{"ID":5935712,"CreatedAt":"2026-07-07T01:22:02.77346169Z","UpdatedAt":"2026-07-07T02:10:06.972658124Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.03376","arxiv_id":"2607.03376","title":"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 $\\mathbb{F}_p$, $p\u003e2$, 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.","short_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 $\\mathbb{F}_p$, $p\u003e2$, by expressing the Wronskian output through the $2\\times 2$ minors of...","url_abs":"https://arxiv.org/abs/2607.03376","url_pdf":"https://arxiv.org/pdf/2607.03376v1","authors":"[\"Hung T. Dang\",\"Diep V. Nguyen\"]","published":"2026-07-03T14:29:46Z","proceeding":"math.NT","tasks":"[\"math.NT\",\"cs.CR\",\"cs.SC\"]","methods":"[]","has_code":false}