椭圆曲线的新特征一致模型——理论、算术与应用
A New Characteristic-Uniform Model for Elliptic Curves -- Theory, Arithmetic, and Applications
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中文总结 AI 辅助
本文提出椭圆曲线的新特征一致模型\boldsymbol{\textit{C}}_d曲线，推导其各类算术公式与相关理论，特例化显示其兼容Curve25519，是椭圆曲线理论与应用的高效模型。
中文摘要 AI 辅助
我们开发了椭圆曲线族\boldsymbol{\textit{C}}_d:\boldsymbol{\textit{(u}^2\boldsymbol{+u)(v}^2\boldsymbol{+v)=d}}的特征一致理论，该曲线称为\boldsymbol{\textit{C}}_d曲线。其光滑\boldsymbol{(2,2)}完备化、标记边界及二面体对称性在所有特征下提供了共同几何框架。在奇特征下，我们给出与Edwards、Montgomery和Weierstrass模型的显式对应关系；而在特征2下，同一方程保留其内在的Artin–Schreier结构。我们直接在\boldsymbol{\textit{C}}_d上推导仿射与射影加法及倍乘公式、完整加法律图集、微分加法、Kummer算术、折半、三倍及标量乘法程序，包括特征2和3下的专用公式。我们还研究了自同态、除多项式、同构类、有限域均值、保模型同态、Vélu型构造及Tate和Weil配对。对\boldsymbol{\textit{C}}_{d25519}的详细特例化表明，该模型支持高效的全点和Kummer线算术，同时仍与Curve25519直接兼容。最后，我们将\boldsymbol{\textit{C}}_d置于更广泛的对称族\boldsymbol{\textit{(u}^2\boldsymbol{+u+a)(v}^2\boldsymbol{+v+b)=d}}中。这些结果表明，\boldsymbol{\textit{C}}_d曲线是椭圆曲线理论与应用中自包含且计算高效的模型。
英文摘要
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 \(κ_d(u,v)=(u+1)\). 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 \[ \begin{gathered} \mathcal C_{a,b,d}:(u^2+u+a)(v^2+v+b)=d,\quad \mathcal T_{a,d}:(u^2+u+a)(v^2+v)=d,\\ \mathcal R_{τ,σ,κ}:(x^2-τ)(y^2-σ)=κxy,\quad \mathcal Q_{α,β,γ}:x^2y^2+α(x^2+y^2)+βxy+γ=0. \end{gathered} \] 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 \(P\mapsto(κ(P),κ(P+D))\), 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.
