3 * Arithmetic in the Goldilocks field GF(2^448 - 2^224 - 1)
5 * (c) 2017 Straylight/Edgeware
8 /*----- Licensing notice --------------------------------------------------*
10 * This file is part of secnet.
11 * See README for full list of copyright holders.
13 * secnet is free software; you can redistribute it and/or modify it
14 * under the terms of the GNU General Public License as published by
15 * the Free Software Foundation; either version d of the License, or
16 * (at your option) any later version.
18 * secnet is distributed in the hope that it will be useful, but
19 * WITHOUT ANY WARRANTY; without even the implied warranty of
20 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
21 * General Public License for more details.
23 * You should have received a copy of the GNU General Public License
24 * version 3 along with secnet; if not, see
25 * https://www.gnu.org/licenses/gpl.html.
27 * This file was originally part of Catacomb, but has been automatically
28 * modified for incorporation into secnet: see `import-catacomb-crypto'
31 * Catacomb is free software; you can redistribute it and/or modify
32 * it under the terms of the GNU Library General Public License as
33 * published by the Free Software Foundation; either version 2 of the
34 * License, or (at your option) any later version.
36 * Catacomb is distributed in the hope that it will be useful,
37 * but WITHOUT ANY WARRANTY; without even the implied warranty of
38 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
39 * GNU Library General Public License for more details.
41 * You should have received a copy of the GNU Library General Public
42 * License along with Catacomb; if not, write to the Free
43 * Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
47 /*----- Header files ------------------------------------------------------*/
51 /*----- Basic setup -------------------------------------------------------*
53 * Let φ = 2^224; then p = φ^2 - φ - 1, and, in GF(p), we have φ^2 = φ + 1
57 /* We represent an element of GF(p) as 16 28-bit signed integer pieces x_i:
58 * x = SUM_{0<=i<16} x_i 2^(28i).
61 typedef int32 piece
; typedef int64 dblpiece
;
62 typedef uint32 upiece
; typedef uint64 udblpiece
;
67 #define B28 0x10000000u
68 #define B27 0x08000000u
69 #define M28 0x0fffffffu
70 #define M27 0x07ffffffu
71 #define M32 0xffffffffu
73 /*----- Debugging machinery -----------------------------------------------*/
75 #if defined(FGOLDI_DEBUG)
82 static mp
*get_pgoldi(void)
84 mp
*p
= MP_NEW
, *t
= MP_NEW
;
86 p
= mp_setbit(p
, MP_ZERO
, 448);
87 t
= mp_setbit(t
, MP_ZERO
, 224);
89 p
= mp_sub(p
, p
, MP_ONE
);
94 DEF_FDUMP(fdump
, piece
, PIECEWD
, NPIECE
, 56, get_pgoldi())
98 /*----- Loading and storing -----------------------------------------------*/
100 /* --- @fgoldi_load@ --- *
102 * Arguments: @fgoldi *z@ = where to store the result
103 * @const octet xv[56]@ = source to read
107 * Use: Reads an element of %$\gf{2^{448} - 2^{224} - 19}$% in
108 * external representation from @xv@ and stores it in @z@.
110 * External representation is little-endian base-256. Some
111 * elements have multiple encodings, which are not produced by
112 * correct software; use of noncanonical encodings is not an
113 * error, and toleration of them is considered a performance
117 void fgoldi_load(fgoldi
*z
, const octet xv
[56])
124 /* First, read the input value as words. */
125 for (i
= 0; i
< 14; i
++) xw
[i
] = LOAD32_L(xv
+ 4*i
);
127 /* Extract unsigned 28-bit pieces from the words. */
128 z
->P
[ 0] = (xw
[ 0] >> 0)&M28
;
129 z
->P
[ 7] = (xw
[ 6] >> 4)&M28
;
130 z
->P
[ 8] = (xw
[ 7] >> 0)&M28
;
131 z
->P
[15] = (xw
[13] >> 4)&M28
;
132 for (i
= 1; i
< 7; i
++) {
133 z
->P
[i
+ 0] = ((xw
[i
+ 0] << (4*i
)) | (xw
[i
- 1] >> (32 - 4*i
)))&M28
;
134 z
->P
[i
+ 8] = ((xw
[i
+ 7] << (4*i
)) | (xw
[i
+ 6] >> (32 - 4*i
)))&M28
;
137 /* Convert the nonnegative pieces into a balanced signed representation, so
138 * each piece ends up in the interval |z_i| <= 2^27. For each piece, if
139 * its top bit is set, lend a bit leftwards; in the case of z_15, reduce
140 * this bit by adding it onto z_0 and z_8, since this is the φ^2 bit, and
141 * φ^2 = φ + 1. We delay this carry until after all of the pieces have
142 * been balanced. If we don't do this, then we have to do a more expensive
143 * test for nonzeroness to decide whether to lend a bit leftwards rather
144 * than just testing a single bit.
146 * Note that we don't try for a canonical representation here: both upper
147 * and lower bounds are achievable.
149 b
= z
->P
[15]&B27
; z
->P
[15] -= b
<< 1; c
= b
>> 27;
150 for (i
= NPIECE
- 1; i
--; )
151 { b
= z
->P
[i
]&B27
; z
->P
[i
] -= b
<< 1; z
->P
[i
+ 1] += b
>> 27; }
152 z
->P
[0] += c
; z
->P
[8] += c
;
155 /* --- @fgoldi_store@ --- *
157 * Arguments: @octet zv[56]@ = where to write the result
158 * @const fgoldi *x@ = the field element to write
162 * Use: Stores a field element in the given octet vector in external
163 * representation. A canonical encoding is always stored.
166 void fgoldi_store(octet zv
[56], const fgoldi
*x
)
169 piece y
[NPIECE
], yy
[NPIECE
], c
, d
;
174 for (i
= 0; i
< NPIECE
; i
++) y
[i
] = x
->P
[i
];
176 /* First, propagate the carries. By the end of this, we'll have all of the
177 * the pieces canonically sized and positive, and maybe there'll be
178 * (signed) carry out. The carry c is in { -1, 0, +1 }, and the remaining
179 * value will be in the half-open interval [0, φ^2). The whole represented
180 * value is then y + φ^2 c.
182 * Assume that we start out with |y_i| <= 2^30. We start off by cutting
183 * off and reducing the carry c_15 from the topmost piece, y_15. This
184 * leaves 0 <= y_15 < 2^28; and we'll have |c_15| <= 4. We'll add this
185 * onto y_0 and y_8, and propagate the carries. It's very clear that we'll
186 * end up with |y + (φ + 1) c_15 - φ^2/2| << φ^2.
188 * Here, the y_i are signed, so we must be cautious about bithacking them.
190 c
= ASR(piece
, y
[15], 28); y
[15] = (upiece
)y
[15]&M28
; y
[8] += c
;
191 for (i
= 0; i
< NPIECE
; i
++)
192 { y
[i
] += c
; c
= ASR(piece
, y
[i
], 28); y
[i
] = (upiece
)y
[i
]&M28
; }
194 /* Now we have a slightly fiddly job to do. If c = +1, or if c = 0 and
195 * y >= p, then we should subtract p from the whole value; if c = -1 then
196 * we should add p; and otherwise we should do nothing.
198 * But conditional behaviour is bad, m'kay. So here's what we do instead.
200 * The first job is to sort out what we wanted to do. If c = -1 then we
201 * want to (a) invert the constant addend and (b) feed in a carry-in;
202 * otherwise, we don't.
207 /* Now do the addition/subtraction. Remember that all of the y_i are
208 * nonnegative, so shifting and masking are safe and easy.
210 d
+= y
[0] + (1 ^ m
); yy
[0] = d
&M28
; d
>>= 28;
211 for (i
= 1; i
< 8; i
++)
212 { d
+= y
[i
] + m
; yy
[i
] = d
&M28
; d
>>= 28; }
213 d
+= y
[8] + (1 ^ m
); yy
[8] = d
&M28
; d
>>= 28;
214 for (i
= 9; i
< 16; i
++)
215 { d
+= y
[i
] + m
; yy
[i
] = d
&M28
; d
>>= 28; }
217 /* The final carry-out is in d; since we only did addition, and the y_i are
218 * nonnegative, then d is in { 0, 1 }. We want to keep y', rather than y,
219 * if (a) c /= 0 (in which case we know that the old value was
220 * unsatisfactory), or (b) if d = 1 (in which case, if c = 0, we know that
221 * the subtraction didn't cause a borrow, so we must be in the case where
224 m
= NONZEROP(c
) | ~NONZEROP(d
- 1);
225 for (i
= 0; i
< NPIECE
; i
++) y
[i
] = (yy
[i
]&m
) | (y
[i
]&~m
);
227 /* Extract 32-bit words from the value. */
228 for (i
= 0; i
< 7; i
++) {
229 u
= ((y
[i
+ 0] >> (4*i
)) | ((uint32
)y
[i
+ 1] << (28 - 4*i
)))&M32
;
230 v
= ((y
[i
+ 8] >> (4*i
)) | ((uint32
)y
[i
+ 9] << (28 - 4*i
)))&M32
;
231 STORE32_L(zv
+ 4*i
, u
);
232 STORE32_L(zv
+ 4*i
+ 28, v
);
236 /* --- @fgoldi_set@ --- *
238 * Arguments: @fgoldi *z@ = where to write the result
239 * @int a@ = a small-ish constant
243 * Use: Sets @z@ to equal @a@.
246 void fgoldi_set(fgoldi
*x
, int a
)
251 for (i
= 1; i
< NPIECE
; i
++) x
->P
[i
] = 0;
254 /*----- Basic arithmetic --------------------------------------------------*/
256 /* --- @fgoldi_add@ --- *
258 * Arguments: @fgoldi *z@ = where to put the result (may alias @x@ or @y@)
259 * @const fgoldi *x, *y@ = two operands
263 * Use: Set @z@ to the sum %$x + y$%.
266 void fgoldi_add(fgoldi
*z
, const fgoldi
*x
, const fgoldi
*y
)
269 for (i
= 0; i
< NPIECE
; i
++) z
->P
[i
] = x
->P
[i
] + y
->P
[i
];
272 /* --- @fgoldi_sub@ --- *
274 * Arguments: @fgoldi *z@ = where to put the result (may alias @x@ or @y@)
275 * @const fgoldi *x, *y@ = two operands
279 * Use: Set @z@ to the difference %$x - y$%.
282 void fgoldi_sub(fgoldi
*z
, const fgoldi
*x
, const fgoldi
*y
)
285 for (i
= 0; i
< NPIECE
; i
++) z
->P
[i
] = x
->P
[i
] - y
->P
[i
];
288 /*----- Constant-time utilities -------------------------------------------*/
290 /* --- @fgoldi_condswap@ --- *
292 * Arguments: @fgoldi *x, *y@ = two operands
293 * @uint32 m@ = a mask
297 * Use: If @m@ is zero, do nothing; if @m@ is all-bits-set, then
298 * exchange @x@ and @y@. If @m@ has some other value, then
299 * scramble @x@ and @y@ in an unhelpful way.
302 void fgoldi_condswap(fgoldi
*x
, fgoldi
*y
, uint32 m
)
305 mask32 mm
= FIX_MASK32(m
);
307 for (i
= 0; i
< NPIECE
; i
++) CONDSWAP(x
->P
[i
], y
->P
[i
], mm
);
310 /*----- Multiplication ----------------------------------------------------*/
312 /* Let B = 2^63 - 1 be the largest value such that +B and -B can be
313 * represented in a double-precision piece. On entry, it must be the case
314 * that |X_i| <= M <= B - 2^27 for some M. If this is the case, then, on
315 * exit, we will have |Z_i| <= 2^27 + M/2^27.
317 #define CARRY_REDUCE(z, x) do { \
318 dblpiece _t[NPIECE], _c; \
321 /* Bias the input pieces. This keeps the carries and so on centred \
322 * around zero rather than biased positive. \
324 for (_i = 0; _i < NPIECE; _i++) _t[_i] = (x)[_i] + B27; \
326 /* Calculate the reduced pieces. Careful with the bithacking. */ \
327 _c = ASR(dblpiece, _t[15], 28); \
328 (z)[0] = (dblpiece)((udblpiece)_t[0]&M28) - B27 + _c; \
329 for (_i = 1; _i < NPIECE; _i++) { \
330 (z)[_i] = (dblpiece)((udblpiece)_t[_i]&M28) - B27 + \
331 ASR(dblpiece, _t[_i - 1], 28); \
336 /* --- @fgoldi_mulconst@ --- *
338 * Arguments: @fgoldi *z@ = where to put the result (may alias @x@)
339 * @const fgoldi *x@ = an operand
340 * @long a@ = a small-ish constant; %$|a| < 2^{20}$%.
344 * Use: Set @z@ to the product %$a x$%.
347 void fgoldi_mulconst(fgoldi
*z
, const fgoldi
*x
, long a
)
350 dblpiece zz
[NPIECE
], aa
= a
;
352 for (i
= 0; i
< NPIECE
; i
++) zz
[i
] = aa
*x
->P
[i
];
353 CARRY_REDUCE(z
->P
, zz
);
356 /* --- @fgoldi_mul@ --- *
358 * Arguments: @fgoldi *z@ = where to put the result (may alias @x@ or @y@)
359 * @const fgoldi *x, *y@ = two operands
363 * Use: Set @z@ to the product %$x y$%.
366 void fgoldi_mul(fgoldi
*z
, const fgoldi
*x
, const fgoldi
*y
)
368 dblpiece zz
[NPIECE
], u
[NPIECE
];
369 piece ab
[NPIECE
/2], cd
[NPIECE
/2];
371 *a
= x
->P
+ NPIECE
/2, *b
= x
->P
,
372 *c
= y
->P
+ NPIECE
/2, *d
= y
->P
;
375 # define M(x,i, y,j) ((dblpiece)(x)[i]*(y)[j])
379 * Write x = a φ + b, and y = c φ + d. Then x y = a c φ^2 +
380 * (a d + b c) φ + b d. Karatsuba and Ofman observed that a d + b c =
381 * (a + b) (c + d) - a c - b d, saving a multiplication, and Hamburg chose
382 * the prime p so that φ^2 = φ + 1. So
384 * x y = ((a + b) (c + d) - b d) φ + a c + b d
387 for (i
= 0; i
< NPIECE
; i
++) zz
[i
] = 0;
389 /* Our first job will be to calculate (1 - φ) b d, and write the result
390 * into z. As we do this, an interesting thing will happen. Write
391 * b d = u φ + v; then (1 - φ) b d = u φ + v - u φ^2 - v φ = (1 - φ) v - u.
392 * So, what we do is to write the product end-swapped and negated, and then
393 * we'll subtract the (negated, remember) high half from the low half.
395 for (i
= 0; i
< NPIECE
/2; i
++) {
396 for (j
= 0; j
< NPIECE
/2 - i
; j
++)
397 zz
[i
+ j
+ NPIECE
/2] -= M(b
,i
, d
,j
);
398 for (; j
< NPIECE
/2; j
++)
399 zz
[i
+ j
- NPIECE
/2] -= M(b
,i
, d
,j
);
401 for (i
= 0; i
< NPIECE
/2; i
++)
402 zz
[i
] -= zz
[i
+ NPIECE
/2];
404 /* Next, we add on a c. There are no surprises here. */
405 for (i
= 0; i
< NPIECE
/2; i
++)
406 for (j
= 0; j
< NPIECE
/2; j
++)
407 zz
[i
+ j
] += M(a
,i
, c
,j
);
409 /* Now, calculate a + b and c + d. */
410 for (i
= 0; i
< NPIECE
/2; i
++)
411 { ab
[i
] = a
[i
] + b
[i
]; cd
[i
] = c
[i
] + d
[i
]; }
413 /* Finally (for the multiplication) we must add on (a + b) (c + d) φ.
414 * Write (a + b) (c + d) as u φ + v; then we actually want u φ^2 + v φ =
415 * v φ + (1 + φ) u. We'll store u in a temporary place and add it on
418 for (i
= 0; i
< NPIECE
; i
++) u
[i
] = 0;
419 for (i
= 0; i
< NPIECE
/2; i
++) {
420 for (j
= 0; j
< NPIECE
/2 - i
; j
++)
421 zz
[i
+ j
+ NPIECE
/2] += M(ab
,i
, cd
,j
);
422 for (; j
< NPIECE
/2; j
++)
423 u
[i
+ j
- NPIECE
/2] += M(ab
,i
, cd
,j
);
425 for (i
= 0; i
< NPIECE
/2; i
++)
426 { zz
[i
] += u
[i
]; zz
[i
+ NPIECE
/2] += u
[i
]; }
430 /* That wraps it up for the multiplication. Let's figure out some bounds.
431 * Fortunately, Karatsuba is a polynomial identity, so all of the pieces
432 * end up the way they'd be if we'd done the thing the easy way, which
433 * simplifies the analysis. Suppose we started with |x_i|, |y_i| <= 9/5
434 * 2^28. The overheads in the result are given by the coefficients of
436 * ((u^16 - 1)/(u - 1))^2 mod u^16 - u^8 - 1
438 * the greatest of which is 38. So |z_i| <= 38*81/25*2^56 < 2^63.
440 * Anyway, a round of `CARRY_REDUCE' will leave us with |z_i| < 2^27 +
441 * 2^36; and a second round will leave us with |z_i| < 2^27 + 512.
443 for (i
= 0; i
< 2; i
++) CARRY_REDUCE(zz
, zz
);
444 for (i
= 0; i
< NPIECE
; i
++) z
->P
[i
] = zz
[i
];
447 /* --- @fgoldi_sqr@ --- *
449 * Arguments: @fgoldi *z@ = where to put the result (may alias @x@ or @y@)
450 * @const fgoldi *x@ = an operand
454 * Use: Set @z@ to the square %$x^2$%.
457 void fgoldi_sqr(fgoldi
*z
, const fgoldi
*x
)
460 dblpiece zz
[NPIECE
], u
[NPIECE
];
462 const piece
*a
= x
->P
+ NPIECE
/2, *b
= x
->P
;
465 # define M(x,i, y,j) ((dblpiece)(x)[i]*(y)[j])
467 /* The magic is basically the same as `fgoldi_mul' above. We write
468 * x = a φ + b and use Karatsuba and the special prime shape. This time,
471 * x^2 = ((a + b)^2 - b^2) φ + a^2 + b^2
474 for (i
= 0; i
< NPIECE
; i
++) zz
[i
] = 0;
476 /* Our first job will be to calculate (1 - φ) b^2, and write the result
477 * into z. Again, this interacts pleasantly with the prime shape.
479 for (i
= 0; i
< NPIECE
/4; i
++) {
480 zz
[2*i
+ NPIECE
/2] -= M(b
,i
, b
,i
);
481 for (j
= i
+ 1; j
< NPIECE
/2 - i
; j
++)
482 zz
[i
+ j
+ NPIECE
/2] -= 2*M(b
,i
, b
,j
);
483 for (; j
< NPIECE
/2; j
++)
484 zz
[i
+ j
- NPIECE
/2] -= 2*M(b
,i
, b
,j
);
486 for (; i
< NPIECE
/2; i
++) {
487 zz
[2*i
- NPIECE
/2] -= M(b
,i
, b
,i
);
488 for (j
= i
+ 1; j
< NPIECE
/2; j
++)
489 zz
[i
+ j
- NPIECE
/2] -= 2*M(b
,i
, b
,j
);
491 for (i
= 0; i
< NPIECE
/2; i
++)
492 zz
[i
] -= zz
[i
+ NPIECE
/2];
494 /* Next, we add on a^2. There are no surprises here. */
495 for (i
= 0; i
< NPIECE
/2; i
++) {
496 zz
[2*i
] += M(a
,i
, a
,i
);
497 for (j
= i
+ 1; j
< NPIECE
/2; j
++)
498 zz
[i
+ j
] += 2*M(a
,i
, a
,j
);
501 /* Now, calculate a + b. */
502 for (i
= 0; i
< NPIECE
/2; i
++)
505 /* Finally (for the multiplication) we must add on (a + b)^2 φ.
506 * Write (a + b)^2 as u φ + v; then we actually want (u + v) φ + u. We'll
507 * store u in a temporary place and add it on twice.
509 for (i
= 0; i
< NPIECE
; i
++) u
[i
] = 0;
510 for (i
= 0; i
< NPIECE
/4; i
++) {
511 zz
[2*i
+ NPIECE
/2] += M(ab
,i
, ab
,i
);
512 for (j
= i
+ 1; j
< NPIECE
/2 - i
; j
++)
513 zz
[i
+ j
+ NPIECE
/2] += 2*M(ab
,i
, ab
,j
);
514 for (; j
< NPIECE
/2; j
++)
515 u
[i
+ j
- NPIECE
/2] += 2*M(ab
,i
, ab
,j
);
517 for (; i
< NPIECE
/2; i
++) {
518 u
[2*i
- NPIECE
/2] += M(ab
,i
, ab
,i
);
519 for (j
= i
+ 1; j
< NPIECE
/2; j
++)
520 u
[i
+ j
- NPIECE
/2] += 2*M(ab
,i
, ab
,j
);
522 for (i
= 0; i
< NPIECE
/2; i
++)
523 { zz
[i
] += u
[i
]; zz
[i
+ NPIECE
/2] += u
[i
]; }
527 /* Finally, carrying. */
528 for (i
= 0; i
< 2; i
++) CARRY_REDUCE(zz
, zz
);
529 for (i
= 0; i
< NPIECE
; i
++) z
->P
[i
] = zz
[i
];
532 /*----- More advanced operations ------------------------------------------*/
534 /* --- @fgoldi_inv@ --- *
536 * Arguments: @fgoldi *z@ = where to put the result (may alias @x@)
537 * @const fgoldi *x@ = an operand
541 * Use: Stores in @z@ the multiplicative inverse %$x^{-1}$%. If
542 * %$x = 0$% then @z@ is set to zero. This is considered a
546 void fgoldi_inv(fgoldi
*z
, const fgoldi
*x
)
551 #define SQRN(z, x, n) do { \
552 fgoldi_sqr((z), (x)); \
553 for (i = 1; i < (n); i++) fgoldi_sqr((z), (z)); \
556 /* Calculate x^-1 = x^(p - 2) = x^(2^448 - 2^224 - 3), which also handles
557 * x = 0 as intended. The addition chain is home-made.
558 */ /* step | value */
559 fgoldi_sqr(&u
, x
); /* 1 | 2 */
560 fgoldi_mul(&t
, &u
, x
); /* 2 | 3 */
561 SQRN(&u
, &t
, 2); /* 4 | 12 */
562 fgoldi_mul(&t
, &u
, &t
); /* 5 | 15 */
563 SQRN(&u
, &t
, 4); /* 9 | 240 */
564 fgoldi_mul(&u
, &u
, &t
); /* 10 | 255 = 2^8 - 1 */
565 SQRN(&u
, &u
, 4); /* 14 | 2^12 - 16 */
566 fgoldi_mul(&t
, &u
, &t
); /* 15 | 2^12 - 1 */
567 SQRN(&u
, &t
, 12); /* 27 | 2^24 - 2^12 */
568 fgoldi_mul(&u
, &u
, &t
); /* 28 | 2^24 - 1 */
569 SQRN(&u
, &u
, 12); /* 40 | 2^36 - 2^12 */
570 fgoldi_mul(&t
, &u
, &t
); /* 41 | 2^36 - 1 */
571 fgoldi_sqr(&t
, &t
); /* 42 | 2^37 - 2 */
572 fgoldi_mul(&t
, &t
, x
); /* 43 | 2^37 - 1 */
573 SQRN(&u
, &t
, 37); /* 80 | 2^74 - 2^37 */
574 fgoldi_mul(&u
, &u
, &t
); /* 81 | 2^74 - 1 */
575 SQRN(&u
, &u
, 37); /* 118 | 2^111 - 2^37 */
576 fgoldi_mul(&t
, &u
, &t
); /* 119 | 2^111 - 1 */
577 SQRN(&u
, &t
, 111); /* 230 | 2^222 - 2^111 */
578 fgoldi_mul(&t
, &u
, &t
); /* 231 | 2^222 - 1 */
579 fgoldi_sqr(&u
, &t
); /* 232 | 2^223 - 2 */
580 fgoldi_mul(&u
, &u
, x
); /* 233 | 2^223 - 1 */
581 SQRN(&u
, &u
, 223); /* 456 | 2^446 - 2^223 */
582 fgoldi_mul(&t
, &u
, &t
); /* 457 | 2^446 - 2^222 - 1 */
583 SQRN(&t
, &t
, 2); /* 459 | 2^448 - 2^224 - 4 */
584 fgoldi_mul(z
, &t
, x
); /* 460 | 2^448 - 2^224 - 3 */
589 /*----- That's all, folks -------------------------------------------------*/