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

Abstract

We develop a characteristic-uniform arithmetic theory for \[ \mathcal C_d:\quad (u^2+u)(v^2+v)=d. \] 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 \[ \begin{gathered} \mathcal C_{a,b,d}:(u^2+u+a)(v^2+v+b)=d,\quad \mathcal T_{a,d}:(u^2+u+a)(v^2+v)=d,\\ \mathcal R_{\tau,\sigma,\kappa}:(x^2-\tau)(y^2-\sigma)=\kappa xy,\quad \mathcal Q_{\alpha,\beta,\gamma}:x^2y^2+\alpha(x^2+y^2)+\beta xy+\gamma=0. \end{gathered} \] 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. We further develop the \(\mathcal C_d\) platform for isogeny-based cryptography.