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76
Algebraic cryptanalysis of hidden field equation (HFE) cryptosystems using Gröbner bases
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On The Complexity Of Computing Determinants
 COMPUTATIONAL COMPLEXITY
, 2001
"... We present new baby steps/giant steps algorithms of asymptotically fast running time for dense matrix problems. Our algorithms compute the determinant, characteristic polynomial, Frobenius normal form and Smith normal form of a dense n n matrix A with integer entries in (n and (n bi ..."
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Cited by 63 (21 self)
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We present new baby steps/giant steps algorithms of asymptotically fast running time for dense matrix problems. Our algorithms compute the determinant, characteristic polynomial, Frobenius normal form and Smith normal form of a dense n n matrix A with integer entries in (n and (n bit operations; here denotes the largest entry in absolute value and the exponent adjustment by "+o(1)" captures additional factors for positive real constants C 1 , C 2 , C 3 . The bit complexity (n results from using the classical cubic matrix multiplication algorithm. Our algorithms are randomized, and we can certify that the output is the determinant of A in a Las Vegas fashion. The second category of problems deals with the setting where the matrix A has elements from an abstract commutative ring, that is, when no divisions in the domain of entries are possible. We present algorithms that deterministically compute the determinant, characteristic polynomial and adjoint of A with n and O(n ) ring additions, subtractions and multiplications.
Factoring Multivariate Polynomials via Partial Differential Equations
 Math. Comput
, 2000
"... A new method is presented for factorization of bivariate polynomials over any field of characteristic zero or of relatively large characteristic. It is based on a simple partial differential equation that gives a system of linear equations. Like Berlekamp's and Niederreiter's algorithms fo ..."
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Cited by 60 (9 self)
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A new method is presented for factorization of bivariate polynomials over any field of characteristic zero or of relatively large characteristic. It is based on a simple partial differential equation that gives a system of linear equations. Like Berlekamp's and Niederreiter's algorithms for factoring univariate polynomials, the dimension of the solution space of the linear system is equal to the number of absolutely irreducible factors of the polynomial to be factored and any basis for the solution space gives a complete factorization by computing gcd's and by factoring univariate polynomials over the ground field. The new method finds absolute and rational factorizations simultaneously and is easy to implement for finite fields, local fields, number fields, and the complex number field. The theory of the new method allows an effective Hilbert irreducibility theorem, thus an efficient reduction of polynomials from multivariate to bivariate.
On the complexity of polynomial matrix computations
 Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation
, 2003
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Factorization of a 768bit RSA modulus
, 2010
"... This paper reports on the factorization of the 768bit number RSA768 by the number field sieve factoring method and discusses some implications for RSA. ..."
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Cited by 38 (13 self)
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This paper reports on the factorization of the 768bit number RSA768 by the number field sieve factoring method and discusses some implications for RSA.
Efficient Matrix Preconditioners for Black Box Linear Algebra
 LINEAR ALGEBRA AND APPLICATIONS 343–344 (2002), 119–146. SPECIAL ISSUE ON STRUCTURED AND INFINITE SYSTEMS OF LINEAR EQUATIONS
, 2001
"... The main idea of the "black box" approach in exact linear algebra is to reduce matrix problems to the computation of minimum polynomials. In most cases preconditioning is necessary to obtain the desired result. Here, good preconditioners will be used to ensure geometrical / algebraic prope ..."
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Cited by 35 (22 self)
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The main idea of the "black box" approach in exact linear algebra is to reduce matrix problems to the computation of minimum polynomials. In most cases preconditioning is necessary to obtain the desired result. Here, good preconditioners will be used to ensure geometrical / algebraic properties on matrices, rather than numerical ones, so we do not address a condition number. We o#er a review of problems for which (algebraic) preconditioning is used, provide a bestiary of preconditioning problems, and discuss several preconditioner types to solve these problems. We present new conditioners, including conditioners to preserve low displacement rank for Toeplitzlike matrices. We also provide new analyses of preconditioner performance and results on the relations among preconditioning problems and with linear algebra problems. Thus improvements are offered for the e#ciency and applicability of preconditioners. The focus is on linear algebra problems over finite fields, but most results are valid for entries from arbitrary fields.
Distributed MatrixFree Solution of Large Sparse Linear Systems over Finite Fields
 Algorithmica
, 1996
"... We describe a coarsegrain parallel software system for the homogeneous solution of linear systems. Our solutions are symbolic, i.e., exact rather than numerical approximations. Our implementation can be run on a network cluster of SPARC20 computers and on an SP2 multiprocessor. Detailed timings a ..."
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Cited by 29 (6 self)
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We describe a coarsegrain parallel software system for the homogeneous solution of linear systems. Our solutions are symbolic, i.e., exact rather than numerical approximations. Our implementation can be run on a network cluster of SPARC20 computers and on an SP2 multiprocessor. Detailed timings are presented for experiments with systems that arise in RSA challenge integer factoring efforts. For example, we can solve a 252; 222 \Theta 252; 222 system with about 11.04 million nonzero entries over the Galois field with 2 elements using 4 processors of an SP2 multiprocessor, in about 26.5 hours CPU time. 1 Introduction The problem of solving large, unstructured, sparse linear systems using exact arithmetic arises in symbolic linear algebra and computational number theory. For example the sievebased factoring of large integers can lead to systems containing over 569,000 equations and variables and over 26.5 million nonzero entries, that need to be solved over the Galois field of two...
A kilobit special number field sieve factorization
 Asiacrypt 2007, volume 4833 of LNCS
, 2007
"... Abstract. We describe how we reached a new factoring milestone by completing the first special number field sieve factorization of a number having more than 1024 bits, namely the Mersenne number 21039 − 1. Although this factorization is orders of magnitude ‘easier ’ than a factorization of a 1024bi ..."
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Cited by 23 (6 self)
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Abstract. We describe how we reached a new factoring milestone by completing the first special number field sieve factorization of a number having more than 1024 bits, namely the Mersenne number 21039 − 1. Although this factorization is orders of magnitude ‘easier ’ than a factorization of a 1024bit RSA modulus is believed to be, the methods we used to obtain our result shed new light on the feasibility of the latter computation. 1
An Efficient MaximumLikelihood Decoding of LDPC Codes Over the Binary Erasure Channel
 IEEE Trans. Inform. Theory
, 2004
"... Abstract — We propose an efficient maximum likelihood decoding algorithm for decoding lowdensity paritycheck codes over the binary erasure channel. We also analyze the computational complexity of the proposed algorithm. Index Terms — Lowdensity paritycheck (LDPC) codes, Binary erasure channel (B ..."
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Cited by 23 (0 self)
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Abstract — We propose an efficient maximum likelihood decoding algorithm for decoding lowdensity paritycheck codes over the binary erasure channel. We also analyze the computational complexity of the proposed algorithm. Index Terms — Lowdensity paritycheck (LDPC) codes, Binary erasure channel (BEC), Iterative decoding, Maximum likelihood (ML) decoding. I.