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# On the Degree of Polynomials Computing Square Roots Mod p

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LIPIcs.CCC.2024.25.pdf
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## Acknowledgements

Both authors acknowledge support from IAS in 2018-19, where initial discussions towards this paper took place. We thank N. Carella and Igor Shparlinski for valuable comments and pointers to the literature.

## 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)
https://doi.org/10.4230/LIPIcs.CCC.2024.25

## 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.

## Subject Classification

##### ACM Subject Classification
• Computing methodologies → Representation of mathematical functions
• Computing methodologies → Number theory algorithms
• Mathematics of computing → Coding theory
##### Keywords
• Algebraic Computation
• Polynomials
• Computing Square roots
• Reed-Solomon Codes

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## References

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