5 * Efficient reduction modulo sparse binary polynomials
7 * (c) 2004 Straylight/Edgeware
10 /*----- Licensing notice --------------------------------------------------*
12 * This file is part of Catacomb.
14 * Catacomb is free software; you can redistribute it and/or modify
15 * it under the terms of the GNU Library General Public License as
16 * published by the Free Software Foundation; either version 2 of the
17 * License, or (at your option) any later version.
19 * Catacomb is distributed in the hope that it will be useful,
20 * but WITHOUT ANY WARRANTY; without even the implied warranty of
21 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
22 * GNU Library General Public License for more details.
24 * You should have received a copy of the GNU Library General Public
25 * License along with Catacomb; if not, write to the Free
26 * Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
30 /*----- Header files ------------------------------------------------------*/
32 #include <mLib/alloc.h>
33 #include <mLib/darray.h>
34 #include <mLib/macros.h>
38 #include "gfreduce-exp.h"
42 /*----- Data structures ---------------------------------------------------*/
44 DA_DECL(instr_v
, gfreduce_instr
);
46 /*----- Main code ---------------------------------------------------------*/
48 /* --- What's going on here? --- *
50 * Let's face it, @gfx_div@ sucks. It works (I hope), but it's not in any
51 * sense fast. Here, we do efficient reduction modulo sparse polynomials.
53 * Suppose we have a polynomial @X@ we're trying to reduce mod @P@. If we
54 * take the topmost nonzero word of @X@, call it @w@, then we can eliminate
55 * it by subtracting off @w P x^{k}@ for an appropriate value of @k@. The
56 * trick is in observing that if @P@ is sparse we can do this multiplication
57 * and subtraction efficiently, just by XORing appropriate shifts of @w@ into
60 * The first tricky bit is in working out when to stop. I'll use eight-bit
61 * words to demonstrate what I'm talking about.
63 * xxxxxx xxxxxxxx xxxxxxxx xxxxxxxx xxxxxxxx xxxxxxxx
64 * 001ppppp pppppppp pppppppp pppppppp
66 * |<------------ bp ------------->|
67 * |<------------ nw --------------->|
69 * The trick of taking whole words off of @X@ stops working when there are
70 * only @nw@ words left. Then we have to mask off the bottom bits of @w@
74 /* --- @gfreduce_create@ --- *
76 * Arguments: @gfreduce *r@ = structure to fill in
77 * @mp *x@ = a (hopefully sparse) polynomial
81 * Use: Initializes a context structure for reduction.
84 void gfreduce_create(gfreduce
*r
, mp
*p
)
93 size_t w
, ww
, wi
, wl
, ll
, bb
;
95 /* --- Sort out the easy stuff --- */
97 d
= mp_bits(p
); assert(d
); d
--;
103 r
->mask
= MPW(((mpw
)-1) << dw
);
108 /* --- Stash a new instruction --- */
110 #define INSTR(op_, arg_) do { \
112 DA(&iv)[DA_LEN(&iv)].op = (op_); \
113 DA(&iv)[DA_LEN(&iv)].arg = (arg_); \
119 w
= (d
+ MPW_BITS
- 1)/MPW_BITS
;
125 for (i
= 0, mp_scan(&sc
, p
); mp_step(&sc
) && i
< d
; i
++) {
128 ww
= (d
- i
+ MPW_BITS
- 1)/MPW_BITS
;
131 INSTR(GFRI_STORE
, w
);
135 INSTR(GFRI_LOAD
, ww
);
137 INSTR(GFRI_LOAD
, w
- 1);
138 for (; wi
< wl
; wi
++) {
140 assert(ip
->op
== GFRI_LSL
);
142 INSTR(GFRI_LSR
, MPW_BITS
- ip
->arg
);
145 INSTR(GFRI_STORE
, w
- 1);
146 INSTR(GFRI_LOAD
, ww
);
153 INSTR(GFRI_LSL
, (bb
+ i
)%MPW_BITS
);
154 if ((bb
+ i
)%MPW_BITS
)
158 INSTR(GFRI_STORE
, w
);
162 INSTR(GFRI_LOAD
, w
- 1);
163 for (; wi
< wl
; wi
++) {
165 assert(ip
->op
== GFRI_LSL
);
167 INSTR(GFRI_LSR
, MPW_BITS
- ip
->arg
);
169 INSTR(GFRI_STORE
, w
- 1);
175 r
->iv
= xmalloc(r
->in
* sizeof(gfreduce_instr
));
177 memcpy(r
->iv
, DA(&iv
), r
->in
* sizeof(gfreduce_instr
));
181 /* --- @gfreduce_destroy@ --- *
183 * Arguments: @gfreduce *r@ = structure to free
187 * Use: Reclaims the resources from a reduction context.
190 void gfreduce_destroy(gfreduce
*r
)
196 /* --- @gfreduce_dump@ --- *
198 * Arguments: @gfreduce *r@ = structure to dump
199 * @FILE *fp@ = file to dump on
203 * Use: Dumps a reduction context.
206 void gfreduce_dump(gfreduce
*r
, FILE *fp
)
210 fprintf(fp
, "poly = "); mp_writefile(r
->p
, fp
, 16);
211 fprintf(fp
, "\n lim = %lu; mask = %lx\n",
212 (unsigned long)r
->lim
, (unsigned long)r
->mask
);
213 for (i
= 0; i
< r
->in
; i
++) {
214 static const char *opname
[] = { "load", "lsl", "lsr", "store" };
215 assert(r
->iv
[i
].op
< N(opname
));
216 fprintf(fp
, " %s %lu\n",
218 (unsigned long)r
->iv
[i
].arg
);
222 /* --- @gfreduce_do@ --- *
224 * Arguments: @gfreduce *r@ = reduction context
225 * @mp *d@ = destination
228 * Returns: Destination, @x@ reduced modulo the reduction poly.
231 static void run(const gfreduce_instr
*i
, const gfreduce_instr
*il
,
236 for (; i
< il
; i
++) {
238 case GFRI_LOAD
: w
= *(v
- i
->arg
); break;
239 case GFRI_LSL
: w
^= z
<< i
->arg
; break;
240 case GFRI_LSR
: w
^= z
>> i
->arg
; break;
241 case GFRI_STORE
: *(v
- i
->arg
) = MPW(w
); break;
247 mp
*gfreduce_do(gfreduce
*r
, mp
*d
, mp
*x
)
250 const gfreduce_instr
*il
;
253 /* --- Try to reuse the source's space --- */
257 MP_DEST(x
, MP_LEN(x
), x
->f
);
259 /* --- Do the reduction --- */
262 if (MP_LEN(x
) >= r
->lim
) {
269 run(r
->iv
, il
, vl
, z
);
273 while (*vl
& r
->mask
) {
276 run(r
->iv
, il
, vl
, z
);
287 /* --- @gfreduce_sqrt@ --- *
289 * Arguments: @gfreduce *r@ = pointer to reduction context
290 * @mp *d@ = destination
291 * @mp *x@ = some polynomial
293 * Returns: The square root of @x@ modulo @r->p@, or null.
296 mp
*gfreduce_sqrt(gfreduce
*r
, mp
*d
, mp
*x
)
299 mp
*z
, *spare
= MP_NEW
;
300 unsigned long m
= mp_bits(r
->p
) - 1;
303 for (i
= 0; i
< m
- 1; i
++) {
304 mp
*t
= gf_sqr(spare
, y
);
306 y
= gfreduce_do(r
, t
, t
);
308 z
= gf_sqr(spare
, y
);
309 z
= gfreduce_do(r
, z
, z
);
319 /* --- @gfreduce_trace@ --- *
321 * Arguments: @gfreduce *r@ = pointer to reduction context
322 * @mp *x@ = some polynomial
324 * Returns: The trace of @x@. (%$\Tr(x)=x + x^2 + \cdots + x^{2^{m-1}}$%
325 * if %$x \in \gf{2^m}$%).
328 int gfreduce_trace(gfreduce
*r
, mp
*x
)
332 unsigned long m
= mp_bits(r
->p
) - 1;
336 for (i
= 0; i
< m
- 1; i
++) {
337 mp
*t
= gf_sqr(spare
, y
);
339 y
= gfreduce_do(r
, t
, t
);
348 /* --- @gfreduce_halftrace@ --- *
350 * Arguments: @gfreduce *r@ = pointer to reduction context
351 * @mp *d@ = destination
352 * @mp *x@ = some polynomial
354 * Returns: The half-trace of @x@.
355 * (%$\HfTr(x)= x + x^{2^2} + \cdots + x^{2^{m-1}}$%
356 * if %$x \in \gf{2^m}$% with %$m$% odd).
359 mp
*gfreduce_halftrace(gfreduce
*r
, mp
*d
, mp
*x
)
363 unsigned long m
= mp_bits(r
->p
) - 1;
367 for (i
= 0; i
< m
- 1; i
+= 2) {
368 mp
*t
= gf_sqr(spare
, y
);
370 y
= gfreduce_do(r
, t
, t
);
371 t
= gf_sqr(spare
, y
);
373 y
= gfreduce_do(r
, t
, t
);
380 /* --- @gfreduce_quadsolve@ --- *
382 * Arguments: @gfreduce *r@ = pointer to reduction context
383 * @mp *d@ = destination
384 * @mp *x@ = some polynomial
386 * Returns: A polynomial @y@ such that %$y^2 + y = x$%, or null.
389 mp
*gfreduce_quadsolve(gfreduce
*r
, mp
*d
, mp
*x
)
391 unsigned long m
= mp_bits(r
->p
) - 1;
396 d
= gfreduce_halftrace(r
, d
, x
);
398 mp
*z
, *w
, *rho
= MP_NEW
;
400 grand
*fr
= fibrand_create(0);
404 rho
= mprand(rho
, m
, fr
, 0);
407 for (i
= 0; i
< m
- 1; i
++) {
408 t
= gf_sqr(spare
, z
); spare
= z
; z
= gfreduce_do(r
, t
, t
);
409 t
= gf_sqr(spare
, w
); spare
= w
; w
= gfreduce_do(r
, t
, t
);
410 t
= gf_mul(spare
, w
, x
); t
= gfreduce_do(r
, t
, t
); spare
= t
;
412 w
= gf_add(w
, w
, rho
);
423 fr
->ops
->destroy(fr
);
427 t
= gf_sqr(MP_NEW
, d
); t
= gfreduce_do(r
, t
, t
); t
= gf_add(t
, t
, d
);
434 if (d
) d
->v
[0] &= ~(mpw
)1;
438 /* --- @gfreduce_exp@ --- *
440 * Arguments: @gfreduce *gr@ = pointer to reduction context
441 * @mp *d@ = fake destination
445 * Returns: Result, %$a^e \bmod m$%.
448 mp
*gfreduce_exp(gfreduce
*gr
, mp
*d
, mp
*a
, mp
*e
)
451 mp
*spare
= (e
->f
& MP_BURN
) ? MP_NEWSEC
: MP_NEW
;
459 a
= gf_modinv(a
, a
, gr
->p
);
460 if (MP_LEN(e
) < EXP_THRESH
)
471 /*----- Test rig ----------------------------------------------------------*/
475 #define MP(x) mp_readstring(MP_NEW, #x, 0, 0)
477 static int vreduce(dstr
*v
)
479 mp
*d
= *(mp
**)v
[0].buf
;
480 mp
*n
= *(mp
**)v
[1].buf
;
481 mp
*r
= *(mp
**)v
[2].buf
;
486 gfreduce_create(&rr
, d
);
487 c
= gfreduce_do(&rr
, MP_NEW
, n
);
489 fprintf(stderr
, "\n*** reduction failed\n*** ");
490 gfreduce_dump(&rr
, stderr
);
491 fprintf(stderr
, "\n*** n = "); mp_writefile(n
, stderr
, 16);
492 fprintf(stderr
, "\n*** r = "); mp_writefile(r
, stderr
, 16);
493 fprintf(stderr
, "\n*** c = "); mp_writefile(c
, stderr
, 16);
494 fprintf(stderr
, "\n");
497 gfreduce_destroy(&rr
);
498 mp_drop(n
); mp_drop(d
); mp_drop(r
); mp_drop(c
);
499 assert(mparena_count(MPARENA_GLOBAL
) == 0);
503 static int vmodexp(dstr
*v
)
505 mp
*p
= *(mp
**)v
[0].buf
;
506 mp
*g
= *(mp
**)v
[1].buf
;
507 mp
*x
= *(mp
**)v
[2].buf
;
508 mp
*r
= *(mp
**)v
[3].buf
;
513 gfreduce_create(&rr
, p
);
514 c
= gfreduce_exp(&rr
, MP_NEW
, g
, x
);
516 fprintf(stderr
, "\n*** modexp failed\n*** ");
517 fprintf(stderr
, "\n*** p = "); mp_writefile(p
, stderr
, 16);
518 fprintf(stderr
, "\n*** g = "); mp_writefile(g
, stderr
, 16);
519 fprintf(stderr
, "\n*** x = "); mp_writefile(x
, stderr
, 16);
520 fprintf(stderr
, "\n*** c = "); mp_writefile(c
, stderr
, 16);
521 fprintf(stderr
, "\n*** r = "); mp_writefile(r
, stderr
, 16);
522 fprintf(stderr
, "\n");
525 gfreduce_destroy(&rr
);
526 mp_drop(p
); mp_drop(g
); mp_drop(r
); mp_drop(x
); mp_drop(c
);
527 assert(mparena_count(MPARENA_GLOBAL
) == 0);
531 static int vsqrt(dstr
*v
)
533 mp
*p
= *(mp
**)v
[0].buf
;
534 mp
*x
= *(mp
**)v
[1].buf
;
535 mp
*r
= *(mp
**)v
[2].buf
;
540 gfreduce_create(&rr
, p
);
541 c
= gfreduce_sqrt(&rr
, MP_NEW
, x
);
543 fprintf(stderr
, "\n*** sqrt failed\n*** ");
544 fprintf(stderr
, "\n*** p = "); mp_writefile(p
, stderr
, 16);
545 fprintf(stderr
, "\n*** x = "); mp_writefile(x
, stderr
, 16);
546 fprintf(stderr
, "\n*** c = "); mp_writefile(c
, stderr
, 16);
547 fprintf(stderr
, "\n*** r = "); mp_writefile(r
, stderr
, 16);
548 fprintf(stderr
, "\n");
551 gfreduce_destroy(&rr
);
552 mp_drop(p
); mp_drop(r
); mp_drop(x
); mp_drop(c
);
553 assert(mparena_count(MPARENA_GLOBAL
) == 0);
557 static int vtr(dstr
*v
)
559 mp
*p
= *(mp
**)v
[0].buf
;
560 mp
*x
= *(mp
**)v
[1].buf
;
561 int r
= *(int *)v
[2].buf
, c
;
565 gfreduce_create(&rr
, p
);
566 c
= gfreduce_trace(&rr
, x
);
568 fprintf(stderr
, "\n*** trace failed\n*** ");
569 fprintf(stderr
, "\n*** p = "); mp_writefile(p
, stderr
, 16);
570 fprintf(stderr
, "\n*** x = "); mp_writefile(x
, stderr
, 16);
571 fprintf(stderr
, "\n*** c = %d", c
);
572 fprintf(stderr
, "\n*** r = %d", r
);
573 fprintf(stderr
, "\n");
576 gfreduce_destroy(&rr
);
577 mp_drop(p
); mp_drop(x
);
578 assert(mparena_count(MPARENA_GLOBAL
) == 0);
582 static int vhftr(dstr
*v
)
584 mp
*p
= *(mp
**)v
[0].buf
;
585 mp
*x
= *(mp
**)v
[1].buf
;
586 mp
*r
= *(mp
**)v
[2].buf
;
591 gfreduce_create(&rr
, p
);
592 c
= gfreduce_halftrace(&rr
, MP_NEW
, x
);
594 fprintf(stderr
, "\n*** halftrace failed\n*** ");
595 fprintf(stderr
, "\n*** p = "); mp_writefile(p
, stderr
, 16);
596 fprintf(stderr
, "\n*** x = "); mp_writefile(x
, stderr
, 16);
597 fprintf(stderr
, "\n*** c = "); mp_writefile(c
, stderr
, 16);
598 fprintf(stderr
, "\n*** r = "); mp_writefile(r
, stderr
, 16);
599 fprintf(stderr
, "\n");
602 gfreduce_destroy(&rr
);
603 mp_drop(p
); mp_drop(r
); mp_drop(x
); mp_drop(c
);
604 assert(mparena_count(MPARENA_GLOBAL
) == 0);
608 static int vquad(dstr
*v
)
610 mp
*p
= *(mp
**)v
[0].buf
;
611 mp
*x
= *(mp
**)v
[1].buf
;
612 mp
*r
= *(mp
**)v
[2].buf
;
617 gfreduce_create(&rr
, p
);
618 c
= gfreduce_quadsolve(&rr
, MP_NEW
, x
);
620 fprintf(stderr
, "\n*** quadsolve failed\n*** ");
621 fprintf(stderr
, "\n*** p = "); mp_writefile(p
, stderr
, 16);
622 fprintf(stderr
, "\n*** x = "); mp_writefile(x
, stderr
, 16);
623 fprintf(stderr
, "\n*** c = "); mp_writefile(c
, stderr
, 16);
624 fprintf(stderr
, "\n*** r = "); mp_writefile(r
, stderr
, 16);
625 fprintf(stderr
, "\n");
628 gfreduce_destroy(&rr
);
629 mp_drop(p
); mp_drop(r
); mp_drop(x
); mp_drop(c
);
630 assert(mparena_count(MPARENA_GLOBAL
) == 0);
634 static test_chunk defs
[] = {
635 { "reduce", vreduce
, { &type_mp
, &type_mp
, &type_mp
, 0 } },
636 { "modexp", vmodexp
, { &type_mp
, &type_mp
, &type_mp
, &type_mp
, 0 } },
637 { "sqrt", vsqrt
, { &type_mp
, &type_mp
, &type_mp
, 0 } },
638 { "trace", vtr
, { &type_mp
, &type_mp
, &type_int
, 0 } },
639 { "halftrace", vhftr
, { &type_mp
, &type_mp
, &type_mp
, 0 } },
640 { "quadsolve", vquad
, { &type_mp
, &type_mp
, &type_mp
, 0 } },
644 int main(int argc
, char *argv
[])
646 test_run(argc
, argv
, defs
, SRCDIR
"/tests/gfreduce");
652 /*----- That's all, folks -------------------------------------------------*/