1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
|
;;;; "modular.scm", modular fixnum arithmetic for Scheme
;;; Copyright (C) 1991, 1993, 1995 Aubrey Jaffer
;
;Permission to copy this software, to modify it, to redistribute it,
;to distribute modified versions, and to use it for any purpose is
;granted, subject to the following restrictions and understandings.
;
;1. Any copy made of this software must include this copyright notice
;in full.
;
;2. I have made no warrantee or representation that the operation of
;this software will be error-free, and I am under no obligation to
;provide any services, by way of maintenance, update, or otherwise.
;
;3. In conjunction with products arising from the use of this
;material, there shall be no use of my name in any advertising,
;promotional, or sales literature without prior written consent in
;each case.
(define (symmetric:modulus n)
(cond ((or (not (number? n)) (not (positive? n)) (even? n))
(slib:error 'symmetric:modulus n))
(else (quotient (+ -1 n) -2))))
(define (modulus->integer m)
(cond ((negative? m) (- 1 m m))
((zero? m) #f)
(else m)))
(define (modular:normalize m k)
(cond ((positive? m) (modulo k m))
((zero? m) k)
((<= m k (- m)) k)
((or (provided? 'bignum)
(<= m (quotient (+ -1 most-positive-fixnum) 2)))
(let* ((pm (+ 1 (* -2 m)))
(s (modulo k pm)))
(if (<= s (- m)) s (- s pm))))
((positive? k) (+ (+ (+ k -1) m) m))
(else (- (- (+ k 1) m) m))))
;;;; NOTE: The rest of these functions assume normalized arguments!
(require 'logical)
(define (modular:extended-euclid x y)
(define q 0)
(do ((r0 x r1) (r1 y (remainder r0 r1))
(u0 1 u1) (u1 0 (- u0 (* q u1)))
(v0 0 v1) (v1 1 (- v0 (* q v1))))
;; (assert (= r0 (+ (* u0 x) (* v0 y))))
;; (assert (= r1 (+ (* u1 x) (* v1 y))))
((zero? r1) (list r0 u0 v0))
(set! q (quotient r0 r1))))
(define (modular:invertable? m a)
(eqv? 1 (gcd (or (modulus->integer m) 0) a)))
(define (modular:invert m a)
(cond ((eqv? 1 (abs a)) a) ; unit
(else
(let ((pm (modulus->integer m)))
(cond
(pm
(let ((d (modular:extended-euclid (modular:normalize pm a) pm)))
(if (= 1 (car d))
(modular:normalize m (cadr d))
(slib:error 'modular:invert "can't invert" m a))))
(else (slib:error 'modular:invert "can't invert" m a)))))))
(define (modular:negate m a)
(if (zero? a) 0
(if (negative? m) (- a)
(- m a))))
;;; Being careful about overflow here
(define (modular:+ m a b)
(cond ((positive? m)
(modulo (+ (- a m) b) m))
((zero? m) (+ a b))
((negative? a)
(if (negative? b)
(let ((s (+ (- a m) b)))
(if (negative? s)
(- s -1 m)
(+ s m)))
(+ a b)))
((negative? b) (+ a b))
(else (let ((s (+ (+ a m) b)))
(if (positive? s)
(+ s -1 m)
(- s m))))))
(define (modular:- m a b)
(cond ((positive? m) (modulo (- a b) m))
((zero? m) (- a b))
(else (modular:+ m a (- b)))))
;;; See: L'Ecuyer, P. and Cote, S. "Implementing a Random Number Package
;;; with Splitting Facilities." ACM Transactions on Mathematical
;;; Software, 17:98-111 (1991)
;;; modular:r = 2**((nb-2)/2) where nb = number of bits in a word.
(define modular:r
(ash 1 (quotient (integer-length most-positive-fixnum) 2)))
(define modular:*
(if (provided? 'bignum)
(lambda (m a b)
(cond ((zero? m) (* a b))
((positive? m) (modulo (* a b) m))
(else (modular:normalize m (* a b)))))
(lambda (m a b)
(let ((a0 a)
(p 0))
(cond
((zero? m) (* a b))
((negative? m)
"This doesn't work for the full range of modulus M;"
"Someone please create or convert the following"
"algorighm to work with symmetric representation"
(modular:normalize m (* a b)))
(else
(cond
((< a modular:r))
((< b modular:r) (set! a b) (set! b a0) (set! a0 a))
(else
(set! a0 (modulo a modular:r))
(let ((a1 (quotient a modular:r))
(qh (quotient m modular:r))
(rh (modulo m modular:r)))
(cond ((>= a1 modular:r)
(set! a1 (- a1 modular:r))
(set! p (modulo (- (* modular:r (modulo b qh))
(* (quotient b qh) rh)) m))))
(cond ((not (zero? a1))
(let ((q (quotient m a1)))
(set! p (- p (* (quotient b q) (modulo m a1))))
(set! p (modulo (+ (if (positive? p) (- p m) p)
(* a1 (modulo b q))) m)))))
(set! p (modulo (- (* modular:r (modulo p qh))
(* (quotient p qh) rh)) m)))))
(if (zero? a0)
p
(let ((q (quotient m a0)))
(set! p (- p (* (quotient b q) (modulo m a0))))
(modulo (+ (if (positive? p) (- p m) p)
(* a0 (modulo b q))) m)))))))))
(define (modular:expt m a b)
(cond ((= a 1) 1)
((= a (- m 1)) (if (odd? b) a 1))
((zero? a) 0)
((zero? m) (integer-expt a b))
(else
(logical:ipow-by-squaring a b 1
(lambda (c d) (modular:* m c d))))))
(define extended-euclid modular:extended-euclid)
|
