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236
Programming Languages and Dimensions
, 1996
"... Scientists and engineers must ensure that the equations and formulae which they use are dimensionally consistent, but existing programming languages treat all numeric values as dimensionless. This thesis investigates the extension of programming languages to support the notion of physical dimension. ..."
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Scientists and engineers must ensure that the equations and formulae which they use are dimensionally consistent, but existing programming languages treat all numeric values as dimensionless. This thesis investigates the extension of programming languages to support the notion of physical dimension. A type system is presented similar to that of the programming language ML but extended with polymorphic dimension types. An algorithm which infers most general dimension types automatically is then described and proved correct. The semantics of the language is given by a translation into an explicitlytyped language in which dimensions are passed as arguments to functions. The operational semantics of this language is specified in the usual way by an evaluation relation defined by a set of rules. This is used to show that if a program is welltyped then no dimension errors can occur during its evaluation. More abstract properties of the language are investigated using a denotational semantics: these include a notion of invariance under changes in the units of measure used, analogous to parametricity in the polymorphic lambda calculus. Finally the dissertation is summarised and many possible directions for future research in dimension types and related type systems are described. i ii
The complexity of decision versus search
 SIAM Journal on Computing
, 1994
"... A basic question about NP is whether or not search reduces in polynomial time to decision. We indicate that the answer is negative: under a complexity assumption (that deterministic and nondeterministic doubleexponential time are unequal) we construct a language in NP for which search does not red ..."
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Cited by 33 (1 self)
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A basic question about NP is whether or not search reduces in polynomial time to decision. We indicate that the answer is negative: under a complexity assumption (that deterministic and nondeterministic doubleexponential time are unequal) we construct a language in NP for which search does not reduce to decision. These ideas extend in a natural way to interactive proofs and program checking. Under similar assumptions we present languages in NP for which it is harder to prove membership interactively than it is to decide this membership, and languages in NP which are not checkable. Keywords: NPcompleteness, selfreducibility, interactive proofs, program checking, sparse sets,
PrivacyEnhanced Searches Using Encrypted Bloom Filters
, 2004
"... It is often necessary for two or more or more parties that do not fully trust each other to selectively share data. We propose a search scheme based on Bloom filters and PohligHellman encryption. A semitrusted third party can transform one party's search queries to a form suitable for queryin ..."
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Cited by 29 (1 self)
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It is often necessary for two or more or more parties that do not fully trust each other to selectively share data. We propose a search scheme based on Bloom filters and PohligHellman encryption. A semitrusted third party can transform one party's search queries to a form suitable for querying the other party's database, in such a way that neither the third party nor the database owner can see the original query. Furthermore, the encryption keys used to construct the Bloom filters are not shared with this third party. Provision can be made for thirdparty "warrant servers", as well as "censorship sets" that limit the data to be shared.
Polynomial identity testing for depth 3 circuits
 in Proceedings of the twentyfirst Annual IEEE Conference on Computational Complexity (CCC
, 2006
"... Abstract — We study ΣΠΣ(k) circuits, i.e., depth three arithmetic circuits with top fanin k. We give the first deterministic polynomial time blackbox identity test for ΣΠΣ(k) circuits over the field Q of rational numbers, thus resolving a question posed by Klivans and Spielman (STOC 2001). Our main ..."
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Cited by 28 (6 self)
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Abstract — We study ΣΠΣ(k) circuits, i.e., depth three arithmetic circuits with top fanin k. We give the first deterministic polynomial time blackbox identity test for ΣΠΣ(k) circuits over the field Q of rational numbers, thus resolving a question posed by Klivans and Spielman (STOC 2001). Our main technical result is a structural theorem for ΣΠΣ(k) circuits that compute the zero polynomial. In particular we show that if a ΣΠΣ(k) circuit C = ∑ i∈[k] Ai
A Cryptographic Solution to Implement Access Control in a Hierarchy and More
 In Proceedings of 7th ACM Symposium on Access Control Models and Technologies (SACMAT’02
, 2002
"... The need for access control in a hierarchy arises in several different contexts. One such context is managing the information of an organization where the users are divided into different security classes depending on who has access to what. Several cryptographic solutions have been proposed to addr ..."
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Cited by 24 (0 self)
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The need for access control in a hierarchy arises in several different contexts. One such context is managing the information of an organization where the users are divided into different security classes depending on who has access to what. Several cryptographic solutions have been proposed to address this problem  the solutions are based on generating cryptographic keys for each security class such that the key for a lower level security class depends on the key for the security class that is higher up in the hierarchy. Most solutions use complex cryptographic techniques: integrating these into existing systems may not be trivial. Others have impractical requirement: if a user at a security level wants to access data at lower levels, then all intermediate nodes must be traversed. Moreover, if there is an access control policy that does not conform to the hierarchical structure, such policy cannot be handled by existing solutions. We propose a new solution that overcomes the above mentioned shortcomings. Our solution not only addresses the problem of access control in a hierarchy but also can be used for general cases. It is a scheme similar to the RSA cryptosystem and can be easily incorporated in existing systems.
Solving the Pell Equation
, 2008
"... We illustrate recent developments in computational number theory by studying their implications for solving the Pell equation. We shall see that, if the solutions to the Pell equation are properly represented, the traditional continued fraction method for solving the equation can be significantly a ..."
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Cited by 23 (0 self)
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We illustrate recent developments in computational number theory by studying their implications for solving the Pell equation. We shall see that, if the solutions to the Pell equation are properly represented, the traditional continued fraction method for solving the equation can be significantly accelerated. The most promising method depends on the use of smooth numbers. As with many algorithms depending on smooth numbers, its run time can presently only conjecturally be established; giving a rigorous analysis is one of the many open problems surrounding the Pell equation.
Elliptic curves and related sequences
, 2003
"... A Somos 4 sequence is a sequence (hn) of rational numbers defined by the quadratic recursion hm+2 hm−2 = λ1 hm+1 hm−1 + λ2 h2 m for all m ∈ Z for some rational constants λ1, λ2. Elliptic divisibility sequences or EDSs are an important special case where λ1 = h2 2, λ2 = −h1 h3, the hn are integers ..."
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Cited by 22 (3 self)
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A Somos 4 sequence is a sequence (hn) of rational numbers defined by the quadratic recursion hm+2 hm−2 = λ1 hm+1 hm−1 + λ2 h2 m for all m ∈ Z for some rational constants λ1, λ2. Elliptic divisibility sequences or EDSs are an important special case where λ1 = h2 2, λ2 = −h1 h3, the hn are integers and hn divides hm whenever n divides m. Somos (4) is the particular Somos 4 sequence whose coefficients λi and initial values are all 1. In this thesis we study the properties of EDSs and Somos 4 sequences reduced modulo a prime power pr. In chapter 2 we collect some results from number theory, and in chapter 3 we give a brief introduction to elliptic curves. In chapter 4 we introduce elliptic divisibility sequences, describe their relationship with elliptic curves, and outline what is known about the properties of an EDS modulo a prime power pr (work by Morgan Ward and Rachel Shipsey). In chapter 5 we extend the EDS “symmetry formulae ” of Ward and Shipsey
An application of number theory to the organization of rastergraphics memory
 Journal of the Association for Computing Machinery
, 1986
"... Abstract. A highresolution rastergraphics display is usually combined with processing power and a memory organization that facilitates basic graphics operations. For many applications, including interactive text processing, the ability to quickly move or copy small rectangles of pixels is essentia ..."
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Cited by 21 (1 self)
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Abstract. A highresolution rastergraphics display is usually combined with processing power and a memory organization that facilitates basic graphics operations. For many applications, including interactive text processing, the ability to quickly move or copy small rectangles of pixels is essential. This paper proposes a novel organization of rastergraphics memory that permits all small rectangles to be moved elBciently. The memory organization is based on a doubly periodic assignment of pixels to M memory chips according to a “Fibonacci ” lattice. The memory organization guarantees that, if a rectilinearly oriented rectangle contains fewer than M/A pixels, then all pixels will reside in different memory chips and thus can be accessed simultaneously. Moreover, any M consecutive pixels, arranged either horizontally or vertically, can be accessed simultaneously. We also define a continuous analog of the problem, which can be posed as: “What is the maximum density of a set of points in the plane such that no two points are contained in the interior of a rectilinearly oriented rectangle of unit area 7 ” We show the existence of such a set with density I/&, and prove this is optimal by giving a matching upper bound.