3 * Efficient reduction modulo sparse binary polynomials
5 * (c) 2004 Straylight/Edgeware
8 /*----- Licensing notice --------------------------------------------------*
10 * This file is part of Catacomb.
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28 /*----- Header files ------------------------------------------------------*/
30 #include <mLib/alloc.h>
31 #include <mLib/darray.h>
32 #include <mLib/macros.h>
36 #include "gfreduce-exp.h"
40 /*----- Data structures ---------------------------------------------------*/
42 DA_DECL(instr_v, gfreduce_instr);
44 /*----- Main code ---------------------------------------------------------*/
46 /* --- What's going on here? --- *
48 * Let's face it, @gfx_div@ sucks. It works (I hope), but it's not in any
49 * sense fast. Here, we do efficient reduction modulo sparse polynomials.
50 * (It works for arbitrary polynomials, but isn't efficient for dense ones.)
52 * Suppose that %$p = x^n + p'$% where %$p' = \sum_{0\le i<n} p_i x^i$%,
53 * hopefully with only a few %$p_i \ne 0$%. We're going to compile %$p$%
54 * into a sequence of instructions which can be used to perform reduction
55 * modulo %$p$%. The important observation is that
56 * %$x^n \equiv p' \pmod p$%.
58 * Suppose we're working with %$w$%-bit words; let %$n = N w + n'$% with
59 * %$0 \le n' < w$%. Let %$u(x)$% be some arbitrary polynomial. Write
60 * %$u = z x^k + u'$% with %$\deg u' < k \ge n$%. Then a reduction step uses
61 * that %$u \equiv u' + z p' x^{k-n} \pmod p$%: the right hand side has
62 * degree %$\max \{ \deg u', k + \deg p' - n + \deg z \} < \deg u$%, so this
63 * makes progress towards a complete reduction.
65 * The compiled instruction sequence computes
66 * %$u' + z p' x^{k-n} = u' + \sum_{0\le i<n} z x^{k-n+i}$%.
69 /* --- @gfreduce_create@ --- *
71 * Arguments: @gfreduce *r@ = structure to fill in
72 * @mp *x@ = a (hopefully sparse) polynomial
76 * Use: Initializes a context structure for reduction.
80 unsigned f; /* Flags */
81 #define f_lsr 1u /* Overflow from previous word */
82 #define f_load 2u /* Outstanding @LOAD@ */
83 #define f_fip 4u /* Final-pass offset is set */
84 instr_v iv; /* Instruction vector */
85 size_t fip; /* Offset for final-pass reduction */
86 size_t w; /* Currently loaded target word */
87 size_t wi; /* Left-shifts for current word */
88 gfreduce *r; /* Reduction context pointer */
91 #define INSTR(g_, op_, arg_) do { \
92 struct gen *_g = (g_); \
93 instr_v *_iv = &_g->iv; \
94 size_t _i = DA_LEN(_iv); \
97 DA(_iv)[_i].op = (op_); \
98 DA(_iv)[_i].arg = (arg_); \
102 static void emit_load(struct gen *g, size_t w)
104 /* --- If this is not the low-order word then note final-pass start --- *
106 * Once we've eliminated the whole high-degree words, there will possibly
107 * remain a few high-degree bits. We can further reduce the subject
108 * polynomial by subtracting an appropriate multiple of %$p'$%, but if we
109 * do this naively we'll end up addressing `low-order' words beyond the
110 * bottom of our input. We solve this problem by storing an alternative
111 * start position for this final pass (which works because we scan bits
115 if (!(g->f & f_fip) && w < g->r->lim) {
116 g->fip = DA_LEN(&g->iv);
120 /* --- Actually emit the instruction --- */
122 INSTR(g, GFRI_LOAD, w);
127 static void emit_right_shifts(struct gen *g)
132 /* --- Close off the current word --- *
134 * If we shifted into this current word with a nonzero bit offset, then
135 * we'll also need to arrange to perform a sequence of right shifts into
136 * the following word, which we might as well do by scanning the
137 * instruction sequence (which starts at @wi@).
139 * Either way, we leave a @LOAD@ unmatched if there was one before, in the
140 * hope that callers have an easier time; @g->w@ is updated to reflect the
141 * currently open word.
148 INSTR(g, GFRI_STORE, g->w);
149 emit_load(g, g->w - 1);
150 for (i = g->wi; i < wl; i++) {
152 assert(ip->op == GFRI_LSL);
154 INSTR(g, GFRI_LSR, MPW_BITS - ip->arg);
159 static void ensure_loaded(struct gen *g, size_t w)
161 if (!(g->f & f_load)) {
163 g->wi = DA_LEN(&g->iv);
164 } else if (w != g->w) {
165 emit_right_shifts(g);
167 INSTR(g, GFRI_STORE, g->w);
170 g->wi = DA_LEN(&g->iv);
174 void gfreduce_create(gfreduce *r, mp *p)
176 struct gen g = { 0, DA_INIT };
183 /* --- Sort out the easy stuff --- */
186 d = mp_bits(p); assert(d); d--;
192 r->mask = MPW(((mpw)-1) << dw);
197 /* --- How this works --- *
199 * The instruction sequence is run with two ambient parameters: a pointer
200 * (usually) just past the most significant word of the polynomial to be
201 * reduced; and a word %$z$% which is the multiple of %$p'$% we are meant
204 * The sequence visits each word of the polynomial at most once. Suppose
205 * %$u = z x^{w N} + u'$%; our pointer points just past the end of %$u'$%.
206 * Word %$I$% of %$u'$% will be affected by modulus bits %$p_i$% where
207 * %$(N - I - 1) w + 1 \le i \le (N - I + 1) w - 1$%, so %$p_i$% affects
208 * word %$I = \lceil (n - i + 1)/w \rceil$% and (if %$i$% is not a multiple
209 * of %$w$%) also word %$I - 1$%.
211 * We have four instructions: @LOAD@ reads a specified word of %$u$% into an
212 * accumulator, and @STORE@ stores it back (we'll always store back to the
213 * same word we most recently read, but this isn't a requirement); and
214 * @LSL@ and @LSR@, which XOR in appropriately shifted copies of %$z$% into
215 * the accumulator. So a typical program will contain sequences of @LSR@
216 * and @LSL@ instructions sandwiched between @LOAD@/@STORE@ pairs.
218 * We do a single right-to-left pass across %$p$%.
223 for (i = 0, mp_scan(&sc, p); mp_step(&sc) && i < d; i++) {
227 /* --- We've found a set bit, so work out which word it affects --- *
229 * In general, a bit affects two words: it needs to be shifted left into
230 * one, and shifted right into the next. We find the former here.
233 w = (d - i + MPW_BITS - 1)/MPW_BITS;
235 /* --- Concentrate on the appropriate word --- */
237 ensure_loaded(&g, w);
239 /* --- Accumulate a new @LSL@ instruction --- *
241 * If this was a nonzero shift, then we'll need to arrange to do right
242 * shifts into the following word.
245 INSTR(&g, GFRI_LSL, (bb + i)%MPW_BITS);
246 if ((bb + i)%MPW_BITS)
250 /* --- Wrapping up --- *
252 * We probably need a final @STORE@, and maybe a sequence of right shifts.
256 emit_right_shifts(&g);
257 INSTR(&g, GFRI_STORE, g.w);
260 /* --- Copy the instruction vector.
262 * If we've not set a final-pass offset yet then now would be an excellent
263 * time. Obviously it should be right at the end, because there's nothing
264 * for a final pass to do.
267 r->in = DA_LEN(&g.iv);
268 r->iv = xmalloc(r->in * sizeof(gfreduce_instr));
269 memcpy(r->iv, DA(&g.iv), r->in * sizeof(gfreduce_instr));
271 if (!(g.f & f_fip)) g.fip = DA_LEN(&g.iv);
272 r->fiv = r->iv + g.fip;
283 /* --- @gfreduce_destroy@ --- *
285 * Arguments: @gfreduce *r@ = structure to free
289 * Use: Reclaims the resources from a reduction context.
292 void gfreduce_destroy(gfreduce *r)
298 /* --- @gfreduce_dump@ --- *
300 * Arguments: @gfreduce *r@ = structure to dump
301 * @FILE *fp@ = file to dump on
305 * Use: Dumps a reduction context.
308 void gfreduce_dump(gfreduce *r, FILE *fp)
312 fprintf(fp, "poly = "); mp_writefile(r->p, fp, 16);
313 fprintf(fp, "\n lim = %lu; mask = %lx\n",
314 (unsigned long)r->lim, (unsigned long)r->mask);
315 for (i = 0; i < r->in; i++) {
316 static const char *opname[] = { "load", "lsl", "lsr", "store" };
317 if (&r->iv[i] == r->fiv)
318 fputs("final:\n", fp);
319 assert(r->iv[i].op < N(opname));
320 fprintf(fp, " %s %lu\n",
322 (unsigned long)r->iv[i].arg);
324 if (&r->iv[i] == r->fiv)
325 fputs("final:\n", fp);
328 /* --- @gfreduce_do@ --- *
330 * Arguments: @gfreduce *r@ = reduction context
331 * @mp *d@ = destination
334 * Returns: Destination, @x@ reduced modulo the reduction poly.
337 static void run(const gfreduce_instr *i, const gfreduce_instr *il,
342 for (; i < il; i++) {
344 case GFRI_LOAD: w = *(v - i->arg); break;
345 case GFRI_LSL: w ^= z << i->arg; break;
346 case GFRI_LSR: w ^= z >> i->arg; break;
347 case GFRI_STORE: *(v - i->arg) = MPW(w); break;
353 mp *gfreduce_do(gfreduce *r, mp *d, mp *x)
356 const gfreduce_instr *il;
359 /* --- Try to reuse the source's space --- */
363 MP_DEST(x, MP_LEN(x), x->f);
365 /* --- Do the reduction --- */
368 if (MP_LEN(x) >= r->lim) {
375 run(r->iv, il, vl, z);
379 while (*vl & r->mask) {
382 run(r->fiv, il, vl, z);
393 /* --- @gfreduce_sqrt@ --- *
395 * Arguments: @gfreduce *r@ = pointer to reduction context
396 * @mp *d@ = destination
397 * @mp *x@ = some polynomial
399 * Returns: The square root of @x@ modulo @r->p@, or null.
402 mp *gfreduce_sqrt(gfreduce *r, mp *d, mp *x)
405 mp *z, *spare = MP_NEW;
406 unsigned long m = mp_bits(r->p) - 1;
409 for (i = 0; i < m - 1; i++) {
410 mp *t = gf_sqr(spare, y);
412 y = gfreduce_do(r, t, t);
414 z = gf_sqr(spare, y);
415 z = gfreduce_do(r, z, z);
425 /* --- @gfreduce_trace@ --- *
427 * Arguments: @gfreduce *r@ = pointer to reduction context
428 * @mp *x@ = some polynomial
430 * Returns: The trace of @x@. (%$\Tr(x)=x + x^2 + \cdots + x^{2^{m-1}}$%
431 * if %$x \in \gf{2^m}$%).
434 int gfreduce_trace(gfreduce *r, mp *x)
438 unsigned long m = mp_bits(r->p) - 1;
442 for (i = 0; i < m - 1; i++) {
443 mp *t = gf_sqr(spare, y);
445 y = gfreduce_do(r, t, t);
454 /* --- @gfreduce_halftrace@ --- *
456 * Arguments: @gfreduce *r@ = pointer to reduction context
457 * @mp *d@ = destination
458 * @mp *x@ = some polynomial
460 * Returns: The half-trace of @x@.
461 * (%$\HfTr(x)= x + x^{2^2} + \cdots + x^{2^{m-1}}$%
462 * if %$x \in \gf{2^m}$% with %$m$% odd).
465 mp *gfreduce_halftrace(gfreduce *r, mp *d, mp *x)
469 unsigned long m = mp_bits(r->p) - 1;
473 for (i = 0; i < m - 1; i += 2) {
474 mp *t = gf_sqr(spare, y);
476 y = gfreduce_do(r, t, t);
477 t = gf_sqr(spare, y);
479 y = gfreduce_do(r, t, t);
486 /* --- @gfreduce_quadsolve@ --- *
488 * Arguments: @gfreduce *r@ = pointer to reduction context
489 * @mp *d@ = destination
490 * @mp *x@ = some polynomial
492 * Returns: A polynomial @y@ such that %$y^2 + y = x$%, or null.
495 mp *gfreduce_quadsolve(gfreduce *r, mp *d, mp *x)
497 unsigned long m = mp_bits(r->p) - 1;
502 d = gfreduce_halftrace(r, d, x);
504 mp *z, *w, *rho = MP_NEW;
506 grand *fr = fibrand_create(0);
510 rho = mprand(rho, m, fr, 0);
513 for (i = 0; i < m - 1; i++) {
514 t = gf_sqr(spare, z); spare = z; z = gfreduce_do(r, t, t);
515 t = gf_sqr(spare, w); spare = w; w = gfreduce_do(r, t, t);
516 t = gf_mul(spare, w, x); t = gfreduce_do(r, t, t); spare = t;
518 w = gf_add(w, w, rho);
529 fr->ops->destroy(fr);
533 t = gf_sqr(MP_NEW, d); t = gfreduce_do(r, t, t); t = gf_add(t, t, d);
540 if (d) d->v[0] &= ~(mpw)1;
544 /* --- @gfreduce_exp@ --- *
546 * Arguments: @gfreduce *gr@ = pointer to reduction context
547 * @mp *d@ = fake destination
551 * Returns: Result, %$a^e \bmod m$%.
554 mp *gfreduce_exp(gfreduce *gr, mp *d, mp *a, mp *e)
557 mp *spare = (e->f & MP_BURN) ? MP_NEWSEC : MP_NEW;
565 a = gf_modinv(a, a, gr->p);
566 if (MP_LEN(e) < EXP_THRESH)
577 /*----- Test rig ----------------------------------------------------------*/
581 static int vreduce(dstr *v)
583 mp *d = *(mp **)v[0].buf;
584 mp *n = *(mp **)v[1].buf;
585 mp *r = *(mp **)v[2].buf;
590 gfreduce_create(&rr, d);
591 c = gfreduce_do(&rr, MP_NEW, n);
593 fprintf(stderr, "\n*** reduction failed\n*** ");
594 gfreduce_dump(&rr, stderr);
595 fprintf(stderr, "\n*** n = "); mp_writefile(n, stderr, 16);
596 fprintf(stderr, "\n*** r = "); mp_writefile(r, stderr, 16);
597 fprintf(stderr, "\n*** c = "); mp_writefile(c, stderr, 16);
598 fprintf(stderr, "\n");
601 gfreduce_destroy(&rr);
602 mp_drop(n); mp_drop(d); mp_drop(r); mp_drop(c);
603 assert(mparena_count(MPARENA_GLOBAL) == 0);
607 static int vmodexp(dstr *v)
609 mp *p = *(mp **)v[0].buf;
610 mp *g = *(mp **)v[1].buf;
611 mp *x = *(mp **)v[2].buf;
612 mp *r = *(mp **)v[3].buf;
617 gfreduce_create(&rr, p);
618 c = gfreduce_exp(&rr, MP_NEW, g, x);
620 fprintf(stderr, "\n*** modexp failed\n*** ");
621 fprintf(stderr, "\n*** p = "); mp_writefile(p, stderr, 16);
622 fprintf(stderr, "\n*** g = "); mp_writefile(g, stderr, 16);
623 fprintf(stderr, "\n*** x = "); mp_writefile(x, stderr, 16);
624 fprintf(stderr, "\n*** c = "); mp_writefile(c, stderr, 16);
625 fprintf(stderr, "\n*** r = "); mp_writefile(r, stderr, 16);
626 fprintf(stderr, "\n");
629 gfreduce_destroy(&rr);
630 mp_drop(p); mp_drop(g); mp_drop(r); mp_drop(x); mp_drop(c);
631 assert(mparena_count(MPARENA_GLOBAL) == 0);
635 static int vsqrt(dstr *v)
637 mp *p = *(mp **)v[0].buf;
638 mp *x = *(mp **)v[1].buf;
639 mp *r = *(mp **)v[2].buf;
644 gfreduce_create(&rr, p);
645 c = gfreduce_sqrt(&rr, MP_NEW, x);
647 fprintf(stderr, "\n*** sqrt failed\n*** ");
648 fprintf(stderr, "\n*** p = "); mp_writefile(p, stderr, 16);
649 fprintf(stderr, "\n*** x = "); mp_writefile(x, stderr, 16);
650 fprintf(stderr, "\n*** c = "); mp_writefile(c, stderr, 16);
651 fprintf(stderr, "\n*** r = "); mp_writefile(r, stderr, 16);
652 fprintf(stderr, "\n");
655 gfreduce_destroy(&rr);
656 mp_drop(p); mp_drop(r); mp_drop(x); mp_drop(c);
657 assert(mparena_count(MPARENA_GLOBAL) == 0);
661 static int vtr(dstr *v)
663 mp *p = *(mp **)v[0].buf;
664 mp *x = *(mp **)v[1].buf;
665 int r = *(int *)v[2].buf, c;
669 gfreduce_create(&rr, p);
670 c = gfreduce_trace(&rr, x);
672 fprintf(stderr, "\n*** trace failed\n*** ");
673 fprintf(stderr, "\n*** p = "); mp_writefile(p, stderr, 16);
674 fprintf(stderr, "\n*** x = "); mp_writefile(x, stderr, 16);
675 fprintf(stderr, "\n*** c = %d", c);
676 fprintf(stderr, "\n*** r = %d", r);
677 fprintf(stderr, "\n");
680 gfreduce_destroy(&rr);
681 mp_drop(p); mp_drop(x);
682 assert(mparena_count(MPARENA_GLOBAL) == 0);
686 static int vhftr(dstr *v)
688 mp *p = *(mp **)v[0].buf;
689 mp *x = *(mp **)v[1].buf;
690 mp *r = *(mp **)v[2].buf;
695 gfreduce_create(&rr, p);
696 c = gfreduce_halftrace(&rr, MP_NEW, x);
698 fprintf(stderr, "\n*** halftrace failed\n*** ");
699 fprintf(stderr, "\n*** p = "); mp_writefile(p, stderr, 16);
700 fprintf(stderr, "\n*** x = "); mp_writefile(x, stderr, 16);
701 fprintf(stderr, "\n*** c = "); mp_writefile(c, stderr, 16);
702 fprintf(stderr, "\n*** r = "); mp_writefile(r, stderr, 16);
703 fprintf(stderr, "\n");
706 gfreduce_destroy(&rr);
707 mp_drop(p); mp_drop(r); mp_drop(x); mp_drop(c);
708 assert(mparena_count(MPARENA_GLOBAL) == 0);
712 static int vquad(dstr *v)
714 mp *p = *(mp **)v[0].buf;
715 mp *x = *(mp **)v[1].buf;
716 mp *r = *(mp **)v[2].buf;
721 gfreduce_create(&rr, p);
722 c = gfreduce_quadsolve(&rr, MP_NEW, x);
724 fprintf(stderr, "\n*** quadsolve failed\n*** ");
725 fprintf(stderr, "\n*** p = "); mp_writefile(p, stderr, 16);
726 fprintf(stderr, "\n*** x = "); mp_writefile(x, stderr, 16);
727 fprintf(stderr, "\n*** c = "); mp_writefile(c, stderr, 16);
728 fprintf(stderr, "\n*** r = "); mp_writefile(r, stderr, 16);
729 fprintf(stderr, "\n");
732 gfreduce_destroy(&rr);
733 mp_drop(p); mp_drop(r); mp_drop(x); mp_drop(c);
734 assert(mparena_count(MPARENA_GLOBAL) == 0);
738 static test_chunk defs[] = {
739 { "reduce", vreduce, { &type_mp, &type_mp, &type_mp, 0 } },
740 { "modexp", vmodexp, { &type_mp, &type_mp, &type_mp, &type_mp, 0 } },
741 { "sqrt", vsqrt, { &type_mp, &type_mp, &type_mp, 0 } },
742 { "trace", vtr, { &type_mp, &type_mp, &type_int, 0 } },
743 { "halftrace", vhftr, { &type_mp, &type_mp, &type_mp, 0 } },
744 { "quadsolve", vquad, { &type_mp, &type_mp, &type_mp, 0 } },
748 int main(int argc, char *argv[])
750 test_run(argc, argv, defs, SRCDIR"/t/gfreduce");
756 /*----- That's all, folks -------------------------------------------------*/