Algorithmica
Solving sparse linear equations over finite fields
IEEE Transactions on Information Theory
The quadratic sieve factoring algorithm
Proc. of the EUROCRYPT 84 workshop on Advances in cryptology: theory and application of cryptographic techniques
Discrete logarithms in finite fields and their cryptographic significance
Proc. of the EUROCRYPT 84 workshop on Advances in cryptology: theory and application of cryptographic techniques
Solving homogeneous linear equations over GF(2) via block Wiedemann algorithm
Mathematics of Computation
Lanczos Algorithms for Large Symmetric Eigenvalue Computations, Vol. 1
Lanczos Algorithms for Large Symmetric Eigenvalue Computations, Vol. 1
Solving Large Sparse Linear Systems over Finite Fields
CRYPTO '90 Proceedings of the 10th Annual International Cryptology Conference on Advances in Cryptology
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
On randomized Lanczos algorithms
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
Designs, Codes and Cryptography - Special issue on towards a quarter-century of public key cryptography
Discrete Logarithms: The Past and the Future
Designs, Codes and Cryptography - Special issue on towards a quarter-century of public key cryptography
Proceedings of the 2001 international symposium on Symbolic and algebraic computation
Journal of Symbolic Computation - Computer algebra: Selected papers from ISSAC 2001
The Security of Hidden Field Equations (HFE)
CT-RSA 2001 Proceedings of the 2001 Conference on Topics in Cryptology: The Cryptographer's Track at RSA
Some Parallel Algorithms for Integer Factorisation
Euro-Par '99 Proceedings of the 5th International Euro-Par Conference on Parallel Processing
Recent Progress and Prospects for Integer Factorisation Algorithms
COCOON '00 Proceedings of the 6th Annual International Conference on Computing and Combinatorics
NFS with Four Large Primes: An Explosive Experiment
CRYPTO '95 Proceedings of the 15th Annual International Cryptology Conference on Advances in Cryptology
Computation of Discrete Logarithms in F2607
ASIACRYPT '01 Proceedings of the 7th International Conference on the Theory and Application of Cryptology and Information Security: Advances in Cryptology
Analysis of Bernstein's Factorization Circuit
ASIACRYPT '02 Proceedings of the 8th International Conference on the Theory and Application of Cryptology and Information Security: Advances in Cryptology
Proceedings of the 8th IMA International Conference on Cryptography and Coding
ANTS-V Proceedings of the 5th International Symposium on Algorithmic Number Theory
Factoring multivariate polynomials via partial differential equations
Mathematics of Computation
Finite field linear algebra subroutines
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
Early termination over small fields
ISSAC '03 Proceedings of the 2003 international symposium on Symbolic and algebraic computation
Computer algebra handbook
Reliable Krylov-based algorithms for matrix null space and rank
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
Parallel Computing - Algorithmic skeletons
Dense Linear Algebra over Word-Size Prime Fields: the FFLAS and FFPACK Packages
ACM Transactions on Mathematical Software (TOMS)
Factorization of a 512-bit RSA modulus
EUROCRYPT'00 Proceedings of the 19th international conference on Theory and application of cryptographic techniques
Experiments on the linear algebra step in the number field sieve
IWSEC'07 Proceedings of the Security 2nd international conference on Advances in information and computer security
A kilobit special number field sieve factorization
ASIACRYPT'07 Proceedings of the Advances in Crypotology 13th international conference on Theory and application of cryptology and information security
Computers & Mathematics with Applications
Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation
Parallelization of the Lanczos algorithm on multi-core platforms
ICDCN'10 Proceedings of the 11th international conference on Distributed computing and networking
Iterative sparse Matrix-Vector multiplication for integer factorization on GPUs
Euro-Par'11 Proceedings of the 17th international conference on Parallel processing - Volume Part II
Load balanced block lanczos algorithm over GF(2) for factorization of large keys
HiPC'06 Proceedings of the 13th international conference on High Performance Computing
EUC'06 Proceedings of the 2006 international conference on Embedded and Ubiquitous Computing
A parallel GNFS algorithm with the biorthogonal block lanczos method for integer factorization
ATC'06 Proceedings of the Third international conference on Autonomic and Trusted Computing
An analysis of XSL Applied to BES
FSE'07 Proceedings of the 14th international conference on Fast Software Encryption
Solving quadratic equations with XL on parallel architectures
CHES'12 Proceedings of the 14th international conference on Cryptographic Hardware and Embedded Systems
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Some integer factorization algorithms require several vectors in the null space of a sparse m × n matrix over the field GF(2). We modify the Lanczos algorithm to produce a sequence of orthogonal subspaces of GF(2)n, each having dimension almost N, where N is the computer word size, by applying the given matrix and its transpose to N binary vectors at once. The resulting algorithm takes about n/(N - 0.76) iterations. It was applied to matrices larger than 106 × 106 during the factorizations of 105-digit and 119-digit numbers via the general number field sieve.