Deterministic polynomial factorisation modulo many primes
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912589126041600 |
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| author | Altman, Daniel |
| author_facet | Altman, Daniel |
| contents | Designing a deterministic polynomial time algorithm for factoring univariate polynomials over finite fields remains a notorious open problem. In this paper, we present an unconditional deterministic algorithm that takes as input an irreducible polynomial $f \in \mathbb{Z}[x]$, and computes the factorisation of its reductions modulo $p$ for all primes $p$ up to a prescribed bound $N$. The \emph{average running time per prime} is polynomial in the size of the input and the degree of the splitting field of $f$ over $\mathbb{Q}$. In particular, if $f$ is Galois, we succeed in factoring in (amortised) deterministic polynomial time. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_12705 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Deterministic polynomial factorisation modulo many primes Altman, Daniel Number Theory Computational Complexity 11Y16, 11T06, 11Y05, 68W30 Designing a deterministic polynomial time algorithm for factoring univariate polynomials over finite fields remains a notorious open problem. In this paper, we present an unconditional deterministic algorithm that takes as input an irreducible polynomial $f \in \mathbb{Z}[x]$, and computes the factorisation of its reductions modulo $p$ for all primes $p$ up to a prescribed bound $N$. The \emph{average running time per prime} is polynomial in the size of the input and the degree of the splitting field of $f$ over $\mathbb{Q}$. In particular, if $f$ is Galois, we succeed in factoring in (amortised) deterministic polynomial time. |
| title | Deterministic polynomial factorisation modulo many primes |
| topic | Number Theory Computational Complexity 11Y16, 11T06, 11Y05, 68W30 |
| url | https://arxiv.org/abs/2509.12705 |