Results 1  10
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33
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.
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.
A study of Coppersmith's block Wiedemann algorithm using matrix polynomials
 LMCIMAG, REPORT # 975 IM
, 1997
"... We analyse a randomized block algorithm proposed by Coppersmith for solving large sparse systems of linear equations, Aw = 0, over a finite field K =GF(q). It is a modification of an algorithm of Wiedemann. Coppersmith has given heuristic arguments to understand why the algorithm works. But it was a ..."
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Cited by 28 (8 self)
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We analyse a randomized block algorithm proposed by Coppersmith for solving large sparse systems of linear equations, Aw = 0, over a finite field K =GF(q). It is a modification of an algorithm of Wiedemann. Coppersmith has given heuristic arguments to understand why the algorithm works. But it was an open question to prove that it may produce a solution, with positive probability, for small finite fields e.g. for K =GF(2). We answer this question nearly completely. The algorithm uses two random matrices X and Y of dimensions m \Theta N and N \Theta n. Over any finite field, we show how the parameters m and n of the algorithm may be tuned so that, for any input system, a solution is computed with high probability. Conversely, for certain particular input systems, we show that the conditions on the input parameters may be relaxed to ensure the success. We also improve the probability bound of Kaltofen in the case of large cardinality fields. Lastly, for the sake of completeness of the...
Subquadratic computation of vector generating polynomials and improvement of the block Wiedemann algorithm
"... This paper describes a new algorithm for computing linear generators (vector generating polynomials) for matrix sequences, running in subquadratic time. This algorithm applies in particular to the sequential stage of Coppersmith's block Wiedemann algorithm. Experiments showed that our metho ..."
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Cited by 26 (4 self)
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This paper describes a new algorithm for computing linear generators (vector generating polynomials) for matrix sequences, running in subquadratic time. This algorithm applies in particular to the sequential stage of Coppersmith's block Wiedemann algorithm. Experiments showed that our method can be substituted in place of the quadratic one proposed by Coppersmith, yielding important speedups even for realistic matrix sizes. The base elds we were interested in were nite elds of large characteristic.
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.
Fast computation of linear generators for matrix sequences and application to the block Wiedemann algorithm
 PROC. ISSAC '2001
, 2001
"... In this paper we describe how the halfgcd algorithm can be adapted in order to speed up the sequential stage of Coppersmith's block Wiedemann algorithm for solving large sparse linear systems over any finite field. This very stage solves a subproblem than can be seen as the computation of a l ..."
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Cited by 17 (2 self)
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In this paper we describe how the halfgcd algorithm can be adapted in order to speed up the sequential stage of Coppersmith's block Wiedemann algorithm for solving large sparse linear systems over any finite field. This very stage solves a subproblem than can be seen as the computation of a linear generator for a matrix sequence. Our primary realm of interest is the field $\GF{q}$ for large prime power $q$. For the solution of a $N\times N$ system, the complexity of this sequential part drops from $O(N²)$ to $O(\mathsf{M}(N)\log N)$ where $\mathsf{M}(d)$ is the cost for multiplying two polynomials of degree $d$. We discuss the implications of this improvement for the overall cost of the block Wiedemann algorithm and how its parameters should be chosen for best efficiency.
An Implementation of the Number Field Sieve
 EXPERIMENTAL MATHEMATICS
, 1996
"... This article describes an implementation of the NFS, including the choice of two quadratic polynomials, both classical sieving and a special form of lattice sieving (line sieving), the block Lanczos method and a new square root algorithm. Finally some data on factorizations obtained with this implem ..."
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Cited by 14 (0 self)
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This article describes an implementation of the NFS, including the choice of two quadratic polynomials, both classical sieving and a special form of lattice sieving (line sieving), the block Lanczos method and a new square root algorithm. Finally some data on factorizations obtained with this implementation are listed, including the record factorization of 12^151 1.