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<p class="MsoNormal">Hello,<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">Sorry for potentially spamming your email reflector. I’m an embedded wireless programmer in need of a very simple Password Authenticated Key Exchange(PAKE). I believe I have created something similar to SPEKE but that works considerable
better over elliptic curves. I would like some help proving that it is secure.<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal"><u>Set up<o:p></o:p></u></p>
<p class="MsoNormal">The Protocol begins with an elliptic curve over F<sub>2m</sub> with parameters T = (m, f(x), a, b, G, n, h) and G<sub>2</sub> as second generator of that group such that v, where vG = G<sub>2</sub> is unknown. Also given P an arbitrary
element of the group generated by G and aP finding a is hard. The curves 163k1 and 283k1 are such curves with these properties. T and G<sub>2</sub> are fixed and known by all implementers of the algorithm.<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">Convention: Capitals will be points on the curve and lower case letters will be integers.<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal"><u>Algorithm<o:p></o:p></u></p>
<p class="MsoListParagraph" style="text-indent:-18.0pt;mso-list:l0 level1 lfo1"><![if !supportLists]><span lang="EN-AU"><span style="mso-list:Ignore">1.<span style="font:7.0pt "Times New Roman"">
</span></span></span><![endif]><span lang="EN-AU">Let Alice and Bob have a shared password s, s is a “smallish” non-negative integer.
<o:p></o:p></span></p>
<p class="MsoListParagraph" style="text-indent:-18.0pt;mso-list:l0 level1 lfo1"><![if !supportLists]><span lang="EN-AU"><span style="mso-list:Ignore">2.<span style="font:7.0pt "Times New Roman"">
</span></span></span><![endif]><span lang="EN-AU">Both Alice and Bob choose a number between 1 and n-2. Let these numbers be a and b. Alice sends the point P<sub>1</sub>=aG + sG<sub>2</sub> to Bob and Bob sends the point P<sub>2</sub> = bG+ sG<sub>2</sub>
to Alice.<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:18.0pt">3a. Alice verifies P<sub>2</sub> is a generator of the group and then computes a(P<sub>2</sub> - sG<sub>2</sub>) = a(bG+ sG<sub>2</sub> - sG<sub>2</sub>) = abG<o:p></o:p></p>
<p class="MsoNormal" style="margin-left:18.0pt">3b. Bob verifies P<sub>1</sub> is a generator of the group and then computes b(P<sub>1</sub> - sG<sub>2</sub>) = b(aG+ sG<sub>2</sub> - sG<sub>2</sub>) = abG<o:p></o:p></p>
<p class="MsoNormal" style="margin-left:18.0pt">4. Alice and Bob verify that they both know the new shared secret abG.<o:p></o:p></p>
<p class="MsoNormal"><span style="font-size:12.0pt;font-family:"Times New Roman","serif""><o:p> </o:p></span></p>
<p class="MsoNormal">If Alice and Bob fail to agree on the new shared secret, abG, they know something has gone wrong.<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal"><span lang="EN-US" style="font-size:10.5pt;font-family:"Arial","sans-serif";color:gray">..................</span><span lang="EN-US" style="font-size:12.0pt;font-family:"Times New Roman","serif""><o:p></o:p></span></p>
<p class="MsoNormal" style="margin-top:6.0pt"><span lang="EN-US" style="font-size:10.5pt;font-family:"Arial","sans-serif";color:gray">Jonathan Cressman</span><span lang="EN-US"><o:p></o:p></span></p>
<p class="MsoNormal" style="margin-top:6.0pt"><span lang="EN-US" style="font-size:8.5pt;font-family:"Arial","sans-serif";color:gray">Firmware Developer</span><span lang="EN-US"><o:p></o:p></span></p>
<p class="MsoNormal" style="margin-bottom:12.0pt"><span style="font-size:8.5pt;font-family:"Arial","sans-serif";color:gray"><img width="293" height="45" id="Picture_x0020_1" src="cid:image001.gif@01CFF821.50C43980" alt="cid:image001.gif@01C91335.F4F940E0"></span><span lang="EN-US" style="font-size:8.5pt;font-family:"Arial","sans-serif";color:gray"><br>
</span><span lang="EN-US" style="font-size:9.0pt;font-family:"Arial","sans-serif";color:gray">Energate Inc. 2379 Holly Lane, Suite 200, Ottawa, Ontario, Canada K1V 7P2<br>
T: 613-482-7928 x226 F: 613-288-0816 <u><a href="http://www.energate.ca/">http://www.energateinc.com</a></u></span><o:p></o:p></p>
<p class="MsoPlainText"><o:p> </o:p></p>
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