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Fuzzy extractors: How to generate strong keys from biometrics and other noisy data. Technical Report 2003/235, Cryptology ePrint archive, http://eprint.iacr.org, 2006. Previous version appeared at EUROCRYPT 2004
 34 [DRS07] [DS05] [EHMS00] [FJ01] Yevgeniy Dodis, Leonid Reyzin, and Adam
, 2004
"... We provide formal definitions and efficient secure techniques for • turning noisy information into keys usable for any cryptographic application, and, in particular, • reliably and securely authenticating biometric data. Our techniques apply not just to biometric information, but to any keying mater ..."
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Cited by 292 (35 self)
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We provide formal definitions and efficient secure techniques for • turning noisy information into keys usable for any cryptographic application, and, in particular, • reliably and securely authenticating biometric data. Our techniques apply not just to biometric information, but to any keying material that, unlike traditional cryptographic keys, is (1) not reproducible precisely and (2) not distributed uniformly. We propose two primitives: a fuzzy extractor reliably extracts nearly uniform randomness R from its input; the extraction is errortolerant in the sense that R will be the same even if the input changes, as long as it remains reasonably close to the original. Thus, R can be used as a key in a cryptographic application. A secure sketch produces public information about its input w that does not reveal w, and yet allows exact recovery of w given another value that is close to w. Thus, it can be used to reliably reproduce errorprone biometric inputs without incurring the security risk inherent in storing them. We define the primitives to be both formally secure and versatile, generalizing much prior work. In addition, we provide nearly optimal constructions of both primitives for various measures of “closeness” of input data, such as Hamming distance, edit distance, and set difference.
On efficient sparse integer matrix Smith normal form computations
, 2001
"... We present a new algorithm to compute the Integer Smith normal form of large sparse matrices. We reduce the computation of the Smith form to independent, and therefore parallel, computations modulo powers of wordsize primes. Consequently, the algorithm does not suffer from coefficient growth. W ..."
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Cited by 37 (15 self)
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We present a new algorithm to compute the Integer Smith normal form of large sparse matrices. We reduce the computation of the Smith form to independent, and therefore parallel, computations modulo powers of wordsize primes. Consequently, the algorithm does not suffer from coefficient growth. We have implemented several variants of this algorithm (Elimination and/or BlackBox techniques) since practical performance depends strongly on the memory available. Our method has proven useful in algebraic topology for the computation of the homology of some large simplicial complexes.
The complexity of class polynomial computation via floating point approximations. ArXiv preprint
, 601
"... Abstract. We analyse the complexity of computing class polynomials, that are an important ingredient for CM constructions of elliptic curves, via complex floating point approximations of their roots. The heart of the algorithm is the evaluation of modular functions in several arguments. The fastest ..."
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Cited by 33 (5 self)
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Abstract. We analyse the complexity of computing class polynomials, that are an important ingredient for CM constructions of elliptic curves, via complex floating point approximations of their roots. The heart of the algorithm is the evaluation of modular functions in several arguments. The fastest one of the presented approaches uses a technique devised by Dupont to evaluate modular functions by Newton iterations on an expression involving the arithmeticgeometric mean. Under the heuristic assumption, justified by experiments, that the correctness of the result is not perturbed by rounding errors, the algorithm runs in time “p “p ”” 3 2 O Dlog D  M Dlog D  ⊆ O ` Dlog 6+ε D  ´ ⊆ O ` h 2+ε´ for any ε> 0, where D is the CM discriminant, h is the degree of the class polynomial and M(n) is the time needed to multiply two nbit numbers. Up to logarithmic factors, this running time matches the size of the constructed polynomials. The estimate also relies on a new result concerning the complexity of enumerating the class group of an imaginary quadratic order and on a rigorously proven upper bound for the height of class polynomials. 1. Motivation and
Computing Simplicial Homology Based on Efficient Smith Normal Form Algorithms
, 2002
"... We recall that the calculation of homology with integer coecients of a simplicial complex reduces to the calculation of the Smith Normal Form of the boundary matrices which in general are sparse. We provide a review of several algorithms for the calculation of Smith Normal Form of sparse matrices an ..."
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Cited by 32 (0 self)
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We recall that the calculation of homology with integer coecients of a simplicial complex reduces to the calculation of the Smith Normal Form of the boundary matrices which in general are sparse. We provide a review of several algorithms for the calculation of Smith Normal Form of sparse matrices and compare their running times for actual boundary matrices. Then we describe alternative approaches to the calculation of simplicial homology. The last section then describes motivating examples and actual experiments with the GAP package that was implemented by the authors. These examples also include as an example of other homology theories some calculations of Lie algebra homology.
THE COMPLETE GENERATING FUNCTION FOR GESSEL WALKS IS ALGEBRAIC
"... Gessel walks are lattice walks in the quarter plane N2 which start at the origin (0, 0) ∈ N2 and consist only of steps chosen from the set {←, ↙, ↗, →}. We prove that if g(n; i, j) denotes the number of Gessel walks of length n which end at the point (i, j) ∈ N2, then the trivariate generating ser ..."
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Cited by 27 (7 self)
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Gessel walks are lattice walks in the quarter plane N2 which start at the origin (0, 0) ∈ N2 and consist only of steps chosen from the set {←, ↙, ↗, →}. We prove that if g(n; i, j) denotes the number of Gessel walks of length n which end at the point (i, j) ∈ N2, then the trivariate generating series G(t; x, y) = X g(n; i, j)x i y j t n is an algebraic function. n,i,j≥0 1.
PRIMES is in P
 Ann. of Math
, 2002
"... We present an unconditional deterministic polynomialtime algorithm that determines whether an input number is prime or composite. 1 ..."
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Cited by 26 (2 self)
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We present an unconditional deterministic polynomialtime algorithm that determines whether an input number is prime or composite. 1
On Lattice Reduction for Polynomial Matrices
 Journal of Symbolic Computation
, 2000
"... A simple algorithm for transformation to weak Popov form  essentially lattice reduction for polynomial matrices  is described and analyzed. The algorithm is adapted and applied to various tasks involving polynomial matrices: rank profile and determinant computation; unimodular triangular factori ..."
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Cited by 23 (1 self)
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A simple algorithm for transformation to weak Popov form  essentially lattice reduction for polynomial matrices  is described and analyzed. The algorithm is adapted and applied to various tasks involving polynomial matrices: rank profile and determinant computation; unimodular triangular factorization; transformation to Hermite and Popov canonical form; rational and diophantine linear system solving; short vector computation.
Fast moment estimation in data streams in optimal space
 In Proceedings of the 43rd ACM Symposium on Theory of Computing (STOC
, 2011
"... We give a spaceoptimal algorithm with update time O(log 2 (1/ε) log log(1/ε)) for (1 ± ε)approximating the pth frequency moment, 0 < p < 2, of a lengthn vector updated in a data stream. This provides a nearly exponential improvement in the update time complexity over the previous spaceoptimal alg ..."
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Cited by 20 (6 self)
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We give a spaceoptimal algorithm with update time O(log 2 (1/ε) log log(1/ε)) for (1 ± ε)approximating the pth frequency moment, 0 < p < 2, of a lengthn vector updated in a data stream. This provides a nearly exponential improvement in the update time complexity over the previous spaceoptimal algorithm of [KaneNelsonWoodruff, SODA 2010], which had update time Ω(1/ε 2). 1
Computing modular polynomials in quasilinear time
 Mathematics of Computation
"... Abstract. We analyse and compare the complexity of several algorithms for computing modular polynomials. Under the assumption that rounding errors do not influence the correctness of the result, which appears to be satisfied in practice, we show that an algorithm relying on floating point evaluation ..."
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Cited by 17 (3 self)
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Abstract. We analyse and compare the complexity of several algorithms for computing modular polynomials. Under the assumption that rounding errors do not influence the correctness of the result, which appears to be satisfied in practice, we show that an algorithm relying on floating point evaluation of modular functions and on interpolation has a complexity that is up to logarithmic factors linear in the size of the computed polynomials. In particular, it obtains the classical modular polynomial Φℓ of prime level ℓ in time O ( ℓ 2 log 3 ℓM(ℓ) ) ⊆ O ( ℓ 3 log 4+ε ℓ), where M(ℓ) is the time needed to multiply two ℓbit numbers. Besides treating modular polynomials for Γ0 (ℓ), which are an important ingredient in many algorithms dealing with isogenies of elliptic curves, the algorithm is easily adapted to more general situations. Composite levels are handled just as easily as prime levels, as well as polynomials between a modular function and its transform of prime level, such as the Schläfli polynomials and their generalisations.
Fast algorithms for zerodimensional polynomial systems using duality
 APPLICABLE ALGEBRA IN ENGINEERING, COMMUNICATION AND COMPUTING
, 2001
"... Many questions concerning a zerodimensional polynomial system can be reduced to linear algebra operations in the quotient algebra A = k[X1,..., Xn]/I, where I is the ideal generated by the input system. Assuming that the multiplicative structure of the algebra A is (partly) known, we address the q ..."
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Cited by 16 (3 self)
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Many questions concerning a zerodimensional polynomial system can be reduced to linear algebra operations in the quotient algebra A = k[X1,..., Xn]/I, where I is the ideal generated by the input system. Assuming that the multiplicative structure of the algebra A is (partly) known, we address the question of speeding up the linear algebra phase for the computation of minimal polynomials and rational parametrizations in A. We present new formulæ for the rational parametrizations, extending those of Rouillier, and algorithms extending ideas introduced by Shoup in the univariate case. Our approach is based on the Amodule structure of the dual space � A. An important feature of our algorithms is that we do not require � A to be free and of rank 1. The complexity of our algorithms for computing the minimal polynomial and the rational parametrizations are O(2 n D 5/2) and O(n2 n D 5/2) respectively, where D is the dimension of A. For fixed n, this is better than algorithms based on linear algebra except when the complexity of the available matrix product has exponent less than 5/2.