log log n or even less this improves the randomized iterate approach by haitner et al 2006 4 which requires seed length o n log n and o n log n calls

how to construct random functions a constructive theory of randomness for functions based on computational complexity is developed and a pseudorandom function generator is presented this generator is a deterministic polynomial time algorithm that transforms pairs g r where g is any one way function and r is a random k bit string to polynomial time computable functions r 1 2 k 1 2 k these r s cannot be distinguished from random functions by any probabilistic polynomial time algorithm that asks and receives the value of a function at arguments of its choice the result has applications in cryptography random constructions and complexity theory

a priori estimates for fractional nonlinear degenerate diffusion equations on bounded domains we investigate quantitative properties of the nonnegative solutions u t x geq 0 to the nonlinear fractional diffusion equation partial t u mathcal l u m 0 posed in a bounded domain x in omega subset mathbb r n with m 1 for t 0 as mathcal l we use one of the most common definitions of the fractional laplacian delta s 0 s 1 in a bounded domain with zero dirichlet boundary conditions we consider a general class of very weak solutions of the equation and obtain a priori estimates in the form of smoothing effects absolute upper bounds lower bounds and harnack inequalities we also investigate the boundary behaviour and we obtain sharp estimates from above and below in addition we obtain similar estimates for fractional semilinear elliptic equations either the standard laplacian case s 1 or the linear case m 1 are recovered as limits the method is quite general suitable to be applied to a number of similar problems

sharp short and long time l bounds for solutions to porous media equations with homogeneous neumann boundary conditions abstract we study a class of nonlinear diffusion equations whose model is the classical porous media equation on domains r n n 3 with homogeneous neumann boundary conditions we improve the known results in such model case proving sharp l q 0 l regularizing properties of the evolution for short time and sharp long time l bounds for convergence of solutions to their mean value the generality of the discussion allows to consider almost at the same time weighted versions of the above equation provided an appropriate weighted sobolev inequality holds in fact we show that the validity of such weighted sobolev inequality is equivalent to the validity of a suitable l q 0 l bound for solutions to the associated weighted porous media equation the long time asymptotic analysis relies on the assumed weighted sobolev inequality only and allows to prove uniform convergence to the mean value with the rate predicted by linearization in such generality

a general fractional porous medium equation we develop a theory of existence and uniqueness for the following porous medium equation with fractional diffusion u t 2 u m 1 u 0 x r n t 0 u x 0 f x x r n we consider data f l 1 r n and all exponents 0 0 existence and uniqueness of a weak solution is established for m m n n giving rise to an l 1 contraction semigroup in addition we obtain the main qualitative properties of these solutions in the lower range 0 m m existence and uniqueness of solutions with good properties happen under some restrictions and the properties are different from the case above m we also study the dependence of solutions on f m and moreover we consider the above questions for the problem posed in a bounded domain

the fractional laplacian in power weighted l p spaces integration by parts formulas and self adjointness abstract we consider the fractional laplacian operator s let s 0 1 on euclidean space and investigate the validity of the classical integration by parts formula that connects the l 2 r d scalar product between a function and its fractional laplacian to the nonlocal norm of the fractional sobolev space h s r d more precisely we focus on functions belonging to some weighted l 2 space whose fractional laplacian belongs to another weighted l 2 space we prove and disprove the validity of the integration by parts formula depending on the behaviour of the weight x at infinity the latter is assumed to be like a power both near the origin and at infinity the two powers being possibly different our results have direct consequences for the self adjointness of the linear operator formally given by 1 s the generality of the techniques developed allows us to deal with weighted l p spaces as well

heat kernel estimates for the fractional laplacian with dirichlet conditions we give sharp estimates for the heat kernel of the fractional laplacian with dirichlet condition for a general class of domains including lipschitz domains

on the conditional renyi entropy the renyi entropy of general order unifies the well known shannon entropy with several other entropy notions like the min entropy or collision entropy in contrast to the shannon entropy there seems to be no commonly accepted definition for the conditional renyi entropy several versions have been proposed and used in the literature in this paper we reconsider the definition for the conditional renyi entropy of general order as proposed by arimoto in the seventies we show that this particular notion satisfies several natural properties in particular we show that it satisfies monotonicity under conditioning meaning that conditioning can only reduce the entropy and a weak form of chain rule which implies that the decrease in entropy due to conditioning is bounded by the number of bits one conditions on none of the other suggestions for the conditional renyi entropy satisfies both these properties finally we show a natural interpretation of the conditional renyi entropy in terms of unconditional renyi divergence and we show consistency with a recently proposed notion of conditional renyi entropy in the quantum setting

entropy analysis and new constructions of biometric key generation systems in this paper a new soft two level approach for the generation of multiple and revocable biometric keys adapted to the analog nature of biometric signals is proposed it consists of a novel randomized soft code offset construction for the euclidean metric at the first level and a code redundancy construction for the hamming metric possibly based on a reed solomon code at the second level the shannon entropy analysis shows that the new construction achieves optimal security with respect to biometric template protection whereas the previously proposed constructions for the euclidean metric are shown to be insecure in the multiple key setting in addition a general code offset construction for the hamming metric is analyzed in terms of the shannon entropy instead of the usual min entropy this allows a nice characterization of the codes that provide biometric template protection in the multiple key scenario as well as a further differentiation among these codes with respect to the achievable security level

non local dirichlet forms and symmetric jump processes we consider the non local symmetric dirichlet form e f given by with f the closure with respect to e 1 of the set of c 1 functions on r d with compact support where e 1 f f e f f f rd f x 2 dx and where the jump kernel j satisfies for 0 2 x y 1 this assumption allows the corresponding jump process to have jump intensities whose sizes depend on the position of the process and the direction of the jump we prove upper and lower estimates on the heat kernel we construct a strong markov process corresponding to e f we prove a parabolic harnack inequality for non negative functions that solve the heat equation with respect to e finally we construct an example where the corresponding harmonic functions need not be continuous

absolute continuity of symmetric diffusions let x and y be symmetric diffusion processes with a common state space and let p m resp q be the law of x resp y with its symmetry measure m resp as initial distribution we study the consequences of the absolute continuity condition q t 1 t the density p also serves to link the dirichlet forms of x and y in a simple way our identification of l relies on notions of even and odd for additive functionals these notions complement fukushima s decomposition and the forward backward martingale decomposition of lyons and zheng

perturbation of dirichlet forms lower semiboundedness closability and form cores we study perturbations e e qu of dirichlet forms e on some l2 space l2 m given by quadratic forms q g g d with a signed borel measure whose positive and negative parts are smooth measures with respect to the given dirichlet form we give necessary and sufficient conditions for e to be a lower semibounded closed quadratic form with a unique associated continuous semigroup pt on l2 m we also study the associated resolvent semigroup as well as the relation between e and the largest closed quadratic form which is smaller than e in particular we exhibit form cores

tight exponential analysis of universally composable privacy amplification and its applications motivated by the desirability of universal composability we analyze in terms of l1 distinguishability the task of secret key generation from a joint random variable under this secrecy criterion using the renyi entropy of order 1 s for s 0 1 we derive a new upper bound of eve s distinguishability under the application of the universal2 hash functions it is also shown that this bound gives the tight exponential rate of decrease in the case of independent and identical distributions the result is applied to the wiretap channel model and to secret key generation distillation by public discussion

unbiased bits from sources of weak randomness and probabilistic communication complexity a new model for weak random physical sources is presented the new model strictly generalizes previous models e g the santha and vazirani model 27 the sources considered output strings according to probability distributions in which no single string is too probable the new model provides a fruitful viewpoint on problems studied previously such as extracting almost perfect bits from sources of weak randomness the question of possibility as well as the question of efficiency of such extraction schemes are addressed probabilistic communication complexity it is shown that most functions have linear communication complexity in a very strong probabilistic sense robustness of bpp with respect to sources of weak randomness generalizing a result of vazirani and vazirani 32 33

non symmetric perturbations of symmetric dirichlet forms abstract we provide a path space integral representation of the semigroup associated with the quadratic form obtained by lower order perturbation of a symmetric local dirichlet form the representation is a combination of feynman kac and girsanov formulas and extends previously known results in the framework of symmetric diffusion processes through the use of the hardy class of smooth measures which contains the kato class of smooth measures

how to construct pseudorandom permutations from pseudorandom functions we show how to efficiently construct a pseudorandom invertible permutation generator from a pseudorandom function generator goldreich goldwasser and micali how to construct random functions p

how to generate cryptographically strong sequences of pseudo random bits we give a set of conditions that allow one to generate 50 50 unpredictable bits based on those conditions we present a general algorithmic scheme for constructing polynomial time deterministic algorithms that stretch a short secret random input into a long sequence of unpredictable pseudo random bits we give an implementation of our scheme and exhibit a pseudo random bit generator for which any efficient strategy for predicting the next output bit with better than 50 50 chance is easily transformable to an equally efficient algorithm for solving the discrete logarithm problem in particular if the discrete logarithm problem cannot be solved in probabilistic polynomial time no probabilistic polynomial time algorithm can guess the next output bit better than by flipping a coin if head guess 0 if tail guess 1

on the potential theory of symmetric markov processes on reprend des resultats de dynkin et on considere un processus de markov fort symetrique continu a droite on etudie la relation existant entre le cone de mesures positives d energie finie et la capacite

uniqueness of very weak solutions for a fractional filtration equation abstract we prove existence and uniqueness of distributional bounded solutions to a fractional filtration equation in r d with regards to uniqueness it was shown even for more general equations in 22 that if two bounded solutions u w of 1 1 satisfy u w l 1 r d 0 t then u w we obtain here that this extra assumption can in fact be removed and establish uniqueness in the class of merely bounded solutions for nonnegative initial data we first show that a minimal solution exists and then that any other solution must coincide with it a similar procedure is carried out for sign changing solutions as a consequence distributional solutions have locally finite energy

a widder s type theorem for the heat equation with nonlocal diffusion the main goal of this work is to prove that every non negative strong solutionu x t to the problem u t delta alpha 2 u 0 rm for x t in mathbb r n times 0 t 0 alpha 2 can be written as u x t int mathbb r n p t x y u y 0 dy where p t x frac 1 t n alpha p left frac x t 1 alpha right and p x int mathbb r n e i x cdot xi xi alpha d xi this result shows uniqueness in the setting of non negative solutions and extends some classical results for the heat equation by widder in 15 to the nonlocal diffusion framework

a fractional porous medium equation abstract we develop a theory of existence uniqueness and regularity for the following porous medium equation with fractional diffusion u t 1 2 u m 1 u 0 x r n t 0 u x 0 f x x r n with m m n 1 n n 1 and f l 1 r n an l 1 contraction semigroup is constructed and the continuous dependence on data and exponent is established nonnegative solutions are proved to be continuous and strictly positive for all x r n t 0

on the existence of pseudorandom generators pseudorandom generators suggested and developed by blum and micali and yao are efficient deterministic programs that expand a randomly selected k bit seed into a much longer pseudorandom bit sequ

green s and dirichlet spaces associated with fine markov processes this is the second paper in a series devoted to green s and dirichlet spaces in the first paper we have investigated green s space k and the dirichlet space h associated with a symmetric markov transition function pt x b now we assume that p is a transition function of a fine markov process x and we prove that a the space h can be built from functions which are right continuous along almost all paths b the positive cone k in k can be identified with a cone m of measures on the state space c the positive cone h in h can be interpreted as the cone of green s potentials of measures m to every measurable set b in the state space e there correspond a subspace k b of k and a subspace h b of h the orthogonal projections of k onto k and of h onto h b can be expressed in terms of the hitting probabilities of b by the markov process x as the main tool we use additive functionals of x corresponding to measures m

markov processes and nonsymmetric dirichlet forms without regularity we study right markov processes in weak duality satisfying the sector condition of silverstein dirichlet space techniques are employed but without the customary c0 regularity hypothesis nonetheless the potential theory of such processes is quite amenable in particular it is proved that semipolar sets are quasi polar this extends results of silverstein and lejan

on the power of the randomized iterate we consider two of the most fundamental theorems in cryptography the first due to hastad et al siam j comput 28 1999 pp 1364 1396 is that pseudorandom generators can be constructed from any one way function the second due to yao proceedings of the 23 rd annual symposium on foundations of computer science focs 1982 pp 80 91 states that the existence of weak one way functions implies the existence of full fledged one way functions these powerful plausibility results shape our understanding of hardness and randomness in cryptography but unfortunately their proofs are not as tight i e security preserving as one may desire this work revisits a technique that we call the randomized iterate introduced by goldreich krawczyk and luby siam j comput 22 1993 pp 1163 1175 this technique was used by goldreich krawczyk and luby siam j comput 22 1993 pp 1163 1175 to give a construction of pseudorandom generators from regular one way functions we simplify and strengthen this technique in order to obtain a similar construction where the seed length of the resulting generators is as short as theta n log n rather than theta n 3 achieved by goldreich krawczyk and luby siam j comput 22 1993 pp 1163 1175 our technique has the potential of implying seed length theta n and the only bottleneck for such a result are the parameters of current generators against bounded space computations we give a construction with similar parameters for security amplification of regular one way functions this improves upon the construction of goldreich et al proceedings of the 31 st annual symposium on foundations of computer science focs 1990 pp 318 326 in that the construction does not need to know the regularity parameter of the functions in terms of security the two reductions are incomparable in addition we use the randomized iterate to show a construction of a pseudorandom generator based on an exponentially hard one way function that has a seed length of only theta n 2 this improves a recent result of holenstein proceedings of the theory of cryptography third theory of cryptography conference tcc 2006 that shows a construction with seed length theta n 5 based on such one way functions finally we show that the randomized iterate may even be useful in the general context of hastad et al siam j comput 28 1999 pp 1364 1396 in particular we use the randomized iterate to replace the basic building block of the hastad et al siam j comput 28 1999 pp 1364 1396 construction interestingly this modification improves efficiency by an theta n 2 factor and reduces the seed length to theta n 7 which also implies improvement in the security of the construction

a pseudorandom generator from any one way function pseudorandom generators are fundamental to many theoretical and applied aspects of computing we show how to construct a pseudorandom generator from any one way function since it is easy to construct a one way function from a pseudorandom generator this result shows that there is a pseudorandom generator if and only if there is a one way function

rsa and rabin functions certain parts are as hard as the whole the rsa and rabin encryption functions e n cdot are respectively defined by raising x in z n to the power e where e is relatively prime to varphi n and squaring modulo n i e e n x x e bmod n e n x x 2 bmod n respectively we prove that for both functions the following problems are computationally equivalent each is probabilistic polynomial time reducible to the other 1 given e n x find x 2 given e n x guess the least significant bit of x with success probability tfrac 1 2 1 operatorname poly n where n is the length of the modulus n this equivalence implies that an adversary given the rsa rabin ciphertext cannot have a non negligible advantage over a random coin flip in guessing the least significant bit of the plaintext unless he can invert rsa factor n the proof techniques also yield the simultaneous security of the log n least significant bits our results improve the efficiency of pseudorandom number generation and

on the cauchy problem for a general fractional porous medium equation with variable density abstract we study the well posedness of the cauchy problem for a fractional porous medium equation with a varying density 0 we establish existence of weak energy solutions uniqueness and nonuniqueness is studied as well according to the behavior of at infinity

degenerate operators of tricomi type in l p spaces and in spaces of continuous functions we study elliptic operators l with dirichlet boundary conditions on a bounded domain whose diffusion coefficients degenerate linearly at in tangential directions we compute the domain of l and establish existence uniqueness and maximal regularity of the elliptic and parabolic problems for l in lp spaces and in spaces of continuous functions moreover the analytic semigroups generated by l are consistent positive compact and exponentially stable

large time behavior for a porous medium equation in a nonhomogeneous medium with critical density abstract we study the large time behavior of solutions to the cauchy problem for the porous medium equation in nonhomogeneous media with critical singular density x 2 t u u m in r n 0 where m 1 and n 3 with nonnegative initial condition u x 0 u 0 x 0 the asymptotic behavior proves to have some interesting and striking properties we show that there are different asymptotic profiles for the solutions depending on whether the continuous initial data u 0 vanishes at x 0 or not moreover when u 0 0 0 we show the convergence towards a peak type profile presenting a jump discontinuity coming from an interesting asymptotic simplification to a conservation law while when u 0 0 0 the limit profile remains continuous these phenomena illustrate the strong effect of the singularity at x 0 we improve the time scale of the convergence in sets avoiding the singularity on the way we also study the large time behavior for a porous medium equation with convection which is interesting for itself

quantitative local and global a priori estimates for fractional nonlinear diffusion equations abstract we establish quantitative estimates for solutions u t x to the fractional nonlinear diffusion equation t u s u m 0 in the whole range of exponents m 0 0 s 1 the equation is posed in the whole space x r d we first obtain weighted global integral estimates that allow to establish existence of solutions for classes of large data in the core of the paper we obtain quantitative pointwise lower estimates of the positivity of the solutions depending only on the norm of the initial data in a certain ball the estimates take a different form in three exponent ranges slow diffusion good range of fast diffusion and very fast diffusion finally we show existence and uniqueness of initial traces

dual universality of hash functions and its applications to quantum cryptography in this paper we introduce the concept of dual universality of hash functions and present its applications to quantum cryptography we begin by establishing the one to one correspondence between a linear function family f and a code family c and thereby defining e almost dual universal2 hash functions as a generalization of the conventional universal2 hash functions then we show that this generalized and thus broader class of hash functions is in fact sufficient for the security of quantum cryptography this result can be explained in two different formalisms first by noting its relation to the biased family introduced by dodis and smith we demonstrate that renner s two universal hashing lemma is generalized to our class of hash functions next we prove that the proof technique by shor and preskill can be applied to quantum key distribution qkd systems that use our generalized class of hash functions for privacy amplification while shor preskill formalism requires an implementer of a qkd system to explicitly construct a linear code of the calderbank shor steane css type this result removes the existing difficulty of the construction of a linear code of css code by replacing it by the combination of an ordinary classical error correcting code and our proposed hash function we also show that a similar result applies to the quantum wire tap channel finally we compare our results in the two formalisms and show that in typical qkd scenarios the shor preskill type argument gives better security bounds in terms of the trace distance and holevo information than the method based on the biased family

one dimensional degenerate operators in lp spaces abstract we comprehensively study a one dimensional elliptic operator degenerating of first order at the boundary in an l p setting here the coefficient of the drift term determines the regularity and the possible boundary conditions of the problem

inferring sequences produced by pseudo random number generators in this paper efficient algorithms are given for inferring sequences produced by certain pseudo random number generators the generators considered are all of the form x n s k j l j j x o x l x n l mod m in each case we assume that the functions j are known and polynomial time computable but that the coefficients aj and the modulus m are unknown using this general method specific examples of generators having this form the linear congruential method linear congruences with n terms in the recurrence and quadratic congruences are shown to be cryptographically insecure