On the degree of polynomials computing square roots mod p
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| Format: | Preprint |
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2023
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| _version_ | 1866917572521230336 |
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| author | Kedlaya, Kiran Kopparty, Swastik |
| author_facet | Kedlaya, Kiran Kopparty, Swastik |
| contents | For an odd prime $p$, we say $f(X) \in {\mathbb F}_p[X]$ computes square roots in $\mathbb F_p$ if, for all nonzero perfect squares $a \in \mathbb F_p$, we have $f(a)^2 = a$.
When $p \equiv 3 \mod 4$, 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 $p \equiv 1 \mod 4$ there was previously no nontrivial bound known on the lowest degree of a polynomial computing square roots in $\mathbb F_p$; it could have been anywhere between $\frac{p}{4}$ and $\frac{p}{2}$.
We show that for all $p \equiv 1 \mod 4$, 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, a result of Agou, Deliglése, and Nicolas (Designs, Codes, and Cryptography, 2003) shows that for infinitely many $p \equiv 1 \mod 4$, the degree of a polynomial computing square roots can be as small as $3p/8$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_10956 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the degree of polynomials computing square roots mod p Kedlaya, Kiran Kopparty, Swastik Number Theory Computational Complexity Combinatorics For an odd prime $p$, we say $f(X) \in {\mathbb F}_p[X]$ computes square roots in $\mathbb F_p$ if, for all nonzero perfect squares $a \in \mathbb F_p$, we have $f(a)^2 = a$. When $p \equiv 3 \mod 4$, 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 $p \equiv 1 \mod 4$ there was previously no nontrivial bound known on the lowest degree of a polynomial computing square roots in $\mathbb F_p$; it could have been anywhere between $\frac{p}{4}$ and $\frac{p}{2}$. We show that for all $p \equiv 1 \mod 4$, 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, a result of Agou, Deliglése, and Nicolas (Designs, Codes, and Cryptography, 2003) shows that for infinitely many $p \equiv 1 \mod 4$, the degree of a polynomial computing square roots can be as small as $3p/8$. |
| title | On the degree of polynomials computing square roots mod p |
| topic | Number Theory Computational Complexity Combinatorics |
| url | https://arxiv.org/abs/2311.10956 |