DA_DECL(instr_v, mpreduce_instr);
+/*----- Theory ------------------------------------------------------------*
+ *
+ * We're given a modulus %$p = 2^n - d$%, where %$d < 2^n$%, and some %$x$%,
+ * and we want to compute %$x \bmod p$%. We work in base %$2^w$%, for some
+ * appropriate %$w$%. The important observation is that
+ * %$d \equiv 2^n \pmod p$%.
+ *
+ * Suppose %$x = x' + z 2^k$%, where %$k \ge n$%; then
+ * %$x \equiv x' + d z 2^{k-n} \pmod p$%. We can use this to trim the
+ * representation of %$x$%; each time, we reduce %$x$% by a mutliple of
+ * %$2^{k-n} p$%. We can do this in two passes: firstly by taking whole
+ * words off the top, and then (if necessary) by trimming the top word.
+ * Finally, if %$p \le x < 2^n$% then %$0 \le x - p < p$% and we're done.
+ *
+ * A common trick, apparently, is to choose %$d$% such that it has a very
+ * sparse non-adjacent form, and, moreover, that this form is nicely aligned
+ * with common word sizes. (That is, write %$d = \sum_{0\le i<m} d_i 2^i$%,
+ * with %$d_i \in \{ -1, 0, +1 \}$% and most %$d_i = 0$%.) Then adding
+ * %$z d$% is a matter of adding and subtracting appropriately shifted copies
+ * of %$z$%.
+ *
+ * Most libraries come with hardwired code for doing this for a few
+ * well-known values of %$p$%. We take a different approach, for two
+ * reasons.
+ *
+ * * Firstly, it privileges built-in numbers over user-selected ones, even
+ * if the latter have the right (or better) properties.
+ *
+ * * Secondly, writing appropriately optimized reduction functions when we
+ * don't know the exact characteristics of the target machine is rather
+ * difficult.
+ *
+ * Our solution, then, is to `compile' the value %$p$% at runtime, into a
+ * sequence of simple instructions for doing the reduction.
+ */
+
/*----- Main code ---------------------------------------------------------*/
/* --- @mpreduce_create@ --- *
* Arguments: @gfreduce *r@ = structure to fill in
* @mp *x@ = an integer
*
- * Returns: Zero if successful; nonzero on failure.
+ * Returns: Zero if successful; nonzero on failure. The current
+ * algorithm always succeeds when given positive @x@. Earlier
+ * versions used to fail on particular kinds of integers, but
+ * this is guaranteed not to happen any more.
*
* Use: Initializes a context structure for reduction.
*/
/* --- Main loop --- *
*
* A simple state machine decomposes @p@ conveniently into positive and
- * negative powers of 2. The pure form of the state machine is left below
- * for reference (and in case I need inspiration for a NAF exponentiator).
+ * negative powers of 2.
+ *
+ * Here's the relevant theory. The important observation is that
+ * %$2^i = 2^{i+1} - 2^i$%, and hence
+ *
+ * * %$\sum_{a\le i<b} 2^i = 2^b - 2^a$%, and
+ *
+ * * %$2^c - 2^{b+1} + 2^b - 2^a = 2^c - 2^b - 2^a$%.
+ *
+ * The first of these gives us a way of combining a run of several one
+ * bits, and the second gives us a way of handling a single-bit
+ * interruption in such a run.
+ *
+ * We start with a number %$p = \sum_{0\le i<n} p_i 2^i$%, and scan
+ * right-to-left using a simple state-machine keeping (approximate) track
+ * of the two previous bits. The @Z@ states denote that we're in a string
+ * of zeros; @Z1@ means that we just saw a 1 bit after a sequence of zeros.
+ * Similarly, the @X@ states denote that we're in a string of ones; and
+ * @X0@ means that we just saw a zero bit after a sequence of ones. The
+ * state machine lets us delay decisions about what to do when we've seen a
+ * change to the status quo (a one after a run of zeros, or vice-versa)
+ * until we've seen the next bit, so we can tell whether this is an
+ * isolated bit or the start of a new sequence.
+ *
+ * More formally: we define two functions %$Z^b_i$% and %$X^b_i$% as
+ * follows.
+ *
+ * * %$Z^0_i(S, 0) = S$%
+ * * %$Z^0_i(S, n) = Z^0_{i+1}(S, n)$%
+ * * %$Z^0_i(S, n + 2^i) = Z^1_{i+1}(S, n)$%
+ * * %$Z^1_i(S, n) = Z^0_{i+1}(S \cup \{ 2^{i-1} \}, n)$%
+ * * %$Z^1_i(S, n + 2^i) = X^1_{i+1}(S \cup \{ -2^{i-1} \}, n)$%
+ * * %$X^0_i(S, n) = Z^0_{i+1}(S, \{ 2^{i-1} \})$%
+ * * %$X^0_i(S, n + 2^i) = X^1_{i+1}(S \cup \{ -2^{i-1} \}, n)$%
+ * * %$X^1_i(S, n) = X^0_{i+1}(S, n)$%
+ * * %$X^1_i(S, n + 2^i) = X^1_{i+1}(S, n)$%
+ *
+ * The reader may verify (by induction on %$n$%) that the following
+ * properties hold.
+ *
+ * * %$Z^0_0(\emptyset, n)$% is well-defined for all %$n \ge 0$%
+ * * %$\sum Z^b_i(S, n) = \sum S + n + b 2^{i-1}$%
+ * * %$\sum X^b_i(S, n) = \sum S + n + (b + 1) 2^{i-1}$%
+ *
+ * From these, of course, we can deduce %$\sum Z^0_0(\emptyset, n) = n$%.
+ *
+ * We apply the above recurrence to build a simple instruction sequence for
+ * adding an appropriate multiple of %$d$% to a given number. Suppose that
+ * %$2^{w(N-1)} \le 2^{n-1} \le p < 2^n \le 2^{wN}$%. The machine which
+ * interprets these instructions does so in the context of a
+ * single-precision multiplicand @z@ and a pointer @v@ to the
+ * %%\emph{most}%% significant word of an %$N + 1$%-word integer, and the
+ * instruction sequence should add %$z d$% to this integer.
+ *
+ * The available instructions are named @MPRI_{ADD,SUB}{,LSL}@; they add
+ * (or subtract) %$z$% (shifted left by some amount, in the @LSL@ variants)
+ * to some word earlier than @v@. The relevant quantities are encoded in
+ * the instruction's immediate operands.
*/
-#ifdef DEBUG
- for (i = 0, mp_scan(&sc, p); mp_step(&sc); i++) {
- switch (st | mp_bit(&sc)) {
- case Z | 1: st = Z1; break;
- case Z1 | 0: st = Z; printf("+ %lu\n", i - 1); break;
- case Z1 | 1: st = X; printf("- %lu\n", i - 1); break;
- case X | 0: st = X0; break;
- case X0 | 1: st = X; printf("- %lu\n", i - 1); break;
- case X0 | 0: st = Z; printf("+ %lu\n", i - 1); break;
- }
- }
- if (st >= X) printf("+ %lu\n", i - 1);
- st = Z;
-#endif
-
bb = MPW_BITS - (d + 1)%MPW_BITS;
for (i = 0, mp_scan(&sc, p); i < d && mp_step(&sc); i++) {
switch (st | mp_bit(&sc)) {
INSTR(op | !!b, w, b);
}
}
- if (DA_LEN(&iv) && (DA(&iv)[DA_LEN(&iv) - 1].op & ~1u) == MPRI_SUB) {
- mp_drop(r->p);
- DA_DESTROY(&iv);
- return (-1);
+
+ /* --- Fix up wrong-sided decompositions --- *
+ *
+ * At this point, we haven't actually finished up the state machine
+ * properly. We stopped scanning just after bit %$n - 1$% -- the most
+ * significant one, which we know in advance must be set (since @x@ is
+ * strictly positive). Therefore we are either in state @X@ or @Z1@. In
+ * the former case, we have nothing to do. In the latter, there are two
+ * subcases to deal with. If there are no other instructions, then @x@ is
+ * a perfect power of two, and %$d = 0$%, so again there is nothing to do.
+ *
+ * In the remaining case, we have decomposed @x@ as %$2^{n-1} + d$%, for
+ * some positive %$d%, which is unfortuante: if we're asked to reduce
+ * %$2^n$%, say, we'll end up with %$-d$% (or would do, if we weren't
+ * sticking to unsigned arithmetic for good performance). So instead, we
+ * rewrite this as %$2^n - 2^{n-1} + d$% and everything will be good.
+ */
+
+ if (st == Z1 && DA_LEN(&iv)) {
+ w = 1;
+ b = (bb + d)%MPW_BITS;
+ INSTR(MPRI_ADD | !!b, w, b);
}
#undef INSTR
- /* --- Wrap up --- */
+ /* --- Wrap up --- *
+ *
+ * Store the generated instruction sequence in our context structure. If
+ * %$p$%'s bit length %$n$% is a multiple of the word size %$w$% then
+ * there's nothing much else to do here. Otherwise, we have an additional
+ * job.
+ *
+ * The reduction operation has three phases. The first trims entire words
+ * from the argument, and the instruction sequence constructed above does
+ * this well; the second phase reduces an integer which has the same number
+ * of words as %$p$%, but strictly more bits. (The third phase is a single
+ * conditional subtraction of %$p$%, in case the argument is the same bit
+ * length as %$p$% but greater; this doesn't concern us here.) To handle
+ * the second phase, we create another copy of the instruction stream, with
+ * all of the target shifts adjusted upwards by %$s = n \bmod w$%.
+ *
+ * In this case, we are acting on an %$(N - 1)$%-word operand, and so
+ * (given the remarks above) must check that this is still valid, but a
+ * moment's reflection shows that this must be fine: the most distant
+ * target must be the bit %$s$% from the top of the least-significant word;
+ * but since we shift all of the targets up by %$s$%, this now addresses
+ * the bottom bit of the next most significant word, and there is no
+ * underflow.
+ */
r->in = DA_LEN(&iv);
if (!r->in)
}
DA_DESTROY(&iv);
-#ifdef DEBUG
- mpreduce_dump(r, stdout);
-#endif
return (0);
}
mpw *v, mpw z)
{
for (; i < il; i++) {
-#ifdef DEBUG
- mp vv;
- mp_build(&vv, v - i->argx, v + 1);
- printf(" 0x"); mp_writefile(&vv, stdout, 16);
- printf(" %c (0x%lx << %u) == 0x",
- (i->op & ~1u) == MPRI_ADD ? '+' : '-',
- (unsigned long)z,
- i->argy);
-#endif
switch (i->op) {
case MPRI_ADD: MPX_UADDN(v - i->argx, v + 1, z); break;
case MPRI_ADDLSL: mpx_uaddnlsl(v - i->argx, v + 1, z, i->argy); break;
default:
abort();
}
-#ifdef DEBUG
- mp_build(&vv, v - i->argx, v + 1);
- mp_writefile(&vv, stdout, 16);
- printf("\n");
-#endif
}
}
const mpreduce_instr *il;
mpw z;
-#ifdef DEBUG
- mp *_r = 0, *_rr = 0;
-#endif
-
/* --- If source is negative, divide --- */
if (MP_NEGP(x)) {
if (d) MP_DROP(d);
MP_DEST(x, MP_LEN(x), x->f);
- /* --- Do the reduction --- */
-
-#ifdef DEBUG
- _r = MP_NEW;
- mp_div(0, &_r, x, r->p);
- MP_PRINTX("x", x);
- _rr = 0;
-#endif
+ /* --- Stage one: trim excess words from the most significant end --- */
il = r->iv + r->in;
if (MP_LEN(x) >= r->lim) {
z = *vl;
*vl = 0;
run(r->iv, il, vl, z);
-#ifdef DEBUG
- MP_PRINTX("x", x);
- mp_div(0, &_rr, x, r->p);
- assert(MP_EQ(_r, _rr));
-#endif
}
}
+
+ /* --- Stage two: trim excess bits from the most significant word --- */
+
if (r->s) {
while (*vl >> r->s) {
z = *vl >> r->s;
*vl &= ((1 << r->s) - 1);
run(r->iv + r->in, il + r->in, vl, z);
-#ifdef DEBUG
- MP_PRINTX("x", x);
- mp_div(0, &_rr, x, r->p);
- assert(MP_EQ(_r, _rr));
-#endif
}
}
}
- /* --- Finishing touches --- */
+ /* --- Stage three: conditional subtraction --- */
MP_SHRINK(x);
if (MP_CMP(x, >=, r->p))
/* --- Done --- */
-#ifdef DEBUG
- assert(MP_EQ(_r, x));
- mp_drop(_r);
- mp_drop(_rr);
-#endif
return (x);
}
projects / u / mdw / catacomb / blobdiff