A New Characteristic-Uniform Model for Elliptic Curves -- Theory, Arithmetic, and Applications

Mathematics > Number Theory [Submitted on 3 Aug 2026 (v1), last revised 10 Aug 2026 (this version, v4)] A New Characteristic-Uniform Model for Elliptic Curves -- Theory, Arithmetic, and Applications Hongfeng Wu We develop a characteristic-uniform arithmetic theory for C d :( u 2 +u)( v 2 +v)=d. For \(d(1-16d)\ne0\), its smooth \((2,2)\)-completion has four rational boundary points forming \(\mathbb Z/4\mathbb Z\), an intrinsic \(D_8\)-action, the inverse \(-(u,v)=(u,-v-1)\), and the native Kummer map \(\kappa_d(u,v)=(u+1)\). Working natively, we derive complete full-point and differential laws, Kummer ladders and recovery, halving, tripling, \(2P+Q\), division polynomials, isogenies, CM endomorphisms, and pairings, with dedicated formulas in characteristics two and three. Classical models provide proof and optimization dictionaries while all endpoints remain native. For Cd25519, the Kummer line is exactly the X25519 line, and a native Segre recoding realizes the optimized complete \(a=-1\) Edwards full-point dependency graph. We further study C a,b,d :( u 2 +u+a)( v 2 +v+b)=d, T a,d :( u 2 +u+a)( v 2 +v)=d, R τ,σ,κ :( x 2 −τ)( y 2 −σ)=κxy, Q α,β,γ : x 2 y 2 +α( x 2 + y 2 )+βxy+γ=0. For these product, one-sided twisted, reciprocal, and QRT families, we determine their genus-one geometry, finite-field forms, arithmetic, and isogenies. On each smooth QRT fibre, the Vieta--McMillan map is a fixed elliptic translation. Marking \(D\) gives the state \(P\mapsto(\kappa(P),\kappa(P+D))\), with maps realizing \(n\mapsto mn+r\). This yields logarithmic ladders and an elliptic Lucas calculus with nonlinear addition, fast-index doubling, state-division polynomials, and bridges to elliptic divisibility sequences, sigma functions, and elliptic nets. In characteristic two, every ordinary pointed elliptic curve over a perfect field admits the binary state model. Subjects: Number Theory (math.NT); Cryptography and Security (cs.CR) Cite as: arXiv:2608.01675 [math.NT]   (or arXiv:2608.01675v4 [math.NT] for this version)   https://doi.org/10.48550/arXiv.2608.01675 Focus to learn more Submission history From: Hongfeng Wu [view email] [v1] Mon, 3 Aug 2026 04:11:39 UTC (186 KB) [v2] Tue, 4 Aug 2026 10:26:52 UTC (186 KB) [v3] Wed, 5 Aug 2026 17:22:06 UTC (195 KB) [v4] Mon, 10 Aug 2026 15:56:35 UTC (376 KB) Access Paper: view license Current browse context: math.NT < prev   |   next > new | recent | 2026-08 Change to browse by: cs cs.CR math References & Citations NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)