Topic

# Integer

About: Integer is a research topic. Over the lifetime, 9350 publications have been published within this topic receiving 112792 citations. The topic is also known as: ℤ & whole number.

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30 Oct 1997

TL;DR: This chapter discusses decision problems and Complexity over a Ring and the Fundamental Theorem of Algebra: Complexity Aspects.

Abstract: 1 Introduction.- 2 Definitions and First Properties of Computation.- 3 Computation over a Ring.- 4 Decision Problems and Complexity over a Ring.- 5 The Class NP and NP-Complete Problems.- 6 Integer Machines.- 7 Algebraic Settings for the Problem "P ? NP?".- 8 Newton's Method.- 9 Fundamental Theorem of Algebra: Complexity Aspects.- 10 Bezout's Theorem.- 11 Condition Numbers and the Loss of Precision of Linear Equations.- 12 The Condition Number for Nonlinear Problems.- 13 The Condition Number in ?(H(d).- 14 Complexity and the Condition Number.- 15 Linear Programming.- 16 Deterministic Lower Bounds.- 17 Probabilistic Machines.- 18 Parallel Computations.- 19 Some Separations of Complexity Classes.- 20 Weak Machines.- 21 Additive Machines.- 22 Nonuniform Complexity Classes.- 23 Descriptive Complexity.- References.

1,594 citations

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30 May 2010

TL;DR: A fully homomorphic encryption scheme, using only elementary modular arithmetic, that reduces the security of the scheme to finding an approximate integer gcd, and investigates the hardness of this task, building on earlier work of Howgrave-Graham.

Abstract: We construct a simple fully homomorphic encryption scheme, using only elementary modular arithmetic. We use Gentry’s technique to construct a fully homomorphic scheme from a “bootstrappable” somewhat homomorphic scheme. However, instead of using ideal lattices over a polynomial ring, our bootstrappable encryption scheme merely uses addition and multiplication over the integers. The main appeal of our scheme is the conceptual simplicity.
We reduce the security of our scheme to finding an approximate integer gcd – i.e., given a list of integers that are near-multiples of a hidden integer, output that hidden integer. We investigate the hardness of this task, building on earlier work of Howgrave-Graham.

1,486 citations

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TL;DR: In this paper, a somewhat homomorphic encryption scheme using elementary modular arithmetic is described. But the main appeal of their approach is the conceptual simplicity. And the security of their scheme is reduced to finding an approximate integer gcd, i.e., given a list of integers that are near-multiples of a hidden integer, output that hidden integer.

Abstract: We describe a very simple “somewhat homomorphic” encryption scheme using only elementary modular arithmetic, and use Gentry’s techniques to convert it into a fully homomorphic scheme. Compared to Gentry’s construction, our somewhat homomorphic scheme merely uses addition and multiplication over the integers rather than working with ideal lattices over a polynomial ring. The main appeal of our approach is the conceptual simplicity. We reduce the security of our somewhat homomorphic scheme to finding an approximate integer gcd – i.e., given a list of integers that are near-multiples of a hidden integer, output that hidden integer. We investigate the hardness of this task, building on earlier work of HowgraveGraham.

1,297 citations

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TL;DR: For the "sphere decoding" algorithm of Fincke and Pohst, a closed-form expression is found for the expected complexity, both for the infinite and finite lattice, which suggests that maximum-likelihood decoding, which was hitherto thought to be computationally intractable, can be implemented in real time.

Abstract: The problem of finding the least-squares solution to a system of linear equations where the unknown vector is comprised of integers, but the matrix coefficient and given vector are comprised of real numbers, arises in many applications: communications, cryptography, GPS, to name a few. The problem is equivalent to finding the closest lattice point to a given point and is known to be NP-hard. In communications applications, however, the given vector is not arbitrary but rather is an unknown lattice point that has been perturbed by an additive noise vector whose statistical properties are known. Therefore, in this paper, rather than dwell on the worst-case complexity of the integer least-squares problem, we study its expected complexity, averaged over the noise and over the lattice. For the "sphere decoding" algorithm of Fincke and Pohst, we find a closed-form expression for the expected complexity, both for the infinite and finite lattice. It is demonstrated in the second part of this paper that, for a wide range of signal-to-noise ratios (SNRs) and numbers of antennas, the expected complexity is polynomial, in fact, often roughly cubic. Since many communications systems operate at noise levels for which the expected complexity turns out to be polynomial, this suggests that maximum-likelihood decoding, which was hitherto thought to be computationally intractable, can, in fact, be implemented in real time-a result with many practical implications.

1,277 citations

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01 Jan 1974

TL;DR: A vocabulary of combinatorial analysis can be found in this paper, where the authors define definitions of partitions of an integer [n]- 22 Generating Functions of p(n) and P(n, m)- 23 Conditional Partitions- 24 Ferrers Diagrams- 25 Special Identities 'Formal' and 'Combinatorial' Proofs- 26 Partitions with Forbidden Summands Denumerants- Supplement and Exercises- III Identities and Expansions- III Identity and Expansion of a Product of Sums Abel Identity- 31

Abstract: I Vocabulary of Combinatorial Analysis- 11 Subsets of a Set Operations- 12 Product Sets- 13 Maps- 14 Arrangements, Permutations- 15 Combinations (without repetitions) or Blocks- 16 Binomial Identity- 17 Combinations with Repetitions- 18 Subsets of [n], Random Walk- 19 Subsets of Z/nZ- 110 Divisions and Partitions of a Set Multinomial Identity- 111 Bound Variables- 112 Formal Series- 113 Generating Functions- 114 List of the Principal Generating Functions- 115 Bracketing Problems- 116 Relations- 117 Graphs- 118 Digraphs Functions from a Finite Set into Itself- Supplement and Exercises- II Partitions of Integers- 21 Definitions of Partitions of an Integer [n]- 22 Generating Functions of p(n) and P(n, m)- 23 Conditional Partitions- 24 Ferrers Diagrams- 25 Special Identities 'Formal' and 'Combinatorial' Proofs- 26 Partitions with Forbidden Summands Denumerants- Supplement and Exercises- III Identities and Expansions- 31 Expansion of a Product of Sums Abel Identity- 32 Product of Formal Series Leibniz Formula- 33 Bell Polynomials- 34 Substitution of One Formal Series into Another Formula of Faa di Bruno- 35 Logarithmic and Potential Polynomials- 36 Inversion Formulas and Matrix Calculus- 37 Fractionary Iterates of Formal Series- 38 Inversion Formula of Lagrange- 39 Finite Summation Formulas- Supplement and Exercises- IV Sieve Formulas- 41 Number of Elements of a Union or Intersection- 42 The 'probleme des rencontres'- 43 The 'probleme des menages'- 44 Boolean Algebra Generated by a System of Subsets- 45 The Method of Renyi for Linear Inequalities- 46 Poincare Formula- 47 Bonferroni Inequalities- 48 Formulas of Ch Jordan- 49 Permanents- Supplement and Exercises- V Stirling Numbers- 51 Stirling Numbers of the Second Kind S(n, k) and Partitions of Sets- 52 Generating Functions for S(n, k)- 53 Recurrence Relations between the S(n, k)- 54 The Number ?(n) of Partitions or Equivalence Relations of a Set with n Elements- 55 Stirling Numbers of the First Kind s(n, k) and their Generating Functions- 56 Recurrence Relations between the s(n, k)- 57 The Values of s(n, k)- 58 Congruence Problems- Supplement and Exercises- VI Permutations- 61 The Symmetric Group- 62 Counting Problems Related to Decomposition in Cycles Return to Stirling Numbers of the First Kind- 63 Multipermutations- 64 Inversions of a Permutation of [n]- 65 Permutations by Number of Rises Eulerian Numbers- 66 Groups of Permutations Cycle Indicator Polynomial Burnside Theorem- 67 Theorem of Polya- Supplement and Exercises- VII Examples of Inequalities and Estimates- 71 Convexity and Unimodality of Combinatorial Sequences- 72 Sperner Systems- 73 Asymptotic Study of the Number of Regular Graphs of Order Two on N- 74 Random Permutations- 75 Theorem of Ramsey- 76 Binary (Bicolour) Ramsey Numbers- 77 Squares in Relations- Supplement and Exercises- Fundamental Numerical Tables- Factorials with Their Prime Factor Decomposition- Binomial Coefficients- Partitions of Integers- Bell Polynomials- Logarithmic Polynomials- Partially Ordinary Bell polynomials- Multinomial Coefficients- Stirling Numbers of the First Kind- Stirling Numbers of the Second Kind and Exponential Numbers

1,155 citations