<html xmlns:v="urn:schemas-microsoft-com:vml" xmlns:o="urn:schemas-microsoft-com:office:office" xmlns:w="urn:schemas-microsoft-com:office:word" xmlns:m="http://schemas.microsoft.com/office/2004/12/omml" xmlns="http://www.w3.org/TR/REC-html40">
<head>
<meta http-equiv=Content-Type content="text/html; charset=us-ascii">
<meta name=Generator content="Microsoft Word 12 (filtered medium)">
<style>
<!--
/* Font Definitions */
@font-face
{font-family:"Cambria Math";
panose-1:2 4 5 3 5 4 6 3 2 4;}
@font-face
{font-family:Calibri;
panose-1:2 15 5 2 2 2 4 3 2 4;}
@font-face
{font-family:Tahoma;
panose-1:2 11 6 4 3 5 4 4 2 4;}
@font-face
{font-family:Verdana;
panose-1:2 11 6 4 3 5 4 4 2 4;}
/* Style Definitions */
p.MsoNormal, li.MsoNormal, div.MsoNormal
{margin:0cm;
margin-bottom:.0001pt;
font-size:10.0pt;
font-family:"Verdana","sans-serif";}
a:link, span.MsoHyperlink
{mso-style-priority:99;
color:blue;
text-decoration:underline;}
a:visited, span.MsoHyperlinkFollowed
{mso-style-priority:99;
color:purple;
text-decoration:underline;}
p.MsoListParagraph, li.MsoListParagraph, div.MsoListParagraph
{mso-style-priority:34;
margin-top:0cm;
margin-right:0cm;
margin-bottom:0cm;
margin-left:36.0pt;
margin-bottom:.0001pt;
font-size:10.0pt;
font-family:"Verdana","sans-serif";}
span.apple-style-span
{mso-style-name:apple-style-span;}
span.EmailStyle18
{mso-style-type:personal-reply;
font-family:"Calibri","sans-serif";
color:#1F497D;}
.MsoChpDefault
{mso-style-type:export-only;
font-size:10.0pt;}
@page WordSection1
{size:612.0pt 792.0pt;
margin:72.0pt 72.0pt 72.0pt 72.0pt;}
div.WordSection1
{page:WordSection1;}
-->
</style>
<!--[if gte mso 9]><xml>
<o:shapedefaults v:ext="edit" spidmax="1026" />
</xml><![endif]--><!--[if gte mso 9]><xml>
<o:shapelayout v:ext="edit">
<o:idmap v:ext="edit" data="1" />
</o:shapelayout></xml><![endif]-->
</head>
<body lang=EN-GB link=blue vlink=purple style='word-wrap: break-word;
-webkit-nbsp-mode: space;-webkit-line-break: after-white-space'>
<div class=WordSection1>
<p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri","sans-serif";
color:#1F497D'>Sorry about the top-post reply, but Outlook is playing me up a
bit…<o:p></o:p></span></p>
<p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri","sans-serif";
color:#1F497D'><o:p> </o:p></span></p>
<p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri","sans-serif";
color:#1F497D'>Excellent, many thanks Ian.<o:p></o:p></span></p>
<p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri","sans-serif";
color:#1F497D'><o:p> </o:p></span></p>
<p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri","sans-serif";
color:#1F497D'>So again to just push this for us non-mathematicians, the
various articles appearing that are talking in terms of this having an impact
on RSA-type crypto are wrong (as my post was) on two grounds:<o:p></o:p></span></p>
<p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri","sans-serif";
color:#1F497D'><o:p> </o:p></span></p>
<p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri","sans-serif";
color:#1F497D'>1) It is not yet certain that crypto is not P.<o:p></o:p></span></p>
<p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri","sans-serif";
color:#1F497D'><o:p> </o:p></span></p>
<p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri","sans-serif";
color:#1F497D'>2) It is not certain that even if it is not P, it is actually
permanently “hard” as our understanding of the maths evolves.<o:p></o:p></span></p>
<p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri","sans-serif";
color:#1F497D'><o:p> </o:p></span></p>
<p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri","sans-serif";
color:#1F497D'>Am I right still in thinking that even if it is permanently hard
as a deterministic problem, this would not in any way preclude a probabilistic
one?<o:p></o:p></span></p>
<p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri","sans-serif";
color:#1F497D'><o:p> </o:p></span></p>
<p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri","sans-serif";
color:#1F497D'>D<o:p></o:p></span></p>
<p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri","sans-serif";
color:#1F497D'><o:p> </o:p></span></p>
<p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri","sans-serif";
color:#1F497D'><o:p> </o:p></span></p>
<p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri","sans-serif";
color:#1F497D'><o:p> </o:p></span></p>
<p class=MsoNormal><a name="_MailEndCompose"><span style='font-size:11.0pt;
font-family:"Calibri","sans-serif";color:#1F497D'><o:p> </o:p></span></a></p>
<div style='border:none;border-left:solid blue 1.5pt;padding:0cm 0cm 0cm 4.0pt'>
<div>
<div style='border:none;border-top:solid #B5C4DF 1.0pt;padding:3.0pt 0cm 0cm 0cm'>
<p class=MsoNormal><b><span lang=EN-US style='font-family:"Tahoma","sans-serif"'>From:</span></b><span
lang=EN-US style='font-family:"Tahoma","sans-serif"'> ukcrypto-bounces@chiark.greenend.org.uk
[mailto:ukcrypto-bounces@chiark.greenend.org.uk] <b>On Behalf Of </b>Ian Batten<br>
<b>Sent:</b> 11 August 2010 22:52<br>
<b>To:</b> UK Cryptography Policy Discussion Group<br>
<b>Subject:</b> Re: HP researcher claims proof that P != NP<o:p></o:p></span></p>
</div>
</div>
<p class=MsoNormal><o:p> </o:p></p>
<p class=MsoNormal><o:p> </o:p></p>
<div>
<div>
<p class=MsoNormal>On 11 Aug 2010, at 21:59, David Biggins wrote:<o:p></o:p></p>
</div>
<p class=MsoNormal><br>
<br>
<o:p></o:p></p>
<p class=MsoNormal><span class=apple-style-span><span style='font-size:11.5pt;
font-family:"Arial","sans-serif";color:black'>So if factoring a large number
that is the product of two primes, to get back the factors, is a problem
that is NP</span></span><o:p></o:p></p>
</div>
<p class=MsoNormal><o:p> </o:p></p>
<div>
<p class=MsoNormal><o:p> </o:p></p>
</div>
<div>
<p class=MsoNormal>Factorisation (or its decision equivalent, more precisely)
is in NP and co-NP, but it's not known if it's in P. It almost certainly
isn't NP-complete, which means that if tomorrow someone were to produce a P
algorithm for factorization, it wouldn't say anything about P=NP, unless
factorization were proved to be NP-complete, which seems unlikely. <o:p></o:p></p>
</div>
<div>
<p class=MsoNormal><o:p> </o:p></p>
</div>
<div>
<p class=MsoNormal>But this is all slightly incidental, because just because
something is in NP doesn't mean it's hard in a practical sense, rather than a
theoretical sense. It's perfectly possible to have tractable solutions
for practical sized problems in harder complexity classes than P whilst not
having tractable solutions for equivalently sized P problems. If you can
verify in O(n) but only solve in O(n^100) it's still in P, but is probably very
hard in reality; if you can verify in O(n) but solve in O((1+epsilon)^n) for a
suitably small n we're in NP, but it may in practical terms be a lot easier to
deal with. There are bigger threats to factorization as the basis for
security than the (highly unlikely) proof that P=NP, and even if P were to be
found to equal NP that wouldn't actually provide P-time algorithms for
factorization. Conversely, a very efficient factorization algorithm
wouldn't say much about P=NP.<o:p></o:p></p>
</div>
<div>
<p class=MsoNormal><o:p> </o:p></p>
</div>
<div>
<p class=MsoNormal>ian<o:p></o:p></p>
</div>
<div>
<p class=MsoNormal><o:p> </o:p></p>
</div>
</div>
</div>
</body>
</html>