An Improvement of the Cipolla-Lehmer Type Algorithms

Let F_q be a finite field with q elements with prime power q and let r>1 be an integer with $q\equiv 1 \pmod{r}$. In this paper, we present a refinement of the Cipolla-Lehmer type algorithm given by H. C. Williams, and subsequently improved by K. S. Williams and K. Hardy. For a given r-th power residue c in F_q where r is an odd prime, the algorithm of H. C. Williams determines a solution of X^r=c in $O(r^3\log q)$ multiplications in F_q, and the algorithm of K. S. Williams and K. Hardy finds a solution in $O(r^4+r^2\log q)$ multiplications in F_q. Our refinement finds a solution in $O(r^3+r^2\log q)$ multiplications in F_q. Therefore our new method is better than the previously proposed algorithms independent of the size of r, and the implementation result via SAGE shows a substantial speed-up compared with the existing algorithms

On r-th Root Extraction Algorithm in F_q For q=lr^s+1 (mod r^(s+1)) with 0 < l < r and Small s

We present an r-th root extraction algorithm over a finite field F_q. Our algorithm precomputes a primitive r^s-th root of unity where s is the largest positive integer satisfying r^s| q-1, and is applicable for the cases when s is small. The proposed algorithm requires one exponentiation for the r-th root computation and is favorably compared to the existing algorithms

Trace Expression of r-th Root over Finite Field

Efficient computation of $r$-th root in $\mathbb F_q$ has many applications in computational number theory and many other related areas. We present a new $r$-th root formula which generalizes Müller\u27s result on square root, and which provides a possible improvement of the Cipolla-Lehmer algorithm for general case. More precisely, for given $r$-th power $c\in \mathbb F_q$, we show that there exists $\alpha \in \mathbb F_{q^r}$ such that $Tr\left(\alpha^\frac{(\sum_{i=0}^{r-1}q^i)-r}{r^2}\right)^r=c$ where $Tr(\alpha)=\alpha+\alpha^q+\alpha^{q^2}+\cdots +\alpha^{q^{r-1}}$ and $\alpha$ is a root of certain irreducible polynomial of degree $r$ over $\mathbb F_q$

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum