On randomized Lanczos algorithms

Eberly, Wayne; Kaltofen, Erich

doi:10.1145/258726.258776

Cited by 35 publications

(26 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…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

View full text Add to dashboard Cite

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.

show abstract

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

View full text Add to dashboard Cite

show abstract

“…The normality of the remainder sequence associated to a given matrix A essentially comes from the genericity of the projections. This may be partly seen in the scalar case for Lanczos algorithm from (Eberly & Kaltofen 1997, Lemma 4.1), cc 13 (2004) (Eberly 2002) or (Kaltofen & Lee 2003; and in the block case from (Kaltofen 1995, Proposition 3) or (Villard 1997b, Proposition 6.1).…”

Section: Is Satisfied If and Only If There Exist Matrices S And T Of mentioning

confidence: 94%

On the complexity of computing determinants

Kaltofen

Villard²

2005

comput. complex.

Self Cite

View full text Add to dashboard Cite

Abstract. 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 3.2 log A ) 1+o(1) and (n 2.697263 log A ) 1+o(1) bit operations; here A denotes the largest entry in absolute value and the exponent adjustment by "+o(1)" captures additional factors C 1 (log n) C 2 (loglog A ) C 3 for positive real constants C 1 , C 2 , C 3 . The bit complexity (n 3.2 log A ) 1+o(1) 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 3.2+o(1) and O(n 2.697263 ) ring additions, subtractions and multiplications.

show abstract

“…В случае если алгоритм на каком-то шаге не срабатывает и если при этом решение системы (1) не лежит в линейной оболочке уже построенных векторов, можно начать сначала с другой матрицей B или рандомизировать матрицу A, применяя соображения описанные в [2,3].…”

Section: м а черепневunclassified

Блочный Алгоритм Типа Ланцоша Решения Разреженных Систем Линейных Уравнений

Cherepnev¹

2008

Дискрет. матем.

View full text Add to dashboard Cite

В статье предлагается новый блочный алгоритм решения разреженных систем ли-2/ -поле из двух элементов. Алгоритм построен с использованием матричных аппроксимаций Паде. Время работы алгоритма при парал-лельной организации вычислений оценивается величиной maxfO.dN 2 =n/; O.N 2 /g; где d -максимальное по строкам матрицы A число ненулевых элементов. Если d < C n для некоторой абсолютной константы C , то эта оценка лучше оценки времени работы известного алгоритма Монтгомери.

show abstract

On randomized Lanczos algorithms

Cited by 35 publications

References 15 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

On the complexity of computing determinants

Блочный Алгоритм Типа Ланцоша Решения Разреженных Систем Линейных Уравнений

Contact Info

Product

Resources

About