Jill Pipher

Brown University · Department of Mathematics

In this paper we present in concise form recent results, with illustrative proofs, on solvability of the \(L^p\) Dirichlet, Regularity and Neumann problems for scalar elliptic equations on Lipschitz domains with coefficients satisfying a variety of Carleson conditions. More precisely, with \(L=\textrm{div}(A\nabla )\), we assume the matrix A is ell...

In this paper we present in concise form recent results, with illustrative proofs, on solvability of the $L^p$ Dirichlet, Regularity and Neumann problems for scalar elliptic equations on Lipschitz domains with coefficients satisfying a variety of Carleson conditions. More precisely, with $L=\mbox{div}(A\nabla)$, we assume the matrix $A$ is elliptic...

In this paper, we continue the study of a class of second order elliptic operators of the form $\mathcal L=\mbox{div}(A\nabla\cdot)$ in a domain above a Lipschitz graph in $\mathbb R^n,$ where the coefficients of the matrix $A$ satisfy a Carleson measure condition, expressed as a condition on the oscillation on Whitney balls. For this class of oper...

The main purpose of this paper is to study Lr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^r$$\end{document} Hölder type estimates for a bi-parameter trilinear Four...

We consider the operator \(L=-\mathrm{div}(A\nabla )\), where A is an \(n\times n\) matrix of real coefficients and satisfies the ellipticity condition, with \(n\ge 2\). We assume that the coefficients of the symmetric part of A are in \(L^\infty ({\mathbb {R}}^n)\), and those of the anti-symmetric part of A only belong to the space \(BMO({\mathbb...

The present paper establishes the first result on the absolute continuity of elliptic measure with respect to the Lebesgue measure for a divergence form elliptic operator with non-smooth coefficients that have a \({{\,\mathrm{BMO}\,}}\) anti-symmetric part. In particular, the coefficients are not necessarily bounded. We prove that the Dirichlet pro...

The notion of p-ellipticity has recently played a significant role in improving our understanding of issues of solvability of boundary value problems for scalar complex valued elliptic PDEs. In particular, the presence of p-ellipticity ensures higher regularity of solutions of such equations. In this work we extend the notion of p-ellipticity to se...

The notion of $p$-ellipticity has recently played a significant role in improving our understanding of issues of solvability of boundary value problems for scalar complex valued elliptic PDEs. In particular, the presence of $p$-ellipticity ensures higher regularity of solutions of such equations. In this work we extend the notion of $p$-ellipticity...

In this paper, we prove an extrapolation result for complex coefficient divergence form operators that satisfy a strong ellipticity condition known as p-ellipticity. Specifically, let Ω be a chord-arc domain in Rn and the operator L=∂i(Aij(x)∂j)+Bi(x)∂i be elliptic, with |Bi(x)|≤Kδ(x)−1 for a small K. Let p0=sup⁡{p>1:Aisp-elliptic}. We establish th...

In this paper we revisit the modular lattice signature scheme and its efficient instantiation known as pqNTRUSign. First, we show that a modular lattice signature scheme can be based on a standard lattice problem. The fundamental problem that needs to be solved by the signer or a potential forger is recovering a lattice vector with a restricted nor...

In this paper, we prove an extrapolation result for complex coefficient divergence form operators that satisfy a strong ellipticity condition known as $p$-{\it ellipticity}. Specifically, let $\Omega$ be a chord-arc domain in $\mathbb R^n$ and the operator $\mathcal L = \partial_{i}\left(A_{ij}(x)\partial_{j}\right) +B_{i}(x)\partial_{i} $ be ellip...

The present paper establishes the first result on the absolute continuity of elliptic measure with respect to the Lebesgue measure for a divergence form elliptic operator with non-smooth coefficients that have a BMO anti-symmetric part. In particular, the coefficients are not necessarily bounded. We prove that the Dirichlet problem for elliptic equ...

We consider the operator $L=-{\rm div}(A\nabla)$, where the $n\times n$ matrix $A$ is real-valued, elliptic, with the symmetric part of $A$ in $L^\infty(\mathbb{R}^n)$, and the anti-symmetric part of $A$ only belongs to the space $BMO(\mathbb{R}^n)$, $n\ge2$. We prove the Gaussian estimates for the kernel of $e^{-tL}$, as well as that of $\partial_...

We establish a Dahlberg-type perturbation theorem for second order divergence form elliptic operators with complex coefficients. In our previous paper, we showed the following result: If L0 = divA⁰(x)∇ + B⁰(x). ∇ is a p-elliptic operator satisfying the assumptions of Theorem 1.1 then the Lp Dirichlet problem for the operator L0 is solvable in the u...

The main purpose of this paper is to establish weighted estimates for singular integrals associated with Zygmund dilations via a discrete Littlewood--Paley theory, and then apply it to obtain the upper bound of the norm of commutators of such singular integrals with a function in the little bmo space associated with Zygmund dilations. Examples of s...

The main purpose of this paper is to study $L^p$ H\"older type estimates for a bi-parameter trilinear Fourier multiplier with flag singularity, and the analogous pseudo-differential operator, when the symbols are in a certain product form. More precisely, for $f,g,h\in \mathcal{S}({\mathcal {R}}^{2})$, the bi-parameter trilinear flag Fourier multip...

The theory of second order complex coefficient operators of the form $\mathcal{L}=\mbox{div} A(x)\nabla$ has recently been developed under the assumption of $p$-ellipticity. In particular, if the matrix $A$ is $p$-elliptic, the solutions $u$ to $\mathcal{L}u = 0$ will satisfy a higher integrability, even though they may not be continuous in the int...

We establish a Dahlberg-type perturbation theorem for second order divergence form elliptic operators with complex coefficients. In our previous paper, we showed the following result: If ${\mathcal L}_0=\mbox{div} A^0(x)\nabla+B^0(x)\cdot\nabla$ is a $p$-elliptic operator satisfying certain Carleson condition on $\nabla A$ and $B$ then the $L^p$ Di...

Let $S$ be the dyadic bi-parameter square function $$Sf(x)^{2} = \sum_{R \in \mathcal{D}} |\langle f, h_{R} \rangle|^{2} \frac{1_{R}(x)}{|R|}.$$ We prove that if $T$ is a bi-parameter martingale transform and $f,g$ are suitable test functions, then there exists a sparse collection of rectangles $\mathcal{S}$ such that $$|\langle Tf, g \rangle| \les...

We describe a method for generating parameter sets, and calculating security estimates, for NTRUEncrypt. Our security analyses consider lattice attacks, the hybrid attack, subfield attacks, and quantum search. Analyses are provided for the IEEE 1363.1-2008 product-form parameter sets, for the NTRU Challenge parameter sets, and for two new parameter...

We establish a new theory of regularity for elliptic complex valued second order equations of the form $\mathcal L=$div$A(\nabla\cdot)$, when the coefficients of the matrix $A$ satisfy a natural algebraic condition, a strengthened version of a condition known in the literature as $L^p$-dissipativity. Precisely, the regularity result is a reverse H\...

We prove that the $A_\infty$ property of parabolic measure for operators in certain time-varying domains is equivalent to a Carleson measure property of bounded solutions. Kircheim, Kenig, Pipher, and T. Toro established this criterion on bounded solutions in the elliptic case, improving an earlier result of Dindos, Kenig and Pipher for solutions w...

In this paper we study the following question related to Diophantine approximations and geometric measure theory: for a given set Ω \Omega find α \alpha such that α − θ \alpha - \theta has bad Diophantine properties simultaneously for all θ ∈ Ω \theta \in \Omega . How do the arising Diophantine inequalities depend on the geometry of the set Ω \Omeg...

We introduce a class of lattice-based digital signature schemes based on modular properties of the coordinates of lattice vectors. We also suggest a method of making such schemes transcript secure via a rejection sampling technique of Lyubashevsky (2009). A particular instantiation of this approach is given, using NTRU lattices. Although the scheme...

In this paper, we provide a new means of establishing solvability of the Dirichlet problem on Lipschitz domains, with measurable data, for second order elliptic, non-symmetric divergence form operators. We show that a certain optimal Carleson measure estimate for bounded solutions of such operators implies a regularity result for the associated ell...

The emphasis of this book has been on the mathematical underpinnings of public key cryptography. We have developed most of the mathematics from scratch and in sufficient depth to enable the reader to understand both the underlying mathematical principles and how they are applied in cryptographic constructions. Unfortunately, in achieving this lauda...

The Diffie–Hellman key exchange method and the Elgamal public key cryptosystem studied in Sects. 2.3 and 2.4 rely on the fact that it is easy to compute powers \(a^{n}\bmod p\), but difficult to recover the exponent n if you know only the values of a and \(a^{n}\bmod p\). An essential result that we used to analyze the security of Diffie–Hellman an...

Encryption schemes, whether symmetric or asymmetric, solve the problem of secure communications over an insecure network. Digital signatures solve a different problem, analogous to the purpose of a pen-and-ink signature on a physical document. It is thus interesting that the tools used to construct digital signatures are very similar to the tools u...

The subject of elliptic curves encompasses a vast amount of mathematics. Our aim in this section is to summarize just enough of the basic theory for cryptographic applications. For additional reading, there are a number of survey articles and books devoted to elliptic curve cryptography [14, 68, 81, 135], and many others that describe the number th...

The security of all of the public key cryptosystems that we have previously studied has been based, either directly or indirectly, on either the difficulty of factoring large numbers or the difficulty of finding discrete logarithms in a finite group. In this chapter we investigate a new type of hard problem arising in the theory of lattices that ca...

As Julius Caesar surveys the unfolding battle from his hilltop outpost, an exhausted and disheveled courier bursts into his presence and hands him a sheet of parchment containing gibberish:

In considering the usefulness and practicality of a cryptographic system, it is necessary to measure its resistance to various forms of attack. Such attacks include simple brute-force searches through the key or message space, somewhat faster searches via collision or meet-in-the-middle algorithms, and more sophisticated methods that are used to co...

In 1976, Whitfield Diffie and Martin Hellman published their now famous paper [38] entitled “New Directions in Cryptography.” In this paper they formulated the concept of a public key encryption system and made several groundbreaking contributions to this new field. A short time earlier, Ralph Merkle had independently isolated one of the fundamenta...

We present PASS RS , a variant of the prior PASS and PASS-2 proposals, as a candidate for a practical post-quantum signature scheme. Its hardness is based on the problem of recovering a ring element with small norm from an incomplete description of its Chinese remainder representation. For our particular instantiation, this corresponds to the recov...

This self-contained introduction to modern cryptography emphasizes the mathematics behind the theory of public key cryptosystems and digital signature schemes. The book focuses on these key topics while developing the mathematical tools needed for the construction and security analysis of diverse cryptosystems. Only basic linear algebra is required...

The present paper establishes a certain duality between the Dirichlet and Regularity problems for elliptic operators with $t$-independent complex bounded measurable coefficients ($t$ being the transversal direction to the boundary). To be precise, we show that the Dirichlet boundary value problem is solvable in $L^{p'}$, subject to the square funct...

Let $\Omega$ be a Lipschitz domain in $\mathbb R^n$ $n\geq 2,$ and $L=\divt A\nabla$ be a second order elliptic operator in divergence form. We establish solvability of the Dirichlet regularity problem with boundary data in $H^{1,p}(\dom)$ and of the Neumann problem with $L^p(\partial\Omega)$ data for the operator $L$ on Lipschitz domains with smal...

We study the boundary regularity of solutions to divergence form operators which are small perturbations of operators for which the boundary regularity of solutions is known. An operator is a small perturbation of another operator if the deviation function of the coefficients satisfies a Carleson measure condition with small norm. We extend Escauri...

We prove that the class of Muckenhoupt A_p weights coincides with the intersection of finitely many suitable translates of dyadic A_p, in both the one-parameter and multiparameter cases, and that the analogous results hold for the reverse H\"older class RH_p, for doubling measures, and for the space VMO of functions of vanishing mean oscillation. W...

We consider divergence form elliptic operators L = - div A(x)\nabla, defined in the half space R^{n+1}_+, n \geq 2, where the coefficient matrix A(x) is bounded, measurable, uniformly elliptic, t-independent, and not necessarily symmetric. We establish square function/non-tangential maximal function estimates for solutions of the homogeneous equati...

In the present paper we study the solvability of the Dirichlet problem for second order divergence form elliptic operators with bounded measurable coefficients which are small perturbations of given operators in rough domains beyond the Lipschitz category. In our approach, the development of the theory of tent spaces on these domains is essential.

We establish a connection between the absolute continuity of elliptic measure associated with a second order divergence form operator with bounded measurable coefficients with the solvability of an end-point BMO Dirichlet problem. We show that these two notions are equivalent. As a consequence we obtain an end-point perturbation result, i.e., the s...

The theory of (Muckenhoupt) weights arises in many areas of analysis, for example in connection with bounds for singular integrals and maximal functions on weighted spaces. We prove that a certain averaging process gives a method for constructing A_p weights from a measurably varying family of dyadic A_p weights. This averaging process is suggested...

We study the possible analogous of the Div-Curl Lemma in classical harmonic analysis and partial differential equations, but from the point of view of the multi-parameter setting. In this context we see two possible Div-Curl lemmas that arise. Extensions to differential forms are also given.

We provide a brief history and overview of lattice based cryptography and cryptanalysis: shortest vector problems, closest vector problems, subset sum problem and knapsack systems, GGH, Ajtai-Dwork and NTRU. A detailed discussion of the algorithms NTRUEncrypt and NTRUSign follows. These algorithms have attractive operating speed and keysize and are...

In the present paper, we study the geometric discrepancy with respect to families of rotated rectangles. The well-known extremal cases are the axis-parallel rectangles (logarithmic discrepancy) and rectangles rotated in all possible directions (polynomial discrepancy). We study several intermediate situations: lacunary sequences of directions, lacu...

We prove a multiparameter version of a classical theorem of Jones and Journe on weak-star convergence in the Hardy space $H^1$.

We give a simple proof of L^p boundedness of iterated commutators of Riesz transforms and a product BMO function. We use a representation of the Riesz transforms by means of simple dyadic operators - dyadic shifts - which in turn reduces the estimate quickly to paraproduct estimates. Comment: 10 pages, submitted to Proceedings of El Escorial 2008

We establish Lp-solvability for 1<p<∞ of the Dirichlet problem on Lipschitz domains with small Lipschitz constants for elliptic divergence and non-divergence type operators with rough coefficients obeying a certain Carleson condition with small norm.

It is shown that product BMO of Chang and Fefferman, defined on the product of Euclidean spaces can be characterized by the multiparameter commutators of Riesz transforms. This extends a classical one-parameter result of Coifman, Rochberg, and Weiss, and at the same time extends the work of Lacey and Ferguson and Lacey and Terwilleger on multiparam...

We generalize to the bidisc a theorem of Garnett and Jones relating the space BMO of functions of bounded mean oscillation to its martingale counterpart, dyadic BMO. Namely, translation-averages of suitable families of dyadic BMO functions belong to BMO. As a corollary, we deduce a biparameter version of a theorem of Burgess Davis connecting the Ha...

The classical operators of harmonic analysis which are important in various elds of analysis and PDE include singular integrals, maximal functions, Hilbert and Riesz trans-forms and other operators de ned by multipliers. Some of these operators are invariant under the usual dilation group of R,; others, like the multiple Hilbert transform, are inva...

We prove a covering lemma for rectangles in R n {\mathbb {R}}^n which has connections to a problem of Zygmund and its solution in three dimensions by Cordoba.

Journé’s Lemma [11] is a critical component of many questions related to the product BMO theory of S.-Y. Chang and R. Fefferman. This article presents several different variants of the Lemma, in two and higher parameters, some known, some implicit in the literature, and some new. 1. Introduction, Journé’s Lemma We begin the discussion in two dimens...

This report explicitly refutes the analysis behind a recent claim that NTRUEncrypt has a bit security of at most 74 bits. We also sum up some existing literature on NTRU and lattices, in order to help explain what should and what should not be classed as an improved at- tack against the hard problem underlying NTRUEncrypt. We also show a connection...

4. This allows a reduction in the size of the public key, while maintaining the security of the key against lattice attacks. This increased lattice security is combined with the use of trinary form for private keys, which increases the possible combinatorial security for a given key size. 2. We note that the structure of signatures in the transpose...

We prove that classical Coifman-Meyer theorem holds on any polidisc Td or arbitrary dimension d = 1

In the first part of the paper we prove a bi-parameter version of a well known multilinear theorem of Coifman and Meyer. As a consequence, we generalize the Kato-Ponce inequality in nonlinear PDE, obtaining a fractional Leibnitz rule for derivatives in the $x_1$ and $x_2$ directions simultaneously. Then, we show that the double bilinear Hilbert tra...

this paper we present a complementary fast authentication and digital signature scheme, which we call NtruSign, based on the same underlying hard problem in the same lattices used by NTRU. Henceforth the original NTRU public key encryption/decryption algorithm will be referred to as NtruEncrypt NOTE, 2013-11: The signature scheme described in this...

We introduce NTRUSign, a new family of signature schemes based on solving the approximate closest vector problem (APPR-CVP) in NTRU-type lattices. We explore the properties of general APPR-CVP based signature schemes (e.g. GGH) and show that they are not immune to transcript attacks even in the random oracle model. We then introduce the idea of usi...

A new authentication and digital signature scheme called the NTRU Signature Scheme (NSS) is introduced. NSS provides an authenti- cation/signature method complementary to the NTRU public key cryp- tosystem. The hard lattice problem underlying NSS is similar to the hard problem underlying NTRU, and NSS similarly features high speed, low footprint, a...

We establish absolute continuity of the elliptic measure associated to certain second order elliptic equations in either divergence or nondivergence form, with drift terms, under minimal smoothness assumptions on the coefficients.

. A new authentication and digital signature scheme called the NTRU Signature Scheme (NSS) is introduced. NSS provides an authentication /signature method complementary to the NTRU public key cryptosystem. The hard lattice problem underlying NSS is similar to the hard problem underlying NTRU, and NSS similarly features high speed, low footprint, an...

In the late 50's and early 60's, the work of De Giorgi (De Gi) and Nash (N), and then Moser (Mo) initiated the study of regularity of solutions to divergence form elliptic equations with merely bounded measurable coecien ts. Weak solutions in a domain , a priori only in a Sobolev space W 2 1;loc (), were shown to be Holder continuous of some order...

this paper is to submit the NTRU public key cryptosystem for consideration for inclusion into the P1363A standard. NTRU was originally presented by Jeffrey Hoffstein in the rump session at CRYPTO '96, and was published in [HPS] in 1998. Since that time, NTRU Cryptosystems, Inc. has issued a number of technical reports. In some cases, these reports...

. We describe NTRU, a new public key cryptosystem. NTRU features reasonably short, easily created keys, high speed, and low memory requirements. NTRU encryption and decryption use a mixing system suggested by polynomial algebra combined with a clustering principle based on elementary probability theory. The security of the NTRU cryptosystem comes f...

We study the inhomogeneous Dirichlet problem for the bi-Laplacian with data given in Sobolev and Besov spaces on non-smooth domains.

This article is concerned with the operators from harmonic analysis which are naturally associated to a multiple parameter family of dilations. We are especially interested here in dealing with questions from the theory of such operators whose answers cannot be obtained by a reduction to the case of product operators. We also introduce a new tool i...

We study boundary value problems for the time-harmonic form of the Maxwell equations, as well as for other related systems of equations, on arbitrary Lipschitz domains in the three-dimensional Euclidean space. The main goal is to develop the corresponding theory for Lp-integrable bounday data for optimal values of p’s. We also discuss a number of...

We study boundary value problems for the time-harmonic form of the Maxwell equations, as well as for other related systems of equations, on arbitrary Lipschitz domains in the three-dimensional Euclidean space. The main goal is to develop the corresponding theory for Lp-integrable bounday data for optimal values of p's. We also discuss a number of r...

The Agmon-Miranda maximum principle for the polyharmonic equations of all orders is shown to hold in Lipschitz domains in ℝ3. In ℝn,n≥4, the Agmon-Miranda maximum principle andL p-Dirichlet estimates for certainp>2 are shown to fail in Lipschitz domains for these equations. In particular if 4≤n≤2m+1 theL p Dirichlet problem for Δ m fails to be solv...

We present an inequality for dyadic martingales (together with its continuous analog for functions on Rn) which is shown to be equivalent to a result of Chang-Wilson-Wolff on exponential square integrability. The analog of this weighted inequality for double dyadic martingales is also proven. Finally, we discuss a possible connection between these...

In recent years there has been a great deal of activity in both the theoretical and applied aspects of partial differential equations, with emphasis on realistic engineering applications, which usually involve lack of smoothness. On March 21-25, 1990, the University of Chicago hosted a workshop that brought together approximately fortyfive experts...

Let D ⊆ Rn be a Lipschitz domain and let u be a function biharmonic in D, i.e., ΔΔu = 0 in D. We prove that the nontangential maximal function and the square function of the gradient of u have equivalent Lp(dµ) norms, where d µ ϵ A∞(do) and dσ is surface measure on ∂D.

Consider the h-paths of Doob, and let logPxh (tD > t) ~ - lD t as t ® ¥\log P_x^h (\tau _D > t) \sim - \lambda _D t as t \to \infty where D is the first positive eigenvalue of-1/2 on D.

Our concern in this paper is to describe a class of Hardy spaces Hp(D) for 1 = p < 2 on a Lipschitz domain D Ì Rn when n = 3, and a certain smooth counterpart of Hp(D) on Rn-1, by providing an atomic decomposition and a description of their duals.

The aim of this paper is to extend the results of Calderón [1] and Kenig-Pipher [12] on solutions to the oblique derivative problem to the case where the data is assumed to be BMO or Hölder continuous.

We give a short proof of the well known Coifman-Meyer theorem on multi-linear operators.