galois field arithmetic.htm

来自「Galois Field Arithmetic algorithm」· HTM 代码 · 共 73 行

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<H1><A name=SECTION00020000000000000000>Galois Field Arithmetic</A></H1>
<P>Galois Fields = fields of finite order (cardinality). <BR>Notation: 
GF(<I>q</I>) = Galois field of order <I>q</I>.<BR>
<P><B>Theorem</B>: The integers <IMG height=27 alt=tex2html_wrap_inline286 
src="Galois Field Arithmetic_ficheiros/img2.gif" width=112 align=middle> where 
<I>p</I> is a prime, form the field GF(<I>p</I>) under <I>modulo p</I> addition 
and multiplication.<BR>
<P><B>Definition</B>: Let <IMG height=29 alt=tex2html_wrap_inline292 
src="Galois Field Arithmetic_ficheiros/img3.gif" width=11 align=middle> be an 
element in GF(<I>q</I>). The order of <IMG height=29 alt=tex2html_wrap_inline292 
src="Galois Field Arithmetic_ficheiros/img3.gif" width=11 align=middle> is the 
smallest positive integer <I>m</I> such that <IMG height=29 
alt=tex2html_wrap_inline300 src="Galois Field Arithmetic_ficheiros/img4.gif" 
width=59 align=middle> . <BR>
<P><B>Theorem</B>: If <IMG height=32 alt=tex2html_wrap_inline302 
src="Galois Field Arithmetic_ficheiros/img5.gif" width=88 align=middle> for some 
<IMG height=32 alt=tex2html_wrap_inline304 
src="Galois Field Arithmetic_ficheiros/img6.gif" width=88 align=middle> , then 
<IMG height=32 alt=tex2html_wrap_inline306 
src="Galois Field Arithmetic_ficheiros/img7.gif" width=81 align=middle> .<BR>
<P><B>Definition</B>: An element with order (<I>q</I>-1) in GF(<I>q</I>) is 
called a <I>primitive element</I> in GF(<I>q</I>).<BR>
<P>Every field GF(<I>q</I>) contains at least one primitive element <IMG 
height=9 alt=tex2html_wrap_inline316 
src="Galois Field Arithmetic_ficheiros/img8.gif" width=10 align=bottom> .<BR>All 
nonzero elements in GF(<I>q</I>) can be represented as (<I>q</I>-1) consecutive 
powers of a primitive element <IMG height=9 alt=tex2html_wrap_inline322 
src="Galois Field Arithmetic_ficheiros/img9.gif" width=20 align=bottom> 
<P><IMG height=21 alt=displaymath278 
src="Galois Field Arithmetic_ficheiros/img10.gif" width=423 align=bottom> 
<P>
<P><B>Theorem</B>: The order <I>q</I> of a Galois Field GF(<I>q</I>) must be a 
power of a prime. 
<P><BR>
<HR>

<UL>
  <LI><A 
  href="http://www.ee.ucla.edu/~matache/rsc/node3.html#SECTION00021000000000000000" 
  name=tex2html34><B>Polynomials over Galois Fields</B></A> 
  <LI><A 
  href="http://www.ee.ucla.edu/~matache/rsc/node4.html#SECTION00022000000000000000" 
  name=tex2html35><B>Construction of Galois Field GF(2^m)</B></A> </LI></UL><BR>
<HR>

<P>
<ADDRESS><I>A. Matache <BR>Sun Oct 20 17:42:25 PDT 1996</I> 
</ADDRESS></BODY></HTML>

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