1% Illustrative diagram (verified constants; layout is schematic): algebraic structure of secp256k1.2\begin{tikzpicture}[font=\small, >=Latex, node distance=0.9cm,3 box/.style={draw, rounded corners=2pt, fill=gray!6, align=center, inner sep=4pt},4 key/.style={draw, rounded corners=2pt, fill=blue!8, align=center, inner sep=4pt}]5 \node[key] (E) {$E/\F_p:\ y^2 = x^3 + 7$,\quad $j(E)=0$,\quad $\#E(\F_p)=n$ prime,\quad $h=1$};6 \node[box, below=of E, xshift=-4.2cm] (cm) {CM by $\Z[\omega]$, $\omega^2+\omega+1=0$\\ $\End(E)=\Z[\omega]$, $\Aut(E)=\langle -\phi\rangle\cong\Z/6$\\ $4p = t^2 + 3f^2$, $n = p+1-t$};7 \node[box, below=of E, xshift=4.2cm] (phi) {$\phi(x,y)=(\beta x,\,y)$, $\beta^3\equiv 1\ (p)$\\ $\phi(P)=[\lambda]P$, $\lambda^3\equiv 1\ (n)$\\ $\lambda^2+\lambda+1\equiv 0\ (n)$};8 \node[box, below=of cm] (orbit) {orbits $\{\pm P,\pm\phi P,\pm\phi^2 P\}$ of size 6\\ $x$-coordinates: $\{x,\beta x,\beta^2 x\}$\\ $\Rightarrow$ rho speed-up $\le\sqrt{6}$};9 \node[box, below=of phi] (glv) {GLV lattice $L=\{(k_1,k_2):k_1+k_2\lambda\equiv 0\ (n)\}$\\ $=(\bar\pi-1)\Z[\omega]$, basis $(a_1,b_1)=\big(\tfrac{t-f}{2}-1,\,-f\big)$,\\ $(a_2,b_2)=(a_1-b_1,\,a_1)$, $\ a_1^2-a_1b_1+b_1^2=n$};10 \node[box, below=of orbit, xshift=4.2cm] (sec) {$n\ne p$ (not anomalous)\quad embedding degree $> 10^{6}$ (no MOV)\\ twist order $=3^2\cdot 13^2\cdot 3319\cdot 22639\cdot q_{220}$\quad no $x=0$ points\\ special prime $p=2^{256}-2^{32}-977$: fast reduction only};11 \draw[->] (E) -- (cm); \draw[->] (E) -- (phi); \draw[->] (cm) -- (orbit); \draw[->] (phi) -- (glv); \draw[->] (phi) -- (orbit);12 \draw[->] (orbit) -- (sec); \draw[->] (glv) -- (sec);13\end{tikzpicture}14