Fast cube roots in Fp2 via the algebraic torus

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Computing cube roots in quadratic extensions of finite fields is a subroutine that arises in elliptic-curve point decompression, hash-to-curve and isogeny-based protocols. While the factorization $p^2-1=(p-1)(p+1)$ suggests a known subgroup decomposition, computing a cube root via separate operations in the two subgroups does not beat a direct $\mathbb{F}_{p^2}$ exponentiation in practice. We propose a carefully engineered algorithm that reduces the $\mathbb{F}_{p^2}$ cube root to a single $\mat