2 Search Results for "Kedlaya, Kiran"

Document

DOI: 10.4230/LIPIcs.CCC.2024.25

On the Degree of Polynomials Computing Square Roots Mod p

Authors: Kiran S. Kedlaya and Swastik Kopparty

Published in: LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

Abstract

For an odd prime p, we say f(X) ∈ F_p[X] computes square roots in F_p if, for all nonzero perfect squares a ∈ F_p, we have f(a)² = a. When p ≡ 3 mod 4, it is well known that f(X) = X^{(p+1)/4} computes square roots. This degree is surprisingly low (and in fact lowest possible), since we have specified (p-1)/2 evaluations (up to sign) of the polynomial f(X). On the other hand, for p ≡ 1 mod 4 there was previously no nontrivial bound known on the lowest degree of a polynomial computing square roots in F_p. We show that for all p ≡ 1 mod 4, the degree of a polynomial computing square roots has degree at least p/3. Our main new ingredient is a general lemma which may be of independent interest: powers of a low degree polynomial cannot have too many consecutive zero coefficients. The proof method also yields a robust version: any polynomial that computes square roots for 99% of the squares also has degree almost p/3. In the other direction, Agou, Deliglése, and Nicolas [Agou et al., 2003] showed that for infinitely many p ≡ 1 mod 4, the degree of a polynomial computing square roots can be as small as 3p/8.

Cite as

Kiran S. Kedlaya and Swastik Kopparty. On the Degree of Polynomials Computing Square Roots Mod p. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 25:1-25:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{kedlaya_et_al:LIPIcs.CCC.2024.25,
  author =	{Kedlaya, Kiran S. and Kopparty, Swastik},
  title =	{{On the Degree of Polynomials Computing Square Roots Mod p}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{25:1--25:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.25},
  URN =		{urn:nbn:de:0030-drops-204219},
  doi =		{10.4230/LIPIcs.CCC.2024.25},
  annote =	{Keywords: Algebraic Computation, Polynomials, Computing Square roots, Reed-Solomon Codes}
}

Document

DOI: 10.4230/DagSemProc.08381.5

Fast polynomial factorization and modular composition

Authors: Kiran Kedlaya and Christopher Umans

Published in: Dagstuhl Seminar Proceedings, Volume 8381, Computational Complexity of Discrete Problems (2008)

Abstract

We obtain randomized algorithms for factoring degree $n$ univariate polynomials over $F_q$ requiring $O(n^{1.5 + o(1)} log^{1+o(1)} q+ n^{1 + o(1)}log^{2+o(1)} q)$ bit operations. When $log q < n$, this is asymptotically faster than the best previous algorithms (von zur Gathen & Shoup (1992) and Kaltofen & Shoup (1998)); for $log q ge n$, it matches the asymptotic running time of the best known algorithms. The improvements come from new algorithms for modular composition of degree $n$ univariate polynomials, which is the asymptotic bottleneck in fast algorithms for factoring polynomials over finite fields. The best previous algorithms for modular composition use $O(n^{(omega + 1)/2})$ field operations, where $omega$ is the exponent of matrix multiplication (Brent & Kung (1978)), with a slight improvement in the exponent achieved by employing fast rectangular matrix multiplication (Huang & Pan (1997)). We show that modular composition and multipoint evaluation of multivariate polynomials are essentially equivalent, in the sense that an algorithm for one achieving exponent $alpha$ implies an algorithm for the other with exponent $alpha + o(1)$, and vice versa. We then give two new algorithms that solve the problem optimally (up to lower order terms): an algebraic algorithm for fields of characteristic at most $n^{o(1)}$, and a nonalgebraic algorithm that works in arbitrary characteristic. The latter algorithm works by lifting to characteristic 0, applying a small number of rounds of {em multimodular reduction}, and finishing with a small number of multidimensional FFTs. The final evaluations are reconstructed using the Chinese Remainder Theorem. As a bonus, this algorithm produces a very efficient data structure supporting polynomial evaluation queries, which is of independent interest. Our algorithms use techniques which are commonly employed in practice, so they may be competitive for real problem sizes. This contrasts with all previous subquadratic algorithsm for these problems, which rely on fast matrix multiplication. This is joint work with Kiran Kedlaya.

Cite as

Kiran Kedlaya and Christopher Umans. Fast polynomial factorization and modular composition. In Computational Complexity of Discrete Problems. Dagstuhl Seminar Proceedings, Volume 8381, pp. 1-33, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{kedlaya_et_al:DagSemProc.08381.5,
  author =	{Kedlaya, Kiran and Umans, Christopher},
  title =	{{Fast polynomial factorization and modular composition}},
  booktitle =	{Computational Complexity of Discrete Problems},
  pages =	{1--33},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2008},
  volume =	{8381},
  editor =	{Peter Bro Miltersen and R\"{u}diger Reischuk and Georg Schnitger and Dieter van Melkebeek},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.08381.5},
  URN =		{urn:nbn:de:0030-drops-17771},
  doi =		{10.4230/DagSemProc.08381.5},
  annote =	{Keywords: Modular composition; polynomial factorization; multipoint evaluation; Chinese Remaindering}
}

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