commit 5ca4c9d8fcfc4d7b7d0e7695e5250fd9ec539d62
parent 1ecbe9f1531db7373b0ddcc902f0a0b54ba656fd
Author: Jared Tobin <jared@jtobin.io>
Date: Fri, 21 Nov 2025 12:49:21 +0400
lib: montgomery modular inverse
Diffstat:
5 files changed, 1097 insertions(+), 3 deletions(-)
diff --git a/.gitignore b/.gitignore
@@ -3,9 +3,9 @@ dist-newstyle/
*.hp
core
.claude/
-etc/
*.ddump*
*dump-llvm
*.s
*.ll
*llvm
+sandbox/
diff --git a/etc/generate_inv.sh b/etc/generate_inv.sh
@@ -0,0 +1,51 @@
+#!/usr/bin/env bash
+
+# generates a constant-time haskell function for performing modular
+# inversion with montgomery arithmetic on a secp256k1-derived field.
+#
+# fermat inversion is used. one proceeds through the (fixed, known)
+# bit-string of the exponent in MSB order, montgomery-squaring an
+# accumulator each time, and montgomery-multiplying on every '1' bit.
+# this script generates a function consisting of this loop, unrolled.
+#
+# since the square-and-multiply schedule is fixed, then given
+# constant-time 'sqr#' and 'mul#", 'inv#' is also constant-time by
+# construction.
+
+# for fermat inversion, we raise an argument to e.g. the secp256k1 field
+# prime - 2. i.e.:
+#
+# a^-1 = a ^ p - 2 mod p
+#
+# or to the secp256k1 scalar group order - 2:
+#
+# a^-1 = a ^ q - 2 mod q
+
+# secp256k1 field prime - 2
+# exponent="1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111011111111111111111111110000101101"
+
+# secp256k1 scalar group order - 2
+exponent="1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111010111010101011101101110011100110101011110100100010100000001110111011111111010010010111101000110011010000001101100100000101000001"
+
+echo "-- generated by etc/generate_inv.sh"
+echo "inv#"
+echo " :: (# Word#, Word#, Word#, Word# #)"
+echo " -> (# Word#, Word#, Word#, Word# #)"
+echo "inv# a ="
+echo " let !t0 = (# 0x1000003D1##, 0##, 0##, 0## #) -- montgomery 'one'"
+
+label=1
+
+for ((i = 0; i < ${#exponent}; i++)); do
+ echo " !t""$label"" = sqr# t""$((label-1))"
+ if [[ "${exponent:i:1}" == "1" ]]; then
+ label=$((label+1))
+ echo " !t""$label"" = mul# a t""$((label-1))"
+ fi
+ label=$((label+1))
+done
+
+echo " !r = t""$((label-1))"
+echo " in r"
+echo '{-# INLINE inv# #-}'
+
diff --git a/generate_inv.sh b/generate_inv.sh
@@ -0,0 +1,51 @@
+#!/usr/bin/env bash
+
+# generates a constant-time haskell function for performing modular
+# inversion with montgomery arithmetic on a secp256k1-derived field.
+#
+# fermat inversion is used. one proceeds through the (fixed, known)
+# bit-string of the exponent in MSB order, montgomery-squaring an
+# accumulator each time, and montgomery-multiplying on every '1' bit.
+# this script generates a function consisting of this loop, unrolled.
+#
+# since the square-and-multiply schedule is fixed, then given
+# constant-time 'sqr#' and 'mul#", 'inv#' is also constant-time by
+# construction.
+
+# for fermat inversion, we raise an argument to e.g. the secp256k1 field
+# prime - 2. i.e.:
+#
+# a^-1 = a ^ p - 2 mod p
+#
+# or to the secp256k1 scalar group order - 2:
+#
+# a^-1 = a ^ q - 2 mod q
+
+# secp256k1 field prime - 2
+# exponent="1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111011111111111111111111110000101101"
+
+# secp256k1 scalar group order - 2
+exponent="1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111010111010101011101101110011100110101011110100100010100000001110111011111111010010010111101000110011010000001101100100000101000001"
+
+echo "-- generated by etc/generate_inv.sh"
+echo "inv#"
+echo " :: (# Word#, Word#, Word#, Word# #)"
+echo " -> (# Word#, Word#, Word#, Word# #)"
+echo "inv# a ="
+echo " let !t0 = (# 0x1000003D1##, 0##, 0##, 0## #) -- montgomery 'one'"
+
+label=1
+
+for ((i = 0; i < ${#exponent}; i++)); do
+ echo " !t""$label"" = sqr# t""$((label-1))"
+ if [[ "${exponent:i:1}" == "1" ]]; then
+ label=$((label+1))
+ echo " !t""$label"" = mul# a t""$((label-1))"
+ fi
+ label=$((label+1))
+done
+
+echo " !r = t""$((label-1))"
+echo " in r"
+echo '{-# INLINE inv# #-}'
+
diff --git a/lib/Numeric/Montgomery/Secp256k1/Curve.hs b/lib/Numeric/Montgomery/Secp256k1/Curve.hs
@@ -286,7 +286,7 @@ mul# a b =
, 0xFFFFFFFFFFFFFFFF##, 0xFFFFFFFFFFFFFFFF## #)
!(# nu, mc #) = mul_inner# a b
in WW.sub_mod_c# nu mc m m
-{-# INLINE mul# #-}
+{-# NOINLINE mul# #-} -- cannot be inlined without exploding comp time
mul
:: Wider -- ^ lhs in montgomery form
@@ -346,3 +346,527 @@ neg# a = sub# (# 0##, 0##, 0##, 0## #) a
neg :: Wider -> Wider
neg (Wider a) = Wider (neg# a)
+sqr# :: (# Word#, Word#, Word#, Word# #) -> (# Word#, Word#, Word#, Word# #)
+sqr# a = mul# a a
+{-# INLINE sqr# #-}
+
+sqr :: Wider -> Wider
+sqr (Wider a) = Wider (mul# a a)
+
+-- generated by etc/generate_inv.sh
+inv#
+ :: (# Word#, Word#, Word#, Word# #)
+ -> (# Word#, Word#, Word#, Word# #)
+inv# a =
+ let !t0 = (# 0x1000003D1##, 0##, 0##, 0## #) -- montgomery 'one'
+ !t1 = sqr# t0
+ !t2 = mul# a t1
+ !t3 = sqr# t2
+ !t4 = mul# a t3
+ !t5 = sqr# t4
+ !t6 = mul# a t5
+ !t7 = sqr# t6
+ !t8 = mul# a t7
+ !t9 = sqr# t8
+ !t10 = mul# a t9
+ !t11 = sqr# t10
+ !t12 = mul# a t11
+ !t13 = sqr# t12
+ !t14 = mul# a t13
+ !t15 = sqr# t14
+ !t16 = mul# a t15
+ !t17 = sqr# t16
+ !t18 = mul# a t17
+ !t19 = sqr# t18
+ !t20 = mul# a t19
+ !t21 = sqr# t20
+ !t22 = mul# a t21
+ !t23 = sqr# t22
+ !t24 = mul# a t23
+ !t25 = sqr# t24
+ !t26 = mul# a t25
+ !t27 = sqr# t26
+ !t28 = mul# a t27
+ !t29 = sqr# t28
+ !t30 = mul# a t29
+ !t31 = sqr# t30
+ !t32 = mul# a t31
+ !t33 = sqr# t32
+ !t34 = mul# a t33
+ !t35 = sqr# t34
+ !t36 = mul# a t35
+ !t37 = sqr# t36
+ !t38 = mul# a t37
+ !t39 = sqr# t38
+ !t40 = mul# a t39
+ !t41 = sqr# t40
+ !t42 = mul# a t41
+ !t43 = sqr# t42
+ !t44 = mul# a t43
+ !t45 = sqr# t44
+ !t46 = mul# a t45
+ !t47 = sqr# t46
+ !t48 = mul# a t47
+ !t49 = sqr# t48
+ !t50 = mul# a t49
+ !t51 = sqr# t50
+ !t52 = mul# a t51
+ !t53 = sqr# t52
+ !t54 = mul# a t53
+ !t55 = sqr# t54
+ !t56 = mul# a t55
+ !t57 = sqr# t56
+ !t58 = mul# a t57
+ !t59 = sqr# t58
+ !t60 = mul# a t59
+ !t61 = sqr# t60
+ !t62 = mul# a t61
+ !t63 = sqr# t62
+ !t64 = mul# a t63
+ !t65 = sqr# t64
+ !t66 = mul# a t65
+ !t67 = sqr# t66
+ !t68 = mul# a t67
+ !t69 = sqr# t68
+ !t70 = mul# a t69
+ !t71 = sqr# t70
+ !t72 = mul# a t71
+ !t73 = sqr# t72
+ !t74 = mul# a t73
+ !t75 = sqr# t74
+ !t76 = mul# a t75
+ !t77 = sqr# t76
+ !t78 = mul# a t77
+ !t79 = sqr# t78
+ !t80 = mul# a t79
+ !t81 = sqr# t80
+ !t82 = mul# a t81
+ !t83 = sqr# t82
+ !t84 = mul# a t83
+ !t85 = sqr# t84
+ !t86 = mul# a t85
+ !t87 = sqr# t86
+ !t88 = mul# a t87
+ !t89 = sqr# t88
+ !t90 = mul# a t89
+ !t91 = sqr# t90
+ !t92 = mul# a t91
+ !t93 = sqr# t92
+ !t94 = mul# a t93
+ !t95 = sqr# t94
+ !t96 = mul# a t95
+ !t97 = sqr# t96
+ !t98 = mul# a t97
+ !t99 = sqr# t98
+ !t100 = mul# a t99
+ !t101 = sqr# t100
+ !t102 = mul# a t101
+ !t103 = sqr# t102
+ !t104 = mul# a t103
+ !t105 = sqr# t104
+ !t106 = mul# a t105
+ !t107 = sqr# t106
+ !t108 = mul# a t107
+ !t109 = sqr# t108
+ !t110 = mul# a t109
+ !t111 = sqr# t110
+ !t112 = mul# a t111
+ !t113 = sqr# t112
+ !t114 = mul# a t113
+ !t115 = sqr# t114
+ !t116 = mul# a t115
+ !t117 = sqr# t116
+ !t118 = mul# a t117
+ !t119 = sqr# t118
+ !t120 = mul# a t119
+ !t121 = sqr# t120
+ !t122 = mul# a t121
+ !t123 = sqr# t122
+ !t124 = mul# a t123
+ !t125 = sqr# t124
+ !t126 = mul# a t125
+ !t127 = sqr# t126
+ !t128 = mul# a t127
+ !t129 = sqr# t128
+ !t130 = mul# a t129
+ !t131 = sqr# t130
+ !t132 = mul# a t131
+ !t133 = sqr# t132
+ !t134 = mul# a t133
+ !t135 = sqr# t134
+ !t136 = mul# a t135
+ !t137 = sqr# t136
+ !t138 = mul# a t137
+ !t139 = sqr# t138
+ !t140 = mul# a t139
+ !t141 = sqr# t140
+ !t142 = mul# a t141
+ !t143 = sqr# t142
+ !t144 = mul# a t143
+ !t145 = sqr# t144
+ !t146 = mul# a t145
+ !t147 = sqr# t146
+ !t148 = mul# a t147
+ !t149 = sqr# t148
+ !t150 = mul# a t149
+ !t151 = sqr# t150
+ !t152 = mul# a t151
+ !t153 = sqr# t152
+ !t154 = mul# a t153
+ !t155 = sqr# t154
+ !t156 = mul# a t155
+ !t157 = sqr# t156
+ !t158 = mul# a t157
+ !t159 = sqr# t158
+ !t160 = mul# a t159
+ !t161 = sqr# t160
+ !t162 = mul# a t161
+ !t163 = sqr# t162
+ !t164 = mul# a t163
+ !t165 = sqr# t164
+ !t166 = mul# a t165
+ !t167 = sqr# t166
+ !t168 = mul# a t167
+ !t169 = sqr# t168
+ !t170 = mul# a t169
+ !t171 = sqr# t170
+ !t172 = mul# a t171
+ !t173 = sqr# t172
+ !t174 = mul# a t173
+ !t175 = sqr# t174
+ !t176 = mul# a t175
+ !t177 = sqr# t176
+ !t178 = mul# a t177
+ !t179 = sqr# t178
+ !t180 = mul# a t179
+ !t181 = sqr# t180
+ !t182 = mul# a t181
+ !t183 = sqr# t182
+ !t184 = mul# a t183
+ !t185 = sqr# t184
+ !t186 = mul# a t185
+ !t187 = sqr# t186
+ !t188 = mul# a t187
+ !t189 = sqr# t188
+ !t190 = mul# a t189
+ !t191 = sqr# t190
+ !t192 = mul# a t191
+ !t193 = sqr# t192
+ !t194 = mul# a t193
+ !t195 = sqr# t194
+ !t196 = mul# a t195
+ !t197 = sqr# t196
+ !t198 = mul# a t197
+ !t199 = sqr# t198
+ !t200 = mul# a t199
+ !t201 = sqr# t200
+ !t202 = mul# a t201
+ !t203 = sqr# t202
+ !t204 = mul# a t203
+ !t205 = sqr# t204
+ !t206 = mul# a t205
+ !t207 = sqr# t206
+ !t208 = mul# a t207
+ !t209 = sqr# t208
+ !t210 = mul# a t209
+ !t211 = sqr# t210
+ !t212 = mul# a t211
+ !t213 = sqr# t212
+ !t214 = mul# a t213
+ !t215 = sqr# t214
+ !t216 = mul# a t215
+ !t217 = sqr# t216
+ !t218 = mul# a t217
+ !t219 = sqr# t218
+ !t220 = mul# a t219
+ !t221 = sqr# t220
+ !t222 = mul# a t221
+ !t223 = sqr# t222
+ !t224 = mul# a t223
+ !t225 = sqr# t224
+ !t226 = mul# a t225
+ !t227 = sqr# t226
+ !t228 = mul# a t227
+ !t229 = sqr# t228
+ !t230 = mul# a t229
+ !t231 = sqr# t230
+ !t232 = mul# a t231
+ !t233 = sqr# t232
+ !t234 = mul# a t233
+ !t235 = sqr# t234
+ !t236 = mul# a t235
+ !t237 = sqr# t236
+ !t238 = mul# a t237
+ !t239 = sqr# t238
+ !t240 = mul# a t239
+ !t241 = sqr# t240
+ !t242 = mul# a t241
+ !t243 = sqr# t242
+ !t244 = mul# a t243
+ !t245 = sqr# t244
+ !t246 = mul# a t245
+ !t247 = sqr# t246
+ !t248 = mul# a t247
+ !t249 = sqr# t248
+ !t250 = mul# a t249
+ !t251 = sqr# t250
+ !t252 = mul# a t251
+ !t253 = sqr# t252
+ !t254 = mul# a t253
+ !t255 = sqr# t254
+ !t256 = mul# a t255
+ !t257 = sqr# t256
+ !t258 = mul# a t257
+ !t259 = sqr# t258
+ !t260 = mul# a t259
+ !t261 = sqr# t260
+ !t262 = mul# a t261
+ !t263 = sqr# t262
+ !t264 = mul# a t263
+ !t265 = sqr# t264
+ !t266 = mul# a t265
+ !t267 = sqr# t266
+ !t268 = mul# a t267
+ !t269 = sqr# t268
+ !t270 = mul# a t269
+ !t271 = sqr# t270
+ !t272 = mul# a t271
+ !t273 = sqr# t272
+ !t274 = mul# a t273
+ !t275 = sqr# t274
+ !t276 = mul# a t275
+ !t277 = sqr# t276
+ !t278 = mul# a t277
+ !t279 = sqr# t278
+ !t280 = mul# a t279
+ !t281 = sqr# t280
+ !t282 = mul# a t281
+ !t283 = sqr# t282
+ !t284 = mul# a t283
+ !t285 = sqr# t284
+ !t286 = mul# a t285
+ !t287 = sqr# t286
+ !t288 = mul# a t287
+ !t289 = sqr# t288
+ !t290 = mul# a t289
+ !t291 = sqr# t290
+ !t292 = mul# a t291
+ !t293 = sqr# t292
+ !t294 = mul# a t293
+ !t295 = sqr# t294
+ !t296 = mul# a t295
+ !t297 = sqr# t296
+ !t298 = mul# a t297
+ !t299 = sqr# t298
+ !t300 = mul# a t299
+ !t301 = sqr# t300
+ !t302 = mul# a t301
+ !t303 = sqr# t302
+ !t304 = mul# a t303
+ !t305 = sqr# t304
+ !t306 = mul# a t305
+ !t307 = sqr# t306
+ !t308 = mul# a t307
+ !t309 = sqr# t308
+ !t310 = mul# a t309
+ !t311 = sqr# t310
+ !t312 = mul# a t311
+ !t313 = sqr# t312
+ !t314 = mul# a t313
+ !t315 = sqr# t314
+ !t316 = mul# a t315
+ !t317 = sqr# t316
+ !t318 = mul# a t317
+ !t319 = sqr# t318
+ !t320 = mul# a t319
+ !t321 = sqr# t320
+ !t322 = mul# a t321
+ !t323 = sqr# t322
+ !t324 = mul# a t323
+ !t325 = sqr# t324
+ !t326 = mul# a t325
+ !t327 = sqr# t326
+ !t328 = mul# a t327
+ !t329 = sqr# t328
+ !t330 = mul# a t329
+ !t331 = sqr# t330
+ !t332 = mul# a t331
+ !t333 = sqr# t332
+ !t334 = mul# a t333
+ !t335 = sqr# t334
+ !t336 = mul# a t335
+ !t337 = sqr# t336
+ !t338 = mul# a t337
+ !t339 = sqr# t338
+ !t340 = mul# a t339
+ !t341 = sqr# t340
+ !t342 = mul# a t341
+ !t343 = sqr# t342
+ !t344 = mul# a t343
+ !t345 = sqr# t344
+ !t346 = mul# a t345
+ !t347 = sqr# t346
+ !t348 = mul# a t347
+ !t349 = sqr# t348
+ !t350 = mul# a t349
+ !t351 = sqr# t350
+ !t352 = mul# a t351
+ !t353 = sqr# t352
+ !t354 = mul# a t353
+ !t355 = sqr# t354
+ !t356 = mul# a t355
+ !t357 = sqr# t356
+ !t358 = mul# a t357
+ !t359 = sqr# t358
+ !t360 = mul# a t359
+ !t361 = sqr# t360
+ !t362 = mul# a t361
+ !t363 = sqr# t362
+ !t364 = mul# a t363
+ !t365 = sqr# t364
+ !t366 = mul# a t365
+ !t367 = sqr# t366
+ !t368 = mul# a t367
+ !t369 = sqr# t368
+ !t370 = mul# a t369
+ !t371 = sqr# t370
+ !t372 = mul# a t371
+ !t373 = sqr# t372
+ !t374 = mul# a t373
+ !t375 = sqr# t374
+ !t376 = mul# a t375
+ !t377 = sqr# t376
+ !t378 = mul# a t377
+ !t379 = sqr# t378
+ !t380 = mul# a t379
+ !t381 = sqr# t380
+ !t382 = mul# a t381
+ !t383 = sqr# t382
+ !t384 = mul# a t383
+ !t385 = sqr# t384
+ !t386 = mul# a t385
+ !t387 = sqr# t386
+ !t388 = mul# a t387
+ !t389 = sqr# t388
+ !t390 = mul# a t389
+ !t391 = sqr# t390
+ !t392 = mul# a t391
+ !t393 = sqr# t392
+ !t394 = mul# a t393
+ !t395 = sqr# t394
+ !t396 = mul# a t395
+ !t397 = sqr# t396
+ !t398 = mul# a t397
+ !t399 = sqr# t398
+ !t400 = mul# a t399
+ !t401 = sqr# t400
+ !t402 = mul# a t401
+ !t403 = sqr# t402
+ !t404 = mul# a t403
+ !t405 = sqr# t404
+ !t406 = mul# a t405
+ !t407 = sqr# t406
+ !t408 = mul# a t407
+ !t409 = sqr# t408
+ !t410 = mul# a t409
+ !t411 = sqr# t410
+ !t412 = mul# a t411
+ !t413 = sqr# t412
+ !t414 = mul# a t413
+ !t415 = sqr# t414
+ !t416 = mul# a t415
+ !t417 = sqr# t416
+ !t418 = mul# a t417
+ !t419 = sqr# t418
+ !t420 = mul# a t419
+ !t421 = sqr# t420
+ !t422 = mul# a t421
+ !t423 = sqr# t422
+ !t424 = mul# a t423
+ !t425 = sqr# t424
+ !t426 = mul# a t425
+ !t427 = sqr# t426
+ !t428 = mul# a t427
+ !t429 = sqr# t428
+ !t430 = mul# a t429
+ !t431 = sqr# t430
+ !t432 = mul# a t431
+ !t433 = sqr# t432
+ !t434 = mul# a t433
+ !t435 = sqr# t434
+ !t436 = mul# a t435
+ !t437 = sqr# t436
+ !t438 = mul# a t437
+ !t439 = sqr# t438
+ !t440 = mul# a t439
+ !t441 = sqr# t440
+ !t442 = mul# a t441
+ !t443 = sqr# t442
+ !t444 = mul# a t443
+ !t445 = sqr# t444
+ !t446 = mul# a t445
+ !t447 = sqr# t446
+ !t448 = sqr# t447
+ !t449 = mul# a t448
+ !t450 = sqr# t449
+ !t451 = mul# a t450
+ !t452 = sqr# t451
+ !t453 = mul# a t452
+ !t454 = sqr# t453
+ !t455 = mul# a t454
+ !t456 = sqr# t455
+ !t457 = mul# a t456
+ !t458 = sqr# t457
+ !t459 = mul# a t458
+ !t460 = sqr# t459
+ !t461 = mul# a t460
+ !t462 = sqr# t461
+ !t463 = mul# a t462
+ !t464 = sqr# t463
+ !t465 = mul# a t464
+ !t466 = sqr# t465
+ !t467 = mul# a t466
+ !t468 = sqr# t467
+ !t469 = mul# a t468
+ !t470 = sqr# t469
+ !t471 = mul# a t470
+ !t472 = sqr# t471
+ !t473 = mul# a t472
+ !t474 = sqr# t473
+ !t475 = mul# a t474
+ !t476 = sqr# t475
+ !t477 = mul# a t476
+ !t478 = sqr# t477
+ !t479 = mul# a t478
+ !t480 = sqr# t479
+ !t481 = mul# a t480
+ !t482 = sqr# t481
+ !t483 = mul# a t482
+ !t484 = sqr# t483
+ !t485 = mul# a t484
+ !t486 = sqr# t485
+ !t487 = mul# a t486
+ !t488 = sqr# t487
+ !t489 = mul# a t488
+ !t490 = sqr# t489
+ !t491 = mul# a t490
+ !t492 = sqr# t491
+ !t493 = sqr# t492
+ !t494 = sqr# t493
+ !t495 = sqr# t494
+ !t496 = sqr# t495
+ !t497 = mul# a t496
+ !t498 = sqr# t497
+ !t499 = sqr# t498
+ !t500 = mul# a t499
+ !t501 = sqr# t500
+ !t502 = mul# a t501
+ !t503 = sqr# t502
+ !t504 = sqr# t503
+ !t505 = mul# a t504
+ !r = t505
+ in r
+{-# INLINE inv# #-}
+
+inv :: Wider -> Wider
+inv (Wider w) = Wider (inv# w)
diff --git a/lib/Numeric/Montgomery/Secp256k1/Scalar.hs b/lib/Numeric/Montgomery/Secp256k1/Scalar.hs
@@ -286,7 +286,7 @@ mul# a b =
, 0xFFFFFFFFFFFFFFFE##, 0xFFFFFFFFFFFFFFFF## #)
!(# nu, mc #) = mul_inner# a b
in WW.sub_mod_c# nu mc m m
-{-# INLINE mul# #-}
+{-# NOINLINE mul# #-} -- cannot be inlined without exploding comp time
mul
:: Wider -- ^ lhs in montgomery form
@@ -347,3 +347,471 @@ neg# a = sub# (# 0##, 0##, 0##, 0## #) a
neg :: Wider -> Wider
neg (Wider a) = Wider (neg# a)
+sqr# :: (# Word#, Word#, Word#, Word# #) -> (# Word#, Word#, Word#, Word# #)
+sqr# a = mul# a a
+{-# INLINE sqr# #-}
+
+sqr :: Wider -> Wider
+sqr (Wider a) = Wider (mul# a a)
+
+-- generated by etc/generate_inv.sh
+inv#
+ :: (# Word#, Word#, Word#, Word# #)
+ -> (# Word#, Word#, Word#, Word# #)
+inv# a =
+ let !t0 = (# 0x1000003D1##, 0##, 0##, 0## #) -- montgomery 'one'
+ !t1 = sqr# t0
+ !t2 = mul# a t1
+ !t3 = sqr# t2
+ !t4 = mul# a t3
+ !t5 = sqr# t4
+ !t6 = mul# a t5
+ !t7 = sqr# t6
+ !t8 = mul# a t7
+ !t9 = sqr# t8
+ !t10 = mul# a t9
+ !t11 = sqr# t10
+ !t12 = mul# a t11
+ !t13 = sqr# t12
+ !t14 = mul# a t13
+ !t15 = sqr# t14
+ !t16 = mul# a t15
+ !t17 = sqr# t16
+ !t18 = mul# a t17
+ !t19 = sqr# t18
+ !t20 = mul# a t19
+ !t21 = sqr# t20
+ !t22 = mul# a t21
+ !t23 = sqr# t22
+ !t24 = mul# a t23
+ !t25 = sqr# t24
+ !t26 = mul# a t25
+ !t27 = sqr# t26
+ !t28 = mul# a t27
+ !t29 = sqr# t28
+ !t30 = mul# a t29
+ !t31 = sqr# t30
+ !t32 = mul# a t31
+ !t33 = sqr# t32
+ !t34 = mul# a t33
+ !t35 = sqr# t34
+ !t36 = mul# a t35
+ !t37 = sqr# t36
+ !t38 = mul# a t37
+ !t39 = sqr# t38
+ !t40 = mul# a t39
+ !t41 = sqr# t40
+ !t42 = mul# a t41
+ !t43 = sqr# t42
+ !t44 = mul# a t43
+ !t45 = sqr# t44
+ !t46 = mul# a t45
+ !t47 = sqr# t46
+ !t48 = mul# a t47
+ !t49 = sqr# t48
+ !t50 = mul# a t49
+ !t51 = sqr# t50
+ !t52 = mul# a t51
+ !t53 = sqr# t52
+ !t54 = mul# a t53
+ !t55 = sqr# t54
+ !t56 = mul# a t55
+ !t57 = sqr# t56
+ !t58 = mul# a t57
+ !t59 = sqr# t58
+ !t60 = mul# a t59
+ !t61 = sqr# t60
+ !t62 = mul# a t61
+ !t63 = sqr# t62
+ !t64 = mul# a t63
+ !t65 = sqr# t64
+ !t66 = mul# a t65
+ !t67 = sqr# t66
+ !t68 = mul# a t67
+ !t69 = sqr# t68
+ !t70 = mul# a t69
+ !t71 = sqr# t70
+ !t72 = mul# a t71
+ !t73 = sqr# t72
+ !t74 = mul# a t73
+ !t75 = sqr# t74
+ !t76 = mul# a t75
+ !t77 = sqr# t76
+ !t78 = mul# a t77
+ !t79 = sqr# t78
+ !t80 = mul# a t79
+ !t81 = sqr# t80
+ !t82 = mul# a t81
+ !t83 = sqr# t82
+ !t84 = mul# a t83
+ !t85 = sqr# t84
+ !t86 = mul# a t85
+ !t87 = sqr# t86
+ !t88 = mul# a t87
+ !t89 = sqr# t88
+ !t90 = mul# a t89
+ !t91 = sqr# t90
+ !t92 = mul# a t91
+ !t93 = sqr# t92
+ !t94 = mul# a t93
+ !t95 = sqr# t94
+ !t96 = mul# a t95
+ !t97 = sqr# t96
+ !t98 = mul# a t97
+ !t99 = sqr# t98
+ !t100 = mul# a t99
+ !t101 = sqr# t100
+ !t102 = mul# a t101
+ !t103 = sqr# t102
+ !t104 = mul# a t103
+ !t105 = sqr# t104
+ !t106 = mul# a t105
+ !t107 = sqr# t106
+ !t108 = mul# a t107
+ !t109 = sqr# t108
+ !t110 = mul# a t109
+ !t111 = sqr# t110
+ !t112 = mul# a t111
+ !t113 = sqr# t112
+ !t114 = mul# a t113
+ !t115 = sqr# t114
+ !t116 = mul# a t115
+ !t117 = sqr# t116
+ !t118 = mul# a t117
+ !t119 = sqr# t118
+ !t120 = mul# a t119
+ !t121 = sqr# t120
+ !t122 = mul# a t121
+ !t123 = sqr# t122
+ !t124 = mul# a t123
+ !t125 = sqr# t124
+ !t126 = mul# a t125
+ !t127 = sqr# t126
+ !t128 = mul# a t127
+ !t129 = sqr# t128
+ !t130 = mul# a t129
+ !t131 = sqr# t130
+ !t132 = mul# a t131
+ !t133 = sqr# t132
+ !t134 = mul# a t133
+ !t135 = sqr# t134
+ !t136 = mul# a t135
+ !t137 = sqr# t136
+ !t138 = mul# a t137
+ !t139 = sqr# t138
+ !t140 = mul# a t139
+ !t141 = sqr# t140
+ !t142 = mul# a t141
+ !t143 = sqr# t142
+ !t144 = mul# a t143
+ !t145 = sqr# t144
+ !t146 = mul# a t145
+ !t147 = sqr# t146
+ !t148 = mul# a t147
+ !t149 = sqr# t148
+ !t150 = mul# a t149
+ !t151 = sqr# t150
+ !t152 = mul# a t151
+ !t153 = sqr# t152
+ !t154 = mul# a t153
+ !t155 = sqr# t154
+ !t156 = mul# a t155
+ !t157 = sqr# t156
+ !t158 = mul# a t157
+ !t159 = sqr# t158
+ !t160 = mul# a t159
+ !t161 = sqr# t160
+ !t162 = mul# a t161
+ !t163 = sqr# t162
+ !t164 = mul# a t163
+ !t165 = sqr# t164
+ !t166 = mul# a t165
+ !t167 = sqr# t166
+ !t168 = mul# a t167
+ !t169 = sqr# t168
+ !t170 = mul# a t169
+ !t171 = sqr# t170
+ !t172 = mul# a t171
+ !t173 = sqr# t172
+ !t174 = mul# a t173
+ !t175 = sqr# t174
+ !t176 = mul# a t175
+ !t177 = sqr# t176
+ !t178 = mul# a t177
+ !t179 = sqr# t178
+ !t180 = mul# a t179
+ !t181 = sqr# t180
+ !t182 = mul# a t181
+ !t183 = sqr# t182
+ !t184 = mul# a t183
+ !t185 = sqr# t184
+ !t186 = mul# a t185
+ !t187 = sqr# t186
+ !t188 = mul# a t187
+ !t189 = sqr# t188
+ !t190 = mul# a t189
+ !t191 = sqr# t190
+ !t192 = mul# a t191
+ !t193 = sqr# t192
+ !t194 = mul# a t193
+ !t195 = sqr# t194
+ !t196 = mul# a t195
+ !t197 = sqr# t196
+ !t198 = mul# a t197
+ !t199 = sqr# t198
+ !t200 = mul# a t199
+ !t201 = sqr# t200
+ !t202 = mul# a t201
+ !t203 = sqr# t202
+ !t204 = mul# a t203
+ !t205 = sqr# t204
+ !t206 = mul# a t205
+ !t207 = sqr# t206
+ !t208 = mul# a t207
+ !t209 = sqr# t208
+ !t210 = mul# a t209
+ !t211 = sqr# t210
+ !t212 = mul# a t211
+ !t213 = sqr# t212
+ !t214 = mul# a t213
+ !t215 = sqr# t214
+ !t216 = mul# a t215
+ !t217 = sqr# t216
+ !t218 = mul# a t217
+ !t219 = sqr# t218
+ !t220 = mul# a t219
+ !t221 = sqr# t220
+ !t222 = mul# a t221
+ !t223 = sqr# t222
+ !t224 = mul# a t223
+ !t225 = sqr# t224
+ !t226 = mul# a t225
+ !t227 = sqr# t226
+ !t228 = mul# a t227
+ !t229 = sqr# t228
+ !t230 = mul# a t229
+ !t231 = sqr# t230
+ !t232 = mul# a t231
+ !t233 = sqr# t232
+ !t234 = mul# a t233
+ !t235 = sqr# t234
+ !t236 = mul# a t235
+ !t237 = sqr# t236
+ !t238 = mul# a t237
+ !t239 = sqr# t238
+ !t240 = mul# a t239
+ !t241 = sqr# t240
+ !t242 = mul# a t241
+ !t243 = sqr# t242
+ !t244 = mul# a t243
+ !t245 = sqr# t244
+ !t246 = mul# a t245
+ !t247 = sqr# t246
+ !t248 = mul# a t247
+ !t249 = sqr# t248
+ !t250 = mul# a t249
+ !t251 = sqr# t250
+ !t252 = mul# a t251
+ !t253 = sqr# t252
+ !t254 = mul# a t253
+ !t255 = sqr# t254
+ !t256 = sqr# t255
+ !t257 = mul# a t256
+ !t258 = sqr# t257
+ !t259 = sqr# t258
+ !t260 = mul# a t259
+ !t261 = sqr# t260
+ !t262 = mul# a t261
+ !t263 = sqr# t262
+ !t264 = mul# a t263
+ !t265 = sqr# t264
+ !t266 = sqr# t265
+ !t267 = mul# a t266
+ !t268 = sqr# t267
+ !t269 = sqr# t268
+ !t270 = mul# a t269
+ !t271 = sqr# t270
+ !t272 = sqr# t271
+ !t273 = mul# a t272
+ !t274 = sqr# t273
+ !t275 = sqr# t274
+ !t276 = mul# a t275
+ !t277 = sqr# t276
+ !t278 = mul# a t277
+ !t279 = sqr# t278
+ !t280 = mul# a t279
+ !t281 = sqr# t280
+ !t282 = sqr# t281
+ !t283 = mul# a t282
+ !t284 = sqr# t283
+ !t285 = mul# a t284
+ !t286 = sqr# t285
+ !t287 = sqr# t286
+ !t288 = mul# a t287
+ !t289 = sqr# t288
+ !t290 = mul# a t289
+ !t291 = sqr# t290
+ !t292 = mul# a t291
+ !t293 = sqr# t292
+ !t294 = sqr# t293
+ !t295 = sqr# t294
+ !t296 = mul# a t295
+ !t297 = sqr# t296
+ !t298 = mul# a t297
+ !t299 = sqr# t298
+ !t300 = mul# a t299
+ !t301 = sqr# t300
+ !t302 = sqr# t301
+ !t303 = sqr# t302
+ !t304 = mul# a t303
+ !t305 = sqr# t304
+ !t306 = mul# a t305
+ !t307 = sqr# t306
+ !t308 = sqr# t307
+ !t309 = mul# a t308
+ !t310 = sqr# t309
+ !t311 = sqr# t310
+ !t312 = mul# a t311
+ !t313 = sqr# t312
+ !t314 = sqr# t313
+ !t315 = mul# a t314
+ !t316 = sqr# t315
+ !t317 = mul# a t316
+ !t318 = sqr# t317
+ !t319 = mul# a t318
+ !t320 = sqr# t319
+ !t321 = mul# a t320
+ !t322 = sqr# t321
+ !t323 = sqr# t322
+ !t324 = mul# a t323
+ !t325 = sqr# t324
+ !t326 = sqr# t325
+ !t327 = sqr# t326
+ !t328 = mul# a t327
+ !t329 = sqr# t328
+ !t330 = sqr# t329
+ !t331 = sqr# t330
+ !t332 = sqr# t331
+ !t333 = mul# a t332
+ !t334 = sqr# t333
+ !t335 = sqr# t334
+ !t336 = mul# a t335
+ !t337 = sqr# t336
+ !t338 = sqr# t337
+ !t339 = sqr# t338
+ !t340 = sqr# t339
+ !t341 = sqr# t340
+ !t342 = sqr# t341
+ !t343 = sqr# t342
+ !t344 = sqr# t343
+ !t345 = mul# a t344
+ !t346 = sqr# t345
+ !t347 = mul# a t346
+ !t348 = sqr# t347
+ !t349 = mul# a t348
+ !t350 = sqr# t349
+ !t351 = sqr# t350
+ !t352 = mul# a t351
+ !t353 = sqr# t352
+ !t354 = mul# a t353
+ !t355 = sqr# t354
+ !t356 = mul# a t355
+ !t357 = sqr# t356
+ !t358 = sqr# t357
+ !t359 = mul# a t358
+ !t360 = sqr# t359
+ !t361 = mul# a t360
+ !t362 = sqr# t361
+ !t363 = mul# a t362
+ !t364 = sqr# t363
+ !t365 = mul# a t364
+ !t366 = sqr# t365
+ !t367 = mul# a t366
+ !t368 = sqr# t367
+ !t369 = mul# a t368
+ !t370 = sqr# t369
+ !t371 = mul# a t370
+ !t372 = sqr# t371
+ !t373 = mul# a t372
+ !t374 = sqr# t373
+ !t375 = sqr# t374
+ !t376 = mul# a t375
+ !t377 = sqr# t376
+ !t378 = sqr# t377
+ !t379 = sqr# t378
+ !t380 = mul# a t379
+ !t381 = sqr# t380
+ !t382 = sqr# t381
+ !t383 = sqr# t382
+ !t384 = mul# a t383
+ !t385 = sqr# t384
+ !t386 = sqr# t385
+ !t387 = mul# a t386
+ !t388 = sqr# t387
+ !t389 = mul# a t388
+ !t390 = sqr# t389
+ !t391 = mul# a t390
+ !t392 = sqr# t391
+ !t393 = mul# a t392
+ !t394 = sqr# t393
+ !t395 = sqr# t394
+ !t396 = mul# a t395
+ !t397 = sqr# t396
+ !t398 = sqr# t397
+ !t399 = sqr# t398
+ !t400 = sqr# t399
+ !t401 = mul# a t400
+ !t402 = sqr# t401
+ !t403 = mul# a t402
+ !t404 = sqr# t403
+ !t405 = sqr# t404
+ !t406 = sqr# t405
+ !t407 = mul# a t406
+ !t408 = sqr# t407
+ !t409 = mul# a t408
+ !t410 = sqr# t409
+ !t411 = sqr# t410
+ !t412 = mul# a t411
+ !t413 = sqr# t412
+ !t414 = sqr# t413
+ !t415 = sqr# t414
+ !t416 = sqr# t415
+ !t417 = sqr# t416
+ !t418 = sqr# t417
+ !t419 = sqr# t418
+ !t420 = mul# a t419
+ !t421 = sqr# t420
+ !t422 = mul# a t421
+ !t423 = sqr# t422
+ !t424 = sqr# t423
+ !t425 = mul# a t424
+ !t426 = sqr# t425
+ !t427 = mul# a t426
+ !t428 = sqr# t427
+ !t429 = sqr# t428
+ !t430 = sqr# t429
+ !t431 = mul# a t430
+ !t432 = sqr# t431
+ !t433 = sqr# t432
+ !t434 = sqr# t433
+ !t435 = sqr# t434
+ !t436 = sqr# t435
+ !t437 = sqr# t436
+ !t438 = mul# a t437
+ !t439 = sqr# t438
+ !t440 = sqr# t439
+ !t441 = mul# a t440
+ !t442 = sqr# t441
+ !t443 = sqr# t442
+ !t444 = sqr# t443
+ !t445 = sqr# t444
+ !t446 = sqr# t445
+ !t447 = sqr# t446
+ !t448 = mul# a t447
+ !r = t448
+ in r
+{-# INLINE inv# #-}
+
+inv :: Wider -> Wider
+inv (Wider w) = Wider (inv# w)
+
