Paper 2025/2145

Derivative-Free Richelot Isogenies via Subresultants: Algebraic Equivalence and Certified Guarded Computation

Abstract

We present a derivative-free Richelot (2,2)-isogeny formulation using first subresultants and a canonical quadratic lift. Over odd characteristic, we prove its algebraic equivalence in Fp[x] to the classical Wronskian under natural normalization. Leveraging this, we introduce the Guarded Subresultant Route (GSR): a deterministic evaluator with constant-size algebraic guards, a lightweight post-check, and at most one affine retry. It returns a certified triple (U, V, W) or rejects non-admissible inputs, eliminating differentiation while enforcing admissibility and auditable control flow. Prototypes show the core is 1.46–1.70 times faster than the classical Wronskian across varied primes, with GSR adding about 5–10 microseconds of constant overhead. The backend-agnostic design suits batched and hierarchical genus-2 isogeny pipelines for reproducible computation.