1 | ;;; -*- Mode: Lisp; Syntax: Common-Lisp; Package: Grobner; Base: 10 -*-
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2 | #|
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3 | $Id: modular.lisp,v 1.4 2009/01/22 04:03:50 marek Exp $
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4 | *--------------------------------------------------------------------------*
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5 | | Copyright (C) 1994, Marek Rychlik (e-mail: rychlik@math.arizona.edu) |
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6 | | Department of Mathematics, University of Arizona, Tucson, AZ 85721 |
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7 | | |
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8 | | Everyone is permitted to copy, distribute and modify the code in this |
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9 | | directory, as long as this copyright note is preserved verbatim. |
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10 | *--------------------------------------------------------------------------*
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11 | |#
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12 |
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13 | (defpackage "MODULAR"
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14 | (:export modular-division make-modular-division)
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15 | (:use "XGCD" "COMMON-LISP"))
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16 |
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17 | (in-package "MODULAR")
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18 |
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19 | (proclaim '(optimize (speed 0) (debug 3)))
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20 |
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21 | (defun modular-inverse (x p)
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22 | "Find the inverse of X modulo prime P, using Euclid algorithm."
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23 | (multiple-value-bind (gcd u v)
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24 | (xgcd x p)
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25 | (declare (ignore gcd v))
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26 | (mod u p)))
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27 |
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28 | (defun modular-division (x y p)
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29 | "Divide X by Y modulo prime P."
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30 | (mod (* x (modular-inverse y p)) p))
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31 |
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32 | (defvar *inverse-by-lookup-limit* 100000
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33 | "If prime modulus is < this number then the division algorithm
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34 | will use a lookup table of inverses created at the time when field-modulo-prime is called.")
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35 |
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36 | (defun make-inverse-table (modulus &aux (table (list 0)))
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37 | "Make a vector of length MODULUS containing all inverses modulo MODULUS,
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38 | which should be a prime number. The inverse of 0 is 0."
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39 | (do ((x 1 (1+ x))) ((>= x modulus) (apply #'vector (nreverse table)))
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40 | (push (modular-inverse x modulus) table)))
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41 |
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42 | (defun make-modular-division (modulus)
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43 | "Return a function of two arguments which will perform division
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44 | modulo MODULUS. Currently, if MODULUS is < *INVERSE-BY-LOOKUP-LIMIT*
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45 | then the returned function does table lookup, otherwise it uses
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46 | the Euclid algorithm to find the inverse."
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47 | (cond
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48 | ((>= modulus *inverse-by-lookup-limit*)
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49 | #'(lambda (x y) (modular-division x y modulus)))
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50 | (t
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51 | (let ((table (make-inverse-table modulus)))
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52 | #'(lambda (x y)
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53 | (mod (* x (svref table y)) modulus))))))
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