[u/mdw/catacomb] / math / gfreduce.c

/* -*-c-*-
 *
 * Efficient reduction modulo sparse binary polynomials
 *
 * (c) 2004 Straylight/Edgeware
 */

/*----- Licensing notice --------------------------------------------------*
 *
 * This file is part of Catacomb.
 *
 * Catacomb is free software; you can redistribute it and/or modify
 * it under the terms of the GNU Library General Public License as
 * published by the Free Software Foundation; either version 2 of the
 * License, or (at your option) any later version.
 *
 * Catacomb is distributed in the hope that it will be useful,
 * but WITHOUT ANY WARRANTY; without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 * GNU Library General Public License for more details.
 *
 * You should have received a copy of the GNU Library General Public
 * License along with Catacomb; if not, write to the Free
 * Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
 * MA 02111-1307, USA.
 */

/*----- Header files ------------------------------------------------------*/

#include <mLib/alloc.h>
#include <mLib/darray.h>
#include <mLib/macros.h>

#include "gf.h"
#include "gfreduce.h"
#include "gfreduce-exp.h"
#include "fibrand.h"
#include "mprand.h"

/*----- Data structures ---------------------------------------------------*/

DA_DECL(instr_v, gfreduce_instr);

/*----- Main code ---------------------------------------------------------*/

/* --- What's going on here? --- *
 *
 * Let's face it, @gfx_div@ sucks.  It works (I hope), but it's not in any
 * sense fast.  Here, we do efficient reduction modulo sparse polynomials.
 * (It works for arbitrary polynomials, but isn't efficient for dense ones.)
 *
 * Suppose that %$p(x) = x^n + p'(x) = \sum_{0\le i<n} p_i x^i$%, hopefully
 * with only a few other %$p_i \ne 0$%.  We're going to compile %$p$% into a
 * sequence of instructions which can be used to perform reduction modulo
 * %$p$%.  The important observation is that %$x^n \equiv p' \pmod p$%.
 *
 * Suppose we're working with %$w$%-bit words; let %$n = N w + n'$% with
 * %$0 \le n' < w$%.  Let %$u(x)$% be some arbitrary polynomial.  Write
 * %$u = z x^k + u'$% with %$\deg u' < k \ge n$%; then a reduction step uses
 * that %$u \equiv u' + z p' x^{k-n} \pmod p$%: the right hand side has
 * degree %$\max \{ \deg u', k + \deg p' - n + \deg z \} < \deg u$%, so this
 * makes progress towards a complete reduction.
 *
 * The compiled instruction sequence computes
 * %$u' + z p' x^{k-n} = u' + \sum_{0\le i<n} z x^{k-n+i}$%.
 */

/* --- @gfreduce_create@ --- *
 *
 * Arguments:   @gfreduce *r@ = structure to fill in
 *              @mp *x@ = a (hopefully sparse) polynomial
 *
 * Returns:     ---
 *
 * Use:         Initializes a context structure for reduction.
 */

struct gen {
  unsigned f;                           /* Flags */
#define f_lsr 1u                        /*   Overflow from previous word */
#define f_load 2u                       /*   Outstanding @LOAD@ */
  instr_v iv;                           /* Instruction vector */
  size_t w;                             /* Currently loaded target word */
  size_t wi;                            /* Left-shifts for current word */
};

#define INSTR(g_, op_, arg_) do {                                       \
  struct gen *_g = (g_);                                                \
  instr_v *_iv = &_g->iv;                                               \
  size_t _i = DA_LEN(_iv);                                              \
                                                                        \
  DA_ENSURE(_iv, 1);                                                    \
  DA(_iv)[_i].op = (op_);                                               \
  DA(_iv)[_i].arg = (arg_);                                             \
  DA_EXTEND(_iv, 1);                                                    \
} while (0)

static void emit_load(struct gen *g, size_t w)
{
  INSTR(g, GFRI_LOAD, w);
  g->f |= f_load;
  g->w = w;
}

static void emit_right_shifts(struct gen *g)
{
  gfreduce_instr *ip;
  size_t i, wl;

  /* --- Close off the current word --- *
   *
   * If we shifted into this current word with a nonzero bit offset, then
   * we'll also need to arrange to perform a sequence of right shifts into
   * the following word, which we might as well do by scanning the
   * instruction sequence (which starts at @wi@).
   *
   * Either way, we leave a @LOAD@ unmatched if there was one before, in the
   * hope that callers have an easier time; @g->w@ is updated to reflect the
   * currently open word.
   */

  if (!(g->f & f_lsr))
    return;

  wl = DA_LEN(&g->iv);
  INSTR(g, GFRI_STORE, g->w);
  emit_load(g, g->w - 1);
  for (i = g->wi; i < wl; i++) {
    ip = &DA(&g->iv)[i];
    assert(ip->op == GFRI_LSL);
    if (ip->arg)
      INSTR(g, GFRI_LSR, MPW_BITS - ip->arg);
  }
  g->f &= ~f_lsr;
}

static void ensure_loaded(struct gen *g, size_t w)
{
  if (!(g->f & f_load)) {
    emit_load(g, w);
    g->wi = DA_LEN(&g->iv);
  } else if (w != g->w) {
    emit_right_shifts(g);
    if (w != g->w) {
      INSTR(g, GFRI_STORE, g->w);
      emit_load(g, w);
    }
    g->wi = DA_LEN(&g->iv);
  }
}

void gfreduce_create(gfreduce *r, mp *p)
{
  struct gen g = { 0, DA_INIT };
  unsigned long d;
  unsigned dw;
  mpscan sc;
  unsigned long i;
  size_t w, bb;

  /* --- Sort out the easy stuff --- */

  d = mp_bits(p); assert(d); d--;
  r->lim = d/MPW_BITS;
  dw = d%MPW_BITS;
  if (!dw)
    r->mask = 0;
  else {
    r->mask = MPW(((mpw)-1) << dw);
    r->lim++;
  }
  r->p = mp_copy(p);

  /* --- How this works --- *
   *
   * The instruction sequence is run with two ambient parameters: a pointer
   * (usually) just past the most significant word of the polynomial to be
   * reduced; and a word %$z$% which is the multiple of %$p'$% we are meant
   * to add.
   *
   * The sequence visits each word of the polynomial at most once.  Suppose
   * %$u = z x^{w N} + u'$%; our pointer points just past the end of %$u'$%.
   * Word %$I$% of %$u'$% will be affected by modulus bits %$p_i$% where
   * %$(N - I - 1) w + 1 \le i \le (N - I + 1) w - 1$%, so %$p_i$% affects
   * word %$I = \lceil (n - i + 1)/w \rceil$% and (if %$i$% is not a multiple
   * of %$w$%) also word %$I - 1$%.
   *
   * We have four instructions: @LOAD@ reads a specified word of %$u$% into an
   * accumulator, and @STORE@ stores it back (we'll always store back to the
   * same word we most recently read, but this isn't a requirement); and
   * @LSL@ and @LSR@, which XOR in appropriately shifted copies of %$z$% into
   * the accumulator.  So a typical program will contain sequences of @LSR@
   * and @LSL@ instructions sandwiched between @LOAD@/@STORE@ pairs.
   *
   * We do a single right-to-left pass across %$p$%.
   */

  bb = MPW_BITS - dw;

  for (i = 0, mp_scan(&sc, p); mp_step(&sc) && i < d; i++) {
    if (!mp_bit(&sc))
      continue;

    /* --- We've found a set bit, so work out which word it affects --- *
     *
     * In general, a bit affects two words: it needs to be shifted left into
     * one, and shifted right into the next.  We find the former here.
     */

    w = (d - i + MPW_BITS - 1)/MPW_BITS;

    /* --- Concentrate on the appropriate word --- */

    ensure_loaded(&g, w);

    /* --- Accumulate a new @LSL@ instruction --- *
     *
     * If this was a nonzero shift, then we'll need to arrange to do right
     * shifts into the following word.
     */

    INSTR(&g, GFRI_LSL, (bb + i)%MPW_BITS);
    if ((bb + i)%MPW_BITS)
      g.f |= f_lsr;
  }

  /* --- Wrapping up --- *
   *
   * We probably need a final @STORE@, and maybe a sequence of right shifts.
   */

  if (g.f & f_load) {
    emit_right_shifts(&g);
    INSTR(&g, GFRI_STORE, g.w);
  }

  r->in = DA_LEN(&g.iv);
  r->iv = xmalloc(r->in * sizeof(gfreduce_instr));
  memcpy(r->iv, DA(&g.iv), r->in * sizeof(gfreduce_instr));
  DA_DESTROY(&g.iv);
}

#undef INSTR

#undef f_lsr
#undef f_load

/* --- @gfreduce_destroy@ --- *
 *
 * Arguments:   @gfreduce *r@ = structure to free
 *
 * Returns:     ---
 *
 * Use:         Reclaims the resources from a reduction context.
 */

void gfreduce_destroy(gfreduce *r)
{
  mp_drop(r->p);
  xfree(r->iv);
}

/* --- @gfreduce_dump@ --- *
 *
 * Arguments:   @gfreduce *r@ = structure to dump
 *              @FILE *fp@ = file to dump on
 *
 * Returns:     ---
 *
 * Use:         Dumps a reduction context.
 */

void gfreduce_dump(gfreduce *r, FILE *fp)
{
  size_t i;

  fprintf(fp, "poly = "); mp_writefile(r->p, fp, 16);
  fprintf(fp, "\n  lim = %lu; mask = %lx\n",
          (unsigned long)r->lim, (unsigned long)r->mask);
  for (i = 0; i < r->in; i++) {
    static const char *opname[] = { "load", "lsl", "lsr", "store" };
    assert(r->iv[i].op < N(opname));
    fprintf(fp, "  %s %lu\n",
            opname[r->iv[i].op],
            (unsigned long)r->iv[i].arg);
  }
}

/* --- @gfreduce_do@ --- *
 *
 * Arguments:   @gfreduce *r@ = reduction context
 *              @mp *d@ = destination
 *              @mp *x@ = source
 *
 * Returns:     Destination, @x@ reduced modulo the reduction poly.
 */

static void run(const gfreduce_instr *i, const gfreduce_instr *il,
                mpw *v, mpw z)
{
  mpw w = 0;

  for (; i < il; i++) {
    switch (i->op) {
      case GFRI_LOAD: w = *(v - i->arg); break;
      case GFRI_LSL: w ^= z << i->arg; break;
      case GFRI_LSR: w ^= z >> i->arg; break;
      case GFRI_STORE: *(v - i->arg) = MPW(w); break;
      default: abort();
    }
  }
}

mp *gfreduce_do(gfreduce *r, mp *d, mp *x)
{
  mpw *v, *vl;
  const gfreduce_instr *il;
  mpw z;

  /* --- Try to reuse the source's space --- */

  MP_COPY(x);
  if (d) MP_DROP(d);
  MP_DEST(x, MP_LEN(x), x->f);

  /* --- Do the reduction --- */

  il = r->iv + r->in;
  if (MP_LEN(x) >= r->lim) {
    v = x->v + r->lim;
    vl = x->vl;
    while (vl-- > v) {
      while (*vl) {
        z = *vl;
        *vl = 0;
        run(r->iv, il, vl, z);
      }
    }
    if (r->mask) {
      while (*vl & r->mask) {
        z = *vl & r->mask;
        *vl &= ~r->mask;
        run(r->iv, il, vl, z);
      }
    }
  }

  /* --- Done --- */

  MP_SHRINK(x);
  return (x);
}

/* --- @gfreduce_sqrt@ --- *
 *
 * Arguments:   @gfreduce *r@ = pointer to reduction context
 *              @mp *d@ = destination
 *              @mp *x@ = some polynomial
 *
 * Returns:     The square root of @x@ modulo @r->p@, or null.
 */

mp *gfreduce_sqrt(gfreduce *r, mp *d, mp *x)
{
  mp *y = MP_COPY(x);
  mp *z, *spare = MP_NEW;
  unsigned long m = mp_bits(r->p) - 1;
  unsigned long i;

  for (i = 0; i < m - 1; i++) {
    mp *t = gf_sqr(spare, y);
    spare = y;
    y = gfreduce_do(r, t, t);
  }
  z = gf_sqr(spare, y);
  z = gfreduce_do(r, z, z);
  if (!MP_EQ(x, z)) {
    mp_drop(y);
    y = 0;
  }
  mp_drop(z);
  mp_drop(d);
  return (y);
}

/* --- @gfreduce_trace@ --- *
 *
 * Arguments:   @gfreduce *r@ = pointer to reduction context
 *              @mp *x@ = some polynomial
 *
 * Returns:     The trace of @x@. (%$\Tr(x)=x + x^2 + \cdots + x^{2^{m-1}}$%
 *              if %$x \in \gf{2^m}$%).
 */

int gfreduce_trace(gfreduce *r, mp *x)
{
  mp *y = MP_COPY(x);
  mp *spare = MP_NEW;
  unsigned long m = mp_bits(r->p) - 1;
  unsigned long i;
  int rc;

  for (i = 0; i < m - 1; i++) {
    mp *t = gf_sqr(spare, y);
    spare = y;
    y = gfreduce_do(r, t, t);
    y = gf_add(y, y, x);
  }
  rc = !MP_ZEROP(y);
  mp_drop(spare);
  mp_drop(y);
  return (rc);
}

/* --- @gfreduce_halftrace@ --- *
 *
 * Arguments:   @gfreduce *r@ = pointer to reduction context
 *              @mp *d@ = destination
 *              @mp *x@ = some polynomial
 *
 * Returns:     The half-trace of @x@.
 *              (%$\HfTr(x)= x + x^{2^2} + \cdots + x^{2^{m-1}}$%
 *              if %$x \in \gf{2^m}$% with %$m$% odd).
 */

mp *gfreduce_halftrace(gfreduce *r, mp *d, mp *x)
{
  mp *y = MP_COPY(x);
  mp *spare = MP_NEW;
  unsigned long m = mp_bits(r->p) - 1;
  unsigned long i;

  mp_drop(d);
  for (i = 0; i < m - 1; i += 2) {
    mp *t = gf_sqr(spare, y);
    spare = y;
    y = gfreduce_do(r, t, t);
    t = gf_sqr(spare, y);
    spare = y;
    y = gfreduce_do(r, t, t);
    y = gf_add(y, y, x);
  }
  mp_drop(spare);
  return (y);
}

/* --- @gfreduce_quadsolve@ --- *
 *
 * Arguments:   @gfreduce *r@ = pointer to reduction context
 *              @mp *d@ = destination
 *              @mp *x@ = some polynomial
 *
 * Returns:     A polynomial @y@ such that %$y^2 + y = x$%, or null.
 */

mp *gfreduce_quadsolve(gfreduce *r, mp *d, mp *x)
{
  unsigned long m = mp_bits(r->p) - 1;
  mp *t;

  MP_COPY(x);
  if (m & 1)
    d = gfreduce_halftrace(r, d, x);
  else {
    mp *z, *w, *rho = MP_NEW;
    mp *spare = MP_NEW;
    grand *fr = fibrand_create(0);
    unsigned long i;

    for (;;) {
      rho = mprand(rho, m, fr, 0);
      z = MP_ZERO;
      w = MP_COPY(rho);
      for (i = 0; i < m - 1; i++) {
        t = gf_sqr(spare, z); spare = z; z = gfreduce_do(r, t, t);
        t = gf_sqr(spare, w); spare = w; w = gfreduce_do(r, t, t);
        t = gf_mul(spare, w, x); t = gfreduce_do(r, t, t); spare = t;
        z = gf_add(z, z, t);
        w = gf_add(w, w, rho);
      }
      if (!MP_ZEROP(w))
        break;
      MP_DROP(z);
      MP_DROP(w);
    }
    if (d) MP_DROP(d);
    MP_DROP(w);
    MP_DROP(spare);
    MP_DROP(rho);
    fr->ops->destroy(fr);
    d = z;
  }

  t = gf_sqr(MP_NEW, d); t = gfreduce_do(r, t, t); t = gf_add(t, t, d);
  if (!MP_EQ(t, x)) {
    MP_DROP(d);
    d = 0;
  }
  MP_DROP(t);
  MP_DROP(x);
  if (d) d->v[0] &= ~(mpw)1;
  return (d);
}

/* --- @gfreduce_exp@ --- *
 *
 * Arguments:   @gfreduce *gr@ = pointer to reduction context
 *              @mp *d@ = fake destination
 *              @mp *a@ = base
 *              @mp *e@ = exponent
 *
 * Returns:     Result, %$a^e \bmod m$%.
 */

mp *gfreduce_exp(gfreduce *gr, mp *d, mp *a, mp *e)
{
  mp *x = MP_ONE;
  mp *spare = (e->f & MP_BURN) ? MP_NEWSEC : MP_NEW;

  MP_SHRINK(e);
  MP_COPY(a);
  if (MP_ZEROP(e))
    ;
  else {
    if (MP_NEGP(e))
      a = gf_modinv(a, a, gr->p);
    if (MP_LEN(e) < EXP_THRESH)
      EXP_SIMPLE(x, a, e);
    else
      EXP_WINDOW(x, a, e);
  }
  mp_drop(d);
  mp_drop(a);
  mp_drop(spare);
  return (x);
}

/*----- Test rig ----------------------------------------------------------*/

#ifdef TEST_RIG

#define MP(x) mp_readstring(MP_NEW, #x, 0, 0)

static int vreduce(dstr *v)
{
  mp *d = *(mp **)v[0].buf;
  mp *n = *(mp **)v[1].buf;
  mp *r = *(mp **)v[2].buf;
  mp *c;
  int ok = 1;
  gfreduce rr;

  gfreduce_create(&rr, d);
  c = gfreduce_do(&rr, MP_NEW, n);
  if (!MP_EQ(c, r)) {
    fprintf(stderr, "\n*** reduction failed\n*** ");
    gfreduce_dump(&rr, stderr);
    fprintf(stderr, "\n*** n = "); mp_writefile(n, stderr, 16);
    fprintf(stderr, "\n*** r = "); mp_writefile(r, stderr, 16);
    fprintf(stderr, "\n*** c = "); mp_writefile(c, stderr, 16);
    fprintf(stderr, "\n");
    ok = 0;
  }
  gfreduce_destroy(&rr);
  mp_drop(n); mp_drop(d); mp_drop(r); mp_drop(c);
  assert(mparena_count(MPARENA_GLOBAL) == 0);
  return (ok);
}

static int vmodexp(dstr *v)
{
  mp *p = *(mp **)v[0].buf;
  mp *g = *(mp **)v[1].buf;
  mp *x = *(mp **)v[2].buf;
  mp *r = *(mp **)v[3].buf;
  mp *c;
  int ok = 1;
  gfreduce rr;

  gfreduce_create(&rr, p);
  c = gfreduce_exp(&rr, MP_NEW, g, x);
  if (!MP_EQ(c, r)) {
    fprintf(stderr, "\n*** modexp failed\n*** ");
    fprintf(stderr, "\n*** p = "); mp_writefile(p, stderr, 16);
    fprintf(stderr, "\n*** g = "); mp_writefile(g, stderr, 16);
    fprintf(stderr, "\n*** x = "); mp_writefile(x, stderr, 16);
    fprintf(stderr, "\n*** c = "); mp_writefile(c, stderr, 16);
    fprintf(stderr, "\n*** r = "); mp_writefile(r, stderr, 16);
    fprintf(stderr, "\n");
    ok = 0;
  }
  gfreduce_destroy(&rr);
  mp_drop(p); mp_drop(g); mp_drop(r); mp_drop(x); mp_drop(c);
  assert(mparena_count(MPARENA_GLOBAL) == 0);
  return (ok);
}

static int vsqrt(dstr *v)
{
  mp *p = *(mp **)v[0].buf;
  mp *x = *(mp **)v[1].buf;
  mp *r = *(mp **)v[2].buf;
  mp *c;
  int ok = 1;
  gfreduce rr;

  gfreduce_create(&rr, p);
  c = gfreduce_sqrt(&rr, MP_NEW, x);
  if (!MP_EQ(c, r)) {
    fprintf(stderr, "\n*** sqrt failed\n*** ");
    fprintf(stderr, "\n*** p = "); mp_writefile(p, stderr, 16);
    fprintf(stderr, "\n*** x = "); mp_writefile(x, stderr, 16);
    fprintf(stderr, "\n*** c = "); mp_writefile(c, stderr, 16);
    fprintf(stderr, "\n*** r = "); mp_writefile(r, stderr, 16);
    fprintf(stderr, "\n");
    ok = 0;
  }
  gfreduce_destroy(&rr);
  mp_drop(p); mp_drop(r); mp_drop(x); mp_drop(c);
  assert(mparena_count(MPARENA_GLOBAL) == 0);
  return (ok);
}

static int vtr(dstr *v)
{
  mp *p = *(mp **)v[0].buf;
  mp *x = *(mp **)v[1].buf;
  int r = *(int *)v[2].buf, c;
  int ok = 1;
  gfreduce rr;

  gfreduce_create(&rr, p);
  c = gfreduce_trace(&rr, x);
  if (c != r) {
    fprintf(stderr, "\n*** trace failed\n*** ");
    fprintf(stderr, "\n*** p = "); mp_writefile(p, stderr, 16);
    fprintf(stderr, "\n*** x = "); mp_writefile(x, stderr, 16);
    fprintf(stderr, "\n*** c = %d", c);
    fprintf(stderr, "\n*** r = %d", r);
    fprintf(stderr, "\n");
    ok = 0;
  }
  gfreduce_destroy(&rr);
  mp_drop(p); mp_drop(x);
  assert(mparena_count(MPARENA_GLOBAL) == 0);
  return (ok);
}

static int vhftr(dstr *v)
{
  mp *p = *(mp **)v[0].buf;
  mp *x = *(mp **)v[1].buf;
  mp *r = *(mp **)v[2].buf;
  mp *c;
  int ok = 1;
  gfreduce rr;

  gfreduce_create(&rr, p);
  c = gfreduce_halftrace(&rr, MP_NEW, x);
  if (!MP_EQ(c, r)) {
    fprintf(stderr, "\n*** halftrace failed\n*** ");
    fprintf(stderr, "\n*** p = "); mp_writefile(p, stderr, 16);
    fprintf(stderr, "\n*** x = "); mp_writefile(x, stderr, 16);
    fprintf(stderr, "\n*** c = "); mp_writefile(c, stderr, 16);
    fprintf(stderr, "\n*** r = "); mp_writefile(r, stderr, 16);
    fprintf(stderr, "\n");
    ok = 0;
  }
  gfreduce_destroy(&rr);
  mp_drop(p); mp_drop(r); mp_drop(x); mp_drop(c);
  assert(mparena_count(MPARENA_GLOBAL) == 0);
  return (ok);
}

static int vquad(dstr *v)
{
  mp *p = *(mp **)v[0].buf;
  mp *x = *(mp **)v[1].buf;
  mp *r = *(mp **)v[2].buf;
  mp *c;
  int ok = 1;
  gfreduce rr;

  gfreduce_create(&rr, p);
  c = gfreduce_quadsolve(&rr, MP_NEW, x);
  if (!MP_EQ(c, r)) {
    fprintf(stderr, "\n*** quadsolve failed\n*** ");
    fprintf(stderr, "\n*** p = "); mp_writefile(p, stderr, 16);
    fprintf(stderr, "\n*** x = "); mp_writefile(x, stderr, 16);
    fprintf(stderr, "\n*** c = "); mp_writefile(c, stderr, 16);
    fprintf(stderr, "\n*** r = "); mp_writefile(r, stderr, 16);
    fprintf(stderr, "\n");
    ok = 0;
  }
  gfreduce_destroy(&rr);
  mp_drop(p); mp_drop(r); mp_drop(x); mp_drop(c);
  assert(mparena_count(MPARENA_GLOBAL) == 0);
  return (ok);
}

static test_chunk defs[] = {
  { "reduce", vreduce, { &type_mp, &type_mp, &type_mp, 0 } },
  { "modexp", vmodexp, { &type_mp, &type_mp, &type_mp, &type_mp, 0 } },
  { "sqrt", vsqrt, { &type_mp, &type_mp, &type_mp, 0 } },
  { "trace", vtr, { &type_mp, &type_mp, &type_int, 0 } },
  { "halftrace", vhftr, { &type_mp, &type_mp, &type_mp, 0 } },
  { "quadsolve", vquad, { &type_mp, &type_mp, &type_mp, 0 } },
  { 0, 0, { 0 } }
};

int main(int argc, char *argv[])
{
  test_run(argc, argv, defs, SRCDIR"/t/gfreduce");
  return (0);
}

#endif

/*----- That's all, folks -------------------------------------------------*/