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Faster addition and doubling on elliptic curves
twisted-20080108.rar
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<html xmlns="http://www.w3.org/1999/xhtml"> <head> <meta charset="utf-8"> <meta name="generator" content="pdf2htmlEX"> <meta http-equiv="X-UA-Compatible" content="IE=edge,chrome=1"> <link rel="stylesheet" href="https://static.pudn.com/base/css/base.min.css"> <link rel="stylesheet" href="https://static.pudn.com/base/css/fancy.min.css"> <link rel="stylesheet" href="https://static.pudn.com/prod/directory_preview_static/622bab1915da9b288b5567fd/raw.css"> <script src="https://static.pudn.com/base/js/compatibility.min.js"></script> <script src="https://static.pudn.com/base/js/pdf2htmlEX.min.js"></script> <script> try{ pdf2htmlEX.defaultViewer = new pdf2htmlEX.Viewer({}); }catch(e){} </script> <title></title> </head> <body> <div id="sidebar" style="display: none"> <div id="outline"> </div> </div> <div id="pf1" class="pf w0 h0" data-page-no="1"><div class="pc pc1 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/622bab1915da9b288b5567fd/bg1.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">Twisted<span class="_ _0"> </span>Edw<span class="_ _1"></span>ards<span class="_ _0"> </span>Curv<span class="_ _1"></span>es</div><div class="t m0 x2 h3 y2 ff2 fs1 fc0 sc0 ls0 ws0">Daniel<span class="_ _2"> </span>J.<span class="_ _2"> </span>Bernstein</div><div class="t m0 x3 h4 y3 ff3 fs2 fc0 sc0 ls0 ws0">1</div><div class="t m0 x4 h3 y2 ff2 fs1 fc0 sc0 ls0 ws0">,<span class="_ _2"> </span>P<span class="_ _1"></span>eter<span class="_ _2"> </span>Birkner</div><div class="t m0 x5 h4 y3 ff3 fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x6 h3 y2 ff2 fs1 fc0 sc0 ls0 ws0">,<span class="_ _2"> </span>T<span class="_ _3"></span>anja<span class="_ _2"> </span>Lange</div><div class="t m0 x7 h4 y3 ff3 fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x8 h3 y2 ff2 fs1 fc0 sc0 ls0 ws0">,<span class="_ _2"> </span>and<span class="_ _2"> </span>Christiane<span class="_ _2"> </span>P<span class="_ _1"></span>eters</div><div class="t m0 x9 h4 y3 ff3 fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 xa h5 y4 ff4 fs3 fc0 sc0 ls0 ws0">1</div><div class="t m0 xb h6 y5 ff5 fs4 fc0 sc0 ls0 ws0">Departmen<span class="_ _1"></span>t<span class="_ _2"> </span>of<span class="_ _4"> </span>Mathematics,<span class="_ _4"> </span>Statistics,<span class="_ _4"> </span>and<span class="_ _4"> </span>Computer<span class="_ _2"> </span>Science<span class="_ _4"> </span>(M/C<span class="_ _4"> </span>249)</div><div class="t m0 xc h6 y6 ff5 fs4 fc0 sc0 ls0 ws0">Univ<span class="_ _1"></span>ersity<span class="_ _4"> </span>of<span class="_ _4"> </span>Illinois<span class="_ _4"> </span>at<span class="_ _4"> </span>Chicago,<span class="_ _2"> </span>Chicago,<span class="_ _4"> </span>IL<span class="_ _4"> </span>60607&#8211;7045,<span class="_ _4"> </span>USA</div><div class="t m0 xd h7 y7 ff6 fs4 fc0 sc0 ls0 ws0">djb@cr.yp.to</div><div class="t m0 xe h5 y8 ff4 fs3 fc0 sc0 ls0 ws0">2</div><div class="t m0 xf h6 y9 ff5 fs4 fc0 sc0 ls0 ws0">Departmen<span class="_ _1"></span>t<span class="_ _4"> </span>of<span class="_ _2"> </span>Mathematics<span class="_ _4"> </span>and<span class="_ _4"> </span>Computer<span class="_ _4"> </span>Science</div><div class="t m0 x10 h6 ya ff5 fs4 fc0 sc0 ls0 ws0">T<span class="_ _3"></span>echnisc<span class="_ _1"></span>he<span class="_ _4"> </span>Universiteit<span class="_ _4"> </span>Eindho<span class="_ _1"></span>ven,<span class="_ _4"> </span>P<span class="_ _3"></span>.O.<span class="_ _4"> </span>Box<span class="_ _4"> </span>513,<span class="_ _4"> </span>5600<span class="_ _4"> </span>MB<span class="_ _4"> </span>Eindhov<span class="_ _1"></span>en,<span class="_ _4"> </span>Netherlands</div><div class="t m0 x11 h7 yb ff6 fs4 fc0 sc0 ls0 ws0">p.birkner@tue.nl,<span class="_ _5"> </span>tanja@hyperelliptic.org,<span class="_ _5"> </span>c.p.peters@tue.nl</div><div class="t m0 x12 h6 yc ff7 fs4 fc0 sc0 ls0 ws0">Abstract.<span class="_ _6"> </span><span class="ff5">This<span class="_ _4"> </span>pap<span class="_ _7"></span>er<span class="_ _4"> </span>introduces<span class="_ _4"> </span>&#8220;twisted<span class="_ _4"> </span>Edw<span class="_ _1"></span>ards<span class="_ _4"> </span>curves,&#8221;<span class="_ _4"> </span>a<span class="_ _4"> </span>general-</span></div><div class="t m0 x12 h6 yd ff5 fs4 fc0 sc0 ls0 ws0">ization<span class="_ _8"> </span>of<span class="_ _8"> </span>the<span class="_ _8"> </span>recen<span class="_ _1"></span>tly<span class="_ _8"> </span>introduced<span class="_ _8"> </span>Edwards<span class="_ _8"> </span>curv<span class="_ _1"></span>es;<span class="_ _8"> </span>shows<span class="_ _8"> </span>that<span class="_ _8"> </span>t<span class="_ _1"></span>wisted</div><div class="t m0 x12 h6 ye ff5 fs4 fc0 sc0 ls0 ws0">Edw<span class="_ _1"></span>ards<span class="_ _2"> </span>curves<span class="_ _4"> </span>i<span class="_"> </span>nclude<span class="_ _2"> </span>more<span class="_ _2"> </span>curves<span class="_ _2"> </span>o<span class="_ _1"></span>ver<span class="_ _4"> </span>&#64257;nite<span class="_ _2"> </span>&#64257;elds,<span class="_ _2"> </span>and<span class="_ _2"> </span>in<span class="_ _2"> </span>particular</div><div class="t m0 x12 h6 yf ff5 fs4 fc0 sc0 ls0 ws0">ev<span class="_ _1"></span>ery<span class="_ _2"> </span>elliptic<span class="_ _4"> </span>curve<span class="_ _4"> </span>in<span class="_ _4"> </span>Montgomery<span class="_ _4"> </span>form;<span class="_ _4"> </span>presents<span class="_ _4"> </span>fast<span class="_ _4"> </span>explicit<span class="_ _2"> </span>form<span class="_ _1"></span>ulas</div><div class="t m0 x12 h6 y10 ff5 fs4 fc0 sc0 ls0 ws0">for<span class="_ _2"> </span>twisted<span class="_ _2"> </span>Edwards<span class="_ _4"> </span>curves<span class="_ _2"> </span>in<span class="_ _9"> </span>pro<span class="_ _7"></span>jective<span class="_ _2"> </span>and<span class="_ _9"> </span>in<span class="_ _1"></span>verted<span class="_ _2"> </span>co<span class="_ _7"></span>ordinates;<span class="_ _2"> </span>and</div><div class="t m0 x12 h6 y11 ff5 fs4 fc0 sc0 ls0 ws0">sho<span class="_ _1"></span>ws<span class="_ _4"> </span>that<span class="_ _4"> </span>t<span class="_ _1"></span>wisted<span class="_ _4"> </span>Edwards<span class="_ _a"> </span>curv<span class="_ _1"></span>es<span class="_ _4"> </span>sav<span class="_ _1"></span>e<span class="_ _a"> </span>time<span class="_ _4"> </span>for<span class="_ _4"> </span>man<span class="_ _1"></span>y<span class="_ _4"> </span>curves<span class="_ _a"> </span>that<span class="_ _a"> </span>were</div><div class="t m0 x12 h6 y12 ff5 fs4 fc0 sc0 ls0 ws0">already<span class="_ _4"> </span>expressible<span class="_ _4"> </span>as<span class="_ _4"> </span>Edwards<span class="_ _4"> </span>curves.</div><div class="t m0 x12 h6 y13 ff7 fs4 fc0 sc0 ls0 ws0">Keyw<span class="_ _1"></span>ords:<span class="_ _b"> </span><span class="ff5">Elliptic<span class="_ _b"> </span>curv<span class="_ _1"></span>es,<span class="_ _b"> </span>Edw<span class="_ _1"></span>ards<span class="_ _b"> </span>curves,<span class="_ _8"> </span>twisted<span class="_ _8"> </span>Edwards<span class="_ _8"> </span>curves,</span></div><div class="t m0 x12 h6 y14 ff5 fs4 fc0 sc0 ls0 ws0">Mon<span class="_ _1"></span>tgomery<span class="_ _4"> </span>curves</div><div class="t m0 x13 h8 y15 ff1 fs5 fc0 sc0 ls0 ws0">1<span class="_ _c"> </span>In<span class="_ _1"></span>tro<span class="_ _7"></span>duction</div><div class="t m0 x13 h3 y16 ff2 fs1 fc0 sc0 ls0 ws0">Edw<span class="_ _1"></span>ards<span class="_ _8"> </span>in<span class="_ _8"> </span>[7],<span class="_ _8"> </span>generalizing<span class="_ _8"> </span>an<span class="_ _8"> </span>example<span class="_ _8"> </span>from<span class="_ _8"> </span>Euler<span class="_ _8"> </span>and<span class="_ _8"> </span>Gauss,<span class="_ _8"> </span>introduced<span class="_ _8"> </span>an</div><div class="t m0 x13 h3 y17 ff2 fs1 fc0 sc0 ls0 ws0">addition<span class="_ _8"> </span>law<span class="_ _8"> </span>for<span class="_ _b"> </span>the<span class="_ _8"> </span>curves<span class="_ _8"> </span><span class="ff8">x</span></div><div class="t m0 x14 h4 y18 ff3 fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x15 h3 y17 ff2 fs1 fc0 sc0 ls0 ws0">+<span class="_ _a"> </span><span class="ff8">y</span></div><div class="t m0 x16 h4 y18 ff3 fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x17 h3 y17 ff2 fs1 fc0 sc0 ls0 ws0">=<span class="_ _8"> </span><span class="ff8">c</span></div><div class="t m0 x18 h4 y18 ff3 fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x19 h3 y17 ff2 fs1 fc0 sc0 ls0 ws0">(1<span class="_ _a"> </span>+<span class="_ _a"> </span><span class="ff8">x</span></div><div class="t m0 x1a h4 y18 ff3 fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x1b h3 y17 ff8 fs1 fc0 sc0 ls0 ws0">y</div><div class="t m0 x1c h4 y18 ff3 fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x1d h3 y17 ff2 fs1 fc0 sc0 ls0 ws0">)<span class="_ _8"> </span>ov<span class="_ _1"></span>er<span class="_ _b"> </span>a<span class="_ _8"> </span>non-binary<span class="_ _b"> </span>&#64257;eld<span class="_ _8"> </span><span class="ff8">k<span class="_ _7"></span></span>.</div><div class="t m0 x13 h3 y19 ff2 fs1 fc0 sc0 ls0 ws0">Edw<span class="_ _1"></span>ards<span class="_ _8"> </span>show<span class="_ _1"></span>ed<span class="_ _8"> </span>that<span class="_ _8"> </span>ev<span class="_ _1"></span>ery<span class="_ _8"> </span>elliptic<span class="_ _8"> </span>curve<span class="_ _9"> </span>ov<span class="_ _1"></span>er<span class="_ _8"> </span><span class="ff8">k<span class="_ _b"> </span></span>can<span class="_ _8"> </span>be<span class="_ _8"> </span>expresse<span class="_"> </span>d<span class="_ _8"> </span>in<span class="_ _8"> </span>the<span class="_ _8"> </span>form</div><div class="t m0 x13 h3 y1a ff8 fs1 fc0 sc0 ls0 ws0">x</div><div class="t m0 x1e h4 y1b ff3 fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x1f h3 y1a ff2 fs1 fc0 sc0 ls0 ws0">+<span class="_ _d"> </span><span class="ff8">y</span></div><div class="t m0 x12 h4 y1b ff3 fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x20 h3 y1a ff2 fs1 fc0 sc0 ls0 ws0">=<span class="_ _2"> </span><span class="ff8">c</span></div><div class="t m0 x21 h4 y1b ff3 fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x22 h3 y1a ff2 fs1 fc0 sc0 ls0 ws0">(1<span class="_ _d"> </span>+<span class="_ _d"> </span><span class="ff8">x</span></div><div class="t m0 x23 h4 y1b ff3 fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x24 h3 y1a ff8 fs1 fc0 sc0 ls0 ws0">y</div><div class="t m0 x3 h4 y1b ff3 fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x4 h3 y1a ff2 fs1 fc0 sc0 ls0 ws0">)<span class="_ _9"> </span>if<span class="_ _8"> </span><span class="ff8">k<span class="_ _8"> </span></span>is<span class="_ _9"> </span>algebraically<span class="_ _8"> </span>closed.<span class="_ _9"> </span>How<span class="_ _1"></span>ev<span class="_ _1"></span>er,<span class="_ _8"> </span>o<span class="_ _1"></span>ver<span class="_ _9"> </span>a<span class="_ _9"> </span>&#64257;nite<span class="_ _8"> </span>&#64257;eld,</div><div class="t m0 x13 h3 y1c ff2 fs1 fc0 sc0 ls0 ws0">only<span class="_ _2"> </span>a<span class="_ _2"> </span>small<span class="_ _2"> </span>fraction<span class="_ _2"> </span>of<span class="_ _2"> </span>elliptic<span class="_ _2"> </span>curv<span class="_ _1"></span>es<span class="_ _2"> </span>can<span class="_ _2"> </span>b<span class="_ _7"></span>e<span class="_ _2"> </span>expressed<span class="_ _2"> </span>in<span class="_ _2"> </span>this<span class="_ _2"> </span>form.</div><div class="t m0 x25 h3 y1d ff2 fs1 fc0 sc0 ls0 ws0">Bernstein<span class="_ _2"> </span>and<span class="_ _9"> </span>Lange<span class="_ _2"> </span>in<span class="_ _9"> </span>[4]<span class="_ _9"> </span>presen<span class="_ _1"></span>ted<span class="_ _9"> </span>fast<span class="_ _2"> </span>explicit<span class="_ _9"> </span>formulas<span class="_ _2"> </span>for<span class="_ _2"> </span>addition<span class="_ _2"> </span>and</div><div class="t m0 x13 h3 y1e ff2 fs1 fc0 sc0 ls0 ws0">doubling<span class="_ _b"> </span>in<span class="_ _b"> </span>co<span class="_ _7"></span>ordinates<span class="_ _b"> </span>(<span class="ff8">X<span class="_ _0"> </span></span>:<span class="_ _b"> </span><span class="ff8">Y<span class="_ _e"> </span></span>:<span class="_ _b"> </span><span class="ff8">Z<span class="_ _f"></span></span>)<span class="_ _b"> </span>represen<span class="_ _1"></span>ting<span class="_ _b"> </span>(<span class="ff8">x,<span class="_ _10"> </span>y<span class="_ _7"></span></span>)<span class="_ _b"> </span>=<span class="_ _5"> </span>(<span class="ff8">X/<span class="_ _3"></span>Z<span class="_ _7"></span>,<span class="_ _10"> </span>Y<span class="_ _f"></span>/<span class="_ _1"></span>Z<span class="_ _f"></span><span class="ff2">)<span class="_ _b"> </span>on<span class="_ _b"> </span>an</span></span></div><div class="t m0 x13 h3 y1f ff2 fs1 fc0 sc0 ls0 ws0">Edw<span class="_ _1"></span>ards<span class="_ _b"> </span>curve,<span class="_ _8"> </span>and<span class="_ _b"> </span>show<span class="_ _1"></span>ed<span class="_ _b"> </span>that<span class="_ _b"> </span>these<span class="_ _b"> </span>explicit<span class="_ _b"> </span>formulas<span class="_ _8"> </span>sav<span class="_ _1"></span>e<span class="_ _b"> </span>time<span class="_ _b"> </span>in<span class="_ _b"> </span>elliptic-</div><div class="t m0 x13 h3 y20 ff2 fs1 fc0 sc0 ls0 ws0">curv<span class="_ _1"></span>e<span class="_ _b"> </span>cryptograph<span class="_ _1"></span>y<span class="_ _3"></span>.<span class="_ _8"> </span>Bernstein<span class="_ _b"> </span>and<span class="_ _8"> </span>Lange<span class="_ _b"> </span>also<span class="_ _8"> </span>generalized<span class="_ _b"> </span>the<span class="_ _8"> </span>addition<span class="_ _8"> </span>law<span class="_ _8"> </span>to</div><div class="t m0 x13 h3 y21 ff2 fs1 fc0 sc0 ls0 ws0">the<span class="_ _4"> </span>curves<span class="_ _a"> </span><span class="ff8">x</span></div><div class="t m0 x26 h4 y22 ff3 fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x27 h3 y21 ff2 fs1 fc0 sc0 ls0 ws0">+<span class="_ _10"> </span><span class="ff8">y</span></div><div class="t m0 xf h4 y22 ff3 fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x28 h3 y21 ff2 fs1 fc0 sc0 ls0 ws0">=<span class="_ _a"> </span><span class="ff8">c</span></div><div class="t m0 x29 h4 y22 ff3 fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x2a h3 y21 ff2 fs1 fc0 sc0 ls0 ws0">(1<span class="_ _10"> </span>+<span class="_ _11"> </span><span class="ff8">dx</span></div><div class="t m0 x2b h4 y22 ff3 fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x2c h3 y21 ff8 fs1 fc0 sc0 ls0 ws0">y</div><div class="t m0 x2d h4 y22 ff3 fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x2e h3 y21 ff2 fs1 fc0 sc0 ls0 ws0">).<span class="_ _4"> </span>This<span class="_ _4"> </span>shap<span class="_ _7"></span>e<span class="_ _4"> </span>cov<span class="_ _1"></span>ers<span class="_ _4"> </span>considerably<span class="_ _4"> </span>more<span class="_ _4"> </span>elliptic</div><div class="t m0 x13 h3 y23 ff2 fs1 fc0 sc0 ls0 ws0">curv<span class="_ _1"></span>es<span class="_ _a"> </span>o<span class="_ _1"></span>ver<span class="_ _d"> </span>a<span class="_ _d"> </span>&#64257;nite<span class="_ _a"> </span>&#64257;eld<span class="_ _d"> </span>than<span class="_ _a"> </span><span class="ff8">x</span></div><div class="t m0 x2f h4 y24 ff3 fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x15 h3 y23 ff2 fs1 fc0 sc0 ls0 ws0">+<span class="_ _f"></span><span class="ff8">y</span></div><div class="t m0 x30 h4 y24 ff3 fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x31 h3 y23 ff2 fs1 fc0 sc0 ls0 ws0">=<span class="_ _a"> </span><span class="ff8">c</span></div><div class="t m0 x32 h4 y24 ff3 fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x18 h3 y23 ff2 fs1 fc0 sc0 ls0 ws0">(1<span class="_ _f"></span>+<span class="_ _f"></span><span class="ff8">x</span></div><div class="t m0 x33 h4 y24 ff3 fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x34 h3 y23 ff8 fs1 fc0 sc0 ls0 ws0">y</div><div class="t m0 x35 h4 y24 ff3 fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x36 h3 y23 ff2 fs1 fc0 sc0 ls0 ws0">).<span class="_ _d"> </span>All<span class="_ _a"> </span>curv<span class="_ _1"></span>es<span class="_ _a"> </span>in<span class="_ _d"> </span>the<span class="_ _a"> </span>generalized</div><div class="t m0 x13 h3 y25 ff2 fs1 fc0 sc0 ls0 ws0">form<span class="_ _2"> </span>are<span class="_ _2"> </span>isomorphic<span class="_ _2"> </span>to<span class="_ _2"> </span>curv<span class="_ _1"></span>es<span class="_ _2"> </span><span class="ff8">x</span></div><div class="t m0 x37 h4 y26 ff3 fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x2e h3 y25 ff2 fs1 fc0 sc0 ls0 ws0">+<span class="_ _12"> </span><span class="ff8">y</span></div><div class="t m0 x38 h4 y26 ff3 fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x39 h3 y25 ff2 fs1 fc0 sc0 ls0 ws0">=<span class="_ _a"> </span>1<span class="_ _12"> </span>+<span class="_ _12"> </span><span class="ff8">dx</span></div><div class="t m0 x3a h4 y26 ff3 fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x35 h3 y25 ff8 fs1 fc0 sc0 ls0 ws0">y</div><div class="t m0 x3b h4 y26 ff3 fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x3c h3 y25 ff2 fs1 fc0 sc0 ls0 ws0">.</div><div class="t m0 x25 h3 y27 ff2 fs1 fc0 sc0 ls0 ws0">In<span class="_ _b"> </span>this<span class="_ _8"> </span>pap<span class="_ _7"></span>er,<span class="_ _b"> </span>w<span class="_ _1"></span>e<span class="_ _b"> </span>further<span class="_ _8"> </span>generalize<span class="_ _b"> </span>the<span class="_ _b"> </span>Edw<span class="_ _1"></span>ards<span class="_ _b"> </span>addition<span class="_ _8"> </span>law<span class="_ _8"> </span>to<span class="_ _b"> </span>cov<span class="_ _1"></span>er<span class="_ _8"> </span>all</div><div class="t m0 x13 h3 y28 ff2 fs1 fc0 sc0 ls0 ws0">curv<span class="_ _1"></span>es<span class="_ _a"> </span><span class="ff8">ax</span></div><div class="t m0 x3d h4 y29 ff3 fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x3e h3 y28 ff2 fs1 fc0 sc0 ls0 ws0">+<span class="_ _f"></span><span class="ff8">y</span></div><div class="t m0 x3f h4 y29 ff3 fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x40 h3 y28 ff2 fs1 fc0 sc0 ls0 ws0">=<span class="_ _a"> </span>1<span class="_ _f"></span>+<span class="_ _13"></span><span class="ff8">dx</span></div><div class="t m0 x41 h4 y29 ff3 fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x42 h3 y28 ff8 fs1 fc0 sc0 ls0 ws0">y</div><div class="t m0 x43 h4 y29 ff3 fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x44 h3 y28 ff2 fs1 fc0 sc0 ls0 ws0">.<span class="_ _d"> </span>Our<span class="_ _a"> </span>fast<span class="_ _a"> </span>explicit<span class="_ _a"> </span>formulas<span class="_ _d"> </span>for<span class="_ _d"> </span>addition<span class="_ _a"> </span>and<span class="_ _a"> </span>doubling</div><div class="t m0 x13 h3 y2a ff2 fs1 fc0 sc0 ls0 ws0">are<span class="_ _2"> </span>almost<span class="_ _2"> </span>as<span class="_ _2"> </span>fast<span class="_ _2"> </span>in<span class="_ _2"> </span>the<span class="_ _9"> </span>general<span class="_ _2"> </span>case<span class="_ _2"> </span>as<span class="_ _2"> </span>they<span class="_ _2"> </span>are<span class="_ _2"> </span>for<span class="_ _2"> </span>the<span class="_ _9"> </span>sp<span class="_ _7"></span>ecial<span class="_ _2"> </span>case<span class="_ _2"> </span><span class="ff8">a<span class="_ _a"> </span></span>=<span class="_ _4"> </span>1.<span class="_ _2"> </span>W<span class="_ _3"></span>e</div><div class="t m0 x10 h5 y2b ff4 fs3 fc0 sc0 ls0 ws0">*</div><div class="t m0 x45 h6 y2c ff5 fs4 fc0 sc0 ls0 ws0">P<span class="_ _1"></span>ermanent<span class="_ _a"> </span>ID<span class="_ _a"> </span>of<span class="_ _a"> </span>this<span class="_ _4"> </span>do<span class="_ _7"></span>cumen<span class="_ _1"></span>t:<span class="_ _4"> </span><span class="ff6">c798703ae3ecfdc375112f19dd0787e4</span>.<span class="_ _a"> </span>Date<span class="_ _a"> </span>of<span class="_ _4"> </span>this</div><div class="t m0 x45 h6 y2d ff5 fs4 fc0 sc0 ls0 ws0">do<span class="_"> </span>cum<span class="_"> </span>ent:<span class="_ _d"> </span>2008.01.08.<span class="_ _a"> </span>This<span class="_ _d"> </span>work<span class="_ _d"> </span>has<span class="_ _d"> </span>b<span class="_ _7"></span>een<span class="_ _a"> </span>supp<span class="_ _7"></span>orted<span class="_ _d"> </span>in<span class="_ _a"> </span>part<span class="_ _a"> </span>b<span class="_ _1"></span>y<span class="_ _a"> </span>the<span class="_ _d"> </span>Europ<span class="_ _7"></span>ean<span class="_ _a"> </span>Com-</div><div class="t m0 x45 h6 y2e ff5 fs4 fc0 sc0 ls0 ws0">mission<span class="_ _9"> </span>through<span class="_ _9"> </span>the<span class="_ _9"> </span>IST<span class="_ _8"> </span>Programme<span class="_ _9"> </span>under<span class="_ _9"> </span>Contract<span class="_ _2"> </span>IST&#8211;2002&#8211;507932<span class="_ _9"> </span>ECR<span class="_ _1"></span>YPT,</div><div class="t m0 x45 h6 y2f ff5 fs4 fc0 sc0 ls0 ws0">and<span class="_ _b"> </span>in<span class="_ _8"> </span>part<span class="_ _b"> </span>b<span class="_ _1"></span>y<span class="_ _b"> </span>the<span class="_ _b"> </span>National<span class="_ _8"> </span>Science<span class="_ _b"> </span>F<span class="_ _3"></span>oundation<span class="_ _b"> </span>under<span class="_ _8"> </span>grant<span class="_ _8"> </span>ITR&#8211;0716498.<span class="_ _b"> </span>This</div><div class="t m0 x45 h6 y30 ff5 fs4 fc0 sc0 ls0 ws0">w<span class="_ _1"></span>ork<span class="_ _2"> </span>w<span class="_ _1"></span>as<span class="_ _4"> </span>carried<span class="_ _2"> </span>out<span class="_ _4"> </span>while<span class="_ _4"> </span>the<span class="_ _4"> </span>authors<span class="_ _4"> </span>were<span class="_ _4"> </span>visiting<span class="_ _4"> </span>INRIA<span class="_ _4"> </span>Lorraine<span class="_ _2"> </span>(LORIA).</div></div><div class="pi" data-data='{"ctm":[1.611850,0.000000,0.000000,1.611850,0.000000,0.000000]}'></div></div> </body> </html>
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