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+<H2><A NAME="SECTION00062000000000000000">
+Security Notes for Public-key Algorithms</A>
+</H2>
+Any of these algorithms can be trivially broken; for example, RSA can be
+broken by factoring the modulus <SPAN  CLASS="textit">n</SPAN> into its two prime factors.
+This is easily done by the following code:
+
+<P>
+<PRE>
+for i in range(2, n): 
+    if (n%i)==0: 
+        print i, 'is a factor' 
+        break
+</PRE>
+
+<P>
+However, <SPAN  CLASS="textit">n</SPAN> is usually a few hundred bits long, so this simple
+program wouldn't find a solution before the universe comes to an end.
+Smarter algorithms can factor numbers more quickly, but it's still
+possible to choose keys so large that they can't be broken in a
+reasonable amount of time.  For ElGamal and DSA, discrete logarithms are
+used instead of factoring, but the principle is the same.
+
+<P>
+Safe key sizes depend on the current state of number theory and
+computer technology.  At the moment, one can roughly define three
+levels of security: low-security commercial, high-security commercial,
+and military-grade.  For RSA, these three levels correspond roughly to
+768, 1024, and 2048-bit keys.
+
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