Derivative-Free Richelot Isogenies via Subresultants with Algebraic Certification

Abstract

The classical Richelot $(2,2)$-isogeny step for genus-$2$ curves constructs a codomain triple $(U,V,W)$ from a factorization $f=uvw$ via Wronskian derivatives. We give a completely derivative-free reformulation over prime fields $\mathbb{F}_p$, $p>2$, by expressing the Wronskian output through the $2\times 2$ minors of the coefficient matrix and recovering them from first subresultants and a linear syzygy. The resulting Remainder-Polynomial Route (RPR) is proven to produce the identical output triple in $\mathbb{F}_p[x]$ not merely up to units, but as an exact polynomial identity. Building on this equivalence, we introduce the Guarded Subresultant Route (GSR), a deterministic evaluator that certifies admissibility through constant-size algebraic guards, a lightweight post-check, and at most one bounded affine retry. All routes execute $O(1)$ field operations per step. A prototype over $10^6$ matched trials per prime confirms a $4.75$--$6\times$ kernel speedup for RPR over the classical Wronskian formula, and the full GSR pipeline remains $1.4$--$3\times$ faster than WRO despite the certification overhead. Correctness is independently verified by a double-Richelot involution test on $2.5\times10^5$ random triples across five primes.