curve25519-fiat32.h

   1 // SPDX-License-Identifier: GPL-2.0 OR MIT
   2 /*
   3  * Copyright (C) 2015-2016 The fiat-crypto Authors.
   4  * Copyright (C) 2018-2020 Jason A. Donenfeld <Jason@zx2c4.com>. All Rights Reserved.
   5  *
   6  * This is a machine-generated formally verified implementation of Curve25519
   7  * ECDH from: <https://github.com/mit-plv/fiat-crypto>. Though originally
   8  * machine generated, it has been tweaked to be suitable for use in the kernel.
   9  * It is optimized for 32-bit machines and machines that cannot work efficiently
  10  * with 128-bit integer types.
  11  */
  12
  13 /* fe means field element. Here the field is \Z/(2^255-19). An element t,
  14  * entries t[0]...t[9], represents the integer t[0]+2^26 t[1]+2^51 t[2]+2^77
  15  * t[3]+2^102 t[4]+...+2^230 t[9].
  16  * fe limbs are bounded by 1.125*2^26,1.125*2^25,1.125*2^26,1.125*2^25,etc.
  17  * Multiplication and carrying produce fe from fe_loose.
  18  */
  19 typedef struct fe { u32 v[10]; } fe;
  20
  21 /* fe_loose limbs are bounded by 3.375*2^26,3.375*2^25,3.375*2^26,3.375*2^25,etc
  22  * Addition and subtraction produce fe_loose from (fe, fe).
  23  */
  24 typedef struct fe_loose { u32 v[10]; } fe_loose;
  25
  26 static __always_inline void fe_frombytes_impl(u32 h[10], const u8 *s)
  27 {
  28         /* Ignores top bit of s. */
  29         u32 a0 = get_unaligned_le32(s);
  30         u32 a1 = get_unaligned_le32(s+4);
  31         u32 a2 = get_unaligned_le32(s+8);
  32         u32 a3 = get_unaligned_le32(s+12);
  33         u32 a4 = get_unaligned_le32(s+16);
  34         u32 a5 = get_unaligned_le32(s+20);
  35         u32 a6 = get_unaligned_le32(s+24);
  36         u32 a7 = get_unaligned_le32(s+28);
  37         h[0] = a0&((1<<26)-1);                    /* 26 used, 32-26 left.   26 */
  38         h[1] = (a0>>26) | ((a1&((1<<19)-1))<< 6); /* (32-26) + 19 =  6+19 = 25 */
  39         h[2] = (a1>>19) | ((a2&((1<<13)-1))<<13); /* (32-19) + 13 = 13+13 = 26 */
  40         h[3] = (a2>>13) | ((a3&((1<< 6)-1))<<19); /* (32-13) +  6 = 19+ 6 = 25 */
  41         h[4] = (a3>> 6);                          /* (32- 6)              = 26 */
  42         h[5] = a4&((1<<25)-1);                    /*                        25 */
  43         h[6] = (a4>>25) | ((a5&((1<<19)-1))<< 7); /* (32-25) + 19 =  7+19 = 26 */
  44         h[7] = (a5>>19) | ((a6&((1<<12)-1))<<13); /* (32-19) + 12 = 13+12 = 25 */
  45         h[8] = (a6>>12) | ((a7&((1<< 6)-1))<<20); /* (32-12) +  6 = 20+ 6 = 26 */
  46         h[9] = (a7>> 6)&((1<<25)-1); /*                                     25 */
  47 }
  48
  49 static __always_inline void fe_frombytes(fe *h, const u8 *s)
  50 {
  51         fe_frombytes_impl(h->v, s);
  52 }
  53
  54 static __always_inline u8 /*bool*/
  55 addcarryx_u25(u8 /*bool*/ c, u32 a, u32 b, u32 *low)
  56 {
  57         /* This function extracts 25 bits of result and 1 bit of carry
  58          * (26 total), so a 32-bit intermediate is sufficient.
  59          */
  60         u32 x = a + b + c;
  61         *low = x & ((1 << 25) - 1);
  62         return (x >> 25) & 1;
  63 }
  64
  65 static __always_inline u8 /*bool*/
  66 addcarryx_u26(u8 /*bool*/ c, u32 a, u32 b, u32 *low)
  67 {
  68         /* This function extracts 26 bits of result and 1 bit of carry
  69          * (27 total), so a 32-bit intermediate is sufficient.
  70          */
  71         u32 x = a + b + c;
  72         *low = x & ((1 << 26) - 1);
  73         return (x >> 26) & 1;
  74 }
  75
  76 static __always_inline u8 /*bool*/
  77 subborrow_u25(u8 /*bool*/ c, u32 a, u32 b, u32 *low)
  78 {
  79         /* This function extracts 25 bits of result and 1 bit of borrow
  80          * (26 total), so a 32-bit intermediate is sufficient.
  81          */
  82         u32 x = a - b - c;
  83         *low = x & ((1 << 25) - 1);
  84         return x >> 31;
  85 }
  86
  87 static __always_inline u8 /*bool*/
  88 subborrow_u26(u8 /*bool*/ c, u32 a, u32 b, u32 *low)
  89 {
  90         /* This function extracts 26 bits of result and 1 bit of borrow
  91          *(27 total), so a 32-bit intermediate is sufficient.
  92          */
  93         u32 x = a - b - c;
  94         *low = x & ((1 << 26) - 1);
  95         return x >> 31;
  96 }
  97
  98 static __always_inline u32 cmovznz32(u32 t, u32 z, u32 nz)
  99 {
 100         t = -!!t; /* all set if nonzero, 0 if 0 */
 101         return (t&nz) | ((~t)&z);
 102 }
 103
 104 static __always_inline void fe_freeze(u32 out[10], const u32 in1[10])
 105 {
 106         { const u32 x17 = in1[9];
 107         { const u32 x18 = in1[8];
 108         { const u32 x16 = in1[7];
 109         { const u32 x14 = in1[6];
 110         { const u32 x12 = in1[5];
 111         { const u32 x10 = in1[4];
 112         { const u32 x8 = in1[3];
 113         { const u32 x6 = in1[2];
 114         { const u32 x4 = in1[1];
 115         { const u32 x2 = in1[0];
 116         { u32 x20; u8/*bool*/ x21 = subborrow_u26(0x0, x2, 0x3ffffed, &x20);
 117         { u32 x23; u8/*bool*/ x24 = subborrow_u25(x21, x4, 0x1ffffff, &x23);
 118         { u32 x26; u8/*bool*/ x27 = subborrow_u26(x24, x6, 0x3ffffff, &x26);
 119         { u32 x29; u8/*bool*/ x30 = subborrow_u25(x27, x8, 0x1ffffff, &x29);
 120         { u32 x32; u8/*bool*/ x33 = subborrow_u26(x30, x10, 0x3ffffff, &x32);
 121         { u32 x35; u8/*bool*/ x36 = subborrow_u25(x33, x12, 0x1ffffff, &x35);
 122         { u32 x38; u8/*bool*/ x39 = subborrow_u26(x36, x14, 0x3ffffff, &x38);
 123         { u32 x41; u8/*bool*/ x42 = subborrow_u25(x39, x16, 0x1ffffff, &x41);
 124         { u32 x44; u8/*bool*/ x45 = subborrow_u26(x42, x18, 0x3ffffff, &x44);
 125         { u32 x47; u8/*bool*/ x48 = subborrow_u25(x45, x17, 0x1ffffff, &x47);
 126         { u32 x49 = cmovznz32(x48, 0x0, 0xffffffff);
 127         { u32 x50 = (x49 & 0x3ffffed);
 128         { u32 x52; u8/*bool*/ x53 = addcarryx_u26(0x0, x20, x50, &x52);
 129         { u32 x54 = (x49 & 0x1ffffff);
 130         { u32 x56; u8/*bool*/ x57 = addcarryx_u25(x53, x23, x54, &x56);
 131         { u32 x58 = (x49 & 0x3ffffff);
 132         { u32 x60; u8/*bool*/ x61 = addcarryx_u26(x57, x26, x58, &x60);
 133         { u32 x62 = (x49 & 0x1ffffff);
 134         { u32 x64; u8/*bool*/ x65 = addcarryx_u25(x61, x29, x62, &x64);
 135         { u32 x66 = (x49 & 0x3ffffff);
 136         { u32 x68; u8/*bool*/ x69 = addcarryx_u26(x65, x32, x66, &x68);
 137         { u32 x70 = (x49 & 0x1ffffff);
 138         { u32 x72; u8/*bool*/ x73 = addcarryx_u25(x69, x35, x70, &x72);
 139         { u32 x74 = (x49 & 0x3ffffff);
 140         { u32 x76; u8/*bool*/ x77 = addcarryx_u26(x73, x38, x74, &x76);
 141         { u32 x78 = (x49 & 0x1ffffff);
 142         { u32 x80; u8/*bool*/ x81 = addcarryx_u25(x77, x41, x78, &x80);
 143         { u32 x82 = (x49 & 0x3ffffff);
 144         { u32 x84; u8/*bool*/ x85 = addcarryx_u26(x81, x44, x82, &x84);
 145         { u32 x86 = (x49 & 0x1ffffff);
 146         { u32 x88; addcarryx_u25(x85, x47, x86, &x88);
 147         out[0] = x52;
 148         out[1] = x56;
 149         out[2] = x60;
 150         out[3] = x64;
 151         out[4] = x68;
 152         out[5] = x72;
 153         out[6] = x76;
 154         out[7] = x80;
 155         out[8] = x84;
 156         out[9] = x88;
 157         }}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}
 158 }
 159
 160 static __always_inline void fe_tobytes(u8 s[32], const fe *f)
 161 {
 162         u32 h[10];
 163         fe_freeze(h, f->v);
 164         s[0] = h[0] >> 0;
 165         s[1] = h[0] >> 8;
 166         s[2] = h[0] >> 16;
 167         s[3] = (h[0] >> 24) | (h[1] << 2);
 168         s[4] = h[1] >> 6;
 169         s[5] = h[1] >> 14;
 170         s[6] = (h[1] >> 22) | (h[2] << 3);
 171         s[7] = h[2] >> 5;
 172         s[8] = h[2] >> 13;
 173         s[9] = (h[2] >> 21) | (h[3] << 5);
 174         s[10] = h[3] >> 3;
 175         s[11] = h[3] >> 11;
 176         s[12] = (h[3] >> 19) | (h[4] << 6);
 177         s[13] = h[4] >> 2;
 178         s[14] = h[4] >> 10;
 179         s[15] = h[4] >> 18;
 180         s[16] = h[5] >> 0;
 181         s[17] = h[5] >> 8;
 182         s[18] = h[5] >> 16;
 183         s[19] = (h[5] >> 24) | (h[6] << 1);
 184         s[20] = h[6] >> 7;
 185         s[21] = h[6] >> 15;
 186         s[22] = (h[6] >> 23) | (h[7] << 3);
 187         s[23] = h[7] >> 5;
 188         s[24] = h[7] >> 13;
 189         s[25] = (h[7] >> 21) | (h[8] << 4);
 190         s[26] = h[8] >> 4;
 191         s[27] = h[8] >> 12;
 192         s[28] = (h[8] >> 20) | (h[9] << 6);
 193         s[29] = h[9] >> 2;
 194         s[30] = h[9] >> 10;
 195         s[31] = h[9] >> 18;
 196 }
 197
 198 /* h = f */
 199 static __always_inline void fe_copy(fe *h, const fe *f)
 200 {
 201         memmove(h, f, sizeof(u32) * 10);
 202 }
 203
 204 static __always_inline void fe_copy_lt(fe_loose *h, const fe *f)
 205 {
 206         memmove(h, f, sizeof(u32) * 10);
 207 }
 208
 209 /* h = 0 */
 210 static __always_inline void fe_0(fe *h)
 211 {
 212         memset(h, 0, sizeof(u32) * 10);
 213 }
 214
 215 /* h = 1 */
 216 static __always_inline void fe_1(fe *h)
 217 {
 218         memset(h, 0, sizeof(u32) * 10);
 219         h->v[0] = 1;
 220 }
 221
 222 static void fe_add_impl(u32 out[10], const u32 in1[10], const u32 in2[10])
 223 {
 224         { const u32 x20 = in1[9];
 225         { const u32 x21 = in1[8];
 226         { const u32 x19 = in1[7];
 227         { const u32 x17 = in1[6];
 228         { const u32 x15 = in1[5];
 229         { const u32 x13 = in1[4];
 230         { const u32 x11 = in1[3];
 231         { const u32 x9 = in1[2];
 232         { const u32 x7 = in1[1];
 233         { const u32 x5 = in1[0];
 234         { const u32 x38 = in2[9];
 235         { const u32 x39 = in2[8];
 236         { const u32 x37 = in2[7];
 237         { const u32 x35 = in2[6];
 238         { const u32 x33 = in2[5];
 239         { const u32 x31 = in2[4];
 240         { const u32 x29 = in2[3];
 241         { const u32 x27 = in2[2];
 242         { const u32 x25 = in2[1];
 243         { const u32 x23 = in2[0];
 244         out[0] = (x5 + x23);
 245         out[1] = (x7 + x25);
 246         out[2] = (x9 + x27);
 247         out[3] = (x11 + x29);
 248         out[4] = (x13 + x31);
 249         out[5] = (x15 + x33);
 250         out[6] = (x17 + x35);
 251         out[7] = (x19 + x37);
 252         out[8] = (x21 + x39);
 253         out[9] = (x20 + x38);
 254         }}}}}}}}}}}}}}}}}}}}
 255 }
 256
 257 /* h = f + g
 258  * Can overlap h with f or g.
 259  */
 260 static __always_inline void fe_add(fe_loose *h, const fe *f, const fe *g)
 261 {
 262         fe_add_impl(h->v, f->v, g->v);
 263 }
 264
 265 static void fe_sub_impl(u32 out[10], const u32 in1[10], const u32 in2[10])
 266 {
 267         { const u32 x20 = in1[9];
 268         { const u32 x21 = in1[8];
 269         { const u32 x19 = in1[7];
 270         { const u32 x17 = in1[6];
 271         { const u32 x15 = in1[5];
 272         { const u32 x13 = in1[4];
 273         { const u32 x11 = in1[3];
 274         { const u32 x9 = in1[2];
 275         { const u32 x7 = in1[1];
 276         { const u32 x5 = in1[0];
 277         { const u32 x38 = in2[9];
 278         { const u32 x39 = in2[8];
 279         { const u32 x37 = in2[7];
 280         { const u32 x35 = in2[6];
 281         { const u32 x33 = in2[5];
 282         { const u32 x31 = in2[4];
 283         { const u32 x29 = in2[3];
 284         { const u32 x27 = in2[2];
 285         { const u32 x25 = in2[1];
 286         { const u32 x23 = in2[0];
 287         out[0] = ((0x7ffffda + x5) - x23);
 288         out[1] = ((0x3fffffe + x7) - x25);
 289         out[2] = ((0x7fffffe + x9) - x27);
 290         out[3] = ((0x3fffffe + x11) - x29);
 291         out[4] = ((0x7fffffe + x13) - x31);
 292         out[5] = ((0x3fffffe + x15) - x33);
 293         out[6] = ((0x7fffffe + x17) - x35);
 294         out[7] = ((0x3fffffe + x19) - x37);
 295         out[8] = ((0x7fffffe + x21) - x39);
 296         out[9] = ((0x3fffffe + x20) - x38);
 297         }}}}}}}}}}}}}}}}}}}}
 298 }
 299
 300 /* h = f - g
 301  * Can overlap h with f or g.
 302  */
 303 static __always_inline void fe_sub(fe_loose *h, const fe *f, const fe *g)
 304 {
 305         fe_sub_impl(h->v, f->v, g->v);
 306 }
 307
 308 static void fe_mul_impl(u32 out[10], const u32 in1[10], const u32 in2[10])
 309 {
 310         { const u32 x20 = in1[9];
 311         { const u32 x21 = in1[8];
 312         { const u32 x19 = in1[7];
 313         { const u32 x17 = in1[6];
 314         { const u32 x15 = in1[5];
 315         { const u32 x13 = in1[4];
 316         { const u32 x11 = in1[3];
 317         { const u32 x9 = in1[2];
 318         { const u32 x7 = in1[1];
 319         { const u32 x5 = in1[0];
 320         { const u32 x38 = in2[9];
 321         { const u32 x39 = in2[8];
 322         { const u32 x37 = in2[7];
 323         { const u32 x35 = in2[6];
 324         { const u32 x33 = in2[5];
 325         { const u32 x31 = in2[4];
 326         { const u32 x29 = in2[3];
 327         { const u32 x27 = in2[2];
 328         { const u32 x25 = in2[1];
 329         { const u32 x23 = in2[0];
 330         { u64 x40 = ((u64)x23 * x5);
 331         { u64 x41 = (((u64)x23 * x7) + ((u64)x25 * x5));
 332         { u64 x42 = ((((u64)(0x2 * x25) * x7) + ((u64)x23 * x9)) + ((u64)x27 * x5));
 333         { u64 x43 = (((((u64)x25 * x9) + ((u64)x27 * x7)) + ((u64)x23 * x11)) + ((u64)x29 * x5));
 334         { u64 x44 = (((((u64)x27 * x9) + (0x2 * (((u64)x25 * x11) + ((u64)x29 * x7)))) + ((u64)x23 * x13)) + ((u64)x31 * x5));
 335         { u64 x45 = (((((((u64)x27 * x11) + ((u64)x29 * x9)) + ((u64)x25 * x13)) + ((u64)x31 * x7)) + ((u64)x23 * x15)) + ((u64)x33 * x5));
 336         { u64 x46 = (((((0x2 * ((((u64)x29 * x11) + ((u64)x25 * x15)) + ((u64)x33 * x7))) + ((u64)x27 * x13)) + ((u64)x31 * x9)) + ((u64)x23 * x17)) + ((u64)x35 * x5));
 337         { u64 x47 = (((((((((u64)x29 * x13) + ((u64)x31 * x11)) + ((u64)x27 * x15)) + ((u64)x33 * x9)) + ((u64)x25 * x17)) + ((u64)x35 * x7)) + ((u64)x23 * x19)) + ((u64)x37 * x5));
 338         { u64 x48 = (((((((u64)x31 * x13) + (0x2 * (((((u64)x29 * x15) + ((u64)x33 * x11)) + ((u64)x25 * x19)) + ((u64)x37 * x7)))) + ((u64)x27 * x17)) + ((u64)x35 * x9)) + ((u64)x23 * x21)) + ((u64)x39 * x5));
 339         { u64 x49 = (((((((((((u64)x31 * x15) + ((u64)x33 * x13)) + ((u64)x29 * x17)) + ((u64)x35 * x11)) + ((u64)x27 * x19)) + ((u64)x37 * x9)) + ((u64)x25 * x21)) + ((u64)x39 * x7)) + ((u64)x23 * x20)) + ((u64)x38 * x5));
 340         { u64 x50 = (((((0x2 * ((((((u64)x33 * x15) + ((u64)x29 * x19)) + ((u64)x37 * x11)) + ((u64)x25 * x20)) + ((u64)x38 * x7))) + ((u64)x31 * x17)) + ((u64)x35 * x13)) + ((u64)x27 * x21)) + ((u64)x39 * x9));
 341         { u64 x51 = (((((((((u64)x33 * x17) + ((u64)x35 * x15)) + ((u64)x31 * x19)) + ((u64)x37 * x13)) + ((u64)x29 * x21)) + ((u64)x39 * x11)) + ((u64)x27 * x20)) + ((u64)x38 * x9));
 342         { u64 x52 = (((((u64)x35 * x17) + (0x2 * (((((u64)x33 * x19) + ((u64)x37 * x15)) + ((u64)x29 * x20)) + ((u64)x38 * x11)))) + ((u64)x31 * x21)) + ((u64)x39 * x13));
 343         { u64 x53 = (((((((u64)x35 * x19) + ((u64)x37 * x17)) + ((u64)x33 * x21)) + ((u64)x39 * x15)) + ((u64)x31 * x20)) + ((u64)x38 * x13));
 344         { u64 x54 = (((0x2 * ((((u64)x37 * x19) + ((u64)x33 * x20)) + ((u64)x38 * x15))) + ((u64)x35 * x21)) + ((u64)x39 * x17));
 345         { u64 x55 = (((((u64)x37 * x21) + ((u64)x39 * x19)) + ((u64)x35 * x20)) + ((u64)x38 * x17));
 346         { u64 x56 = (((u64)x39 * x21) + (0x2 * (((u64)x37 * x20) + ((u64)x38 * x19))));
 347         { u64 x57 = (((u64)x39 * x20) + ((u64)x38 * x21));
 348         { u64 x58 = ((u64)(0x2 * x38) * x20);
 349         { u64 x59 = (x48 + (x58 << 0x4));
 350         { u64 x60 = (x59 + (x58 << 0x1));
 351         { u64 x61 = (x60 + x58);
 352         { u64 x62 = (x47 + (x57 << 0x4));
 353         { u64 x63 = (x62 + (x57 << 0x1));
 354         { u64 x64 = (x63 + x57);
 355         { u64 x65 = (x46 + (x56 << 0x4));
 356         { u64 x66 = (x65 + (x56 << 0x1));
 357         { u64 x67 = (x66 + x56);
 358         { u64 x68 = (x45 + (x55 << 0x4));
 359         { u64 x69 = (x68 + (x55 << 0x1));
 360         { u64 x70 = (x69 + x55);
 361         { u64 x71 = (x44 + (x54 << 0x4));
 362         { u64 x72 = (x71 + (x54 << 0x1));
 363         { u64 x73 = (x72 + x54);
 364         { u64 x74 = (x43 + (x53 << 0x4));
 365         { u64 x75 = (x74 + (x53 << 0x1));
 366         { u64 x76 = (x75 + x53);
 367         { u64 x77 = (x42 + (x52 << 0x4));
 368         { u64 x78 = (x77 + (x52 << 0x1));
 369         { u64 x79 = (x78 + x52);
 370         { u64 x80 = (x41 + (x51 << 0x4));
 371         { u64 x81 = (x80 + (x51 << 0x1));
 372         { u64 x82 = (x81 + x51);
 373         { u64 x83 = (x40 + (x50 << 0x4));
 374         { u64 x84 = (x83 + (x50 << 0x1));
 375         { u64 x85 = (x84 + x50);
 376         { u64 x86 = (x85 >> 0x1a);
 377         { u32 x87 = ((u32)x85 & 0x3ffffff);
 378         { u64 x88 = (x86 + x82);
 379         { u64 x89 = (x88 >> 0x19);
 380         { u32 x90 = ((u32)x88 & 0x1ffffff);
 381         { u64 x91 = (x89 + x79);
 382         { u64 x92 = (x91 >> 0x1a);
 383         { u32 x93 = ((u32)x91 & 0x3ffffff);
 384         { u64 x94 = (x92 + x76);
 385         { u64 x95 = (x94 >> 0x19);
 386         { u32 x96 = ((u32)x94 & 0x1ffffff);
 387         { u64 x97 = (x95 + x73);
 388         { u64 x98 = (x97 >> 0x1a);
 389         { u32 x99 = ((u32)x97 & 0x3ffffff);
 390         { u64 x100 = (x98 + x70);
 391         { u64 x101 = (x100 >> 0x19);
 392         { u32 x102 = ((u32)x100 & 0x1ffffff);
 393         { u64 x103 = (x101 + x67);
 394         { u64 x104 = (x103 >> 0x1a);
 395         { u32 x105 = ((u32)x103 & 0x3ffffff);
 396         { u64 x106 = (x104 + x64);
 397         { u64 x107 = (x106 >> 0x19);
 398         { u32 x108 = ((u32)x106 & 0x1ffffff);
 399         { u64 x109 = (x107 + x61);
 400         { u64 x110 = (x109 >> 0x1a);
 401         { u32 x111 = ((u32)x109 & 0x3ffffff);
 402         { u64 x112 = (x110 + x49);
 403         { u64 x113 = (x112 >> 0x19);
 404         { u32 x114 = ((u32)x112 & 0x1ffffff);
 405         { u64 x115 = (x87 + (0x13 * x113));
 406         { u32 x116 = (u32) (x115 >> 0x1a);
 407         { u32 x117 = ((u32)x115 & 0x3ffffff);
 408         { u32 x118 = (x116 + x90);
 409         { u32 x119 = (x118 >> 0x19);
 410         { u32 x120 = (x118 & 0x1ffffff);
 411         out[0] = x117;
 412         out[1] = x120;
 413         out[2] = (x119 + x93);
 414         out[3] = x96;
 415         out[4] = x99;
 416         out[5] = x102;
 417         out[6] = x105;
 418         out[7] = x108;
 419         out[8] = x111;
 420         out[9] = x114;
 421         }}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}
 422 }
 423
 424 static __always_inline void fe_mul_ttt(fe *h, const fe *f, const fe *g)
 425 {
 426         fe_mul_impl(h->v, f->v, g->v);
 427 }
 428
 429 static __always_inline void fe_mul_tlt(fe *h, const fe_loose *f, const fe *g)
 430 {
 431         fe_mul_impl(h->v, f->v, g->v);
 432 }
 433
 434 static __always_inline void
 435 fe_mul_tll(fe *h, const fe_loose *f, const fe_loose *g)
 436 {
 437         fe_mul_impl(h->v, f->v, g->v);
 438 }
 439
 440 static void fe_sqr_impl(u32 out[10], const u32 in1[10])
 441 {
 442         { const u32 x17 = in1[9];
 443         { const u32 x18 = in1[8];
 444         { const u32 x16 = in1[7];
 445         { const u32 x14 = in1[6];
 446         { const u32 x12 = in1[5];
 447         { const u32 x10 = in1[4];
 448         { const u32 x8 = in1[3];
 449         { const u32 x6 = in1[2];
 450         { const u32 x4 = in1[1];
 451         { const u32 x2 = in1[0];
 452         { u64 x19 = ((u64)x2 * x2);
 453         { u64 x20 = ((u64)(0x2 * x2) * x4);
 454         { u64 x21 = (0x2 * (((u64)x4 * x4) + ((u64)x2 * x6)));
 455         { u64 x22 = (0x2 * (((u64)x4 * x6) + ((u64)x2 * x8)));
 456         { u64 x23 = ((((u64)x6 * x6) + ((u64)(0x4 * x4) * x8)) + ((u64)(0x2 * x2) * x10));
 457         { u64 x24 = (0x2 * ((((u64)x6 * x8) + ((u64)x4 * x10)) + ((u64)x2 * x12)));
 458         { u64 x25 = (0x2 * (((((u64)x8 * x8) + ((u64)x6 * x10)) + ((u64)x2 * x14)) + ((u64)(0x2 * x4) * x12)));
 459         { u64 x26 = (0x2 * (((((u64)x8 * x10) + ((u64)x6 * x12)) + ((u64)x4 * x14)) + ((u64)x2 * x16)));
 460         { u64 x27 = (((u64)x10 * x10) + (0x2 * ((((u64)x6 * x14) + ((u64)x2 * x18)) + (0x2 * (((u64)x4 * x16) + ((u64)x8 * x12))))));
 461         { u64 x28 = (0x2 * ((((((u64)x10 * x12) + ((u64)x8 * x14)) + ((u64)x6 * x16)) + ((u64)x4 * x18)) + ((u64)x2 * x17)));
 462         { u64 x29 = (0x2 * (((((u64)x12 * x12) + ((u64)x10 * x14)) + ((u64)x6 * x18)) + (0x2 * (((u64)x8 * x16) + ((u64)x4 * x17)))));
 463         { u64 x30 = (0x2 * (((((u64)x12 * x14) + ((u64)x10 * x16)) + ((u64)x8 * x18)) + ((u64)x6 * x17)));
 464         { u64 x31 = (((u64)x14 * x14) + (0x2 * (((u64)x10 * x18) + (0x2 * (((u64)x12 * x16) + ((u64)x8 * x17))))));
 465         { u64 x32 = (0x2 * ((((u64)x14 * x16) + ((u64)x12 * x18)) + ((u64)x10 * x17)));
 466         { u64 x33 = (0x2 * ((((u64)x16 * x16) + ((u64)x14 * x18)) + ((u64)(0x2 * x12) * x17)));
 467         { u64 x34 = (0x2 * (((u64)x16 * x18) + ((u64)x14 * x17)));
 468         { u64 x35 = (((u64)x18 * x18) + ((u64)(0x4 * x16) * x17));
 469         { u64 x36 = ((u64)(0x2 * x18) * x17);
 470         { u64 x37 = ((u64)(0x2 * x17) * x17);
 471         { u64 x38 = (x27 + (x37 << 0x4));
 472         { u64 x39 = (x38 + (x37 << 0x1));
 473         { u64 x40 = (x39 + x37);
 474         { u64 x41 = (x26 + (x36 << 0x4));
 475         { u64 x42 = (x41 + (x36 << 0x1));
 476         { u64 x43 = (x42 + x36);
 477         { u64 x44 = (x25 + (x35 << 0x4));
 478         { u64 x45 = (x44 + (x35 << 0x1));
 479         { u64 x46 = (x45 + x35);
 480         { u64 x47 = (x24 + (x34 << 0x4));
 481         { u64 x48 = (x47 + (x34 << 0x1));
 482         { u64 x49 = (x48 + x34);
 483         { u64 x50 = (x23 + (x33 << 0x4));
 484         { u64 x51 = (x50 + (x33 << 0x1));
 485         { u64 x52 = (x51 + x33);
 486         { u64 x53 = (x22 + (x32 << 0x4));
 487         { u64 x54 = (x53 + (x32 << 0x1));
 488         { u64 x55 = (x54 + x32);
 489         { u64 x56 = (x21 + (x31 << 0x4));
 490         { u64 x57 = (x56 + (x31 << 0x1));
 491         { u64 x58 = (x57 + x31);
 492         { u64 x59 = (x20 + (x30 << 0x4));
 493         { u64 x60 = (x59 + (x30 << 0x1));
 494         { u64 x61 = (x60 + x30);
 495         { u64 x62 = (x19 + (x29 << 0x4));
 496         { u64 x63 = (x62 + (x29 << 0x1));
 497         { u64 x64 = (x63 + x29);
 498         { u64 x65 = (x64 >> 0x1a);
 499         { u32 x66 = ((u32)x64 & 0x3ffffff);
 500         { u64 x67 = (x65 + x61);
 501         { u64 x68 = (x67 >> 0x19);
 502         { u32 x69 = ((u32)x67 & 0x1ffffff);
 503         { u64 x70 = (x68 + x58);
 504         { u64 x71 = (x70 >> 0x1a);
 505         { u32 x72 = ((u32)x70 & 0x3ffffff);
 506         { u64 x73 = (x71 + x55);
 507         { u64 x74 = (x73 >> 0x19);
 508         { u32 x75 = ((u32)x73 & 0x1ffffff);
 509         { u64 x76 = (x74 + x52);
 510         { u64 x77 = (x76 >> 0x1a);
 511         { u32 x78 = ((u32)x76 & 0x3ffffff);
 512         { u64 x79 = (x77 + x49);
 513         { u64 x80 = (x79 >> 0x19);
 514         { u32 x81 = ((u32)x79 & 0x1ffffff);
 515         { u64 x82 = (x80 + x46);
 516         { u64 x83 = (x82 >> 0x1a);
 517         { u32 x84 = ((u32)x82 & 0x3ffffff);
 518         { u64 x85 = (x83 + x43);
 519         { u64 x86 = (x85 >> 0x19);
 520         { u32 x87 = ((u32)x85 & 0x1ffffff);
 521         { u64 x88 = (x86 + x40);
 522         { u64 x89 = (x88 >> 0x1a);
 523         { u32 x90 = ((u32)x88 & 0x3ffffff);
 524         { u64 x91 = (x89 + x28);
 525         { u64 x92 = (x91 >> 0x19);
 526         { u32 x93 = ((u32)x91 & 0x1ffffff);
 527         { u64 x94 = (x66 + (0x13 * x92));
 528         { u32 x95 = (u32) (x94 >> 0x1a);
 529         { u32 x96 = ((u32)x94 & 0x3ffffff);
 530         { u32 x97 = (x95 + x69);
 531         { u32 x98 = (x97 >> 0x19);
 532         { u32 x99 = (x97 & 0x1ffffff);
 533         out[0] = x96;
 534         out[1] = x99;
 535         out[2] = (x98 + x72);
 536         out[3] = x75;
 537         out[4] = x78;
 538         out[5] = x81;
 539         out[6] = x84;
 540         out[7] = x87;
 541         out[8] = x90;
 542         out[9] = x93;
 543         }}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}
 544 }
 545
 546 static __always_inline void fe_sq_tl(fe *h, const fe_loose *f)
 547 {
 548         fe_sqr_impl(h->v, f->v);
 549 }
 550
 551 static __always_inline void fe_sq_tt(fe *h, const fe *f)
 552 {
 553         fe_sqr_impl(h->v, f->v);
 554 }
 555
 556 static __always_inline void fe_loose_invert(fe *out, const fe_loose *z)
 557 {
 558         fe t0;
 559         fe t1;
 560         fe t2;
 561         fe t3;
 562         int i;
 563
 564         fe_sq_tl(&t0, z);
 565         fe_sq_tt(&t1, &t0);
 566         for (i = 1; i < 2; ++i)
 567                 fe_sq_tt(&t1, &t1);
 568         fe_mul_tlt(&t1, z, &t1);
 569         fe_mul_ttt(&t0, &t0, &t1);
 570         fe_sq_tt(&t2, &t0);
 571         fe_mul_ttt(&t1, &t1, &t2);
 572         fe_sq_tt(&t2, &t1);
 573         for (i = 1; i < 5; ++i)
 574                 fe_sq_tt(&t2, &t2);
 575         fe_mul_ttt(&t1, &t2, &t1);
 576         fe_sq_tt(&t2, &t1);
 577         for (i = 1; i < 10; ++i)
 578                 fe_sq_tt(&t2, &t2);
 579         fe_mul_ttt(&t2, &t2, &t1);
 580         fe_sq_tt(&t3, &t2);
 581         for (i = 1; i < 20; ++i)
 582                 fe_sq_tt(&t3, &t3);
 583         fe_mul_ttt(&t2, &t3, &t2);
 584         fe_sq_tt(&t2, &t2);
 585         for (i = 1; i < 10; ++i)
 586                 fe_sq_tt(&t2, &t2);
 587         fe_mul_ttt(&t1, &t2, &t1);
 588         fe_sq_tt(&t2, &t1);
 589         for (i = 1; i < 50; ++i)
 590                 fe_sq_tt(&t2, &t2);
 591         fe_mul_ttt(&t2, &t2, &t1);
 592         fe_sq_tt(&t3, &t2);
 593         for (i = 1; i < 100; ++i)
 594                 fe_sq_tt(&t3, &t3);
 595         fe_mul_ttt(&t2, &t3, &t2);
 596         fe_sq_tt(&t2, &t2);
 597         for (i = 1; i < 50; ++i)
 598                 fe_sq_tt(&t2, &t2);
 599         fe_mul_ttt(&t1, &t2, &t1);
 600         fe_sq_tt(&t1, &t1);
 601         for (i = 1; i < 5; ++i)
 602                 fe_sq_tt(&t1, &t1);
 603         fe_mul_ttt(out, &t1, &t0);
 604 }
 605
 606 static __always_inline void fe_invert(fe *out, const fe *z)
 607 {
 608         fe_loose l;
 609         fe_copy_lt(&l, z);
 610         fe_loose_invert(out, &l);
 611 }
 612
 613 /* Replace (f,g) with (g,f) if b == 1;
 614  * replace (f,g) with (f,g) if b == 0.
 615  *
 616  * Preconditions: b in {0,1}
 617  */
 618 static __always_inline void fe_cswap(fe *f, fe *g, unsigned int b)
 619 {
 620         unsigned i;
 621         b = 0 - b;
 622         for (i = 0; i < 10; i++) {
 623                 u32 x = f->v[i] ^ g->v[i];
 624                 x &= b;
 625                 f->v[i] ^= x;
 626                 g->v[i] ^= x;
 627         }
 628 }
 629
 630 /* NOTE: based on fiat-crypto fe_mul, edited for in2=121666, 0, 0.*/
 631 static __always_inline void fe_mul_121666_impl(u32 out[10], const u32 in1[10])
 632 {
 633         { const u32 x20 = in1[9];
 634         { const u32 x21 = in1[8];
 635         { const u32 x19 = in1[7];
 636         { const u32 x17 = in1[6];
 637         { const u32 x15 = in1[5];
 638         { const u32 x13 = in1[4];
 639         { const u32 x11 = in1[3];
 640         { const u32 x9 = in1[2];
 641         { const u32 x7 = in1[1];
 642         { const u32 x5 = in1[0];
 643         { const u32 x38 = 0;
 644         { const u32 x39 = 0;
 645         { const u32 x37 = 0;
 646         { const u32 x35 = 0;
 647         { const u32 x33 = 0;
 648         { const u32 x31 = 0;
 649         { const u32 x29 = 0;
 650         { const u32 x27 = 0;
 651         { const u32 x25 = 0;
 652         { const u32 x23 = 121666;
 653         { u64 x40 = ((u64)x23 * x5);
 654         { u64 x41 = (((u64)x23 * x7) + ((u64)x25 * x5));
 655         { u64 x42 = ((((u64)(0x2 * x25) * x7) + ((u64)x23 * x9)) + ((u64)x27 * x5));
 656         { u64 x43 = (((((u64)x25 * x9) + ((u64)x27 * x7)) + ((u64)x23 * x11)) + ((u64)x29 * x5));
 657         { u64 x44 = (((((u64)x27 * x9) + (0x2 * (((u64)x25 * x11) + ((u64)x29 * x7)))) + ((u64)x23 * x13)) + ((u64)x31 * x5));
 658         { u64 x45 = (((((((u64)x27 * x11) + ((u64)x29 * x9)) + ((u64)x25 * x13)) + ((u64)x31 * x7)) + ((u64)x23 * x15)) + ((u64)x33 * x5));
 659         { u64 x46 = (((((0x2 * ((((u64)x29 * x11) + ((u64)x25 * x15)) + ((u64)x33 * x7))) + ((u64)x27 * x13)) + ((u64)x31 * x9)) + ((u64)x23 * x17)) + ((u64)x35 * x5));
 660         { u64 x47 = (((((((((u64)x29 * x13) + ((u64)x31 * x11)) + ((u64)x27 * x15)) + ((u64)x33 * x9)) + ((u64)x25 * x17)) + ((u64)x35 * x7)) + ((u64)x23 * x19)) + ((u64)x37 * x5));
 661         { u64 x48 = (((((((u64)x31 * x13) + (0x2 * (((((u64)x29 * x15) + ((u64)x33 * x11)) + ((u64)x25 * x19)) + ((u64)x37 * x7)))) + ((u64)x27 * x17)) + ((u64)x35 * x9)) + ((u64)x23 * x21)) + ((u64)x39 * x5));
 662         { u64 x49 = (((((((((((u64)x31 * x15) + ((u64)x33 * x13)) + ((u64)x29 * x17)) + ((u64)x35 * x11)) + ((u64)x27 * x19)) + ((u64)x37 * x9)) + ((u64)x25 * x21)) + ((u64)x39 * x7)) + ((u64)x23 * x20)) + ((u64)x38 * x5));
 663         { u64 x50 = (((((0x2 * ((((((u64)x33 * x15) + ((u64)x29 * x19)) + ((u64)x37 * x11)) + ((u64)x25 * x20)) + ((u64)x38 * x7))) + ((u64)x31 * x17)) + ((u64)x35 * x13)) + ((u64)x27 * x21)) + ((u64)x39 * x9));
 664         { u64 x51 = (((((((((u64)x33 * x17) + ((u64)x35 * x15)) + ((u64)x31 * x19)) + ((u64)x37 * x13)) + ((u64)x29 * x21)) + ((u64)x39 * x11)) + ((u64)x27 * x20)) + ((u64)x38 * x9));
 665         { u64 x52 = (((((u64)x35 * x17) + (0x2 * (((((u64)x33 * x19) + ((u64)x37 * x15)) + ((u64)x29 * x20)) + ((u64)x38 * x11)))) + ((u64)x31 * x21)) + ((u64)x39 * x13));
 666         { u64 x53 = (((((((u64)x35 * x19) + ((u64)x37 * x17)) + ((u64)x33 * x21)) + ((u64)x39 * x15)) + ((u64)x31 * x20)) + ((u64)x38 * x13));
 667         { u64 x54 = (((0x2 * ((((u64)x37 * x19) + ((u64)x33 * x20)) + ((u64)x38 * x15))) + ((u64)x35 * x21)) + ((u64)x39 * x17));
 668         { u64 x55 = (((((u64)x37 * x21) + ((u64)x39 * x19)) + ((u64)x35 * x20)) + ((u64)x38 * x17));
 669         { u64 x56 = (((u64)x39 * x21) + (0x2 * (((u64)x37 * x20) + ((u64)x38 * x19))));
 670         { u64 x57 = (((u64)x39 * x20) + ((u64)x38 * x21));
 671         { u64 x58 = ((u64)(0x2 * x38) * x20);
 672         { u64 x59 = (x48 + (x58 << 0x4));
 673         { u64 x60 = (x59 + (x58 << 0x1));
 674         { u64 x61 = (x60 + x58);
 675         { u64 x62 = (x47 + (x57 << 0x4));
 676         { u64 x63 = (x62 + (x57 << 0x1));
 677         { u64 x64 = (x63 + x57);
 678         { u64 x65 = (x46 + (x56 << 0x4));
 679         { u64 x66 = (x65 + (x56 << 0x1));
 680         { u64 x67 = (x66 + x56);
 681         { u64 x68 = (x45 + (x55 << 0x4));
 682         { u64 x69 = (x68 + (x55 << 0x1));
 683         { u64 x70 = (x69 + x55);
 684         { u64 x71 = (x44 + (x54 << 0x4));
 685         { u64 x72 = (x71 + (x54 << 0x1));
 686         { u64 x73 = (x72 + x54);
 687         { u64 x74 = (x43 + (x53 << 0x4));
 688         { u64 x75 = (x74 + (x53 << 0x1));
 689         { u64 x76 = (x75 + x53);
 690         { u64 x77 = (x42 + (x52 << 0x4));
 691         { u64 x78 = (x77 + (x52 << 0x1));
 692         { u64 x79 = (x78 + x52);
 693         { u64 x80 = (x41 + (x51 << 0x4));
 694         { u64 x81 = (x80 + (x51 << 0x1));
 695         { u64 x82 = (x81 + x51);
 696         { u64 x83 = (x40 + (x50 << 0x4));
 697         { u64 x84 = (x83 + (x50 << 0x1));
 698         { u64 x85 = (x84 + x50);
 699         { u64 x86 = (x85 >> 0x1a);
 700         { u32 x87 = ((u32)x85 & 0x3ffffff);
 701         { u64 x88 = (x86 + x82);
 702         { u64 x89 = (x88 >> 0x19);
 703         { u32 x90 = ((u32)x88 & 0x1ffffff);
 704         { u64 x91 = (x89 + x79);
 705         { u64 x92 = (x91 >> 0x1a);
 706         { u32 x93 = ((u32)x91 & 0x3ffffff);
 707         { u64 x94 = (x92 + x76);
 708         { u64 x95 = (x94 >> 0x19);
 709         { u32 x96 = ((u32)x94 & 0x1ffffff);
 710         { u64 x97 = (x95 + x73);
 711         { u64 x98 = (x97 >> 0x1a);
 712         { u32 x99 = ((u32)x97 & 0x3ffffff);
 713         { u64 x100 = (x98 + x70);
 714         { u64 x101 = (x100 >> 0x19);
 715         { u32 x102 = ((u32)x100 & 0x1ffffff);
 716         { u64 x103 = (x101 + x67);
 717         { u64 x104 = (x103 >> 0x1a);
 718         { u32 x105 = ((u32)x103 & 0x3ffffff);
 719         { u64 x106 = (x104 + x64);
 720         { u64 x107 = (x106 >> 0x19);
 721         { u32 x108 = ((u32)x106 & 0x1ffffff);
 722         { u64 x109 = (x107 + x61);
 723         { u64 x110 = (x109 >> 0x1a);
 724         { u32 x111 = ((u32)x109 & 0x3ffffff);
 725         { u64 x112 = (x110 + x49);
 726         { u64 x113 = (x112 >> 0x19);
 727         { u32 x114 = ((u32)x112 & 0x1ffffff);
 728         { u64 x115 = (x87 + (0x13 * x113));
 729         { u32 x116 = (u32) (x115 >> 0x1a);
 730         { u32 x117 = ((u32)x115 & 0x3ffffff);
 731         { u32 x118 = (x116 + x90);
 732         { u32 x119 = (x118 >> 0x19);
 733         { u32 x120 = (x118 & 0x1ffffff);
 734         out[0] = x117;
 735         out[1] = x120;
 736         out[2] = (x119 + x93);
 737         out[3] = x96;
 738         out[4] = x99;
 739         out[5] = x102;
 740         out[6] = x105;
 741         out[7] = x108;
 742         out[8] = x111;
 743         out[9] = x114;
 744         }}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}
 745 }
 746
 747 static __always_inline void fe_mul121666(fe *h, const fe_loose *f)
 748 {
 749         fe_mul_121666_impl(h->v, f->v);
 750 }
 751
 752 static void curve25519_generic(u8 out[CURVE25519_KEY_SIZE],
 753                                const u8 scalar[CURVE25519_KEY_SIZE],
 754                                const u8 point[CURVE25519_KEY_SIZE])
 755 {
 756         fe x1, x2, z2, x3, z3;
 757         fe_loose x2l, z2l, x3l;
 758         unsigned swap = 0;
 759         int pos;
 760         u8 e[32];
 761
 762         memcpy(e, scalar, 32);
 763         curve25519_clamp_secret(e);
 764
 765         /* The following implementation was transcribed to Coq and proven to
 766          * correspond to unary scalar multiplication in affine coordinates given
 767          * that x1 != 0 is the x coordinate of some point on the curve. It was
 768          * also checked in Coq that doing a ladderstep with x1 = x3 = 0 gives
 769          * z2' = z3' = 0, and z2 = z3 = 0 gives z2' = z3' = 0. The statement was
 770          * quantified over the underlying field, so it applies to Curve25519
 771          * itself and the quadratic twist of Curve25519. It was not proven in
 772          * Coq that prime-field arithmetic correctly simulates extension-field
 773          * arithmetic on prime-field values. The decoding of the byte array
 774          * representation of e was not considered.
 775          *
 776          * Specification of Montgomery curves in affine coordinates:
 777          * <https://github.com/mit-plv/fiat-crypto/blob/2456d821825521f7e03e65882cc3521795b0320f/src/Spec/MontgomeryCurve.v#L27>
 778          *
 779          * Proof that these form a group that is isomorphic to a Weierstrass
 780          * curve:
 781          * <https://github.com/mit-plv/fiat-crypto/blob/2456d821825521f7e03e65882cc3521795b0320f/src/Curves/Montgomery/AffineProofs.v#L35>
 782          *
 783          * Coq transcription and correctness proof of the loop
 784          * (where scalarbits=255):
 785          * <https://github.com/mit-plv/fiat-crypto/blob/2456d821825521f7e03e65882cc3521795b0320f/src/Curves/Montgomery/XZ.v#L118>
 786          * <https://github.com/mit-plv/fiat-crypto/blob/2456d821825521f7e03e65882cc3521795b0320f/src/Curves/Montgomery/XZProofs.v#L278>
 787          * preconditions: 0 <= e < 2^255 (not necessarily e < order),
 788          * fe_invert(0) = 0
 789          */
 790         fe_frombytes(&x1, point);
 791         fe_1(&x2);
 792         fe_0(&z2);
 793         fe_copy(&x3, &x1);
 794         fe_1(&z3);
 795
 796         for (pos = 254; pos >= 0; --pos) {
 797                 fe tmp0, tmp1;
 798                 fe_loose tmp0l, tmp1l;
 799                 /* loop invariant as of right before the test, for the case
 800                  * where x1 != 0:
 801                  *   pos >= -1; if z2 = 0 then x2 is nonzero; if z3 = 0 then x3
 802                  *   is nonzero
 803                  *   let r := e >> (pos+1) in the following equalities of
 804                  *   projective points:
 805                  *   to_xz (r*P)     === if swap then (x3, z3) else (x2, z2)
 806                  *   to_xz ((r+1)*P) === if swap then (x2, z2) else (x3, z3)
 807                  *   x1 is the nonzero x coordinate of the nonzero
 808                  *   point (r*P-(r+1)*P)
 809                  */
 810                 unsigned b = 1 & (e[pos / 8] >> (pos & 7));
 811                 swap ^= b;
 812                 fe_cswap(&x2, &x3, swap);
 813                 fe_cswap(&z2, &z3, swap);
 814                 swap = b;
 815                 /* Coq transcription of ladderstep formula (called from
 816                  * transcribed loop):
 817                  * <https://github.com/mit-plv/fiat-crypto/blob/2456d821825521f7e03e65882cc3521795b0320f/src/Curves/Montgomery/XZ.v#L89>
 818                  * <https://github.com/mit-plv/fiat-crypto/blob/2456d821825521f7e03e65882cc3521795b0320f/src/Curves/Montgomery/XZProofs.v#L131>
 819                  * x1 != 0 <https://github.com/mit-plv/fiat-crypto/blob/2456d821825521f7e03e65882cc3521795b0320f/src/Curves/Montgomery/XZProofs.v#L217>
 820                  * x1  = 0 <https://github.com/mit-plv/fiat-crypto/blob/2456d821825521f7e03e65882cc3521795b0320f/src/Curves/Montgomery/XZProofs.v#L147>
 821                  */
 822                 fe_sub(&tmp0l, &x3, &z3);
 823                 fe_sub(&tmp1l, &x2, &z2);
 824                 fe_add(&x2l, &x2, &z2);
 825                 fe_add(&z2l, &x3, &z3);
 826                 fe_mul_tll(&z3, &tmp0l, &x2l);
 827                 fe_mul_tll(&z2, &z2l, &tmp1l);
 828                 fe_sq_tl(&tmp0, &tmp1l);
 829                 fe_sq_tl(&tmp1, &x2l);
 830                 fe_add(&x3l, &z3, &z2);
 831                 fe_sub(&z2l, &z3, &z2);
 832                 fe_mul_ttt(&x2, &tmp1, &tmp0);
 833                 fe_sub(&tmp1l, &tmp1, &tmp0);
 834                 fe_sq_tl(&z2, &z2l);
 835                 fe_mul121666(&z3, &tmp1l);
 836                 fe_sq_tl(&x3, &x3l);
 837                 fe_add(&tmp0l, &tmp0, &z3);
 838                 fe_mul_ttt(&z3, &x1, &z2);
 839                 fe_mul_tll(&z2, &tmp1l, &tmp0l);
 840         }
 841         /* here pos=-1, so r=e, so to_xz (e*P) === if swap then (x3, z3)
 842          * else (x2, z2)
 843          */
 844         fe_cswap(&x2, &x3, swap);
 845         fe_cswap(&z2, &z3, swap);
 846
 847         fe_invert(&z2, &z2);
 848         fe_mul_ttt(&x2, &x2, &z2);
 849         fe_tobytes(out, &x2);
 850
 851         memzero_explicit(&x1, sizeof(x1));
 852         memzero_explicit(&x2, sizeof(x2));
 853         memzero_explicit(&z2, sizeof(z2));
 854         memzero_explicit(&x3, sizeof(x3));
 855         memzero_explicit(&z3, sizeof(z3));
 856         memzero_explicit(&x2l, sizeof(x2l));
 857         memzero_explicit(&z2l, sizeof(z2l));
 858         memzero_explicit(&x3l, sizeof(x3l));
 859         memzero_explicit(&e, sizeof(e));
 860 }