<div dir="ltr"><div class="gmail_extra">Here's a simplified way to see the equivalence between ellipses and Edwards curves.  First, here's two forms of the group law, which are equivalent, though the Edwards form is faster:</div><div class="gmail_extra"><br></div><div class="gmail_extra">Edwards form:</div><div class="gmail_extra"><br></div><div class="gmail_extra"><div style="font-size:12.8000001907349px">    sn3 = (sn1(cn2/dn2) + sn2)/(1 + d*sn1(cn1/sn1)sn2(cn2/dn2)</div><div style="font-size:12.8000001907349px">    cn3/dn3 = ((cn1/dn1)(cn2/dn2) - sn1*sn2)/(1 - d*sn1(cn1/dn1)sn2(cn2/dn2))</div><div class="" style="font-size:12.8000001907349px"><div id=":s3" class="" tabindex="0"><img class="" src="https://ssl.gstatic.com/ui/v1/icons/mail/images/cleardot.gif"></div><div id=":s3" class="" tabindex="0">Old form:</div><div id=":s3" class="" tabindex="0"><br></div><div id=":s3" class="" tabindex="0"><span style="font-size:12.8000001907349px">    cn3 = (cn1*cn2 - sn1*sn2*dn1*dn2) / (</span><span style="font-size:12.8000001907349px">1 - d*sn1^2*sn2^2)</span><br></div><div id=":s3" class="" tabindex="0">    sn3 = (sn1*cn2*dn2 + sn2*cn1*dn1) / <span style="font-size:12.8000001907349px">(</span><span style="font-size:12.8000001907349px">1 - d*sn1^2*sn2^2)</span></div><div id=":s3" class="" tabindex="0">    dn3 = (dn1*dn2 - d*sn1*sn2*cn1*cn2) / <span style="font-size:12.8000001907349px">(</span><span style="font-size:12.8000001907349px">1 - d*sn1^2*sn2^2)</span></div><div id=":s3" class="" tabindex="0"><span style="font-size:12.8000001907349px"><br></span></div><div id=":s3" class="" tabindex="0"><span style="font-size:12.8000001907349px">Given either of these equivalent addition laws, we can compute a point on the Edwards curve as:</span></div><div id=":s3" class="" tabindex="0"><span style="font-size:12.8000001907349px"><br></span></div><div id=":s3" class="" tabindex="0"><span style="font-size:12.8000001907349px">    x = cn/dn</span></div><div id=":s3" class="" tabindex="0"><span style="font-size:12.8000001907349px">    y = sn</span></div><div id=":s3" class="" tabindex="0"><span style="font-size:12.8000001907349px"><br></span></div><div id=":s3" class="" tabindex="0">and we can compute teh point on the ellipse as:</div><div id=":s3" class="" tabindex="0"><br></div><div id=":s3" class="" tabindex="0">    x = cn</div><div id=":s3" class="" tabindex="0">    y = sn/b</div><div id=":s3" class="" tabindex="0"><br></div><div id=":s3" class="" tabindex="0">The Edwards equation is:</div><div id=":s3" class="" tabindex="0"><br></div><div id=":s3" class="" tabindex="0">    x^2 + y^2 = 1 + d*x^2*y^2</div><div id=":s3" class="" tabindex="0"><br></div><div id=":s3" class="" tabindex="0">and the ellipse equation is:</div><div id=":s3" class="" tabindex="0"><br></div><div id=":s3" class="" tabindex="0">    x^2 + y^2/b = 1</div><div id=":s3" class="" tabindex="0"><br></div><div id=":s3" class="" tabindex="0">where:</div><div id=":s3" class="" tabindex="0"><br></div><div id=":s3" class="" tabindex="0">    b^2 = 1/(1-d)</div><div id=":s3" class="" tabindex="0"><br></div><div id=":s3" class="" tabindex="0">Bill</div></div></div></div>