3 * Arithmetic modulo 2^255 - 19
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 typedef f25519_piece piece
;
55 /* Elements x of GF(2^255 - 19) are represented by ten signed integers x_i: x
56 * = SUM_{0<=i<10} x_i 2^ceil(51i/2), mostly following Bernstein's original
60 typedef int64 dblpiece
;
61 typedef uint32 upiece
; typedef uint64 udblpiece
;
63 #define PIECEWD(i) ((i)%2 ? 25 : 26)
66 #define M26 0x03ffffffu
67 #define M25 0x01ffffffu
68 #define B25 0x02000000u
69 #define B24 0x01000000u
71 #define PIECES(v) v##0, v##1, v##2, v##3, v##4, v##5, v##6, v##7, v##8, v##9
72 #define FETCH(v, w) do { \
73 v##0 = (w)->P[0]; v##1 = (w)->P[1]; \
74 v##2 = (w)->P[2]; v##3 = (w)->P[3]; \
75 v##4 = (w)->P[4]; v##5 = (w)->P[5]; \
76 v##6 = (w)->P[6]; v##7 = (w)->P[7]; \
77 v##8 = (w)->P[8]; v##9 = (w)->P[9]; \
79 #define STASH(w, v) do { \
80 (w)->P[0] = v##0; (w)->P[1] = v##1; \
81 (w)->P[2] = v##2; (w)->P[3] = v##3; \
82 (w)->P[4] = v##4; (w)->P[5] = v##5; \
83 (w)->P[6] = v##6; (w)->P[7] = v##7; \
84 (w)->P[8] = v##8; (w)->P[9] = v##9; \
87 /*----- Debugging machinery -----------------------------------------------*/
89 #if defined(F25519_DEBUG)
96 static mp
*get_2p255m91(void)
101 p
= mp_setbit(p
, MP_ZERO
, 255);
102 mp_build(&m19
, &w19
, &w19
+ 1);
103 p
= mp_sub(p
, p
, &m19
);
107 DEF_FDUMP(fdump
, piece
, PIECEWD
, NPIECE
, 32, get_2p255m91())
111 /*----- Loading and storing -----------------------------------------------*/
113 /* --- @f25519_load@ --- *
115 * Arguments: @f25519 *z@ = where to store the result
116 * @const octet xv[32]@ = source to read
120 * Use: Reads an element of %$\gf{2^{255} - 19}$% in external
121 * representation from @xv@ and stores it in @z@.
123 * External representation is little-endian base-256. Some
124 * elements have multiple encodings, which are not produced by
125 * correct software; use of noncanonical encodings is not an
126 * error, and toleration of them is considered a performance
130 void f25519_load(f25519
*z
, const octet xv
[32])
133 uint32 xw0
= LOAD32_L(xv
+ 0), xw1
= LOAD32_L(xv
+ 4),
134 xw2
= LOAD32_L(xv
+ 8), xw3
= LOAD32_L(xv
+ 12),
135 xw4
= LOAD32_L(xv
+ 16), xw5
= LOAD32_L(xv
+ 20),
136 xw6
= LOAD32_L(xv
+ 24), xw7
= LOAD32_L(xv
+ 28);
137 piece
PIECES(x
), b
, c
;
139 /* First, split the 32-bit words into the irregularly-sized pieces we need
140 * for the field representation. These pieces are all positive. We'll do
141 * the sign correction afterwards.
143 * It may be that the top bit of the input is set. Avoid trouble by
144 * folding that back round into the bottom piece of the representation.
146 * Here, we briefly have 0 <= x_0 < 2^26 + 19, but will resolve this later.
147 * Otherwise, we have 0 <= x_{2i} < 2^26, and 0 <= x_{2i+1} < 2^25.
149 x0
= ((xw0
<< 0)&0x03ffffff) + 19*((xw7
>> 31)&0x00000001);
150 x1
= ((xw1
<< 6)&0x01ffffc0) | ((xw0
>> 26)&0x0000003f);
151 x2
= ((xw2
<< 13)&0x03ffe000) | ((xw1
>> 19)&0x00001fff);
152 x3
= ((xw3
<< 19)&0x01f80000) | ((xw2
>> 13)&0x0007ffff);
153 x4
= ((xw3
>> 6)&0x03ffffff);
154 x5
= (xw4
<< 0)&0x01ffffff;
155 x6
= ((xw5
<< 7)&0x03ffff80) | ((xw4
>> 25)&0x0000007f);
156 x7
= ((xw6
<< 13)&0x01ffe000) | ((xw5
>> 19)&0x00001fff);
157 x8
= ((xw7
<< 20)&0x03f00000) | ((xw6
>> 12)&0x000fffff);
158 x9
= ((xw7
>> 6)&0x01ffffff);
160 /* Next, we convert these pieces into a roughly balanced signed
161 * representation. For each piece, check to see if its top bit is set. If
162 * it is, then lend a bit to the next piece over. For x_9, this needs to
163 * be carried around, which is a little fiddly. In particular, we delay
164 * the carry until after all of the pieces have been balanced. If we don't
165 * do this, then we have to do a more expensive test for nonzeroness to
166 * decide whether to lend a bit leftwards rather than just testing a single
169 * This fixes up the anomalous size of x_0: the loan of a bit becomes an
170 * actual carry if x_0 >= 2^26. By the end, then, we have:
176 * Note that we don't try for a canonical representation here: both upper
177 * and lower bounds are achievable.
179 * All of the x_i at this point are positive, so we don't need to do
180 * anything wierd when masking them.
182 b
= x9
&B24
; c
= 19&((b
>> 19) - (b
>> 24)); x9
-= b
<< 1;
183 b
= x8
&B25
; x9
+= b
>> 25; x8
-= b
<< 1;
184 b
= x7
&B24
; x8
+= b
>> 24; x7
-= b
<< 1;
185 b
= x6
&B25
; x7
+= b
>> 25; x6
-= b
<< 1;
186 b
= x5
&B24
; x6
+= b
>> 24; x5
-= b
<< 1;
187 b
= x4
&B25
; x5
+= b
>> 25; x4
-= b
<< 1;
188 b
= x3
&B24
; x4
+= b
>> 24; x3
-= b
<< 1;
189 b
= x2
&B25
; x3
+= b
>> 25; x2
-= b
<< 1;
190 b
= x1
&B24
; x2
+= b
>> 24; x1
-= b
<< 1;
191 b
= x0
&B25
; x1
+= (b
>> 25) + (x0
>> 26); x0
= (x0
&M26
) - (b
<< 1);
194 /* And with that, we're done. */
198 /* --- @f25519_store@ --- *
200 * Arguments: @octet zv[32]@ = where to write the result
201 * @const f25519 *x@ = the field element to write
205 * Use: Stores a field element in the given octet vector in external
206 * representation. A canonical encoding is always stored, so,
207 * in particular, the top bit of @xv[31]@ is always left clear.
210 void f25519_store(octet zv
[32], const f25519
*x
)
213 piece
PIECES(x
), PIECES(y
), c
, d
;
214 uint32 zw0
, zw1
, zw2
, zw3
, zw4
, zw5
, zw6
, zw7
;
219 /* First, propagate the carries throughout the pieces. By the end of this,
220 * we'll have all of the pieces canonically sized and positive, and maybe
221 * there'll be (signed) carry out. The carry c is in { -1, 0, +1 }, and
222 * the remaining value will be in the half-open interval [0, 2^255). The
223 * whole represented value is then x + 2^255 c.
225 * It's worth paying careful attention to the bounds. We assume that we
226 * start out with |x_i| <= 2^30. We start by cutting off and reducing the
227 * carry c_9 from the topmost piece, x_9. This leaves 0 <= x_9 < 2^25; and
228 * we'll have |c_9| <= 2^5. We multiply this by 19 and we'll add it onto
229 * x_0 and propagate the carries: but what bounds can we calculate on x
232 * Let o_i = floor(51 i/2). We have X_i = SUM_{0<=j<i} x_j 2^{o_i}, so
233 * x = X_10. We see, inductively, that |X_i| < 2^{31+o_{i-1}}: X_0 = 0;
234 * |x_i| <= 2^30; and |X_{i+1}| = |X_i + x_i 2^{o_i}| <= |X_i| + 2^{30+o_i}
235 * < 2^{31+o_i}. Then x = X_9 + 2^230 x_9, and we have better bounds for
238 * -2^235 < x + 19 c_9 < 2^255 + 2^235
240 * Here, the x_i are signed, so we must be cautious about bithacking them.
242 c
= ASR(piece
, x9
, 25); x9
= (upiece
)x9
&M25
;
243 x0
+= 19*c
; c
= ASR(piece
, x0
, 26); x0
= (upiece
)x0
&M26
;
244 x1
+= c
; c
= ASR(piece
, x1
, 25); x1
= (upiece
)x1
&M25
;
245 x2
+= c
; c
= ASR(piece
, x2
, 26); x2
= (upiece
)x2
&M26
;
246 x3
+= c
; c
= ASR(piece
, x3
, 25); x3
= (upiece
)x3
&M25
;
247 x4
+= c
; c
= ASR(piece
, x4
, 26); x4
= (upiece
)x4
&M26
;
248 x5
+= c
; c
= ASR(piece
, x5
, 25); x5
= (upiece
)x5
&M25
;
249 x6
+= c
; c
= ASR(piece
, x6
, 26); x6
= (upiece
)x6
&M26
;
250 x7
+= c
; c
= ASR(piece
, x7
, 25); x7
= (upiece
)x7
&M25
;
251 x8
+= c
; c
= ASR(piece
, x8
, 26); x8
= (upiece
)x8
&M26
;
252 x9
+= c
; c
= ASR(piece
, x9
, 25); x9
= (upiece
)x9
&M25
;
254 /* Now we have a slightly fiddly job to do. If c = +1, or if c = 0 and
255 * x >= 2^255 - 19, then we should subtract 2^255 - 19 from the whole
256 * value; if c = -1 then we should add 2^255 - 19; and otherwise we should
259 * But conditional behaviour is bad, m'kay. So here's what we do instead.
261 * The first job is to sort out what we wanted to do. If c = -1 then we
262 * want to (a) invert the constant addend and (b) feed in a carry-in;
263 * otherwise, we don't.
268 /* Now do the addition/subtraction. Remember that all of the x_i are
269 * nonnegative, so shifting and masking are safe and easy.
271 d
+= x0
+ (19 ^ (M26
&m
)); y0
= d
&M26
; d
>>= 26;
272 d
+= x1
+ (M25
&m
); y1
= d
&M25
; d
>>= 25;
273 d
+= x2
+ (M26
&m
); y2
= d
&M26
; d
>>= 26;
274 d
+= x3
+ (M25
&m
); y3
= d
&M25
; d
>>= 25;
275 d
+= x4
+ (M26
&m
); y4
= d
&M26
; d
>>= 26;
276 d
+= x5
+ (M25
&m
); y5
= d
&M25
; d
>>= 25;
277 d
+= x6
+ (M26
&m
); y6
= d
&M26
; d
>>= 26;
278 d
+= x7
+ (M25
&m
); y7
= d
&M25
; d
>>= 25;
279 d
+= x8
+ (M26
&m
); y8
= d
&M26
; d
>>= 26;
280 d
+= x9
+ (M25
&m
); y9
= d
&M25
; d
>>= 25;
282 /* The final carry-out is in d; since we only did addition, and the x_i are
283 * nonnegative, then d is in { 0, 1 }. We want to keep y, rather than x,
284 * if (a) c /= 0 (in which case we know that the old value was
285 * unsatisfactory), or (b) if d = 1 (in which case, if c = 0, we know that
286 * the subtraction didn't cause a borrow, so we must be in the case where
287 * 2^255 - 19 <= x < 2^255).
289 m
= NONZEROP(c
) | ~NONZEROP(d
- 1);
290 x0
= (y0
&m
) | (x0
&~m
); x1
= (y1
&m
) | (x1
&~m
);
291 x2
= (y2
&m
) | (x2
&~m
); x3
= (y3
&m
) | (x3
&~m
);
292 x4
= (y4
&m
) | (x4
&~m
); x5
= (y5
&m
) | (x5
&~m
);
293 x6
= (y6
&m
) | (x6
&~m
); x7
= (y7
&m
) | (x7
&~m
);
294 x8
= (y8
&m
) | (x8
&~m
); x9
= (y9
&m
) | (x9
&~m
);
296 /* Extract 32-bit words from the value. */
297 zw0
= ((x0
>> 0)&0x03ffffff) | (((uint32
)x1
<< 26)&0xfc000000);
298 zw1
= ((x1
>> 6)&0x0007ffff) | (((uint32
)x2
<< 19)&0xfff80000);
299 zw2
= ((x2
>> 13)&0x00001fff) | (((uint32
)x3
<< 13)&0xffffe000);
300 zw3
= ((x3
>> 19)&0x0000003f) | (((uint32
)x4
<< 6)&0xffffffc0);
301 zw4
= ((x5
>> 0)&0x01ffffff) | (((uint32
)x6
<< 25)&0xfe000000);
302 zw5
= ((x6
>> 7)&0x0007ffff) | (((uint32
)x7
<< 19)&0xfff80000);
303 zw6
= ((x7
>> 13)&0x00000fff) | (((uint32
)x8
<< 12)&0xfffff000);
304 zw7
= ((x8
>> 20)&0x0000003f) | (((uint32
)x9
<< 6)&0x7fffffc0);
306 /* Store the result as an octet string. */
307 STORE32_L(zv
+ 0, zw0
); STORE32_L(zv
+ 4, zw1
);
308 STORE32_L(zv
+ 8, zw2
); STORE32_L(zv
+ 12, zw3
);
309 STORE32_L(zv
+ 16, zw4
); STORE32_L(zv
+ 20, zw5
);
310 STORE32_L(zv
+ 24, zw6
); STORE32_L(zv
+ 28, zw7
);
313 /* --- @f25519_set@ --- *
315 * Arguments: @f25519 *z@ = where to write the result
316 * @int a@ = a small-ish constant
320 * Use: Sets @z@ to equal @a@.
323 void f25519_set(f25519
*x
, int a
)
328 for (i
= 1; i
< NPIECE
; i
++) x
->P
[i
] = 0;
331 /*----- Basic arithmetic --------------------------------------------------*/
333 /* --- @f25519_add@ --- *
335 * Arguments: @f25519 *z@ = where to put the result (may alias @x@ or @y@)
336 * @const f25519 *x, *y@ = two operands
340 * Use: Set @z@ to the sum %$x + y$%.
343 void f25519_add(f25519
*z
, const f25519
*x
, const f25519
*y
)
345 z
->P
[0] = x
->P
[0] + y
->P
[0]; z
->P
[1] = x
->P
[1] + y
->P
[1];
346 z
->P
[2] = x
->P
[2] + y
->P
[2]; z
->P
[3] = x
->P
[3] + y
->P
[3];
347 z
->P
[4] = x
->P
[4] + y
->P
[4]; z
->P
[5] = x
->P
[5] + y
->P
[5];
348 z
->P
[6] = x
->P
[6] + y
->P
[6]; z
->P
[7] = x
->P
[7] + y
->P
[7];
349 z
->P
[8] = x
->P
[8] + y
->P
[8]; z
->P
[9] = x
->P
[9] + y
->P
[9];
352 /* --- @f25519_sub@ --- *
354 * Arguments: @f25519 *z@ = where to put the result (may alias @x@ or @y@)
355 * @const f25519 *x, *y@ = two operands
359 * Use: Set @z@ to the difference %$x - y$%.
362 void f25519_sub(f25519
*z
, const f25519
*x
, const f25519
*y
)
364 z
->P
[0] = x
->P
[0] - y
->P
[0]; z
->P
[1] = x
->P
[1] - y
->P
[1];
365 z
->P
[2] = x
->P
[2] - y
->P
[2]; z
->P
[3] = x
->P
[3] - y
->P
[3];
366 z
->P
[4] = x
->P
[4] - y
->P
[4]; z
->P
[5] = x
->P
[5] - y
->P
[5];
367 z
->P
[6] = x
->P
[6] - y
->P
[6]; z
->P
[7] = x
->P
[7] - y
->P
[7];
368 z
->P
[8] = x
->P
[8] - y
->P
[8]; z
->P
[9] = x
->P
[9] - y
->P
[9];
371 /* --- @f25519_neg@ --- *
373 * Arguments: @f25519 *z@ = where to put the result (may alias @x@)
374 * @const f25519 *x@ = an operand
381 void f25519_neg(f25519
*z
, const f25519
*x
)
383 z
->P
[0] = -x
->P
[0]; z
->P
[1] = -x
->P
[1];
384 z
->P
[2] = -x
->P
[2]; z
->P
[3] = -x
->P
[3];
385 z
->P
[4] = -x
->P
[4]; z
->P
[5] = -x
->P
[5];
386 z
->P
[6] = -x
->P
[6]; z
->P
[7] = -x
->P
[7];
387 z
->P
[8] = -x
->P
[8]; z
->P
[9] = -x
->P
[9];
390 /*----- Constant-time utilities -------------------------------------------*/
392 /* --- @f25519_pick2@ --- *
394 * Arguments: @f25519 *z@ = where to put the result (may alias @x@ or @y@)
395 * @const f25519 *x, *y@ = two operands
396 * @uint32 m@ = a mask
400 * Use: If @m@ is zero, set @z = y@; if @m@ is all-bits-set, then set
401 * @z = x@. If @m@ has some other value, then scramble @z@ in
405 void f25519_pick2(f25519
*z
, const f25519
*x
, const f25519
*y
, uint32 m
)
407 mask32 mm
= FIX_MASK32(m
);
409 z
->P
[0] = PICK2(x
->P
[0], y
->P
[0], mm
);
410 z
->P
[1] = PICK2(x
->P
[1], y
->P
[1], mm
);
411 z
->P
[2] = PICK2(x
->P
[2], y
->P
[2], mm
);
412 z
->P
[3] = PICK2(x
->P
[3], y
->P
[3], mm
);
413 z
->P
[4] = PICK2(x
->P
[4], y
->P
[4], mm
);
414 z
->P
[5] = PICK2(x
->P
[5], y
->P
[5], mm
);
415 z
->P
[6] = PICK2(x
->P
[6], y
->P
[6], mm
);
416 z
->P
[7] = PICK2(x
->P
[7], y
->P
[7], mm
);
417 z
->P
[8] = PICK2(x
->P
[8], y
->P
[8], mm
);
418 z
->P
[9] = PICK2(x
->P
[9], y
->P
[9], mm
);
421 /* --- @f25519_pickn@ --- *
423 * Arguments: @f25519 *z@ = where to put the result
424 * @const f25519 *v@ = a table of entries
425 * @size_t n@ = the number of entries in @v@
426 * @size_t i@ = an index
430 * Use: If @0 <= i < n < 32@ then set @z = v[i]@. If @n >= 32@ then
431 * do something unhelpful; otherwise, if @i >= n@ then set @z@
435 void f25519_pickn(f25519
*z
, const f25519
*v
, size_t n
, size_t i
)
437 uint32 b
= (uint32
)1 << (31 - i
);
440 z
->P
[0] = z
->P
[1] = z
->P
[2] = z
->P
[3] = z
->P
[4] =
441 z
->P
[5] = z
->P
[6] = z
->P
[7] = z
->P
[8] = z
->P
[9] = 0;
444 CONDPICK(z
->P
[0], v
->P
[0], m
);
445 CONDPICK(z
->P
[1], v
->P
[1], m
);
446 CONDPICK(z
->P
[2], v
->P
[2], m
);
447 CONDPICK(z
->P
[3], v
->P
[3], m
);
448 CONDPICK(z
->P
[4], v
->P
[4], m
);
449 CONDPICK(z
->P
[5], v
->P
[5], m
);
450 CONDPICK(z
->P
[6], v
->P
[6], m
);
451 CONDPICK(z
->P
[7], v
->P
[7], m
);
452 CONDPICK(z
->P
[8], v
->P
[8], m
);
453 CONDPICK(z
->P
[9], v
->P
[9], m
);
458 /* --- @f25519_condswap@ --- *
460 * Arguments: @f25519 *x, *y@ = two operands
461 * @uint32 m@ = a mask
465 * Use: If @m@ is zero, do nothing; if @m@ is all-bits-set, then
466 * exchange @x@ and @y@. If @m@ has some other value, then
467 * scramble @x@ and @y@ in an unhelpful way.
470 void f25519_condswap(f25519
*x
, f25519
*y
, uint32 m
)
472 mask32 mm
= FIX_MASK32(m
);
474 CONDSWAP(x
->P
[0], y
->P
[0], mm
);
475 CONDSWAP(x
->P
[1], y
->P
[1], mm
);
476 CONDSWAP(x
->P
[2], y
->P
[2], mm
);
477 CONDSWAP(x
->P
[3], y
->P
[3], mm
);
478 CONDSWAP(x
->P
[4], y
->P
[4], mm
);
479 CONDSWAP(x
->P
[5], y
->P
[5], mm
);
480 CONDSWAP(x
->P
[6], y
->P
[6], mm
);
481 CONDSWAP(x
->P
[7], y
->P
[7], mm
);
482 CONDSWAP(x
->P
[8], y
->P
[8], mm
);
483 CONDSWAP(x
->P
[9], y
->P
[9], mm
);
486 /* --- @f25519_condneg@ --- *
488 * Arguments: @f25519 *z@ = where to put the result (may alias @x@)
489 * @const f25519 *x@ = an operand
490 * @uint32 m@ = a mask
494 * Use: If @m@ is zero, set @z = x@; if @m@ is all-bits-set, then set
495 * @z = -x@. If @m@ has some other value then scramble @z@ in
499 void f25519_condneg(f25519
*z
, const f25519
*x
, uint32 m
)
501 mask32 m_xor
= FIX_MASK32(m
);
503 # define CONDNEG(x) (((x) ^ m_xor) + m_add)
505 z
->P
[0] = CONDNEG(x
->P
[0]);
506 z
->P
[1] = CONDNEG(x
->P
[1]);
507 z
->P
[2] = CONDNEG(x
->P
[2]);
508 z
->P
[3] = CONDNEG(x
->P
[3]);
509 z
->P
[4] = CONDNEG(x
->P
[4]);
510 z
->P
[5] = CONDNEG(x
->P
[5]);
511 z
->P
[6] = CONDNEG(x
->P
[6]);
512 z
->P
[7] = CONDNEG(x
->P
[7]);
513 z
->P
[8] = CONDNEG(x
->P
[8]);
514 z
->P
[9] = CONDNEG(x
->P
[9]);
519 /*----- Multiplication ----------------------------------------------------*/
521 /* Let B = 2^63 - 1 be the largest value such that +B and -B can be
522 * represented in a double-precision piece. On entry, it must be the case
523 * that |X_i| <= M <= B - 2^25 for some M. If this is the case, then, on
524 * exit, we will have |Z_i| <= 2^25 + 19 M/2^25.
526 #define CARRYSTEP(z, x, m, b, f, xx, n) do { \
527 (z) = (dblpiece)((udblpiece)(x)&(m)) - (b) + \
528 (f)*ASR(dblpiece, (xx), (n)); \
530 #define CARRY_REDUCE(z, x) do { \
531 dblpiece PIECES(_t); \
533 /* Bias the input pieces. This keeps the carries and so on centred \
534 * around zero rather than biased positive. \
536 _t0 = (x##0) + B25; _t1 = (x##1) + B24; \
537 _t2 = (x##2) + B25; _t3 = (x##3) + B24; \
538 _t4 = (x##4) + B25; _t5 = (x##5) + B24; \
539 _t6 = (x##6) + B25; _t7 = (x##7) + B24; \
540 _t8 = (x##8) + B25; _t9 = (x##9) + B24; \
542 /* Calculate the reduced pieces. Careful with the bithacking. */ \
543 CARRYSTEP(z##0, _t0, M26, B25, 19, _t9, 25); \
544 CARRYSTEP(z##1, _t1, M25, B24, 1, _t0, 26); \
545 CARRYSTEP(z##2, _t2, M26, B25, 1, _t1, 25); \
546 CARRYSTEP(z##3, _t3, M25, B24, 1, _t2, 26); \
547 CARRYSTEP(z##4, _t4, M26, B25, 1, _t3, 25); \
548 CARRYSTEP(z##5, _t5, M25, B24, 1, _t4, 26); \
549 CARRYSTEP(z##6, _t6, M26, B25, 1, _t5, 25); \
550 CARRYSTEP(z##7, _t7, M25, B24, 1, _t6, 26); \
551 CARRYSTEP(z##8, _t8, M26, B25, 1, _t7, 25); \
552 CARRYSTEP(z##9, _t9, M25, B24, 1, _t8, 26); \
555 /* --- @f25519_mulconst@ --- *
557 * Arguments: @f25519 *z@ = where to put the result (may alias @x@)
558 * @const f25519 *x@ = an operand
559 * @long a@ = a small-ish constant; %$|a| < 2^{20}$%.
563 * Use: Set @z@ to the product %$a x$%.
566 void f25519_mulconst(f25519
*z
, const f25519
*x
, long a
)
570 dblpiece
PIECES(z
), aa
= a
;
574 /* Suppose that |x_i| <= 2^27, and |a| <= 2^23. Then we'll have
577 z0
= aa
*x0
; z1
= aa
*x1
; z2
= aa
*x2
; z3
= aa
*x3
; z4
= aa
*x4
;
578 z5
= aa
*x5
; z6
= aa
*x6
; z7
= aa
*x7
; z8
= aa
*x8
; z9
= aa
*x9
;
580 /* Following `CARRY_REDUCE', we'll have |z_i| <= 2^26. */
585 /* --- @f25519_mul@ --- *
587 * Arguments: @f25519 *z@ = where to put the result (may alias @x@ or @y@)
588 * @const f25519 *x, *y@ = two operands
592 * Use: Set @z@ to the product %$x y$%.
595 void f25519_mul(f25519
*z
, const f25519
*x
, const f25519
*y
)
598 piece
PIECES(x
), PIECES(y
);
602 FETCH(x
, x
); FETCH(y
, y
);
604 /* Suppose that |x_i|, |y_i| <= 2^27. Then we'll have
617 * all of which are less than 2^63 - 2^25.
620 #define M(a, b) ((dblpiece)(a)*(b))
622 19*(M(x2
, y8
) + M(x4
, y6
) + M(x6
, y4
) + M(x8
, y2
)) +
623 38*(M(x1
, y9
) + M(x3
, y7
) + M(x5
, y5
) + M(x7
, y3
) + M(x9
, y1
));
624 z1
= M(x0
, y1
) + M(x1
, y0
) +
625 19*(M(x2
, y9
) + M(x3
, y8
) + M(x4
, y7
) + M(x5
, y6
) +
626 M(x6
, y5
) + M(x7
, y4
) + M(x8
, y3
) + M(x9
, y2
));
627 z2
= M(x0
, y2
) + M(x2
, y0
) +
629 19*(M(x4
, y8
) + M(x6
, y6
) + M(x8
, y4
)) +
630 38*(M(x3
, y9
) + M(x5
, y7
) + M(x7
, y5
) + M(x9
, y3
));
631 z3
= M(x0
, y3
) + M(x1
, y2
) + M(x2
, y1
) + M(x3
, y0
) +
632 19*(M(x4
, y9
) + M(x5
, y8
) + M(x6
, y7
) +
633 M(x7
, y6
) + M(x8
, y5
) + M(x9
, y4
));
634 z4
= M(x0
, y4
) + M(x2
, y2
) + M(x4
, y0
) +
635 2*(M(x1
, y3
) + M(x3
, y1
)) +
636 19*(M(x6
, y8
) + M(x8
, y6
)) +
637 38*(M(x5
, y9
) + M(x7
, y7
) + M(x9
, y5
));
638 z5
= M(x0
, y5
) + M(x1
, y4
) + M(x2
, y3
) +
639 M(x3
, y2
) + M(x4
, y1
) + M(x5
, y0
) +
640 19*(M(x6
, y9
) + M(x7
, y8
) + M(x8
, y7
) + M(x9
, y6
));
641 z6
= M(x0
, y6
) + M(x2
, y4
) + M(x4
, y2
) + M(x6
, y0
) +
642 2*(M(x1
, y5
) + M(x3
, y3
) + M(x5
, y1
)) +
644 38*(M(x7
, y9
) + M(x9
, y7
));
645 z7
= M(x0
, y7
) + M(x1
, y6
) + M(x2
, y5
) + M(x3
, y4
) +
646 M(x4
, y3
) + M(x5
, y2
) + M(x6
, y1
) + M(x7
, y0
) +
647 19*(M(x8
, y9
) + M(x9
, y8
));
648 z8
= M(x0
, y8
) + M(x2
, y6
) + M(x4
, y4
) + M(x6
, y2
) + M(x8
, y0
) +
649 2*(M(x1
, y7
) + M(x3
, y5
) + M(x5
, y3
) + M(x7
, y1
)) +
651 z9
= M(x0
, y9
) + M(x1
, y8
) + M(x2
, y7
) + M(x3
, y6
) + M(x4
, y5
) +
652 M(x5
, y4
) + M(x6
, y3
) + M(x7
, y2
) + M(x8
, y1
) + M(x9
, y0
);
655 /* From above, we have |z_i| <= 2^63 - 2^25. A pass of `CARRY_REDUCE' will
656 * leave |z_i| <= 2^38 + 2^25; and a second pass will leave |z_i| <= 2^25 +
657 * 2^13, which is comfortable for an addition prior to the next
660 for (i
= 0; i
< 2; i
++) CARRY_REDUCE(z
, z
);
664 /* --- @f25519_sqr@ --- *
666 * Arguments: @f25519 *z@ = where to put the result (may alias @x@ or @y@)
667 * @const f25519 *x@ = an operand
671 * Use: Set @z@ to the square %$x^2$%.
674 void f25519_sqr(f25519
*z
, const f25519
*x
)
683 /* See `f25519_mul' for bounds. */
685 #define M(a, b) ((dblpiece)(a)*(b))
687 38*(M(x2
, x8
) + M(x4
, x6
) + M(x5
, x5
)) +
688 76*(M(x1
, x9
) + M(x3
, x7
));
690 38*(M(x2
, x9
) + M(x3
, x8
) + M(x4
, x7
) + M(x5
, x6
));
691 z2
= 2*(M(x0
, x2
) + M(x1
, x1
)) +
694 76*(M(x3
, x9
) + M(x5
, x7
));
695 z3
= 2*(M(x0
, x3
) + M(x1
, x2
)) +
696 38*(M(x4
, x9
) + M(x5
, x8
) + M(x6
, x7
));
700 38*(M(x6
, x8
) + M(x7
, x7
)) +
702 z5
= 2*(M(x0
, x5
) + M(x1
, x4
) + M(x2
, x3
)) +
703 38*(M(x6
, x9
) + M(x7
, x8
));
704 z6
= 2*(M(x0
, x6
) + M(x2
, x4
) + M(x3
, x3
)) +
708 z7
= 2*(M(x0
, x7
) + M(x1
, x6
) + M(x2
, x5
) + M(x3
, x4
)) +
711 2*(M(x0
, x8
) + M(x2
, x6
)) +
712 4*(M(x1
, x7
) + M(x3
, x5
)) +
714 z9
= 2*(M(x0
, x9
) + M(x1
, x8
) + M(x2
, x7
) + M(x3
, x6
) + M(x4
, x5
));
717 /* See `f25519_mul' for details. */
718 for (i
= 0; i
< 2; i
++) CARRY_REDUCE(z
, z
);
722 /*----- More complicated things -------------------------------------------*/
724 /* --- @f25519_inv@ --- *
726 * Arguments: @f25519 *z@ = where to put the result (may alias @x@)
727 * @const f25519 *x@ = an operand
731 * Use: Stores in @z@ the multiplicative inverse %$x^{-1}$%. If
732 * %$x = 0$% then @z@ is set to zero. This is considered a
736 void f25519_inv(f25519
*z
, const f25519
*x
)
738 f25519 t
, u
, t2
, t11
, t2p10m1
, t2p50m1
;
741 #define SQRN(z, x, n) do { \
742 f25519_sqr((z), (x)); \
743 for (i = 1; i < (n); i++) f25519_sqr((z), (z)); \
746 /* Calculate x^-1 = x^(p - 2) = x^(2^255 - 21), which also handles x = 0 as
747 * intended. The addition chain here is from Bernstein's implementation; I
748 * couldn't find a better one.
749 */ /* step | value */
750 f25519_sqr(&t2
, x
); /* 1 | 2 */
751 SQRN(&u
, &t2
, 2); /* 3 | 8 */
752 f25519_mul(&t
, &u
, x
); /* 4 | 9 */
753 f25519_mul(&t11
, &t
, &t2
); /* 5 | 11 = 2^5 - 21 */
754 f25519_sqr(&u
, &t11
); /* 6 | 22 */
755 f25519_mul(&t
, &t
, &u
); /* 7 | 31 = 2^5 - 1 */
756 SQRN(&u
, &t
, 5); /* 12 | 2^10 - 2^5 */
757 f25519_mul(&t2p10m1
, &t
, &u
); /* 13 | 2^10 - 1 */
758 SQRN(&u
, &t2p10m1
, 10); /* 23 | 2^20 - 2^10 */
759 f25519_mul(&t
, &t2p10m1
, &u
); /* 24 | 2^20 - 1 */
760 SQRN(&u
, &t
, 20); /* 44 | 2^40 - 2^20 */
761 f25519_mul(&t
, &t
, &u
); /* 45 | 2^40 - 1 */
762 SQRN(&u
, &t
, 10); /* 55 | 2^50 - 2^10 */
763 f25519_mul(&t2p50m1
, &t2p10m1
, &u
); /* 56 | 2^50 - 1 */
764 SQRN(&u
, &t2p50m1
, 50); /* 106 | 2^100 - 2^50 */
765 f25519_mul(&t
, &t2p50m1
, &u
); /* 107 | 2^100 - 1 */
766 SQRN(&u
, &t
, 100); /* 207 | 2^200 - 2^100 */
767 f25519_mul(&t
, &t
, &u
); /* 208 | 2^200 - 1 */
768 SQRN(&u
, &t
, 50); /* 258 | 2^250 - 2^50 */
769 f25519_mul(&t
, &t2p50m1
, &u
); /* 259 | 2^250 - 1 */
770 SQRN(&u
, &t
, 5); /* 264 | 2^255 - 2^5 */
771 f25519_mul(z
, &u
, &t11
); /* 265 | 2^255 - 21 */
776 /* --- @f25519_quosqrt@ --- *
778 * Arguments: @f25519 *z@ = where to put the result (may alias @x@ or @y@)
779 * @const f25519 *x, *y@ = two operands
781 * Returns: Zero if successful, @-1@ if %$x/y$% is not a square.
783 * Use: Stores in @z@ the one of the square roots %$\pm\sqrt{x/y}$%.
784 * If %$x = y = 0% then the result is zero; if %$y = 0$% but %$x
785 * \ne 0$% then the operation fails. If you wanted a specific
786 * square root then you'll have to pick it yourself.
789 static const piece sqrtm1_pieces
[NPIECE
] = {
790 -32595792, -7943725, 9377950, 3500415, 12389472,
791 -272473, -25146209, -2005654, 326686, 11406482
793 #define SQRTM1 ((const f25519 *)sqrtm1_pieces)
795 int f25519_quosqrt(f25519
*z
, const f25519
*x
, const f25519
*y
)
797 f25519 t
, u
, v
, w
, t15
;
798 octet xb
[32], b0
[32], b1
[32];
803 #define SQRN(z, x, n) do { \
804 f25519_sqr((z), (x)); \
805 for (i = 1; i < (n); i++) f25519_sqr((z), (z)); \
808 /* This is a bit tricky; the algorithm is loosely based on Bernstein, Duif,
809 * Lange, Schwabe, and Yang, `High-speed high-security signatures',
810 * 2011-09-26, https://ed25519.cr.yp.to/ed25519-20110926.pdf.
812 f25519_mul(&v
, x
, y
);
814 /* Now for an addition chain. */ /* step | value */
815 f25519_sqr(&u
, &v
); /* 1 | 2 */
816 f25519_mul(&t
, &u
, &v
); /* 2 | 3 */
817 SQRN(&u
, &t
, 2); /* 4 | 12 */
818 f25519_mul(&t15
, &u
, &t
); /* 5 | 15 */
819 f25519_sqr(&u
, &t15
); /* 6 | 30 */
820 f25519_mul(&t
, &u
, &v
); /* 7 | 31 = 2^5 - 1 */
821 SQRN(&u
, &t
, 5); /* 12 | 2^10 - 2^5 */
822 f25519_mul(&t
, &u
, &t
); /* 13 | 2^10 - 1 */
823 SQRN(&u
, &t
, 10); /* 23 | 2^20 - 2^10 */
824 f25519_mul(&u
, &u
, &t
); /* 24 | 2^20 - 1 */
825 SQRN(&u
, &u
, 10); /* 34 | 2^30 - 2^10 */
826 f25519_mul(&t
, &u
, &t
); /* 35 | 2^30 - 1 */
827 f25519_sqr(&u
, &t
); /* 36 | 2^31 - 2 */
828 f25519_mul(&t
, &u
, &v
); /* 37 | 2^31 - 1 */
829 SQRN(&u
, &t
, 31); /* 68 | 2^62 - 2^31 */
830 f25519_mul(&t
, &u
, &t
); /* 69 | 2^62 - 1 */
831 SQRN(&u
, &t
, 62); /* 131 | 2^124 - 2^62 */
832 f25519_mul(&t
, &u
, &t
); /* 132 | 2^124 - 1 */
833 SQRN(&u
, &t
, 124); /* 256 | 2^248 - 2^124 */
834 f25519_mul(&t
, &u
, &t
); /* 257 | 2^248 - 1 */
835 f25519_sqr(&u
, &t
); /* 258 | 2^249 - 2 */
836 f25519_mul(&t
, &u
, &v
); /* 259 | 2^249 - 1 */
837 SQRN(&t
, &t
, 3); /* 262 | 2^252 - 8 */
838 f25519_sqr(&u
, &t
); /* 263 | 2^253 - 16 */
839 f25519_mul(&t
, &u
, &t
); /* 264 | 3*2^252 - 24 */
840 f25519_mul(&t
, &t
, &t15
); /* 265 | 3*2^252 - 9 */
841 f25519_mul(&w
, &t
, &v
); /* 266 | 3*2^252 - 8 */
843 /* Awesome. Now let me explain. Let v be a square in GF(p), and let w =
844 * v^(3*2^252 - 8). In particular, let's consider
846 * v^2 w^4 = v^2 v^{3*2^254 - 32} = (v^{2^254 - 10})^3
848 * But 2^254 - 10 = ((2^255 - 19) - 1)/2 = (p - 1)/2. Since v is a square,
849 * it has order dividing (p - 1)/2, and therefore v^2 w^4 = 1 and
853 * That in turn implies that w^2 = ±1/v. Now, recall that v = x y, and let
854 * w' = w x. Then w'^2 = ±x^2/v = ±x/y. If y w'^2 = x then we set
855 * z = w', since we have z^2 = x/y; otherwise let z = i w', where i^2 = -1,
856 * so z^2 = -w^2 = x/y, and we're done.
858 * The easiest way to compare is to encode. This isn't as wasteful as it
859 * sounds: the hard part is normalizing the representations, which we have
862 f25519_mul(&w
, &w
, x
);
864 f25519_mul(&t
, &t
, y
);
867 f25519_store(b0
, &t
);
868 f25519_store(b1
, &u
);
869 f25519_mul(&u
, &w
, SQRTM1
);
871 m
= -consttime_memeq(b0
, xb
, 32);
872 rc
= PICK2(0, rc
, m
);
873 f25519_pick2(z
, &w
, &u
, m
);
874 m
= -consttime_memeq(b1
, xb
, 32);
875 rc
= PICK2(0, rc
, m
);
877 /* And we're done. */
881 /*----- That's all, folks -------------------------------------------------*/