A subset $V \subseteq \mathbb{F}_2^n$ is a tile if $\mathbb{F}_2^n$ can be covered by disjoint tr... more A subset $V \subseteq \mathbb{F}_2^n$ is a tile if $\mathbb{F}_2^n$ can be covered by disjoint translates of $V$. In other words, $V$ is a tile if and only if there is a subset $A \subseteq \mathbb{F}_2^n$ such that $V+A = \mathbb{F}_2^n$ uniquely (i.e., $v + a = v' + a'$ implies that $v=v'$ and $a=a'$, where $v,v' \in V$ and $a,a' \in A$). In some problems in coding theory and hashing we are given a putative tile $V$ and wish to know whether or not it is a tile. In this paper we give two computational criteria for certifying that $V$ is not a tile. The first involves the impossibility of a bin-packing problem, and the second involves the infeasibility of a linear program. We apply both criteria to a list of putative tiles given by Gordon, Miller, and Ostapenko [IEEE Trans. Inform. Theory, 56 (2010), pp. 984-991] in the context of hashing to find close matches, to show that none of them are, in fact, tiles.
E. D. F. Meissel, a German astronomer, found in the 1870’s a method for computing individual valu... more E. D. F. Meissel, a German astronomer, found in the 1870’s a method for computing individual values of π ( x ) \pi (x) , the counting function for the number of primes ⩽ x \leqslant x . His method was based on recurrences for partial sieving functions, and he used it to compute π ( 10 9 ) \pi ({10^9}) . D. H. Lehmer simplified and extended Meissel’s method. We present further refinements of the Meissel-Lehmer method which incorporate some new sieving techniques. We give an asymptotic running time analysis of the resulting algorithm, showing that for every ε > 0 \varepsilon > 0 it computes π ( x ) \pi (x) using at most O ( x 2 / 3 + ε ) O({x^{2/3 + \varepsilon }}) arithmetic operations and using at most O ( x 1 / 3 + ε ) O({x^{1/3 + \varepsilon }}) storage locations on a Random Access Machine (RAM) using words of length [ log 2 x ] + 1 [{\log _2}x] + 1 bits. The algorithm can be further speeded up using parallel processors. We show that there is an algorithm which, when given M...
A subset V of GF(2)^n is a tile if GF(2)^n can be covered by disjoint translates of V. In other w... more A subset V of GF(2)^n is a tile if GF(2)^n can be covered by disjoint translates of V. In other words, V is a tile if and only if there is a subset A of GF(2)^n such that V+A = GF(2)^n uniquely (i.e., v + a = v' + a' implies that v=v' and a=a' where v,v' in V and
ABSTRACT We answer a recent challenge by Benaloh, Lauter, Horvitz, and Chase [1] concerning patie... more ABSTRACT We answer a recent challenge by Benaloh, Lauter, Horvitz, and Chase [1] concerning patient privacy in electronic medical records. Our approach offers strong privacy and confidential-ity, and enables autonomous delegation of priviliges in a distributed setting. We instantiate our constructions using the recent results of Gentry [4] via a framework already known in the early sixties [3].
One way to find closest pairs in large datasets is to use hash functions. In recent years localit... more One way to find closest pairs in large datasets is to use hash functions. In recent years locality-sensitive hash functions for various metrics have been given: projecting an n-cube onto k bits is simple hash function that performs well. In this paper we investigate alternatives to projection. For various parameters hash functions given by complete decoding algorithms for codes work better, and asymptotically random codes perform better than projection.
ABSTRACT In this paper we describe a new algorithm for fully factoring polynomials defined over t... more ABSTRACT In this paper we describe a new algorithm for fully factoring polynomials defined over the rationals or over number-fields. This algorithm uses as an essential subroutine, any fast relation finding algorithm for vectors of real numbers. Unlike previous algorithms which work on one factor at a time, the new algorithm finds all factors at once. Let P be a polynomial of degree n, height H(P) (=sum of the absolute values of P's coefficients), logarthmic height h(P)=log H(P). If we use the HJLS relation-finding algorithm of Hastad, Just, Lagarias and Schnorr, our algorithm has running time O(n 5+Ch(P)) if fast multiplication is used, and O(n 6+Ch(P)) if ordinary multiplication is used. This is an improvement by a factor of n over the algorithm of Schönhage, the previously best known.
A subset $V \subseteq \mathbb{F}_2^n$ is a tile if $\mathbb{F}_2^n$ can be covered by disjoint tr... more A subset $V \subseteq \mathbb{F}_2^n$ is a tile if $\mathbb{F}_2^n$ can be covered by disjoint translates of $V$. In other words, $V$ is a tile if and only if there is a subset $A \subseteq \mathbb{F}_2^n$ such that $V+A = \mathbb{F}_2^n$ uniquely (i.e., $v + a = v' + a'$ implies that $v=v'$ and $a=a'$, where $v,v' \in V$ and $a,a' \in A$). In some problems in coding theory and hashing we are given a putative tile $V$ and wish to know whether or not it is a tile. In this paper we give two computational criteria for certifying that $V$ is not a tile. The first involves the impossibility of a bin-packing problem, and the second involves the infeasibility of a linear program. We apply both criteria to a list of putative tiles given by Gordon, Miller, and Ostapenko [IEEE Trans. Inform. Theory, 56 (2010), pp. 984-991] in the context of hashing to find close matches, to show that none of them are, in fact, tiles.
E. D. F. Meissel, a German astronomer, found in the 1870’s a method for computing individual valu... more E. D. F. Meissel, a German astronomer, found in the 1870’s a method for computing individual values of π ( x ) \pi (x) , the counting function for the number of primes ⩽ x \leqslant x . His method was based on recurrences for partial sieving functions, and he used it to compute π ( 10 9 ) \pi ({10^9}) . D. H. Lehmer simplified and extended Meissel’s method. We present further refinements of the Meissel-Lehmer method which incorporate some new sieving techniques. We give an asymptotic running time analysis of the resulting algorithm, showing that for every ε > 0 \varepsilon > 0 it computes π ( x ) \pi (x) using at most O ( x 2 / 3 + ε ) O({x^{2/3 + \varepsilon }}) arithmetic operations and using at most O ( x 1 / 3 + ε ) O({x^{1/3 + \varepsilon }}) storage locations on a Random Access Machine (RAM) using words of length [ log 2 x ] + 1 [{\log _2}x] + 1 bits. The algorithm can be further speeded up using parallel processors. We show that there is an algorithm which, when given M...
A subset V of GF(2)^n is a tile if GF(2)^n can be covered by disjoint translates of V. In other w... more A subset V of GF(2)^n is a tile if GF(2)^n can be covered by disjoint translates of V. In other words, V is a tile if and only if there is a subset A of GF(2)^n such that V+A = GF(2)^n uniquely (i.e., v + a = v' + a' implies that v=v' and a=a' where v,v' in V and
ABSTRACT We answer a recent challenge by Benaloh, Lauter, Horvitz, and Chase [1] concerning patie... more ABSTRACT We answer a recent challenge by Benaloh, Lauter, Horvitz, and Chase [1] concerning patient privacy in electronic medical records. Our approach offers strong privacy and confidential-ity, and enables autonomous delegation of priviliges in a distributed setting. We instantiate our constructions using the recent results of Gentry [4] via a framework already known in the early sixties [3].
One way to find closest pairs in large datasets is to use hash functions. In recent years localit... more One way to find closest pairs in large datasets is to use hash functions. In recent years locality-sensitive hash functions for various metrics have been given: projecting an n-cube onto k bits is simple hash function that performs well. In this paper we investigate alternatives to projection. For various parameters hash functions given by complete decoding algorithms for codes work better, and asymptotically random codes perform better than projection.
ABSTRACT In this paper we describe a new algorithm for fully factoring polynomials defined over t... more ABSTRACT In this paper we describe a new algorithm for fully factoring polynomials defined over the rationals or over number-fields. This algorithm uses as an essential subroutine, any fast relation finding algorithm for vectors of real numbers. Unlike previous algorithms which work on one factor at a time, the new algorithm finds all factors at once. Let P be a polynomial of degree n, height H(P) (=sum of the absolute values of P's coefficients), logarthmic height h(P)=log H(P). If we use the HJLS relation-finding algorithm of Hastad, Just, Lagarias and Schnorr, our algorithm has running time O(n 5+Ch(P)) if fast multiplication is used, and O(n 6+Ch(P)) if ordinary multiplication is used. This is an improvement by a factor of n over the algorithm of Schönhage, the previously best known.
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Papers by Victor Miller