ECCC-Report TR17-016https://eccc.weizmann.ac.il/report/2017/016Comments and Revisions published for TR17-016en-usSun, 05 Feb 2017 19:06:26 +0200
Paper TR17-016
| Irreducibility and deterministic r-th root finding over finite fields |
Vishwas Bhargava,
GĂˇbor Ivanyos,
Rajat Mittal,
Nitin Saxena
https://eccc.weizmann.ac.il/report/2017/016Constructing $r$-th nonresidue over a finite field is a fundamental computational problem. A related problem is to construct an irreducible polynomial of degree $r^e$ (where $r$ is a prime) over a given finite field $\F_q$ of characteristic $p$ (equivalently, constructing the bigger field $\F_{q^{r^e}}$). Both these problems have famous randomized algorithms but the derandomization is an open question. We give some new connections between these two problems and their variants.
In 1897, Stickelberger proved that if a polynomial has an odd number of even degree factors, then its discriminant is a quadratic nonresidue in the field. We give an extension of Stickelberger's Lemma;
we construct $r$-th nonresidues from a polynomial $f$ for which there is a $d$, such that, $r|d$ and $r\nmid\,$\#(irreducible factor of $f(x)$ of degree $d$). Our theorem has the following interesting consequences: (1) we can construct $\F_{q^m}$ in deterministic poly($\deg(f),m\log q$)-time if $m$ is an $r$-power and $f$ is known; (2) we can find $r$-th roots in $\F_{p^m}$ in deterministic poly($m\log p$)-time if $r$ is constant and $r|\gcd(m,p-1)$.
We also discuss a conjecture significantly weaker than the Generalized Riemann hypothesis to get a deterministic poly-time algorithm for $r$-th root finding.Sun, 05 Feb 2017 19:06:26 +0200https://eccc.weizmann.ac.il/report/2017/016