2 * fgoldi.c: arithmetic modulo 2^448 - 2^224 - 1
5 * This file is Free Software. It has been modified to as part of its
6 * incorporation into secnet.
8 * Copyright 2017 Mark Wooding
10 * You may redistribute this file and/or modify it under the terms of
11 * the permissive licence shown below.
13 * You may redistribute secnet as a whole and/or modify it under the
14 * terms of the GNU General Public License as published by the Free
15 * Software Foundation; either version 3, or (at your option) any
18 * This program is distributed in the hope that it will be useful,
19 * but WITHOUT ANY WARRANTY; without even the implied warranty of
20 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
21 * GNU General Public License for more details.
23 * You should have received a copy of the GNU General Public License
24 * along with this program; if not, see
25 * https://www.gnu.org/licenses/gpl.html.
28 * Imported from Catacomb, and modified for Secnet (2017-04-30):
30 * * Use `fake-mLib-bits.h' in place of the real <mLib/bits.h>.
32 * * Remove the 16/32-bit implementation, since C99 always has 64-bit
35 * * Remove the test rig code: a replacement is in a separate source file.
37 * The file's original comment headers are preserved below.
41 * Arithmetic in the Goldilocks field GF(2^448 - 2^224 - 1)
43 * (c) 2017 Straylight/Edgeware
46 /*----- Licensing notice --------------------------------------------------*
48 * This file is part of Catacomb.
50 * Catacomb is free software; you can redistribute it and/or modify
51 * it under the terms of the GNU Library General Public License as
52 * published by the Free Software Foundation; either version 2 of the
53 * License, or (at your option) any later version.
55 * Catacomb is distributed in the hope that it will be useful,
56 * but WITHOUT ANY WARRANTY; without even the implied warranty of
57 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
58 * GNU Library General Public License for more details.
60 * You should have received a copy of the GNU Library General Public
61 * License along with Catacomb; if not, write to the Free
62 * Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
66 /*----- Header files ------------------------------------------------------*/
70 /*----- Basic setup -------------------------------------------------------*
72 * Let φ = 2^224; then p = φ^2 - φ - 1, and, in GF(p), we have φ^2 = φ + 1
76 /* We represent an element of GF(p) as 16 28-bit signed integer pieces x_i:
77 * x = SUM_{0<=i<16} x_i 2^(28i).
80 typedef int32 piece
; typedef int64 dblpiece
;
81 typedef uint32 upiece
; typedef uint64 udblpiece
;
86 #define B28 0x10000000u
87 #define B27 0x08000000u
88 #define M28 0x0fffffffu
89 #define M27 0x07ffffffu
90 #define M32 0xffffffffu
92 /*----- Debugging machinery -----------------------------------------------*/
94 #if defined(FGOLDI_DEBUG)
101 static mp
*get_pgoldi(void)
103 mp
*p
= MP_NEW
, *t
= MP_NEW
;
105 p
= mp_setbit(p
, MP_ZERO
, 448);
106 t
= mp_setbit(t
, MP_ZERO
, 224);
108 p
= mp_sub(p
, p
, MP_ONE
);
113 DEF_FDUMP(fdump
, piece
, PIECEWD
, NPIECE
, 56, get_pgoldi())
117 /*----- Loading and storing -----------------------------------------------*/
119 /* --- @fgoldi_load@ --- *
121 * Arguments: @fgoldi *z@ = where to store the result
122 * @const octet xv[56]@ = source to read
126 * Use: Reads an element of %$\gf{2^{448} - 2^{224} - 19}$% in
127 * external representation from @xv@ and stores it in @z@.
129 * External representation is little-endian base-256. Some
130 * elements have multiple encodings, which are not produced by
131 * correct software; use of noncanonical encodings is not an
132 * error, and toleration of them is considered a performance
136 void fgoldi_load(fgoldi
*z
, const octet xv
[56])
142 /* First, read the input value as words. */
143 for (i
= 0; i
< 14; i
++) xw
[i
] = LOAD32_L(xv
+ 4*i
);
145 /* Extract unsigned 28-bit pieces from the words. */
146 z
->P
[ 0] = (xw
[ 0] >> 0)&M28
;
147 z
->P
[ 7] = (xw
[ 6] >> 4)&M28
;
148 z
->P
[ 8] = (xw
[ 7] >> 0)&M28
;
149 z
->P
[15] = (xw
[13] >> 4)&M28
;
150 for (i
= 1; i
< 7; i
++) {
151 z
->P
[i
+ 0] = ((xw
[i
+ 0] << (4*i
)) | (xw
[i
- 1] >> (32 - 4*i
)))&M28
;
152 z
->P
[i
+ 8] = ((xw
[i
+ 7] << (4*i
)) | (xw
[i
+ 6] >> (32 - 4*i
)))&M28
;
155 /* Convert the nonnegative pieces into a balanced signed representation, so
156 * each piece ends up in the interval |z_i| <= 2^27. For each piece, if
157 * its top bit is set, lend a bit leftwards; in the case of z_15, reduce
158 * this bit by adding it onto z_0 and z_8, since this is the φ^2 bit, and
159 * φ^2 = φ + 1. We delay this carry until after all of the pieces have
160 * been balanced. If we don't do this, then we have to do a more expensive
161 * test for nonzeroness to decide whether to lend a bit leftwards rather
162 * than just testing a single bit.
164 * Note that we don't try for a canonical representation here: both upper
165 * and lower bounds are achievable.
167 b
= z
->P
[15]&B27
; z
->P
[15] -= b
<< 1; c
= b
>> 27;
168 for (i
= NPIECE
- 1; i
--; )
169 { b
= z
->P
[i
]&B27
; z
->P
[i
] -= b
<< 1; z
->P
[i
+ 1] += b
>> 27; }
170 z
->P
[0] += c
; z
->P
[8] += c
;
173 /* --- @fgoldi_store@ --- *
175 * Arguments: @octet zv[56]@ = where to write the result
176 * @const fgoldi *x@ = the field element to write
180 * Use: Stores a field element in the given octet vector in external
181 * representation. A canonical encoding is always stored.
184 void fgoldi_store(octet zv
[56], const fgoldi
*x
)
186 piece y
[NPIECE
], yy
[NPIECE
], c
, d
;
191 for (i
= 0; i
< NPIECE
; i
++) y
[i
] = x
->P
[i
];
193 /* First, propagate the carries. By the end of this, we'll have all of the
194 * the pieces canonically sized and positive, and maybe there'll be
195 * (signed) carry out. The carry c is in { -1, 0, +1 }, and the remaining
196 * value will be in the half-open interval [0, φ^2). The whole represented
197 * value is then y + φ^2 c.
199 * Assume that we start out with |y_i| <= 2^30. We start off by cutting
200 * off and reducing the carry c_15 from the topmost piece, y_15. This
201 * leaves 0 <= y_15 < 2^28; and we'll have |c_15| <= 4. We'll add this
202 * onto y_0 and y_8, and propagate the carries. It's very clear that we'll
203 * end up with |y + (φ + 1) c_15 - φ^2/2| << φ^2.
205 * Here, the y_i are signed, so we must be cautious about bithacking them.
207 c
= ASR(piece
, y
[15], 28); y
[15] = (upiece
)y
[15]&M28
; y
[8] += c
;
208 for (i
= 0; i
< NPIECE
; i
++)
209 { y
[i
] += c
; c
= ASR(piece
, y
[i
], 28); y
[i
] = (upiece
)y
[i
]&M28
; }
211 /* Now we have a slightly fiddly job to do. If c = +1, or if c = 0 and
212 * y >= p, then we should subtract p from the whole value; if c = -1 then
213 * we should add p; and otherwise we should do nothing.
215 * But conditional behaviour is bad, m'kay. So here's what we do instead.
217 * The first job is to sort out what we wanted to do. If c = -1 then we
218 * want to (a) invert the constant addend and (b) feed in a carry-in;
219 * otherwise, we don't.
224 /* Now do the addition/subtraction. Remember that all of the y_i are
225 * nonnegative, so shifting and masking are safe and easy.
227 d
+= y
[0] + (1 ^ m
); yy
[0] = d
&M28
; d
>>= 28;
228 for (i
= 1; i
< 8; i
++)
229 { d
+= y
[i
] + m
; yy
[i
] = d
&M28
; d
>>= 28; }
230 d
+= y
[8] + (1 ^ m
); yy
[8] = d
&M28
; d
>>= 28;
231 for (i
= 9; i
< 16; i
++)
232 { d
+= y
[i
] + m
; yy
[i
] = d
&M28
; d
>>= 28; }
234 /* The final carry-out is in d; since we only did addition, and the y_i are
235 * nonnegative, then d is in { 0, 1 }. We want to keep y', rather than y,
236 * if (a) c /= 0 (in which case we know that the old value was
237 * unsatisfactory), or (b) if d = 1 (in which case, if c = 0, we know that
238 * the subtraction didn't cause a borrow, so we must be in the case where
241 m
= NONZEROP(c
) | ~NONZEROP(d
- 1);
242 for (i
= 0; i
< NPIECE
; i
++) y
[i
] = (yy
[i
]&m
) | (y
[i
]&~m
);
244 /* Extract 32-bit words from the value. */
245 for (i
= 0; i
< 7; i
++) {
246 u
= ((y
[i
+ 0] >> (4*i
)) | ((uint32
)y
[i
+ 1] << (28 - 4*i
)))&M32
;
247 v
= ((y
[i
+ 8] >> (4*i
)) | ((uint32
)y
[i
+ 9] << (28 - 4*i
)))&M32
;
248 STORE32_L(zv
+ 4*i
, u
);
249 STORE32_L(zv
+ 4*i
+ 28, v
);
253 /* --- @fgoldi_set@ --- *
255 * Arguments: @fgoldi *z@ = where to write the result
256 * @int a@ = a small-ish constant
260 * Use: Sets @z@ to equal @a@.
263 void fgoldi_set(fgoldi
*x
, int a
)
268 for (i
= 1; i
< NPIECE
; i
++) x
->P
[i
] = 0;
271 /*----- Basic arithmetic --------------------------------------------------*/
273 /* --- @fgoldi_add@ --- *
275 * Arguments: @fgoldi *z@ = where to put the result (may alias @x@ or @y@)
276 * @const fgoldi *x, *y@ = two operands
280 * Use: Set @z@ to the sum %$x + y$%.
283 void fgoldi_add(fgoldi
*z
, const fgoldi
*x
, const fgoldi
*y
)
286 for (i
= 0; i
< NPIECE
; i
++) z
->P
[i
] = x
->P
[i
] + y
->P
[i
];
289 /* --- @fgoldi_sub@ --- *
291 * Arguments: @fgoldi *z@ = where to put the result (may alias @x@ or @y@)
292 * @const fgoldi *x, *y@ = two operands
296 * Use: Set @z@ to the difference %$x - y$%.
299 void fgoldi_sub(fgoldi
*z
, const fgoldi
*x
, const fgoldi
*y
)
302 for (i
= 0; i
< NPIECE
; i
++) z
->P
[i
] = x
->P
[i
] - y
->P
[i
];
305 /*----- Constant-time utilities -------------------------------------------*/
307 /* --- @fgoldi_condswap@ --- *
309 * Arguments: @fgoldi *x, *y@ = two operands
310 * @uint32 m@ = a mask
314 * Use: If @m@ is zero, do nothing; if @m@ is all-bits-set, then
315 * exchange @x@ and @y@. If @m@ has some other value, then
316 * scramble @x@ and @y@ in an unhelpful way.
319 void fgoldi_condswap(fgoldi
*x
, fgoldi
*y
, uint32 m
)
322 mask32 mm
= FIX_MASK32(m
);
324 for (i
= 0; i
< NPIECE
; i
++) CONDSWAP(x
->P
[i
], y
->P
[i
], mm
);
327 /*----- Multiplication ----------------------------------------------------*/
329 /* Let B = 2^63 - 1 be the largest value such that +B and -B can be
330 * represented in a double-precision piece. On entry, it must be the case
331 * that |X_i| <= M <= B - 2^27 for some M. If this is the case, then, on
332 * exit, we will have |Z_i| <= 2^27 + M/2^27.
334 #define CARRY_REDUCE(z, x) do { \
335 dblpiece _t[NPIECE], _c; \
338 /* Bias the input pieces. This keeps the carries and so on centred \
339 * around zero rather than biased positive. \
341 for (_i = 0; _i < NPIECE; _i++) _t[_i] = (x)[_i] + B27; \
343 /* Calculate the reduced pieces. Careful with the bithacking. */ \
344 _c = ASR(dblpiece, _t[15], 28); \
345 (z)[0] = (dblpiece)((udblpiece)_t[0]&M28) - B27 + _c; \
346 for (_i = 1; _i < NPIECE; _i++) { \
347 (z)[_i] = (dblpiece)((udblpiece)_t[_i]&M28) - B27 + \
348 ASR(dblpiece, _t[_i - 1], 28); \
353 /* --- @fgoldi_mulconst@ --- *
355 * Arguments: @fgoldi *z@ = where to put the result (may alias @x@)
356 * @const fgoldi *x@ = an operand
357 * @long a@ = a small-ish constant; %$|a| < 2^{20}$%.
361 * Use: Set @z@ to the product %$a x$%.
364 void fgoldi_mulconst(fgoldi
*z
, const fgoldi
*x
, long a
)
367 dblpiece zz
[NPIECE
], aa
= a
;
369 for (i
= 0; i
< NPIECE
; i
++) zz
[i
] = aa
*x
->P
[i
];
370 CARRY_REDUCE(z
->P
, zz
);
373 /* --- @fgoldi_mul@ --- *
375 * Arguments: @fgoldi *z@ = where to put the result (may alias @x@ or @y@)
376 * @const fgoldi *x, *y@ = two operands
380 * Use: Set @z@ to the product %$x y$%.
383 void fgoldi_mul(fgoldi
*z
, const fgoldi
*x
, const fgoldi
*y
)
385 dblpiece zz
[NPIECE
], u
[NPIECE
];
386 piece ab
[NPIECE
/2], cd
[NPIECE
/2];
388 *a
= x
->P
+ NPIECE
/2, *b
= x
->P
,
389 *c
= y
->P
+ NPIECE
/2, *d
= y
->P
;
392 # define M(x,i, y,j) ((dblpiece)(x)[i]*(y)[j])
396 * Write x = a φ + b, and y = c φ + d. Then x y = a c φ^2 +
397 * (a d + b c) φ + b d. Karatsuba and Ofman observed that a d + b c =
398 * (a + b) (c + d) - a c - b d, saving a multiplication, and Hamburg chose
399 * the prime p so that φ^2 = φ + 1. So
401 * x y = ((a + b) (c + d) - b d) φ + a c + b d
404 for (i
= 0; i
< NPIECE
; i
++) zz
[i
] = 0;
406 /* Our first job will be to calculate (1 - φ) b d, and write the result
407 * into z. As we do this, an interesting thing will happen. Write
408 * b d = u φ + v; then (1 - φ) b d = u φ + v - u φ^2 - v φ = (1 - φ) v - u.
409 * So, what we do is to write the product end-swapped and negated, and then
410 * we'll subtract the (negated, remember) high half from the low half.
412 for (i
= 0; i
< NPIECE
/2; i
++) {
413 for (j
= 0; j
< NPIECE
/2 - i
; j
++)
414 zz
[i
+ j
+ NPIECE
/2] -= M(b
,i
, d
,j
);
415 for (; j
< NPIECE
/2; j
++)
416 zz
[i
+ j
- NPIECE
/2] -= M(b
,i
, d
,j
);
418 for (i
= 0; i
< NPIECE
/2; i
++)
419 zz
[i
] -= zz
[i
+ NPIECE
/2];
421 /* Next, we add on a c. There are no surprises here. */
422 for (i
= 0; i
< NPIECE
/2; i
++)
423 for (j
= 0; j
< NPIECE
/2; j
++)
424 zz
[i
+ j
] += M(a
,i
, c
,j
);
426 /* Now, calculate a + b and c + d. */
427 for (i
= 0; i
< NPIECE
/2; i
++)
428 { ab
[i
] = a
[i
] + b
[i
]; cd
[i
] = c
[i
] + d
[i
]; }
430 /* Finally (for the multiplication) we must add on (a + b) (c + d) φ.
431 * Write (a + b) (c + d) as u φ + v; then we actually want u φ^2 + v φ =
432 * v φ + (1 + φ) u. We'll store u in a temporary place and add it on
435 for (i
= 0; i
< NPIECE
; i
++) u
[i
] = 0;
436 for (i
= 0; i
< NPIECE
/2; i
++) {
437 for (j
= 0; j
< NPIECE
/2 - i
; j
++)
438 zz
[i
+ j
+ NPIECE
/2] += M(ab
,i
, cd
,j
);
439 for (; j
< NPIECE
/2; j
++)
440 u
[i
+ j
- NPIECE
/2] += M(ab
,i
, cd
,j
);
442 for (i
= 0; i
< NPIECE
/2; i
++)
443 { zz
[i
] += u
[i
]; zz
[i
+ NPIECE
/2] += u
[i
]; }
447 /* That wraps it up for the multiplication. Let's figure out some bounds.
448 * Fortunately, Karatsuba is a polynomial identity, so all of the pieces
449 * end up the way they'd be if we'd done the thing the easy way, which
450 * simplifies the analysis. Suppose we started with |x_i|, |y_i| <= 9/5
451 * 2^28. The overheads in the result are given by the coefficients of
453 * ((u^16 - 1)/(u - 1))^2 mod u^16 - u^8 - 1
455 * the greatest of which is 38. So |z_i| <= 38*81/25*2^56 < 2^63.
457 * Anyway, a round of `CARRY_REDUCE' will leave us with |z_i| < 2^27 +
458 * 2^36; and a second round will leave us with |z_i| < 2^27 + 512.
460 for (i
= 0; i
< 2; i
++) CARRY_REDUCE(zz
, zz
);
461 for (i
= 0; i
< NPIECE
; i
++) z
->P
[i
] = zz
[i
];
464 /* --- @fgoldi_sqr@ --- *
466 * Arguments: @fgoldi *z@ = where to put the result (may alias @x@ or @y@)
467 * @const fgoldi *x@ = an operand
471 * Use: Set @z@ to the square %$x^2$%.
474 void fgoldi_sqr(fgoldi
*z
, const fgoldi
*x
)
476 dblpiece zz
[NPIECE
], u
[NPIECE
];
478 const piece
*a
= x
->P
+ NPIECE
/2, *b
= x
->P
;
481 # define M(x,i, y,j) ((dblpiece)(x)[i]*(y)[j])
483 /* The magic is basically the same as `fgoldi_mul' above. We write
484 * x = a φ + b and use Karatsuba and the special prime shape. This time,
487 * x^2 = ((a + b)^2 - b^2) φ + a^2 + b^2
490 for (i
= 0; i
< NPIECE
; i
++) zz
[i
] = 0;
492 /* Our first job will be to calculate (1 - φ) b^2, and write the result
493 * into z. Again, this interacts pleasantly with the prime shape.
495 for (i
= 0; i
< NPIECE
/4; i
++) {
496 zz
[2*i
+ NPIECE
/2] -= M(b
,i
, b
,i
);
497 for (j
= i
+ 1; j
< NPIECE
/2 - i
; j
++)
498 zz
[i
+ j
+ NPIECE
/2] -= 2*M(b
,i
, b
,j
);
499 for (; j
< NPIECE
/2; j
++)
500 zz
[i
+ j
- NPIECE
/2] -= 2*M(b
,i
, b
,j
);
502 for (; i
< NPIECE
/2; i
++) {
503 zz
[2*i
- NPIECE
/2] -= M(b
,i
, b
,i
);
504 for (j
= i
+ 1; j
< NPIECE
/2; j
++)
505 zz
[i
+ j
- NPIECE
/2] -= 2*M(b
,i
, b
,j
);
507 for (i
= 0; i
< NPIECE
/2; i
++)
508 zz
[i
] -= zz
[i
+ NPIECE
/2];
510 /* Next, we add on a^2. There are no surprises here. */
511 for (i
= 0; i
< NPIECE
/2; i
++) {
512 zz
[2*i
] += M(a
,i
, a
,i
);
513 for (j
= i
+ 1; j
< NPIECE
/2; j
++)
514 zz
[i
+ j
] += 2*M(a
,i
, a
,j
);
517 /* Now, calculate a + b. */
518 for (i
= 0; i
< NPIECE
/2; i
++)
521 /* Finally (for the multiplication) we must add on (a + b)^2 φ.
522 * Write (a + b)^2 as u φ + v; then we actually want (u + v) φ + u. We'll
523 * store u in a temporary place and add it on twice.
525 for (i
= 0; i
< NPIECE
; i
++) u
[i
] = 0;
526 for (i
= 0; i
< NPIECE
/4; i
++) {
527 zz
[2*i
+ NPIECE
/2] += M(ab
,i
, ab
,i
);
528 for (j
= i
+ 1; j
< NPIECE
/2 - i
; j
++)
529 zz
[i
+ j
+ NPIECE
/2] += 2*M(ab
,i
, ab
,j
);
530 for (; j
< NPIECE
/2; j
++)
531 u
[i
+ j
- NPIECE
/2] += 2*M(ab
,i
, ab
,j
);
533 for (; i
< NPIECE
/2; i
++) {
534 u
[2*i
- NPIECE
/2] += M(ab
,i
, ab
,i
);
535 for (j
= i
+ 1; j
< NPIECE
/2; j
++)
536 u
[i
+ j
- NPIECE
/2] += 2*M(ab
,i
, ab
,j
);
538 for (i
= 0; i
< NPIECE
/2; i
++)
539 { zz
[i
] += u
[i
]; zz
[i
+ NPIECE
/2] += u
[i
]; }
543 /* Finally, carrying. */
544 for (i
= 0; i
< 2; i
++) CARRY_REDUCE(zz
, zz
);
545 for (i
= 0; i
< NPIECE
; i
++) z
->P
[i
] = zz
[i
];
549 /*----- More advanced operations ------------------------------------------*/
551 /* --- @fgoldi_inv@ --- *
553 * Arguments: @fgoldi *z@ = where to put the result (may alias @x@)
554 * @const fgoldi *x@ = an operand
558 * Use: Stores in @z@ the multiplicative inverse %$x^{-1}$%. If
559 * %$x = 0$% then @z@ is set to zero. This is considered a
563 void fgoldi_inv(fgoldi
*z
, const fgoldi
*x
)
568 #define SQRN(z, x, n) do { \
569 fgoldi_sqr((z), (x)); \
570 for (i = 1; i < (n); i++) fgoldi_sqr((z), (z)); \
573 /* Calculate x^-1 = x^(p - 2) = x^(2^448 - 2^224 - 3), which also handles
574 * x = 0 as intended. The addition chain is home-made.
575 */ /* step | value */
576 fgoldi_sqr(&u
, x
); /* 1 | 2 */
577 fgoldi_mul(&t
, &u
, x
); /* 2 | 3 */
578 SQRN(&u
, &t
, 2); /* 4 | 12 */
579 fgoldi_mul(&t
, &u
, &t
); /* 5 | 15 */
580 SQRN(&u
, &t
, 4); /* 9 | 240 */
581 fgoldi_mul(&u
, &u
, &t
); /* 10 | 255 = 2^8 - 1 */
582 SQRN(&u
, &u
, 4); /* 14 | 2^12 - 16 */
583 fgoldi_mul(&t
, &u
, &t
); /* 15 | 2^12 - 1 */
584 SQRN(&u
, &t
, 12); /* 27 | 2^24 - 2^12 */
585 fgoldi_mul(&u
, &u
, &t
); /* 28 | 2^24 - 1 */
586 SQRN(&u
, &u
, 12); /* 40 | 2^36 - 2^12 */
587 fgoldi_mul(&t
, &u
, &t
); /* 41 | 2^36 - 1 */
588 fgoldi_sqr(&t
, &t
); /* 42 | 2^37 - 2 */
589 fgoldi_mul(&t
, &t
, x
); /* 43 | 2^37 - 1 */
590 SQRN(&u
, &t
, 37); /* 80 | 2^74 - 2^37 */
591 fgoldi_mul(&u
, &u
, &t
); /* 81 | 2^74 - 1 */
592 SQRN(&u
, &u
, 37); /* 118 | 2^111 - 2^37 */
593 fgoldi_mul(&t
, &u
, &t
); /* 119 | 2^111 - 1 */
594 SQRN(&u
, &t
, 111); /* 230 | 2^222 - 2^111 */
595 fgoldi_mul(&t
, &u
, &t
); /* 231 | 2^222 - 1 */
596 fgoldi_sqr(&u
, &t
); /* 232 | 2^223 - 2 */
597 fgoldi_mul(&u
, &u
, x
); /* 233 | 2^223 - 1 */
598 SQRN(&u
, &u
, 223); /* 456 | 2^446 - 2^223 */
599 fgoldi_mul(&t
, &u
, &t
); /* 457 | 2^446 - 2^222 - 1 */
600 SQRN(&t
, &t
, 2); /* 459 | 2^448 - 2^224 - 4 */
601 fgoldi_mul(z
, &t
, x
); /* 460 | 2^448 - 2^224 - 3 */
606 /*----- That's all, folks -------------------------------------------------*/