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Pseudo-Deterministic Construction of Irreducible Polynomials over Finite Fields

Author Shanthanu S. Rai

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Subject Classification

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

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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

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