New cube root algorithm based on the third order linear recurrence relations in finite fields

Abstract

In this paper, we present a new cube root algorithm in the finite field $$\mathbb {F}_{q}$$ F q with $$q$$ q a power of prime, which extends the Cipolla---Lehmer type algorithms (Cipolla, Un metodo per la risolutione della congruenza di secondo grado, 1903; Lehmer, Computer technology applied to the theory of numbers, 1969). Our cube root method is inspired by the work of Muller (Des Codes Cryptogr 31:301---312, 2004) on the quadratic case. For a given cubic residue $$c \in \mathbb {F}_{q}$$ c ? F q with $$q \equiv 1 \pmod {9}$$ q ? 1 ( mod 9 ) , we show that there is an irreducible polynomial $$f(x)$$ f ( x ) with root $$\alpha \in \mathbb {F}_{q^{3}}$$ ? ? F q 3 such that $$Tr\left( \alpha ^{\frac{q^{2}+q-2}{9}}\right) $$ T r ? q 2 + q - 2 9 is a cube root of $$c$$ c . Consequently we find an efficient cube root algorithm based on the third order linear recurrence sequences arising from $$f(x)$$ f ( x ) . The complexity estimation shows that our algorithm is better than the previously proposed Cipolla---Lehmer type algorithms.