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<td><img src="/images/white.gif" width="5" height="1" alt="" border="0"></td><!-- end of ITK left NAV --><!-- begin main content --><td width="100%" valign="top" align="left"><!-- END SUB HEADER --><!--Begin Content Column --><FONT FACE="Arial,Helvetica" SIZE="-1">To access the contents, click the chapter and section titles.</FONT><P><B>Applied Cryptography, Second Edition: Protocols, Algorthms, and Source Code in C (cloth)</B><FONT SIZE="-1"><BR><I>(Publisher: John Wiley & Sons, Inc.)</I><BR>Author(s): Bruce Schneier<BR>ISBN: 0471128457<BR>Publication Date: 01/01/96</FONT><P><form name="Search" method="GET" action="http://search.earthweb.com/search97/search_redir.cgi"><INPUT TYPE="hidden" NAME="Action" VALUE="Search"><INPUT TYPE="hidden" NAME="SearchPage" VALUE="http://search.earthweb.com/search97/samples/forms/srchdemo.htm"><INPUT TYPE="hidden" NAME="Collection" VALUE="ITK"><INPUT TYPE="hidden" NAME="ResultTemplate" VALUE="itk-full.hts"><INPUT TYPE="hidden" NAME="ViewTemplate" VALUE="view.hts"><font face="arial, helvetica" size=2><b>Search this book:</b></font><br><INPUT NAME="queryText" size=50 VALUE=""> <input type="submit" name="submitbutton" value="Go!"><INPUT type=hidden NAME="section_on" VALUE="on"><INPUT type=hidden NAME="section" VALUE="http://www.itknowledge.com/reference/standard/0471128457/"></form><!-- Empty Reference Subhead --><!--ISBN=0471128457//--><!--TITLE=APPLIED CRYPTOGRAPHY, SECOND EDITION: Protocols, Algorithms, and Source Code in C//--><!--AUTHOR=Bruce Schneier//--><!--PUBLISHER=Wiley Computer Publishing//--><!--CHAPTER=20//--><!--PAGES=489-491//--><!--UNASSIGNED1//--><!--UNASSIGNED2//--><CENTER><TABLE BORDER><TR><TD><A HREF="20-03.html">Previous</A></TD><TD><A HREF="../ewtoc.html">Table of Contents</A></TD><TD><A HREF="20-05.html">Next</A></TD></TR></TABLE></CENTER><P><BR></P><DL><DD><B>(1)</B> Choose an arbitrary sequence of at least 160 bits and call it <I>S</I>. Let <I>g</I> be the length of <I>S</I> in bits.<DD><B>(2)</B> Compute <I>U</I> = SHA(<I>S</I>) ⊕ SHA ((<I>S</I> + 1) mod 2<SUP><I>g</I></SUP>), where SHA is the Secure Hash Algorithm (see Section 18.7).<DD><B>(3)</B> Form <I>q</I> by setting the most significant bit and the least significant bit of <I>U</I> to 1.<DD><B>(4)</B> Check whether <I>q</I> is prime.<DD><B>(5)</B> If <I>q</I> is not prime, go back to step (1).<DD><B>(6)</B> Let <I>C</I> = 0 and <I>N</I> = 2.<DD><B>(7)</B> For <I>k</I> = 0, 1,..., <I>n</I>, let <I>V</I><SUB>k</SUB> = SHA ((<I>S</I> + <I>N</I> + <I>k</I>) mod 2<SUP><I>g</I></SUP>)<DD><B>(8)</B> Let <I>W</I> be the integer<DL><DD><I>W</I> = <I>V</I><SUB>0</SUB> + 2<SUP>160</SUP><I>V</I><SUB>1</SUB> +...+ 2<SUP>160(<I>n</I> - 1)</SUP><I>V</I><SUB>n - 1</SUB> + 2<SUP>160<I>n</I></SUP>(<I>V</I><SUB>n</SUB> mod 2<SUP><I>b</I></SUP>)</DL><BR>and let<DL><DD><I>X</I> = <I>W</I> + 2<SUP><I>L</I> - 1</SUP></DL><BR>Note that <I>X</I> is an <I>L</I>-bit number.<DD><B>(9)</B> Let <I>p</I> = <I>X</I> – ((<I>X</I> mod <I>2</I><SUB>q</SUB>) – 1). Note that <I>p</I> is congruent to 1 mod <I>2</I><SUB>q</SUB>.<DD><B>(10)</B> If <I>p</I> < 2<SUP><I>L</I> - 1</SUP>, then go to step (13).<DD><B>(11)</B> Check whether <I>p</I> is prime.<DD><B>(12)</B> If <I>p</I> is prime, go to step (15).<DD><B>(13)</B> Let <I>C</I> = <I>C</I> + 1 and <I>N</I> = <I>N</I> + <I>n</I> + 1.<DD><B>(14)</B> If <I>C</I> = 4096, then go to step (1). Otherwise, go to step (7).<DD><B>(15)</B> Save the value of <I>S</I> and the value of <I>C</I> used to generate <I>p</I> and <I>q</I>.</DL><P>In [1154], the variable <I>S</I> is called the “seed,” <I>C</I> is called the “counter,” and <I>N</I> the “offset.”</P><P>The point of this exercise is that there is a public means of generating <I>p</I> and <I>q</I>. For all practical purposes, this method prevents cooked values of <I>p</I> and <I>q</I>. If someone hands you a <I>p</I> and a <I>q</I>, you might wonder where that person got them. However, if someone hands you a value for <I>S</I> and <I>C</I> that generated the random <I>p</I> and <I>q</I>, you can go through this routine yourself. Using a one-way hash function, SHA in the standard, prevents someone from working backwards from a <I>p</I> and <I>q</I> to generate an <I>S</I> and <I>C</I>.</P><P>This security is better than what you get with RSA. In RSA, the prime numbers are kept secret. Someone could generate a fake prime or one of a special form that makes factoring easier. Unless you know the private key, you won’t know that. Here, even if you don’t know a person’s private key, you can confirm that <I>p</I> and <I>q</I> have been generated randomly.</P><P><FONT SIZE="+1"><B><I>ElGamal Encryption with DSA</I></B></FONT></P><P>There have been allegations that the government likes the DSA because it is only a digital signature algorithm and can’t be used for encryption. It is, however, possible to use the DSA function call to do ElGamal encryption.</P><P>Assume that the DSA algorithm is implemented with a single function call:</P><!-- CODE SNIP //--><PRE> DSAsign (p,q,g,k,x,h,r,s)</PRE><!-- END CODE SNIP //--><P>You supply the numbers <I>p, q, g, k, x</I>, and <I>h</I>, and the function returns the signature parameters: <I>r</I> and <I>s</I>.</P><P>To do ElGamal encryption of message <I>m</I> with public key <I>y</I>, choose a random number, <I>k</I>, and call</P><!-- CODE SNIP //--><PRE> DSAsign (p,p,g,k,0,0,r,s)</PRE><!-- END CODE SNIP //--><P>The value of <I>r</I> returned is <I>a</I> in the ElGamal scheme. Throw <I>s</I> away. Then, call</P><!-- CODE SNIP //--><PRE> DSAsign (p,p,y,k,0,0,r,s)</PRE><!-- END CODE SNIP //--><P>Rename the value of <I>r</I> to be <I>u</I>; throw <I>s</I> away. Call</P><!-- CODE SNIP //--><PRE> DSAsign (p,p,m,1,u,0,r,s)</PRE><!-- END CODE SNIP //--><P>Throw <I>r</I> away. The value of <I>s</I> returned is <I>b</I> in the ElGamal scheme. You now have the ciphertext, <I>a</I> and <I>b</I>.</P><P>Decryption is just as easy. Using secret key <I>x</I>, and ciphertext messages <I>a</I> and <I>b</I>, call</P><!-- CODE SNIP //--><PRE> DSAsign (p,p,a,x,0,0,r,s)</PRE><!-- END CODE SNIP //--><P>The value <I>r</I> is <I>a<SUP>x</I></SUP> mod <I>p</I>. Call that <I>e</I>. Then call</P><!-- CODE SNIP //--><PRE> DSAsign (p,p,1,e,b,0,r,s)</PRE><!-- END CODE SNIP //--><P>The value <I>s</I> is the plaintext message, <I>m</I>.</P><P>This method will not work with all implementations of DSA. Some may fix the values of <I>p</I> and <I>q</I>, or the lengths of some of the other parameters. Still, if the implementation is general enough, this is a way to encrypt using nothing more than digital signature function.</P><P><FONT SIZE="+1"><B><I>RSA Encryption with DSA</I></B></FONT></P><P>RSA encryption is even easier. With a modulus <I>n</I>, message <I>m</I>, and public key <I>e</I>, call</P><!-- CODE SNIP //--><PRE> DSAsign (n,n,m,e,0,0,r,s)</PRE><!-- END CODE SNIP //--><P>The value of <I>r</I> returned is the ciphertext.</P><P>RSA decryption is the same thing. If <I>d</I> is the private key, then</P><!-- CODE SNIP //--><PRE> DSAsign (n,n,m,d,0,0,r,s)</PRE><!-- END CODE SNIP //--><P>returns the plaintext as the value of <I>r</I>.</P><P><FONT SIZE="+1"><B><I>Security of DSA</I></B></FONT></P><P>At 512-bits, DSA wasn’t strong enough for long-term security. At 1024 bits, it is.</P><P><BR></P><CENTER><TABLE BORDER><TR><TD><A HREF="20-03.html">Previous</A></TD><TD><A HREF="../ewtoc.html">Table of Contents</A></TD><TD><A HREF="20-05.html">Next</A></TD></TR></TABLE></CENTER>[an error occurred while processing this directive]<!-- all of the reference materials (books) have the footer and subfoot reveresed --><!-- reference_subfoot = footer --><!-- reference_footer = subfoot --><!-- BEGIN SUB FOOTER --> <br><br> </TD> </TR> </TABLE> <table width="640" border=0 cellpadding=0 cellspacing=0> <tr> <td align="left" width=135><img src="/images/white.gif" width=100 height="1" alt="" border="0"></td> <!-- END SUB FOOTER --><!-- all of the books have the footer and subfoot reveresed --><!-- reference_subfoot = footer --><!-- reference_footer = subfoot --><!-- FOOTER --> <td width="515" align="left" bgcolor="#FFFFFF"><font face="arial, helvetica" size="1"><b><a href="/products.html"><font color="#006666">Products</font></a> | <a href="/contactus.html"><font color="#006666">Contact Us</font></a> | <a href="/aboutus.html"><font color="#006666">About Us</font></a> | <a href="http://www.earthweb.com/corporate/privacy.html" target="_blank"><font color="#006666">Privacy</font></a> | <a href="http://www.itmarketer.com/" target="_blank"><font color="#006666">Ad Info</font></a> | <a href="/"><font color="#006666">Home</font></a></b> <br><br> Use of this site is subject to certain <a href="/agreement.html">Terms & Conditions</a>, <a href="/copyright.html">Copyright © 1996-1999 EarthWeb Inc.</a><br> All rights reserved. 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