Schönhage–Strassen algorithm (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Schönhage–Strassen algorithm" in English language version.

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2archive.org

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acm.org

dl.acm.org

Fürer's algorithm has asymptotic complexity ${\textstyle O{\bigl (}n\cdot \log n\cdot 2^{\Theta (\log ^{*}n)}{\bigr )}.}$
Fürer, Martin (2007). "Faster Integer Multiplication" (PDF). Proc. STOC '07. Symposium on Theory of Computing, San Diego, Jun 2007. pp. 57–66. Archived from the original (PDF) on 2007-03-05.
Fürer, Martin (2009). "Faster Integer Multiplication". SIAM Journal on Computing. 39 (3): 979–1005. doi:10.1137/070711761. ISSN 0097-5397.

Fürer's algorithm is used in the Basic Polynomial Algebra Subprograms (BPAS) open source library. See: Covanov, Svyatoslav; Mohajerani, Davood; Moreno Maza, Marc; Wang, Linxiao (2019-07-08). "Big Prime Field FFT on Multi-core Processors". Proceedings of the 2019 on International Symposium on Symbolic and Algebraic Computation (PDF). Beijing China: ACM. pp. 106–113. doi:10.1145/3326229.3326273. ISBN 978-1-4503-6084-5. S2CID 195848601.

ams.org

mathscinet.ams.org

Harvey, David; van der Hoeven, Joris (2021). "Integer multiplication in time $O(n\log n)$ " (PDF). Annals of Mathematics. Second Series. 193 (2): 563–617. doi:10.4007/annals.2021.193.2.4. MR 4224716. S2CID 109934776.

archive.org

Kleinberg, Jon; Tardos, Eva (2005). Algorithm Design (1 ed.). Pearson. p. 237. ISBN 0-321-29535-8.
Knuth, Donald E. (1997). "§ 4.3.3.C: Discrete Fourier transforms". The Art of Computer Programming. Vol. 2: Seminumerical Algorithms (3rd ed.). Addison-Wesley. pp. 305–311. ISBN 0-201-89684-2.

archives-ouvertes.fr

hal.archives-ouvertes.fr

Harvey, David; van der Hoeven, Joris (2021). "Integer multiplication in time $O(n\log n)$ " (PDF). Annals of Mathematics. Second Series. 193 (2): 563–617. doi:10.4007/annals.2021.193.2.4. MR 4224716. S2CID 109934776.

arxiv.org

Karatsuba multiplication has asymptotic complexity of about $O(n^{1.58})$ and Toom–Cook multiplication has asymptotic complexity of about $O(n^{1.46}).$

Van Meter, Rodney; Itoh, Kohei M. (2005). "Fast Quantum Modular Exponentiation". Physical Review. 71 (5): 052320. arXiv:quant-ph/0408006. Bibcode:2005PhRvA..71e2320V. doi:10.1103/PhysRevA.71.052320. S2CID 14983569.

A discussion of practical crossover points between various algorithms can be found in: Overview of Magma V2.9 Features, arithmetic section Archived 2006-08-20 at the Wayback Machine

Luis Carlos Coronado García, "Can Schönhage multiplication speed up the RSA encryption or decryption? Archived", University of Technology, Darmstadt (2005)

The GNU Multi-Precision Library uses it for values of at least 1728 to 7808 64-bit words (33,000 to 150,000 decimal digits), depending on architecture. See:

"FFT Multiplication (GNU MP 6.2.1)". gmplib.org. Retrieved 2021-07-20.

"MUL_FFT_THRESHOLD". GMP developers' corner. Archived from the original on 24 November 2010. Retrieved 3 November 2011.

"MUL_FFT_THRESHOLD". gmplib.org. Retrieved 2021-07-20.

doi.org

Schönhage, Arnold; Strassen, Volker (1971). "Schnelle Multiplikation großer Zahlen" [Fast multiplication of large numbers]. Computing (in German). 7 (3–4): 281–292. doi:10.1007/BF02242355. S2CID 9738629.
Karatsuba multiplication has asymptotic complexity of about $O(n^{1.58})$ and Toom–Cook multiplication has asymptotic complexity of about $O(n^{1.46}).$

Van Meter, Rodney; Itoh, Kohei M. (2005). "Fast Quantum Modular Exponentiation". Physical Review. 71 (5): 052320. arXiv:quant-ph/0408006. Bibcode:2005PhRvA..71e2320V. doi:10.1103/PhysRevA.71.052320. S2CID 14983569.

A discussion of practical crossover points between various algorithms can be found in: Overview of Magma V2.9 Features, arithmetic section Archived 2006-08-20 at the Wayback Machine

Luis Carlos Coronado García, "Can Schönhage multiplication speed up the RSA encryption or decryption? Archived", University of Technology, Darmstadt (2005)

The GNU Multi-Precision Library uses it for values of at least 1728 to 7808 64-bit words (33,000 to 150,000 decimal digits), depending on architecture. See:

"FFT Multiplication (GNU MP 6.2.1)". gmplib.org. Retrieved 2021-07-20.

"MUL_FFT_THRESHOLD". GMP developers' corner. Archived from the original on 24 November 2010. Retrieved 3 November 2011.

"MUL_FFT_THRESHOLD". gmplib.org. Retrieved 2021-07-20.
Fürer's algorithm has asymptotic complexity ${\textstyle O{\bigl (}n\cdot \log n\cdot 2^{\Theta (\log ^{*}n)}{\bigr )}.}$
Fürer, Martin (2007). "Faster Integer Multiplication" (PDF). Proc. STOC '07. Symposium on Theory of Computing, San Diego, Jun 2007. pp. 57–66. Archived from the original (PDF) on 2007-03-05.
Fürer, Martin (2009). "Faster Integer Multiplication". SIAM Journal on Computing. 39 (3): 979–1005. doi:10.1137/070711761. ISSN 0097-5397.

Fürer's algorithm is used in the Basic Polynomial Algebra Subprograms (BPAS) open source library. See: Covanov, Svyatoslav; Mohajerani, Davood; Moreno Maza, Marc; Wang, Linxiao (2019-07-08). "Big Prime Field FFT on Multi-core Processors". Proceedings of the 2019 on International Symposium on Symbolic and Algebraic Computation (PDF). Beijing China: ACM. pp. 106–113. doi:10.1145/3326229.3326273. ISBN 978-1-4503-6084-5. S2CID 195848601.
Harvey, David; van der Hoeven, Joris (2021). "Integer multiplication in time $O(n\log n)$ " (PDF). Annals of Mathematics. Second Series. 193 (2): 563–617. doi:10.4007/annals.2021.193.2.4. MR 4224716. S2CID 109934776.

dx.doi.org

Fürer's algorithm has asymptotic complexity ${\textstyle O{\bigl (}n\cdot \log n\cdot 2^{\Theta (\log ^{*}n)}{\bigr )}.}$
Fürer, Martin (2007). "Faster Integer Multiplication" (PDF). Proc. STOC '07. Symposium on Theory of Computing, San Diego, Jun 2007. pp. 57–66. Archived from the original (PDF) on 2007-03-05.
Fürer, Martin (2009). "Faster Integer Multiplication". SIAM Journal on Computing. 39 (3): 979–1005. doi:10.1137/070711761. ISSN 0097-5397.

Fürer's algorithm is used in the Basic Polynomial Algebra Subprograms (BPAS) open source library. See: Covanov, Svyatoslav; Mohajerani, Davood; Moreno Maza, Marc; Wang, Linxiao (2019-07-08). "Big Prime Field FFT on Multi-core Processors". Proceedings of the 2019 on International Symposium on Symbolic and Algebraic Computation (PDF). Beijing China: ACM. pp. 106–113. doi:10.1145/3326229.3326273. ISBN 978-1-4503-6084-5. S2CID 195848601.

gmplib.org

Karatsuba multiplication has asymptotic complexity of about $O(n^{1.58})$ and Toom–Cook multiplication has asymptotic complexity of about $O(n^{1.46}).$

Van Meter, Rodney; Itoh, Kohei M. (2005). "Fast Quantum Modular Exponentiation". Physical Review. 71 (5): 052320. arXiv:quant-ph/0408006. Bibcode:2005PhRvA..71e2320V. doi:10.1103/PhysRevA.71.052320. S2CID 14983569.

A discussion of practical crossover points between various algorithms can be found in: Overview of Magma V2.9 Features, arithmetic section Archived 2006-08-20 at the Wayback Machine

Luis Carlos Coronado García, "Can Schönhage multiplication speed up the RSA encryption or decryption? Archived", University of Technology, Darmstadt (2005)

The GNU Multi-Precision Library uses it for values of at least 1728 to 7808 64-bit words (33,000 to 150,000 decimal digits), depending on architecture. See:

"FFT Multiplication (GNU MP 6.2.1)". gmplib.org. Retrieved 2021-07-20.

"MUL_FFT_THRESHOLD". GMP developers' corner. Archived from the original on 24 November 2010. Retrieved 3 November 2011.

"MUL_FFT_THRESHOLD". gmplib.org. Retrieved 2021-07-20.

hal.science

inria.hal.science

Gaudry, Pierrick; Alexander, Kruppa; Paul, Zimmermann (2007). "A GMP-based implementation of Schönhage-Strassen's large integer multiplication algorithm" (PDF). p. 6.
Gaudry, Pierrick; Kruppa, Alexander; Zimmermann, Paul (2007). "A GMP-based Implementation of Schönhage-Strassen's Large Integer Multiplication Algorithm" (PDF). p. 2.
Gaudry, Pierrick; Kruppa, Alexander; Zimmermann, Paul (2007). "A GMP-based implementation of Schönhage-Strassen's large integer multiplication algorithm" (PDF). p. 7.
Gaudry, Pierrick; Kruppa, Alexander; Zimmermann, Paul (2007). "A GMP-based implementation of Schönhage-Strassen's large integer multiplication algorithm" (PDF). p. 6.
Gaudry, Pierrick; Kruppa, Alexander; Zimmermann, Paul (2007). "A GMP-based implementation of Schönhage-Strassen's large integer multiplication algorithm" (PDF). p. 6.

harvard.edu

ui.adsabs.harvard.edu

Karatsuba multiplication has asymptotic complexity of about $O(n^{1.58})$ and Toom–Cook multiplication has asymptotic complexity of about $O(n^{1.46}).$

Van Meter, Rodney; Itoh, Kohei M. (2005). "Fast Quantum Modular Exponentiation". Physical Review. 71 (5): 052320. arXiv:quant-ph/0408006. Bibcode:2005PhRvA..71e2320V. doi:10.1103/PhysRevA.71.052320. S2CID 14983569.

A discussion of practical crossover points between various algorithms can be found in: Overview of Magma V2.9 Features, arithmetic section Archived 2006-08-20 at the Wayback Machine

Luis Carlos Coronado García, "Can Schönhage multiplication speed up the RSA encryption or decryption? Archived", University of Technology, Darmstadt (2005)

The GNU Multi-Precision Library uses it for values of at least 1728 to 7808 64-bit words (33,000 to 150,000 decimal digits), depending on architecture. See:

"FFT Multiplication (GNU MP 6.2.1)". gmplib.org. Retrieved 2021-07-20.

"MUL_FFT_THRESHOLD". GMP developers' corner. Archived from the original on 24 November 2010. Retrieved 3 November 2011.

"MUL_FFT_THRESHOLD". gmplib.org. Retrieved 2021-07-20.

iacr.org

eprint.iacr.org

Becker, Hanno; Hwang, Vincent; J. Kannwischer, Matthias; Panny, Lorenz (2022). "Efficient Multiplication of Somewhat Small Integers using Number-Theoretic Transforms" (PDF).

inria.fr

hal.inria.fr

Fürer's algorithm has asymptotic complexity ${\textstyle O{\bigl (}n\cdot \log n\cdot 2^{\Theta (\log ^{*}n)}{\bigr )}.}$
Fürer, Martin (2007). "Faster Integer Multiplication" (PDF). Proc. STOC '07. Symposium on Theory of Computing, San Diego, Jun 2007. pp. 57–66. Archived from the original (PDF) on 2007-03-05.
Fürer, Martin (2009). "Faster Integer Multiplication". SIAM Journal on Computing. 39 (3): 979–1005. doi:10.1137/070711761. ISSN 0097-5397.

Fürer's algorithm is used in the Basic Polynomial Algebra Subprograms (BPAS) open source library. See: Covanov, Svyatoslav; Mohajerani, Davood; Moreno Maza, Marc; Wang, Linxiao (2019-07-08). "Big Prime Field FFT on Multi-core Processors". Proceedings of the 2019 on International Symposium on Symbolic and Algebraic Computation (PDF). Beijing China: ACM. pp. 106–113. doi:10.1145/3326229.3326273. ISBN 978-1-4503-6084-5. S2CID 195848601.

gitlab.inria.fr

This method is used in INRIA's ECM library.

loria.fr

members.loria.fr

"ECMNET". members.loria.fr. Retrieved 2023-04-09.

psu.edu

cse.psu.edu

Fürer's algorithm has asymptotic complexity ${\textstyle O{\bigl (}n\cdot \log n\cdot 2^{\Theta (\log ^{*}n)}{\bigr )}.}$
Fürer, Martin (2007). "Faster Integer Multiplication" (PDF). Proc. STOC '07. Symposium on Theory of Computing, San Diego, Jun 2007. pp. 57–66. Archived from the original (PDF) on 2007-03-05.
Fürer, Martin (2009). "Faster Integer Multiplication". SIAM Journal on Computing. 39 (3): 979–1005. doi:10.1137/070711761. ISSN 0097-5397.

Fürer's algorithm is used in the Basic Polynomial Algebra Subprograms (BPAS) open source library. See: Covanov, Svyatoslav; Mohajerani, Davood; Moreno Maza, Marc; Wang, Linxiao (2019-07-08). "Big Prime Field FFT on Multi-core Processors". Proceedings of the 2019 on International Symposium on Symbolic and Algebraic Computation (PDF). Beijing China: ACM. pp. 106–113. doi:10.1145/3326229.3326273. ISBN 978-1-4503-6084-5. S2CID 195848601.

researchgate.net

Lüders, Christoph (2014). "Fast Multiplication of Large Integers: Implementation and Analysis of the DKSS Algorithm". p. 26.
S. Dimitrov, Vassil; V. Cooklev, Todor; D. Donevsky, Borislav (1994). "Generalized Fermat-Mersenne Number Theoretic Transform". p. 2.
S. Dimitrov, Vassil; V. Cooklev, Todor; D. Donevsky, Borislav (1994). "Generalized Fermat-Mersenne Number Theoretic Transform". p. 3.
Lüders, Christoph (2014). "Fast Multiplication of Large Integers: Implementation and Analysis of the DKSS Algorithm". p. 28.

semanticscholar.org

api.semanticscholar.org

Schönhage, Arnold; Strassen, Volker (1971). "Schnelle Multiplikation großer Zahlen" [Fast multiplication of large numbers]. Computing (in German). 7 (3–4): 281–292. doi:10.1007/BF02242355. S2CID 9738629.
Karatsuba multiplication has asymptotic complexity of about $O(n^{1.58})$ and Toom–Cook multiplication has asymptotic complexity of about $O(n^{1.46}).$

Van Meter, Rodney; Itoh, Kohei M. (2005). "Fast Quantum Modular Exponentiation". Physical Review. 71 (5): 052320. arXiv:quant-ph/0408006. Bibcode:2005PhRvA..71e2320V. doi:10.1103/PhysRevA.71.052320. S2CID 14983569.

A discussion of practical crossover points between various algorithms can be found in: Overview of Magma V2.9 Features, arithmetic section Archived 2006-08-20 at the Wayback Machine

Luis Carlos Coronado García, "Can Schönhage multiplication speed up the RSA encryption or decryption? Archived", University of Technology, Darmstadt (2005)

The GNU Multi-Precision Library uses it for values of at least 1728 to 7808 64-bit words (33,000 to 150,000 decimal digits), depending on architecture. See:

"FFT Multiplication (GNU MP 6.2.1)". gmplib.org. Retrieved 2021-07-20.

"MUL_FFT_THRESHOLD". GMP developers' corner. Archived from the original on 24 November 2010. Retrieved 3 November 2011.

"MUL_FFT_THRESHOLD". gmplib.org. Retrieved 2021-07-20.
Fürer's algorithm has asymptotic complexity ${\textstyle O{\bigl (}n\cdot \log n\cdot 2^{\Theta (\log ^{*}n)}{\bigr )}.}$
Fürer, Martin (2007). "Faster Integer Multiplication" (PDF). Proc. STOC '07. Symposium on Theory of Computing, San Diego, Jun 2007. pp. 57–66. Archived from the original (PDF) on 2007-03-05.
Fürer, Martin (2009). "Faster Integer Multiplication". SIAM Journal on Computing. 39 (3): 979–1005. doi:10.1137/070711761. ISSN 0097-5397.

Fürer's algorithm is used in the Basic Polynomial Algebra Subprograms (BPAS) open source library. See: Covanov, Svyatoslav; Mohajerani, Davood; Moreno Maza, Marc; Wang, Linxiao (2019-07-08). "Big Prime Field FFT on Multi-core Processors". Proceedings of the 2019 on International Symposium on Symbolic and Algebraic Computation (PDF). Beijing China: ACM. pp. 106–113. doi:10.1145/3326229.3326273. ISBN 978-1-4503-6084-5. S2CID 195848601.
Harvey, David; van der Hoeven, Joris (2021). "Integer multiplication in time $O(n\log n)$ " (PDF). Annals of Mathematics. Second Series. 193 (2): 563–617. doi:10.4007/annals.2021.193.2.4. MR 4224716. S2CID 109934776.

tu-darmstadt.de

cdc.informatik.tu-darmstadt.de

Karatsuba multiplication has asymptotic complexity of about $O(n^{1.58})$ and Toom–Cook multiplication has asymptotic complexity of about $O(n^{1.46}).$

Van Meter, Rodney; Itoh, Kohei M. (2005). "Fast Quantum Modular Exponentiation". Physical Review. 71 (5): 052320. arXiv:quant-ph/0408006. Bibcode:2005PhRvA..71e2320V. doi:10.1103/PhysRevA.71.052320. S2CID 14983569.

A discussion of practical crossover points between various algorithms can be found in: Overview of Magma V2.9 Features, arithmetic section Archived 2006-08-20 at the Wayback Machine

Luis Carlos Coronado García, "Can Schönhage multiplication speed up the RSA encryption or decryption? Archived", University of Technology, Darmstadt (2005)

The GNU Multi-Precision Library uses it for values of at least 1728 to 7808 64-bit words (33,000 to 150,000 decimal digits), depending on architecture. See:

"FFT Multiplication (GNU MP 6.2.1)". gmplib.org. Retrieved 2021-07-20.

"MUL_FFT_THRESHOLD". GMP developers' corner. Archived from the original on 24 November 2010. Retrieved 3 November 2011.

"MUL_FFT_THRESHOLD". gmplib.org. Retrieved 2021-07-20.

usyd.edu.au

magma.maths.usyd.edu.au

Karatsuba multiplication has asymptotic complexity of about $O(n^{1.58})$ and Toom–Cook multiplication has asymptotic complexity of about $O(n^{1.46}).$

Van Meter, Rodney; Itoh, Kohei M. (2005). "Fast Quantum Modular Exponentiation". Physical Review. 71 (5): 052320. arXiv:quant-ph/0408006. Bibcode:2005PhRvA..71e2320V. doi:10.1103/PhysRevA.71.052320. S2CID 14983569.

A discussion of practical crossover points between various algorithms can be found in: Overview of Magma V2.9 Features, arithmetic section Archived 2006-08-20 at the Wayback Machine

Luis Carlos Coronado García, "Can Schönhage multiplication speed up the RSA encryption or decryption? Archived", University of Technology, Darmstadt (2005)

The GNU Multi-Precision Library uses it for values of at least 1728 to 7808 64-bit words (33,000 to 150,000 decimal digits), depending on architecture. See:

"FFT Multiplication (GNU MP 6.2.1)". gmplib.org. Retrieved 2021-07-20.

"MUL_FFT_THRESHOLD". GMP developers' corner. Archived from the original on 24 November 2010. Retrieved 3 November 2011.

"MUL_FFT_THRESHOLD". gmplib.org. Retrieved 2021-07-20.

web.archive.org

Karatsuba multiplication has asymptotic complexity of about $O(n^{1.58})$ and Toom–Cook multiplication has asymptotic complexity of about $O(n^{1.46}).$

Van Meter, Rodney; Itoh, Kohei M. (2005). "Fast Quantum Modular Exponentiation". Physical Review. 71 (5): 052320. arXiv:quant-ph/0408006. Bibcode:2005PhRvA..71e2320V. doi:10.1103/PhysRevA.71.052320. S2CID 14983569.

A discussion of practical crossover points between various algorithms can be found in: Overview of Magma V2.9 Features, arithmetic section Archived 2006-08-20 at the Wayback Machine

Luis Carlos Coronado García, "Can Schönhage multiplication speed up the RSA encryption or decryption? Archived", University of Technology, Darmstadt (2005)

The GNU Multi-Precision Library uses it for values of at least 1728 to 7808 64-bit words (33,000 to 150,000 decimal digits), depending on architecture. See:

"FFT Multiplication (GNU MP 6.2.1)". gmplib.org. Retrieved 2021-07-20.

"MUL_FFT_THRESHOLD". GMP developers' corner. Archived from the original on 24 November 2010. Retrieved 3 November 2011.

"MUL_FFT_THRESHOLD". gmplib.org. Retrieved 2021-07-20.
Fürer's algorithm has asymptotic complexity ${\textstyle O{\bigl (}n\cdot \log n\cdot 2^{\Theta (\log ^{*}n)}{\bigr )}.}$
Fürer, Martin (2007). "Faster Integer Multiplication" (PDF). Proc. STOC '07. Symposium on Theory of Computing, San Diego, Jun 2007. pp. 57–66. Archived from the original (PDF) on 2007-03-05.
Fürer, Martin (2009). "Faster Integer Multiplication". SIAM Journal on Computing. 39 (3): 979–1005. doi:10.1137/070711761. ISSN 0097-5397.

Fürer's algorithm is used in the Basic Polynomial Algebra Subprograms (BPAS) open source library. See: Covanov, Svyatoslav; Mohajerani, Davood; Moreno Maza, Marc; Wang, Linxiao (2019-07-08). "Big Prime Field FFT on Multi-core Processors". Proceedings of the 2019 on International Symposium on Symbolic and Algebraic Computation (PDF). Beijing China: ACM. pp. 106–113. doi:10.1145/3326229.3326273. ISBN 978-1-4503-6084-5. S2CID 195848601.

worldcat.org

Fürer's algorithm has asymptotic complexity ${\textstyle O{\bigl (}n\cdot \log n\cdot 2^{\Theta (\log ^{*}n)}{\bigr )}.}$
Fürer, Martin (2007). "Faster Integer Multiplication" (PDF). Proc. STOC '07. Symposium on Theory of Computing, San Diego, Jun 2007. pp. 57–66. Archived from the original (PDF) on 2007-03-05.
Fürer, Martin (2009). "Faster Integer Multiplication". SIAM Journal on Computing. 39 (3): 979–1005. doi:10.1137/070711761. ISSN 0097-5397.

Fürer's algorithm is used in the Basic Polynomial Algebra Subprograms (BPAS) open source library. See: Covanov, Svyatoslav; Mohajerani, Davood; Moreno Maza, Marc; Wang, Linxiao (2019-07-08). "Big Prime Field FFT on Multi-core Processors". Proceedings of the 2019 on International Symposium on Symbolic and Algebraic Computation (PDF). Beijing China: ACM. pp. 106–113. doi:10.1145/3326229.3326273. ISBN 978-1-4503-6084-5. S2CID 195848601.