Solving sparse linear equations over finite fields

Wiedemann, Doug

doi:10.1109/tit.1986.1057137

Cited by 441 publications

(313 citation statements)

References 8 publications

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“…On the other hand, then there is no way to beat sparse-matrix methods for finding a unique solution to a sparse system of equations. Standard estimates for Lanczos, Conjugate Gradients or Wiedemann methods ( [17,24,38]) resemble…”

Section: A Framework For Estimating Security Levelsmentioning

confidence: 99%

On Asymptotic Security Estimates in XL and Gröbner Bases-Related Algebraic Cryptanalysis

Yang

Chen²,

Courtois³

2004

Information and Communications Security

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Abstract. "Algebraic Cryptanalysis" against a cryptosystem often comprises finding enough relations that are generally or probabilistically valid, then solving the resultant system. The security of many schemes (most important being AES) thus depends on the difficulty of solving multivariate polynomial equations. Generically, this is NP-hard. The related methods of XL (eXtended Linearization), Gröbner Bases, and their variants (of which a large number has been proposed) form a unified approach to solving equations and thus affect our assessment and understanding of many cryptosystems. Building on prior theory, we analyze these XL variants and derive asymptotic formulas giving better security estimates under XL-related algebraic attacks; through this examination we have hopefully improved our understanding of such variants. In particular, guessing a portion of variables is a good idea for both XL and Gröbner Bases methods.

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Section: A Framework For Estimating Security Levelsmentioning

confidence: 99%

On Asymptotic Security Estimates in XL and Gröbner Bases-Related Algebraic Cryptanalysis

Yang

Chen²,

Courtois³

2004

Information and Communications Security

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“….} has degree at most r. Denoting its coefficients by F i ∈ F 2 and assuming that F 0 = 1 we have [34] determines x in three steps. For any j with 1 ≤ j ≤ r the j-th coordinates of the vectors M i v for i = 0, 1, 2, .…”

Section: The Block Wiedemann Algorithmmentioning

confidence: 99%

Mersenne Factorization Factory

Kleinjung

Bos

Lenstra

2014

Lecture Notes in Computer Science

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Abstract. We present new factors of seventeen Mersenne numbers, obtained using a variant of the special number field sieve where sieving on the algebraic side is shared among the numbers. It reduced the overall factoring effort by more than 50%. As far as we know this is the first practical application of Coppersmith's "factorization factory" idea. Most factorizations used a new double-product approach that led to additional savings in the matrix step.

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“…One is the discovery of exact sparse iterative algorithms based on the numeric Krylov and Lanczos algorithms [Wiedemann 1986;Kaltofen and Saunders 1991] and their block versions [Coppersmith 1994;Kaltofen 1995;Villard 1997] whose probabilistic analysis for small coefficient fields is being completed today [Eberly 2010 [Ferguson and Forcade 1982;Lenstra et al 1982] that today have greatly improved implementations [Novocin et al 2011] and are used extensively for discovery of exact identities from numeric approximations ( [Håstad et al 1989], "the inverse symbolic calculator" [URL http://oldweb. cecm.sfu.ca/projects/ISC/ISCmain.html]).…”

Section: Exact Linear Algebra Integer Latticesmentioning

confidence: 99%

The “Seven Dwarfs” of Symbolic Computation

Kaltofen

2011

Texts &Amp; Monographs in Symbolic Computation

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We present the Seven Dwarfs of Symbolic Computation, which are sequential and parallel algorithmic methods that today carry a great majority of all exact and hybrid symbolic compute cycles. SymDwf 1. Exact linear algebra, integer lattices SymDwf 2. Exact polynomial and differential algebra, Gröbner bases SymDwf 3. Inverse symbolic problems, e.g., interpolation and parameterization SymDwf 4. Tarski's algebraic theory of real geometry SymDwf 5. Hybrid symbolic-numeric computation SymDwf 6. Computation of closed form solutions SymDwf 7. Rewrite rule systems and computational group theory We will elaborate on each dwarf and compare with Colella's seven and the Berkeley team's thirteen dwarfs of scientific computing.

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Solving sparse linear equations over finite fields

Cited by 441 publications

References 8 publications

On Asymptotic Security Estimates in XL and Gröbner Bases-Related Algebraic Cryptanalysis

On Asymptotic Security Estimates in XL and Gröbner Bases-Related Algebraic Cryptanalysis

Mersenne Factorization Factory

The “Seven Dwarfs” of Symbolic Computation

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