Document Open Access Logo

Pseudo-Deterministic Construction of Irreducible Polynomials over Finite Fields

Author Shanthanu S. Rai

PDF

File

Thumbnail PDF
PDF
LIPIcs.FSTTCS.2024.33.pdf
  • Filesize: 0.69 MB
  • 12 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
Keywords
  • Algebra and Computation
  • Finite fields
  • Factorization
  • Pseudo-deterministic
  • Polynomials

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

We present a polynomial-time pseudo-deterministic algorithm for constructing irreducible polynomial of degree d over finite field 𝔽_q. A pseudo-deterministic algorithm is allowed to use randomness, but with high probability it must output a canonical irreducible polynomial. Our construction runs in time Õ(d⁴log⁴q). 
Our construction extends Shoup’s deterministic algorithm (FOCS 1988) for the same problem, which runs in time Õ(d⁴p^{1/2}log⁴q) (where p is the characteristic of the field 𝔽_q). Shoup had shown a reduction from constructing irreducible polynomials to factoring polynomials over finite fields. We show that by using a fast randomized factoring algorithm, the above reduction yields an efficient pseudo-deterministic algorithm for constructing irreducible polynomials over finite fields.

Cite As Get BibTex

Shanthanu S. Rai. Pseudo-Deterministic Construction of Irreducible Polynomials over Finite Fields. In 44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 323, pp. 33:1-33:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.FSTTCS.2024.33

Author Details

Shanthanu S. Rai
  • Tata Institute of Fundamental Research, Mumbai, India

Funding

Research supported by the Department of Atomic Energy, Government of India, under project 12-R&D-TFR-5.01-0500.

Acknowledgements

The author would like to thank Mrinal Kumar and Ramprasad Saptharishi for introducing him to the question of pseudo-deterministic construction of irreducible polynomials and for the many insightful discussions along the way.

References

  1. Leonard M. Adleman and Hendrik W. Lenstra Jr. Finding irreducible polynomials over finite fields. In Juris Hartmanis, editor, Proceedings of the 18th Annual ACM Symposium on Theory of Computing, May 28-30, 1986, Berkeley, California, USA, STOC '86, pages 350-355, New York, NY, USA, 1986. ACM. URL: https://doi.org/10.1145/12130.12166.
  2. Manindra Agrawal, Neeraj Kayal, and Nitin Saxena. PRIMES is in P. Annals of Mathematics, 160(2):781-793, 2004. URL: http://www.jstor.org/stable/3597229.
  3. David G. Cantor and Hans Zassenhaus. A new algorithm for factoring polynomials over finite fields. Mathematics of Computation, 36(154):587-592, 1981. URL: http://www.jstor.org/stable/2007663.
  4. Lijie Chen, Zhenjian Lu, Igor C. Oliveira, Hanlin Ren, and Rahul Santhanam. Polynomial-time pseudodeterministic construction of primes. In 64th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2023, Santa Cruz, CA, USA, November 6-9, 2023, pages 1261-1270, Los Alamitos, CA, USA, November 2023. IEEE. URL: https://doi.org/10.1109/FOCS57990.2023.00074.
  5. Jean-Marc Couveignes and Reynald Lercier. Fast construction of irreducible polynomials over finite fields. Israel Journal of Mathematics, 194(1):77-105, March 2013. URL: https://doi.org/10.1007/s11856-012-0070-8.
  6. Eran Gat and Shafi Goldwasser. Probabilistic search algorithms with unique answers and their cryptographic applications. Electron. Colloquium Comput. Complex., TR11-136(TR11-136), October 2011. URL: https://eccc.weizmann.ac.il/report/2011/136.
  7. Kiran S. Kedlaya and Christopher Umans. Fast polynomial factorization and modular composition. SIAM J. Comput., 40(6):1767-1802, 2011. URL: https://doi.org/10.1137/08073408X.
  8. H. W. Lenstra. Finding isomorphisms between finite fields. Mathematics of Computation, 56(193):329-347, 1991. URL: https://doi.org/10.1090/S0025-5718-1991-1052099-2.
  9. Rudolf Lidl and Harald Niederreiter. Introduction to Finite Fields and their Applications. Cambridge University Press, Cambridge, 2 edition, 1994. URL: https://doi.org/10.1017/CBO9781139172769.
  10. Michael O. Rabin. Probabilistic algorithms in finite fields. SIAM J. Comput., 9(2):273-280, 1980. URL: https://doi.org/10.1137/0209024.
  11. Daniel Shanks. Five number-theoretic algorithms. In Proceedings of the Second Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1972), volume No. VII of Congress. Numer., pages 51-70. Utilitas Math., Winnipeg, MB, 1973. Google Scholar
  12. Victor Shoup. New algorithms for finding irreducible polynomials over finite fields. In 29th Annual Symposium on Foundations of Computer Science, White Plains, New York, USA, 24-26 October 1988, pages 283-290. IEEE Computer Society, 1988. URL: https://doi.org/10.1109/SFCS.1988.21944.
  13. Terence Tao, Ernest Croot III, and Harald Helfgott. Deterministic methods to find primes. Math. Comput., 81(278):1233-1246, 2012. URL: https://doi.org/10.1090/S0025-5718-2011-02542-1.
  14. Alberto Tonelli. Bemerkung über die auflösung quadratischer congruenzen. Nachrichten von der Königl. Gesellschaft der Wissenschaften und der Georg-Augusts-Universität zu Göttingen, 1891:344-346, 1891. URL: http://eudml.org/doc/180329.
  15. Joachim von zur Gathen and Victor Shoup. Computing frobenius maps and factoring polynomials. Comput. Complex., 2(3):187-224, September 1992. URL: https://doi.org/10.1007/BF01272074.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact