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POPicking a bounded uniform random value in GF(2^M) without exponentiation or tables
 

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  1. POPicking a bounded uniform random value in GF(2^M) without exponentiation or tables
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    <p>I am looking for a way to uniformly choose values in GF(2^M) between two bounds. </p> <p>GF(2^M) is a Galois Field - See GF(4) as defined on this page - <a href="http://www.math.umbc.edu/~campbell/Math413Spr09/Notes/12-13_Finite_Fields.html" rel="nofollow noreferrer">http://www.math.umbc.edu/~campbell/Math413Spr09/Notes/12-13_Finite_Fields.html</a><br> From a technical, non-math perspective, this is most similar to CRC operations. </p> <p>For example: </p> <pre><code>ulong gf2step( ulong x, int bits, ulong p ) { x = x &lt;&lt; 1; // "multiply" by x if ( x &gt;= (1 &lt;&lt; bits)) x = x ^ p; return x; } </code></pre> <p>Expanding the example from below:</p> <blockquote> <pre><code> 12 is '1100 '1100 shifted left by 1 becomes `11000. Since bit 4 is set, xor with `10011 (p). Next is `1011 or 11. </code></pre> </blockquote> <p>Similarly,</p> <blockquote> <pre><code> 9 is '1001 '1001 shifted left by 1 becomes `10010. Since bit 4 is set, xor with `10011 (p). Next is `0001. </code></pre> </blockquote> <p>My obvious method is to start with the integer exponents corresponding to the bounds, pick a random exponent between those and generate the value from that. </p> <p>However, this has two problems --<br> 1. Given arbitrary bounds, I can't find the corresponding integer exponent..<br> 2. This will be repeated many, many times, so I am concerned about exponentiation speed.</p> <p>Example: </p> <pre><code> int gf2random( ulong low, ulong high, ulong p); gf2random( 12, 13, 19) should return evenly from the set {12, 11,5,10,7,14,15, 13} gf2random( 9, 1, 19) should return either 9 or 1 </code></pre> <p>I can step the values in GF(2^M) fairly easily - but I'm not sure how to avoid overshooting the upper bound.</p> <p><hr /> Does it simplify the problem if the low bound was always '1' ? </p>
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      1. POPicking a bounded uniform random value in GF(2^M) without exponentiation or tables
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    1. COWhat is the GF you refer to. http://en.wikipedia.org/wiki/GF_method or something else? You might get more answers if people with stats experience but lack of familiarity with the function/method/distribution/whatever
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    2. COAh I assume it is http://en.wikipedia.org/wiki/Finite_field
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    3. COI think it will be much easier to answer your question if you look at the existing answers, comment on them and clarify the original question.
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