| 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 | /*----- Debugging utilties ------------------------------------------------*/ |
| 35 | |
| 36 | #ifdef SCAF_DEBUG |
| 37 | |
| 38 | #include <stdio.h> |
| 39 | |
| 40 | #include "mp.h" |
| 41 | #include "mpint.h" |
| 42 | #include "mptext.h" |
| 43 | |
| 44 | static void scaf_dump(const char *what, const scaf_piece *x, |
| 45 | size_t npiece, size_t piecewd) |
| 46 | { |
| 47 | mp *y = MP_ZERO, *t = MP_NEW; |
| 48 | size_t i; |
| 49 | unsigned o = 0; |
| 50 | |
| 51 | for (i = 0; i < npiece; i++) { |
| 52 | t = mp_fromuint64(t, x[i]); |
| 53 | t = mp_lsl(t, t, o); |
| 54 | y = mp_add(y, y, t); |
| 55 | o += piecewd; |
| 56 | } |
| 57 | printf(";; %s", what); MP_PRINT("", y); putchar('\n'); |
| 58 | mp_drop(y); mp_drop(t); |
| 59 | } |
| 60 | |
| 61 | static void scaf_dumpdbl(const char *what, const scaf_dblpiece *x, |
| 62 | size_t npiece, size_t piecewd) |
| 63 | { |
| 64 | mp *y = MP_ZERO, *t = MP_NEW; |
| 65 | size_t i; |
| 66 | unsigned o = 0; |
| 67 | |
| 68 | for (i = 0; i < npiece; i++) { |
| 69 | t = mp_fromuint64(t, x[i]); |
| 70 | t = mp_lsl(t, t, o); |
| 71 | y = mp_add(y, y, t); |
| 72 | o += piecewd; |
| 73 | } |
| 74 | printf(";; %s", what); MP_PRINT("", y); putchar('\n'); |
| 75 | mp_drop(y); mp_drop(t); |
| 76 | } |
| 77 | |
| 78 | #endif |
| 79 | |
| 80 | /*----- Main code ---------------------------------------------------------*/ |
| 81 | |
| 82 | /* --- @scaf_load@ --- * |
| 83 | * |
| 84 | * Arguments: @scaf_piece *z@ = where to write the result |
| 85 | * @const octet *b@ = source buffer to read |
| 86 | * @size_t sz@ = size of the source buffer |
| 87 | * @size_t npiece@ = number of pieces to read |
| 88 | * @unsigned piecewd@ = nominal width of pieces in bits |
| 89 | * |
| 90 | * Returns: --- |
| 91 | * |
| 92 | * Use: Loads a little-endian encoded scalar into a vector @z@ of |
| 93 | * single-precision pieces. |
| 94 | */ |
| 95 | |
| 96 | void scaf_load(scaf_piece *z, const octet *b, size_t sz, |
| 97 | size_t npiece, unsigned piecewd) |
| 98 | { |
| 99 | uint32 a, m = ((scaf_piece)1 << piecewd) - 1; |
| 100 | unsigned i, j, n; |
| 101 | |
| 102 | for (i = j = n = 0, a = 0; i < sz; i++) { |
| 103 | a |= b[i] << n; n += 8; |
| 104 | if (n >= piecewd) { |
| 105 | z[j++] = a&m; a >>= piecewd; n -= piecewd; |
| 106 | if (j >= npiece) return; |
| 107 | } |
| 108 | } |
| 109 | z[j++] = a; |
| 110 | while (j < npiece) z[j++] = 0; |
| 111 | } |
| 112 | |
| 113 | /* --- @scaf_loaddbl@ --- * |
| 114 | * |
| 115 | * Arguments: @scaf_dblpiece *z@ = where to write the result |
| 116 | * @const octet *b@ = source buffer to read |
| 117 | * @size_t sz@ = size of the source buffer |
| 118 | * @size_t npiece@ = number of pieces to read |
| 119 | * @unsigned piecewd@ = nominal width of pieces in bits |
| 120 | * |
| 121 | * Returns: --- |
| 122 | * |
| 123 | * Use: Loads a little-endian encoded scalar into a vector @z@ of |
| 124 | * double-precision pieces. |
| 125 | */ |
| 126 | |
| 127 | void scaf_loaddbl(scaf_dblpiece *z, const octet *b, size_t sz, |
| 128 | size_t npiece, unsigned piecewd) |
| 129 | { |
| 130 | uint32 a, m = ((scaf_piece)1 << piecewd) - 1; |
| 131 | unsigned i, j, n; |
| 132 | |
| 133 | for (i = j = n = 0, a = 0; i < sz; i++) { |
| 134 | a |= b[i] << n; n += 8; |
| 135 | if (n >= piecewd) { |
| 136 | z[j++] = a&m; a >>= piecewd; n -= piecewd; |
| 137 | if (j >= npiece) return; |
| 138 | } |
| 139 | } |
| 140 | z[j++] = a; |
| 141 | while (j < npiece) z[j++] = 0; |
| 142 | } |
| 143 | |
| 144 | /* --- @scaf_store@ --- * |
| 145 | * |
| 146 | * Arguments: @octet *b@ = buffer to fill in |
| 147 | * @size_t sz@ = size of the buffer |
| 148 | * @const scaf_piece *x@ = scalar to store |
| 149 | * @size_t npiece@ = number of pieces in @x@ |
| 150 | * @unsigned piecewd@ = nominal width of pieces in bits |
| 151 | * |
| 152 | * Returns: --- |
| 153 | * |
| 154 | * Use: Stores a scalar in a vector of single-precison pieces as a |
| 155 | * little-endian vector of bytes. |
| 156 | */ |
| 157 | |
| 158 | void scaf_store(octet *b, size_t sz, const scaf_piece *x, |
| 159 | size_t npiece, unsigned piecewd) |
| 160 | { |
| 161 | uint32 a; |
| 162 | unsigned i, j, n; |
| 163 | |
| 164 | for (i = j = n = 0, a = 0; i < npiece; i++) { |
| 165 | a |= x[i] << n; n += piecewd; |
| 166 | while (n >= 8) { |
| 167 | b[j++] = a&0xffu; a >>= 8; n -= 8; |
| 168 | if (j >= sz) return; |
| 169 | } |
| 170 | } |
| 171 | b[j++] = a; |
| 172 | memset(b + j, 0, sz - j); |
| 173 | } |
| 174 | |
| 175 | /* --- @scaf_mul@ --- * |
| 176 | * |
| 177 | * Arguments: @scaf_dblpiece *z@ = where to put the answer |
| 178 | * @const scaf_piece *x, *y@ = the operands |
| 179 | * @size_t npiece@ = the length of the operands |
| 180 | * |
| 181 | * Returns: --- |
| 182 | * |
| 183 | * Use: Multiply two scalars. The destination must have space for |
| 184 | * @2*npiece@ pieces (though the last one will always be zero). |
| 185 | * The result is not reduced. |
| 186 | */ |
| 187 | |
| 188 | void scaf_mul(scaf_dblpiece *z, const scaf_piece *x, const scaf_piece *y, |
| 189 | size_t npiece) |
| 190 | { |
| 191 | unsigned i, j; |
| 192 | |
| 193 | for (i = 0; i < 2*npiece; i++) z[i] = 0; |
| 194 | |
| 195 | for (i = 0; i < npiece; i++) |
| 196 | for (j = 0; j < npiece; j++) |
| 197 | z[i + j] += (scaf_dblpiece)x[i]*y[j]; |
| 198 | } |
| 199 | |
| 200 | /* --- @scaf_reduce@ --- * |
| 201 | * |
| 202 | * Arguments: @scaf_piece *z@ = where to write the result |
| 203 | * @const scaf_dblpiece *x@ = the operand to reduce |
| 204 | * @const scaf_piece *l@ = the modulus, in internal format |
| 205 | * @const scaf_piece *mu@ = scaled approximation to @1/l@ |
| 206 | * @size_t npiece@ = number of pieces in @l@ |
| 207 | * @unsigned piecewd@ = nominal width of a piece in bits |
| 208 | * @scaf_piece *scratch@ = @3*npiece@ scratch pieces |
| 209 | * |
| 210 | * Returns: --- |
| 211 | * |
| 212 | * Use: Reduce @x@ (a vector of @2*npiece@ double-precision pieces) |
| 213 | * modulo @l@ (a vector of @npiece@ single-precision pieces), |
| 214 | * writing the result to @z@. |
| 215 | * |
| 216 | * Write @n = npiece@, @w = piecewd@, and %$B = 2^w$%. The |
| 217 | * operand @mu@ must contain %$\lfloor B^{2n}/l \rfloor$%, in |
| 218 | * @npiece + 1@ pieces. Furthermore, we must have |
| 219 | * %$3 l < B^n$%. (Fiddle with %$w$% if necessary.) |
| 220 | */ |
| 221 | |
| 222 | void scaf_reduce(scaf_piece *z, const scaf_dblpiece *x, |
| 223 | const scaf_piece *l, const scaf_piece *mu, |
| 224 | size_t npiece, unsigned piecewd, scaf_piece *scratch) |
| 225 | { |
| 226 | unsigned i, j; |
| 227 | scaf_piece *t = scratch, *q = scratch + 2*npiece; |
| 228 | scaf_piece u, m = ((scaf_piece)1 << piecewd) - 1; |
| 229 | scaf_dblpiece c; |
| 230 | |
| 231 | /* This here is the hard part. |
| 232 | * |
| 233 | * Let w = PIECEWD, let n = NPIECE, and let B = 2^w. We must have |
| 234 | * B^(n-1) <= l < B^n. |
| 235 | * |
| 236 | * The argument MU contains pieces of the quantity µ = floor(B^2n/l), which |
| 237 | * is a scaled approximation to 1/l. We'll calculate |
| 238 | * |
| 239 | * q = floor(µ floor(x/B^(n-1))/B^(n+1)) |
| 240 | * |
| 241 | * which is an underestimate of x/l. |
| 242 | * |
| 243 | * With a bit more precision: by definition, u - 1 < floor(u) <= u. Hence, |
| 244 | * |
| 245 | * B^2n/l - 1 < µ <= B^2/l |
| 246 | * |
| 247 | * and |
| 248 | * |
| 249 | * x/B^(n-1) - 1 < floor(x/B^(n-1)) <= x/B^(n-1) |
| 250 | * |
| 251 | * Multiplying these together, and dividing through by B^(n+1), gives |
| 252 | * |
| 253 | * floor(x/l - B^(n-1)/l - x/B^2n + 1/B^(n+1)) <= |
| 254 | * q <= µ floor(x/B^(n-1))/B^(n+1) <= floor(x/l) |
| 255 | * |
| 256 | * Now, noticing that x < B^2n and l > B^(n-1) shows that x/B^2n and |
| 257 | * B^(n-1)/l are each less than 1; hence |
| 258 | * |
| 259 | * floor(x/l) - 2 <= q <= floor(x/l) <= x/l |
| 260 | * |
| 261 | * Now we set r = x - q l. Certainly, r == x (mod l); and |
| 262 | * |
| 263 | * 0 <= r < x - l floor(x/l) + 2 l < 3 l < B^n |
| 264 | */ |
| 265 | |
| 266 | /* Before we start on the fancy stuff, we need to resolve the pending |
| 267 | * carries in x. We'll be doing the floor division by just ignoring some |
| 268 | * of the pieces, and it would be bad if we missed some significant bits. |
| 269 | * Of course, this means that we don't actually have to store the low |
| 270 | * NPIECE - 1 pieces of the result. |
| 271 | */ |
| 272 | for (i = 0, c = 0; i < 2*npiece; i++) |
| 273 | { c += x[i]; t[i] = c&m; c >>= piecewd; } |
| 274 | |
| 275 | /* Now we calculate q. If we calculate this in product-scanning order, we |
| 276 | * can avoid having to store the low NPIECE + 1 pieces of the product as |
| 277 | * long as we keep track of the carry out properly. Conveniently, NMU = |
| 278 | * NPIECE + 1, which keeps the loop bounds easy in the first pass. |
| 279 | * |
| 280 | * Furthermore, because we know that r fits in NPIECE pieces, we only need |
| 281 | * the low NPIECE pieces of q. |
| 282 | */ |
| 283 | for (i = 0, c = 0; i < npiece + 1; i++) { |
| 284 | for (j = 0; j <= i; j++) |
| 285 | c += (scaf_dblpiece)t[j + npiece - 1]*mu[i - j]; |
| 286 | c >>= piecewd; |
| 287 | } |
| 288 | for (i = 0; i < npiece; i++) { |
| 289 | for (j = i + 1; j < npiece + 1; j++) |
| 290 | c += (scaf_dblpiece)t[j + npiece - 1]*mu[npiece + 1 + i - j]; |
| 291 | q[i] = c&m; c >>= piecewd; |
| 292 | } |
| 293 | |
| 294 | /* Next, we calculate r - q l in z. Product-scanning seems to be working |
| 295 | * out for us, and this time it will save us needing a large temporary |
| 296 | * space for the product q l as we go. On the downside, we have to track |
| 297 | * the carries from the multiplication and subtraction separately. |
| 298 | * |
| 299 | * Notice that the result r is at most NPIECE pieces long, so we can stop |
| 300 | * once we have that many. |
| 301 | */ |
| 302 | u = 1; c = 0; |
| 303 | for (i = 0; i < npiece; i++) { |
| 304 | for (j = 0; j <= i; j++) c += (scaf_dblpiece)q[j]*l[i - j]; |
| 305 | u += t[i] + ((scaf_piece)(c&m) ^ m); |
| 306 | z[i] = u&m; u >>= piecewd; c >>= piecewd; |
| 307 | } |
| 308 | |
| 309 | /* Finally, two passes of conditional subtraction. Calculate t = z - l; if |
| 310 | * there's no borrow out the top, then update z = t; otherwise leave t |
| 311 | * alone. |
| 312 | */ |
| 313 | for (i = 0; i < 2; i++) { |
| 314 | for (j = 0, u = 1; j < npiece; j++) { |
| 315 | u += z[j] + (l[j] ^ m); |
| 316 | t[j] = u&m; u >>= piecewd; |
| 317 | } |
| 318 | for (j = 0, u = -u; j < npiece; j++) z[j] = (t[j]&u) | (z[j]&~u); |
| 319 | } |
| 320 | } |
| 321 | |
| 322 | /*----- That's all, folks -------------------------------------------------*/ |