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d56fd9d1 MW |
1 | /* -*-c-*- |
2 | * | |
3 | * Simple scalar fields | |
4 | * | |
5 | * (c) 2017 Straylight/Edgeware | |
6 | */ | |
7 | ||
8 | /*----- Licensing notice --------------------------------------------------* | |
9 | * | |
10 | * This file is part of Catacomb. | |
11 | * | |
12 | * Catacomb is free software; you can redistribute it and/or modify | |
13 | * it under the terms of the GNU Library General Public License as | |
14 | * published by the Free Software Foundation; either version 2 of the | |
15 | * License, or (at your option) any later version. | |
16 | * | |
17 | * Catacomb is distributed in the hope that it will be useful, | |
18 | * but WITHOUT ANY WARRANTY; without even the implied warranty of | |
19 | * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the | |
20 | * GNU Library General Public License for more details. | |
21 | * | |
22 | * You should have received a copy of the GNU Library General Public | |
23 | * License along with Catacomb; if not, write to the Free | |
24 | * Software Foundation, Inc., 59 Temple Place - Suite 330, Boston, | |
25 | * MA 02111-1307, USA. | |
26 | */ | |
27 | ||
28 | /*----- Header files ------------------------------------------------------*/ | |
29 | ||
30 | #include <string.h> | |
31 | ||
32 | #include "scaf.h" | |
33 | ||
34 | /*----- Main code ---------------------------------------------------------*/ | |
35 | ||
36 | /* --- @scaf_load@ --- * | |
37 | * | |
38 | * Arguments: @scaf_piece *z@ = where to write the result | |
39 | * @const octet *b@ = source buffer to read | |
40 | * @size_t sz@ = size of the source buffer | |
41 | * @size_t npiece@ = number of pieces to read | |
42 | * @unsigned piecewd@ = nominal width of pieces in bits | |
43 | * | |
44 | * Returns: --- | |
45 | * | |
46 | * Use: Loads a little-endian encoded scalar into a vector @z@ of | |
47 | * single-precision pieces. | |
48 | */ | |
49 | ||
50 | void scaf_load(scaf_piece *z, const octet *b, size_t sz, | |
51 | size_t npiece, unsigned piecewd) | |
52 | { | |
53 | uint32 a, m = ((scaf_piece)1 << piecewd) - 1; | |
54 | unsigned i, j, n; | |
55 | ||
56 | for (i = j = n = 0, a = 0; i < sz; i++) { | |
57 | a |= b[i] << n; n += 8; | |
58 | if (n >= piecewd) { | |
59 | z[j++] = a&m; a >>= piecewd; n -= piecewd; | |
60 | if (j >= npiece) return; | |
61 | } | |
62 | } | |
63 | z[j++] = a; | |
64 | while (j < npiece) z[j++] = 0; | |
65 | } | |
66 | ||
67 | /* --- @scaf_loaddbl@ --- * | |
68 | * | |
69 | * Arguments: @scaf_dblpiece *z@ = where to write the result | |
70 | * @const octet *b@ = source buffer to read | |
71 | * @size_t sz@ = size of the source buffer | |
72 | * @size_t npiece@ = number of pieces to read | |
73 | * @unsigned piecewd@ = nominal width of pieces in bits | |
74 | * | |
75 | * Returns: --- | |
76 | * | |
77 | * Use: Loads a little-endian encoded scalar into a vector @z@ of | |
78 | * double-precision pieces. | |
79 | */ | |
80 | ||
81 | void scaf_loaddbl(scaf_dblpiece *z, const octet *b, size_t sz, | |
82 | size_t npiece, unsigned piecewd) | |
83 | { | |
84 | uint32 a, m = ((scaf_piece)1 << piecewd) - 1; | |
85 | unsigned i, j, n; | |
86 | ||
87 | for (i = j = n = 0, a = 0; i < sz; i++) { | |
88 | a |= b[i] << n; n += 8; | |
89 | if (n >= piecewd) { | |
90 | z[j++] = a&m; a >>= piecewd; n -= piecewd; | |
91 | if (j >= npiece) return; | |
92 | } | |
93 | } | |
94 | z[j++] = a; | |
95 | while (j < npiece) z[j++] = 0; | |
96 | } | |
97 | ||
98 | /* --- @scaf_store@ --- * | |
99 | * | |
100 | * Arguments: @octet *b@ = buffer to fill in | |
101 | * @size_t sz@ = size of the buffer | |
102 | * @const scaf_piece *x@ = scalar to store | |
103 | * @size_t npiece@ = number of pieces in @x@ | |
104 | * @unsigned piecewd@ = nominal width of pieces in bits | |
105 | * | |
106 | * Returns: --- | |
107 | * | |
108 | * Use: Stores a scalar in a vector of single-precison pieces as a | |
109 | * little-endian vector of bytes. | |
110 | */ | |
111 | ||
112 | void scaf_store(octet *b, size_t sz, const scaf_piece *x, | |
113 | size_t npiece, unsigned piecewd) | |
114 | { | |
115 | uint32 a; | |
116 | unsigned i, j, n; | |
117 | ||
118 | for (i = j = n = 0, a = 0; i < npiece; i++) { | |
119 | a |= x[i] << n; n += piecewd; | |
120 | while (n >= 8) { | |
121 | b[j++] = a&0xffu; a >>= 8; n -= 8; | |
122 | if (j >= sz) return; | |
123 | } | |
124 | } | |
125 | b[j++] = a; | |
126 | memset(b + j, 0, sz - j); | |
127 | } | |
128 | ||
129 | /* --- @scaf_mul@ --- * | |
130 | * | |
131 | * Arguments: @scaf_dblpiece *z@ = where to put the answer | |
132 | * @const scaf_piece *x, *y@ = the operands | |
133 | * @size_t npiece@ = the length of the operands | |
134 | * | |
135 | * Returns: --- | |
136 | * | |
137 | * Use: Multiply two scalars. The destination must have space for | |
138 | * @2*npiece@ pieces (though the last one will always be zero). | |
139 | * The result is not reduced. | |
140 | */ | |
141 | ||
142 | void scaf_mul(scaf_dblpiece *z, const scaf_piece *x, const scaf_piece *y, | |
143 | size_t npiece) | |
144 | { | |
145 | unsigned i, j; | |
146 | ||
147 | for (i = 0; i < 2*npiece; i++) z[i] = 0; | |
148 | ||
149 | for (i = 0; i < npiece; i++) | |
150 | for (j = 0; j < npiece; j++) | |
151 | z[i + j] += (scaf_dblpiece)x[i]*y[j]; | |
152 | } | |
153 | ||
154 | /* --- @scaf_reduce@ --- * | |
155 | * | |
156 | * Arguments: @scaf_piece *z@ = where to write the result | |
157 | * @const scaf_dblpiece *x@ = the operand to reduce | |
158 | * @const scaf_piece *l@ = the modulus, in internal format | |
159 | * @const scaf_piece *mu@ = scaled approximation to @1/l@ | |
160 | * @size_t npiece@ = number of pieces in @l@ | |
161 | * @unsigned piecewd@ = nominal width of a piece in bits | |
162 | * @scaf_piece *scratch@ = @3*npiece + 1@ scratch pieces | |
163 | * | |
164 | * Returns: --- | |
165 | * | |
166 | * Use: Reduce @x@ (a vector of @2*npiece@ double-precision pieces) | |
167 | * modulo @l@ (a vector of @npiece@ single-precision pieces), | |
168 | * writing the result to @z@. | |
169 | * | |
170 | * Write @n = npiece@, @w = piecewd@, and %$B = 2^w$%. The | |
171 | * operand @mu@ must contain %$\lfloor B^{2n}/l \rfloor$%, in | |
172 | * @npiece + 1@ pieces. Furthermore, we must have | |
173 | * %$3 l < B^n$%. (Fiddle with %$w$% if necessary.) | |
174 | */ | |
175 | ||
176 | void scaf_reduce(scaf_piece *z, const scaf_dblpiece *x, | |
177 | const scaf_piece *l, const scaf_piece *mu, | |
178 | size_t npiece, unsigned piecewd, scaf_piece *scratch) | |
179 | { | |
180 | unsigned i, j; | |
181 | scaf_piece *t = scratch, *q = scratch + 2*npiece; | |
182 | scaf_piece u, m = ((scaf_piece)1 << piecewd) - 1; | |
183 | scaf_dblpiece c; | |
184 | ||
185 | /* This here is the hard part. | |
186 | * | |
187 | * Let w = PIECEWD, let n = NPIECE, and let B = 2^w. Wwe must have | |
188 | * B^(n-1) <= l < B^n. | |
189 | * | |
190 | * The argument MU contains pieces of the quantity µ = floor(B^2n/l), which | |
191 | * is a scaled approximation to 1/l. We'll calculate | |
192 | * | |
193 | * q = floor(µ floor(x/B^(n-1))/B^(n+1)) | |
194 | * | |
195 | * which is an underestimate of x/l. | |
196 | * | |
197 | * With a bit more precision: by definition, u - 1 < floor(u) <= u. Hence, | |
198 | * | |
199 | * B^2n/l - 1 < µ <= B^2/l | |
200 | * | |
201 | * and | |
202 | * | |
203 | * x/B^(n-1) - 1 < floor(x/B^(n-1)) <= x/B^(n-1) | |
204 | * | |
205 | * Multiplying these together, and dividing through by B^(n+1), gives | |
206 | * | |
207 | * floor(x/l - B^(n-1)/l - x/B^2n + 1/B^(n+1)) <= | |
208 | * q <= µ floor(x/B^(n-1))/B^(n+1) <= floor(x/l) | |
209 | * | |
210 | * Now, noticing that x < B^2n and l > B^(n-1) shows that x/B^2n and | |
211 | * B^(n-1)/l are each less than 1; hence | |
212 | * | |
213 | * floor(x/l) - 2 <= q <= floor(x/l) <= x/l | |
214 | * | |
215 | * Now we set r = x - q l. Certainly, r == x (mod l); and | |
216 | * | |
217 | * 0 <= r < x - l floor(x/l) + 2 l < 3 l < B^n | |
218 | */ | |
219 | ||
220 | /* Before we start on the fancy stuff, we need to resolve the pending | |
221 | * carries in x. We'll be doing the floor division by just ignoring some | |
222 | * of the pieces, and it would be bad if we missed some significant bits. | |
223 | * Of course, this means that we don't actually have to store the low | |
224 | * NPIECE - 1 pieces of the result. | |
225 | */ | |
226 | for (i = 0, c = 0; i < 2*npiece; i++) | |
227 | { c += x[i]; t[i] = c&m; c >>= piecewd; } | |
228 | ||
229 | /* Now we calculate q. If we calculate this in product-scanning order, we | |
230 | * can avoid having to store the low NPIECE + 1 pieces of the product as | |
231 | * long as we keep track of the carry out properly. Conveniently, NMU = | |
232 | * NPIECE + 1, which keeps the loop bounds easy in the first pass. | |
233 | * | |
234 | * Furthermore, because we know that r fits in NPIECE pieces, we only need | |
235 | * the low NPIECE pieces of q. | |
236 | */ | |
237 | for (i = 0, c = 0; i < npiece + 1; i++) { | |
238 | for (j = 0; j <= i; j++) | |
239 | c += (scaf_dblpiece)t[j + npiece - 1]*mu[i - j]; | |
240 | c >>= piecewd; | |
241 | } | |
242 | for (i = 0; i < npiece; i++) { | |
243 | for (j = i + 1; j < npiece + 1; j++) | |
244 | c += (scaf_dblpiece)t[j + npiece - 1]*mu[npiece + 1 + i - j]; | |
245 | q[i] = c&m; c >>= piecewd; | |
246 | } | |
247 | ||
248 | /* Next, we calculate r - q l in z. Product-scanning seems to be working | |
249 | * out for us, and this time it will save us needing a large temporary | |
250 | * space for the product q l as we go. On the downside, we have to track | |
251 | * the carries from the multiplication and subtraction separately. | |
252 | * | |
253 | * Notice that the result r is at most NPIECE pieces long, so we can stop | |
254 | * once we have that many. | |
255 | */ | |
256 | u = 1; c = 0; | |
257 | for (i = 0; i < npiece; i++) { | |
258 | for (j = 0; j <= i; j++) c += (scaf_dblpiece)q[j]*l[i - j]; | |
259 | u += t[i] + ((scaf_piece)(c&m) ^ m); | |
260 | z[i] = u&m; u >>= piecewd; c >>= piecewd; | |
261 | } | |
262 | ||
263 | /* Finally, two passes of conditional subtraction. Calculate t = z - l; if | |
264 | * there's no borrow out the top, then update z = t; otherwise leave t | |
265 | * alone. | |
266 | */ | |
267 | for (i = 0; i < 2; i++) { | |
268 | for (j = 0, u = 1; j < npiece; j++) { | |
269 | u += z[j] + (l[j] ^ m); | |
270 | t[j] = u&m; u >>= piecewd; | |
271 | } | |
06325636 | 272 | for (j = 0, u = -u; j < npiece; j++) z[j] = (t[j]&u) | (z[j]&~u); |
d56fd9d1 MW |
273 | } |
274 | } | |
275 | ||
276 | /*----- That's all, folks -------------------------------------------------*/ |