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<?php |
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if (!$STDNWPSTYLE_INCLUDED) |
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{ |
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include("stdnwpstyle.inc"); |
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$STDNWPSTYLE_INCLUDED=1; |
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} |
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?> |
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<?php |
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//******************************************************************************** |
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//Copyright (C) 2003 David T. Ashley |
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//******************************************************************************** |
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//This program or source file is free software; you can redistribute it and/or |
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//modify it under the terms of the GNU General Public License as published by |
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//the Free Software Foundation; either version 2 of the License, or (at your |
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//option) any later version. |
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// |
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//This program or source file is distributed in the hope that it will |
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//be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of |
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//MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
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//GNU General Public License for more details. |
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// |
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//You may have received a copy of the GNU General Public License |
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//along with this program; if not, write to the Free Software |
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//Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA |
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//******************************************************************************** |
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// |
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//Returns TRUE if the passed argument is a digit, or FALSE otherwise. If the |
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//passed string has length > 1, only the least significant digit is |
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//considered. |
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function miller_rabin_isdigit($Digit) |
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{ |
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$Digit = (string)$Digit; |
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|
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//If the string has zero length, this is an error condition and return 0. |
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$Length = strlen($Digit); |
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|
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if ($Length==0) |
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{ |
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return(0); |
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} |
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|
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//Obtain the trailing character of the string. |
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$Trailing_char = Substr($Digit, $Length-1, 1); |
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|
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//Do a switch and return an integer based on the character. |
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switch($Trailing_char) |
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{ |
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case "9": |
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return 1; |
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break; |
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case "8": |
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return 1; |
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break; |
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case "7": |
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return 1; |
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break; |
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case "6": |
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return 1; |
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break; |
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case "5": |
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return 1; |
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break; |
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case "4": |
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return 1; |
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break; |
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case "3": |
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return 1; |
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break; |
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case "2": |
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return 1; |
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break; |
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case "1": |
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return 1; |
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break; |
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case "0": |
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return 1; |
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break; |
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default: |
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return 0; |
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break; |
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} |
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} |
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// |
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// |
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//Returns TRUE if the passed argument is a simple integer (no error |
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//codes, no commas), or FALSE otherwise. |
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function miller_rabin_issimpleint($Intnum) |
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{ |
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$Intnum = (string)$Intnum; |
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|
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//If the string has zero length, this is an error condition and return 0. |
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$Length = strlen($Intnum); |
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|
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if ($Length==0) |
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{ |
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return(0); |
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} |
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|
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//If the first digit is "0", this must be the only character in the string, |
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//otherwise there is a leading-zero violation. |
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if (Substr($Intnum, 0, 1)=="0") |
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{ |
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if ($Length==1) |
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{ |
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return 1; |
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} |
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else |
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{ |
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return 0; |
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} |
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} |
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|
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//If the first character is a "-" sign, it may not be the only character, |
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//and it may not precede a "0" (which would again be a leading 0 violation). |
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//If it is a "-", we can exclude "0" and parse as a positive. |
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if (Substr($Intnum, 0, 1)=="-") |
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{ |
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if ($Length==1) |
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{ |
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return 0; |
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} |
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else |
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{ |
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if (Substr($Intnum, 1, 1) == "0") |
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{ |
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//This is illegal (a "-" followed by a "0"). |
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return 0; |
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} |
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else |
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{ |
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//We can just remove the "-" and parse as a positive. |
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$Intnum = Substr($Intnum, 1, $Length - 1); |
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$Length--; |
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} |
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} |
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} |
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|
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//It is known at this point that the length is at least 1 and the leading |
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//character is not zero. All we need to do at this point is check that each |
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//and every character is a digit. |
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for ($index=0; $index < $Length; $index++) |
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{ |
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if (!miller_rabin_isdigit(Substr($Intnum, $index, 1))) |
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return 0; |
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} |
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|
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return 1; |
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} |
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// |
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// |
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//Returns the number of digits in the argument, which must be a valid simple |
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//integer. |
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function miller_rabin_ndigits($Intnum) |
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{ |
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$Intnum = (string)$Intnum; |
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|
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if (Substr($Intnum, 0, 1) == "-") |
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{ |
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$rv = strlen($Intnum) - 1; |
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} |
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else |
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{ |
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$rv = strlen($Intnum); |
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} |
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|
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return($rv); |
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} |
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// |
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// |
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//Given a string, converts the final character to an integer. If the character |
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//is not a digit, returns 0. |
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function miller_rabin_digittoint($Digit) |
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{ |
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//echo "Digit is: "; |
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//echo $Digit; |
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//echo "<br>"; |
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|
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$Digit = (string)$Digit; |
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|
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//echo "Digit is: "; |
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//echo $Digit; |
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//echo "<br>"; |
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|
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//If the string has zero length, this is an error condition and return 0. |
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$Length = strlen($Digit); |
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|
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//echo "Length is: "; |
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//echo $Length; |
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//echo "<br>"; |
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|
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if ($Length==0) |
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{ |
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return(0); |
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} |
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|
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//Obtain the trailing character of the string. |
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$Trailing_char = Substr($Digit, $Length-1, 1); |
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|
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//echo "Trailing char is: "; |
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//echo $Trailing_char; |
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//echo "<br>"; |
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|
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//Do a switch and return an integer based on the character. |
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//The default case gives 0. |
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|
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switch($Trailing_char) |
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{ |
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case "9": |
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return 9; |
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break; |
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case "8": |
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return 8; |
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break; |
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case "7": |
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return 7; |
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break; |
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case "6": |
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return 6; |
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break; |
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case "5": |
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return 5; |
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break; |
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case "4": |
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return 4; |
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break; |
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case "3": |
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return 3; |
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break; |
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case "2": |
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return 2; |
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break; |
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case "1": |
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return 1; |
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break; |
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default: |
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return 0; |
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break; |
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} |
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} |
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|
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//Decommanates an integer and removes all other crap without respect for |
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//whether the commas are in the right place or the integer is syntactically valid. |
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function miller_rabin_decommanate(&$arg) |
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{ |
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$temp = ""; |
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|
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$len = strlen($arg); |
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|
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for ($i=0; $i<$len; $i++) |
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{ |
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$c = (string)Substr($arg, $i, 1); |
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|
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if (miller_rabin_isdigit($c) || ($c == "-")) |
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$temp = $temp . $c; |
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} |
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|
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$arg = (string)$temp; |
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} |
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|
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//Commanates an integer which is assumed to by syntactically correct. |
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function miller_rabin_commanate(&$arg) |
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{ |
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miller_rabin_decommanate($arg); |
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|
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$temp = ""; |
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|
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$len = strlen($arg); |
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|
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for ($i=0; $i<$len; $i++) |
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{ |
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$c = (string)Substr($arg, $len - $i - 1, 1); |
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|
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$temp = $c . $temp; |
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|
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if ($i != ($len-1)) |
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{ |
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if (Substr($arg, $len-$i-2, 1) != "-") |
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{ |
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if (($i % 3) == 2) |
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$temp = "," . $temp; |
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} |
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} |
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} |
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|
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$arg = (string)$temp; |
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} |
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|
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function do_header(&$style, $miller_rabin_nrounds) |
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{ |
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$style->header_title("Probable Primality Test (Miller-Rabin)", |
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"Miller-Rabin Probable Primality Test", |
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"This utility uses " . $miller_rabin_nrounds . " applications of the " . |
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"<a href=\"http://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html\">" . |
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"Miller-Rabin test</a> to determine " . |
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"with high probabilistic certainty whether an integer " . |
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"is " . |
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"<a href=\"http://mathworld.wolfram.com/PrimeNumber.html\">prime</a> or " . |
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"<a href=\"http://mathworld.wolfram.com/CompositeNumber.html\">composite</a>." . |
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" This utility does <i>not</i> " . |
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"factor composite numbers."); |
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} |
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|
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function do_footer(&$style) |
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{ |
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$style->hrule_std(); |
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echo "<p align=\"center\">This utility is powered by the "; |
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echo "<a href=\"http://www.swox.com/gmp\"><img hspace=\"5\" vspace=\"5\" src=\"../genimages/gmplogo1.png\" width=\"232\" height=\"104\" align=\"absmiddle\"></a>. All source code is available "; |
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echo "<a href=\"../howtos/nth_web_gmp_src_code_dist/obtain_all_source.htm\">here</a>.</p>\n"; |
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$style->footer_std(); |
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} |
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|
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function do_form($N_refresh, $Buttontext, $max_digits) |
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{ |
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echo "<form method=get action=\"miller_rabin.php\" width=\"100%\">\n"; |
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echo "<table align=\"center\">\n"; |
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echo "<tr>\n"; |
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echo " <td width=\"10%\">\n"; |
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echo " <p align=\"right\"><b>N</b> (" . number_format($max_digits) . "-digit maximum):</td>\n"; |
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echo " <td width=\"18%\"><p align=\"left\"><input type=\"text\" "; |
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if (strlen($N_refresh)) |
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{ |
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echo "value=\"" . $N_refresh . "\" "; |
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} |
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echo "name=\"N\" size=\"50\"></p></td>\n"; |
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echo " </tr>\n"; |
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echo " <tr>\n"; |
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echo " <td width=\"20%\" colspan=\"2\">\n"; |
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echo " <p align=\"center\"><input type=\"submit\" value=\""; |
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echo $Buttontext; |
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echo "\" name=\"B1\" style=\"margin-top: 12\"></td>\n"; |
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echo " </tr>\n"; |
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echo "</table>\n"; |
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echo "</form>\n"; |
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} |
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|
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function do_err_msg($Err_msg) |
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{ |
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echo "<p align=\"center\"><b><font color=\"#FF0000\">"; |
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echo $Err_msg; |
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echo "</font></b></p>\n"; |
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} |
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|
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|
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//Main script begins here. |
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// |
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// |
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//Let's agree on a maximum for number of digits in the number to test. It is known that |
| 348 |
//Apache won't process URL's larger than 1024, so let's be conservative and say 500 |
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//digits. |
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$max_digits = 500; |
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// |
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//Let's agree on a number of rounds for the Miller-Rabin test. This influences the |
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//probabilities. |
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$miller_rabin_nrounds = 25; |
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// |
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//We need to establish the probability that a number which passes the rounds specified |
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//above is still in fact composite. This values need to be calculated by hand whenever |
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//the value above is changed. |
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$miller_rabin_p_real = "0.99999999999999911182"; |
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$miller_rabin_p_percent = "99.999999999999911182"; |
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$miller_rabin_residual_real = "0.00000000000000088818"; |
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$miller_rabin_residual_percent = "0.000000000000088818"; |
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$miller_rabin_residual_lt_reciprocal = "1,125,899,906,842,624"; |
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// |
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$style = new Stdnwpstyle; |
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//Assign the current style in force. Also, starts the CPU |
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//usage clock. |
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|
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//Do the header unconditionally. The header is always used on this page. |
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do_header($style, $miller_rabin_nrounds); |
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// |
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// |
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//If N was supplied, decommanate it and be sure it is a string. |
| 374 |
if (isset($N)) |
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miller_rabin_decommanate($N); |
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// |
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//There are a few cases to break into here, and this affects the overall layout of the page. |
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//The header and footer are fixed, but the "guts" will change. |
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if (!isset($N)) |
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{ |
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//In this case, we are probably visiting the form for the first time, and have not done a submit. |
| 382 |
//Just display the form itself. |
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do_form("", "Is This Prime?", $max_digits); |
| 384 |
} |
| 385 |
elseif (!miller_rabin_issimpleint($N)) |
| 386 |
{ |
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do_err_msg("N is a malformed integer. Choose a new value for N and try again."); |
| 388 |
$style->hrule_std(); |
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miller_rabin_commanate($N); |
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do_form($N, "Is This Prime?", $max_digits); |
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} |
| 392 |
elseif ($N == "1") |
| 393 |
{ |
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do_err_msg("It is not a valid question to ask if 1 is prime. It is agreed " . |
| 395 |
"in number theory (largely to minimize exception cases in proofs and results) " . |
| 396 |
"that 1 is neither prime nor composite. " . |
| 397 |
"Choose a new value for N and try again."); |
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$style->hrule_std(); |
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miller_rabin_commanate($N); |
| 400 |
do_form($N, "Is This Prime?", $max_digits); |
| 401 |
} |
| 402 |
elseif (strlen($N) > $max_digits) |
| 403 |
{ |
| 404 |
do_err_msg("N is too long (a " . number_format($max_digits) . "-digit maximum is " . |
| 405 |
"in effect). Choose a shorter value and try again."); |
| 406 |
$style->hrule_std(); |
| 407 |
miller_rabin_commanate($N); |
| 408 |
do_form($N, "Is This Prime?", $max_digits); |
| 409 |
} |
| 410 |
else |
| 411 |
{ |
| 412 |
//N seems to be a valid integer. |
| 413 |
// |
| 414 |
//If N is negative, make it positive. |
| 415 |
if (Substr($N, 0, 1)=="-") |
| 416 |
$N = Substr($N, 1, strlen($N)-1); |
| 417 |
|
| 418 |
//If N is "0", make it "2". |
| 419 |
if ($N=="0") |
| 420 |
$N = (string)"2"; |
| 421 |
|
| 422 |
//We can now run the external program to do the calculation. |
| 423 |
$cgi_command_string = "./arith_large_cgi gmp_prob_prime " . $N . " " . $miller_rabin_nrounds; |
| 424 |
|
| 425 |
//echo "CGI command string is : " . $cgi_command_string . "<br>\n"; |
| 426 |
|
| 427 |
exec($cgi_command_string, $cgi_result); |
| 428 |
$cgi_result_nelem = count($cgi_result); |
| 429 |
|
| 430 |
//echo "Result is : " . $cgi_result_nelem . "<br>\n"; |
| 431 |
|
| 432 |
//for ($i=0; $i<$cgi_result_nelem; $i++) |
| 433 |
//{ |
| 434 |
// echo $cgi_result[$i] . "<br>\n"; |
| 435 |
//} |
| 436 |
|
| 437 |
//We need to perform some sanity checks on the CGI output to be sure it |
| 438 |
//is what we want. If it seems wrong, we need to generate an error message |
| 439 |
//and not try to display the output. |
| 440 |
$cgi_output_is_sane = 1; |
| 441 |
if ($cgi_result_nelem == 0) |
| 442 |
$cgi_output_is_sane = 0; |
| 443 |
if ($cgi_result_nelem > 0) |
| 444 |
{ |
| 445 |
if ($cgi_result_nelem != 5) |
| 446 |
$cgi_output_is_sane = 0; |
| 447 |
if ($cgi_result[0] != "S") |
| 448 |
$cgi_output_is_sane = 0; |
| 449 |
if (($cgi_result[3] != "0") && ($cgi_result[3] != "1") && ($cgi_result[3] != "2")) |
| 450 |
$cgi_output_is_sane = 0; |
| 451 |
if ($cgi_result[4] != "S") |
| 452 |
$cgi_output_is_sane = 0; |
| 453 |
} |
| 454 |
|
| 455 |
if (!$cgi_output_is_sane) |
| 456 |
{ |
| 457 |
do_err_msg("An unspecified error occurred when interacting with the CGI-BIN program. Please advise the webmaster."); |
| 458 |
$style->hrule_std(); |
| 459 |
miller_rabin_commanate($N); |
| 460 |
do_form($N, "Is This Prime", $max_digits); |
| 461 |
} |
| 462 |
else |
| 463 |
{ |
| 464 |
miller_rabin_commanate($cgi_result[1]); |
| 465 |
|
| 466 |
if ($cgi_result[3] == 0) |
| 467 |
{ |
| 468 |
echo "<p align=\"center\"><font size=\"4\">The integer "; |
| 469 |
echo $cgi_result[1]; |
| 470 |
echo " is definitely (with 100% certainty) composite. Note, however, that although "; |
| 471 |
echo "it was relatively computationally inexpensive to determine that this integer is composite, "; |
| 472 |
echo "factoring it may be computationally <i>much</i> more expensive.</font></p>\n"; |
| 473 |
} |
| 474 |
elseif ($cgi_result[3] == 1) |
| 475 |
{ |
| 476 |
echo "<p align=\"center\"><font size=\"4\">Based on " . $miller_rabin_nrounds . " applications of "; |
| 477 |
echo "the Miller-Rabin test, the integer "; |
| 478 |
echo $cgi_result[1]; |
| 479 |
echo " is probably (with " . $miller_rabin_p_percent . "% certainty) "; |
| 480 |
echo "prime. (This means that there is no more than 1 chance in "; |
| 481 |
echo $miller_rabin_residual_lt_reciprocal . " that this integer is "; |
| 482 |
echo "composite.)</font></p>\n"; |
| 483 |
} |
| 484 |
else |
| 485 |
{ |
| 486 |
echo "<p align=\"center\"><font size=\"4\">The integer "; |
| 487 |
echo $cgi_result[1]; |
| 488 |
echo " is prime (with 100% certainty). This means that the "; |
| 489 |
echo "GMP"; |
| 490 |
echo " library routine was able to attempt trial division with divisors "; |
| 491 |
echo "up to <i>floor</i>(<i>sqrt</i>(N)).</font></p>\n"; |
| 492 |
} |
| 493 |
|
| 494 |
$style->hrule_std(); |
| 495 |
?> |
| 496 |
<p align="left"><b>Notes on the algorithm:</b> |
| 497 |
<ul> |
| 498 |
<li>It is agreed in number theory that |
| 499 |
<a href="http://www.mth.uct.ac.za/~digest/faq/prime.html">'1' is neither prime nor |
| 500 |
composite</a>. This definition |
| 501 |
is largely <a href="http://dictionary.reference.com/search?q=arbitrary">arbitrary</a>, |
| 502 |
and there is some discussion in Usenet newsgroups |
| 503 |
such as <i>sci.math</i> from time to time about this topic. It substantially shortens |
| 504 |
and neatens many results and proofs to define '1' in this way. For this reason, |
| 505 |
this utility refuses to <a href="http://dictionary.reference.com/search?q=opine">opine</a> on '1'. |
| 506 |
</li> |
| 507 |
<li>The <a href="http://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html">Miller-Rabin |
| 508 |
test</a> and other |
| 509 |
<a href="http://mathworld.wolfram.com/StrongPseudoprime.html">strong pseudoprime</a> tests |
| 510 |
are part of the backbone of popular |
| 511 |
<a href="http://mathworld.wolfram.com/Public-KeyCryptography.html">public-key cryptography</a>, |
| 512 |
specifically |
| 513 |
<a href="http://mathworld.wolfram.com/RSAEncryption.html">RSA public-key cryptography</a>, |
| 514 |
which relies on the fact that it is much easier to create a large composite number with |
| 515 |
large prime components than to factor it. |
| 516 |
</li> |
| 517 |
<li>This utility uses the <i>mpz_prob_prime_p()</i> function from |
| 518 |
the <a href="http://www.swox.com/gmp">GMP</a> |
| 519 |
to make its primality assessment. This function has not been proofread |
| 520 |
to verify how it selects bases and applies the Miller-Rabin test. It is assumed |
| 521 |
that the standard upper bound of 0.25^N can be applied where N is the <i>reps</i> |
| 522 |
parameter to the function (in the case of this utility, <i>reps</i> is |
| 523 |
<?php echo " " . $miller_rabin_nrounds ; ?>). |
| 524 |
</li> |
| 525 |
<li>This web site also contains a <a href="pfact18digit.php">factorization |
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utility</a> which will factor integers subject to size and CPU time constraints. |
| 527 |
</ul> |
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<?php |
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$style->hrule_std(); |
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|
| 531 |
miller_rabin_commanate($N); |
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do_form($N, "Is This Prime?", $max_digits); |
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} |
| 534 |
} |
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|
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//Now do the footer unconditionally. The footer is always used on this web page. |
| 537 |
do_footer($style); |
| 538 |
?> |
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