Curve.hs (43715B)
1 {-# LANGUAGE BangPatterns #-} 2 {-# LANGUAGE MagicHash #-} 3 {-# LANGUAGE NumericUnderscores #-} 4 {-# LANGUAGE PatternSynonyms #-} 5 {-# LANGUAGE ViewPatterns #-} 6 {-# LANGUAGE UnboxedSums #-} 7 {-# LANGUAGE UnboxedTuples #-} 8 {-# LANGUAGE UnliftedNewtypes #-} 9 10 -- | 11 -- Module: Numeric.Montgomery.Secp256k1.Curve 12 -- Copyright: (c) 2025 Jared Tobin 13 -- License: MIT 14 -- Maintainer: Jared Tobin <jared@ppad.tech> 15 -- 16 -- Montgomery form 'Wider' words, as well as arithmetic operations, with 17 -- domain derived from the secp256k1 elliptic curve field prime. 18 19 module Numeric.Montgomery.Secp256k1.Curve ( 20 -- * Montgomery form, secp256k1 field prime modulus 21 Montgomery(..) 22 , render 23 , to 24 , from 25 , zero 26 , one 27 28 -- * Comparison 29 , eq 30 , eq_vartime 31 32 -- * Reduction and retrieval 33 , redc 34 , redc# 35 , retr 36 , retr# 37 38 -- * Constant-time selection 39 , select 40 , select# 41 42 -- * Montgomery arithmetic 43 , add 44 , add# 45 , sub 46 , sub# 47 , mul 48 , mul# 49 , sqr 50 , sqr# 51 , neg 52 , neg# 53 , inv 54 , inv# 55 , sqrt_vartime 56 , sqrt# 57 , exp 58 , exp# 59 , odd# 60 , odd_vartime 61 ) where 62 63 import Control.DeepSeq 64 import qualified Data.Choice as C 65 import Data.Word.Limb (Limb(..)) 66 import qualified Data.Word.Limb as L 67 import qualified Data.Word.Wide as W 68 import Data.Word.Wider (Wider(..)) 69 import qualified Data.Word.Wider as WW 70 import GHC.Exts (Word(..), Word#) 71 import Prelude hiding (or, and, not, sqrt, exp) 72 73 -- montgomery arithmetic, specialized to the secp256k1 field prime modulus 74 -- 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F 75 76 -- | Montgomery-form 'Wider' words, on the Montgomery domain defined by 77 -- the secp256k1 field prime. 78 -- 79 -- >>> let one = 1 :: Montgomery 80 -- >>> one 81 -- 1 82 -- >>> putStrLn (render one) 83 -- (4294968273, 0, 0, 0) 84 data Montgomery = Montgomery !Limb4 85 86 -- | Render a 'Montgomery' value as a 'String', showing its individual 87 -- 'Limb's. 88 -- 89 -- >>> putStrLn (render 1) 90 -- (4294968273, 0, 0, 0) 91 render :: Montgomery -> String 92 render (Montgomery (L4 a b c d)) = 93 "(" <> show (W# a) <> ", " <> show (W# b) <> ", " 94 <> show (W# c) <> ", " <> show (W# d) <> ")" 95 96 instance Show Montgomery where 97 show = show . from 98 99 -- | Note that 'fromInteger' necessarily runs in variable time due 100 -- to conversion from the variable-size, potentially heap-allocated 101 -- 'Integer' type. 102 instance Num Montgomery where 103 a + b = add a b 104 a - b = sub a b 105 a * b = mul a b 106 negate a = neg a 107 abs = id 108 fromInteger = to . WW.to_vartime 109 signum (Montgomery (# l0, l1, l2, l3 #)) = 110 let !(Limb l) = l0 `L.or#` l1 `L.or#` l2 `L.or#` l3 111 !n = C.from_word_nonzero# l 112 !b = C.to_word# n 113 in Montgomery (# Limb b, Limb 0##, Limb 0##, Limb 0## #) 114 115 instance NFData Montgomery where 116 rnf (Montgomery a) = case a of (# _, _, _, _ #) -> () 117 118 -- utilities ------------------------------------------------------------------ 119 120 type Limb2 = (# Limb, Limb #) 121 122 type Limb4 = (# Limb, Limb, Limb, Limb #) 123 124 pattern L4 :: Word# -> Word# -> Word# -> Word# -> Limb4 125 pattern L4 w0 w1 w2 w3 = (# Limb w0, Limb w1, Limb w2, Limb w3 #) 126 {-# COMPLETE L4 #-} 127 128 -- Wide wrapping addition, when addend is only a limb. 129 wadd_w# :: Limb2 -> Limb -> Limb2 130 wadd_w# (# x_lo, x_hi #) y_lo = 131 let !(# s0, c0 #) = L.add_o# x_lo y_lo 132 !(# s1, _ #) = L.add_o# x_hi c0 133 in (# s0, s1 #) 134 {-# INLINE wadd_w# #-} 135 136 -- Truncate a wide word to a 'Limb'. 137 lo :: Limb2 -> Limb 138 lo (# l, _ #) = l 139 {-# INLINE lo #-} 140 141 -- comparison ----------------------------------------------------------------- 142 143 -- | Constant-time equality comparison. 144 eq :: Montgomery -> Montgomery -> C.Choice 145 eq (Montgomery (L4 a0 a1 a2 a3)) (Montgomery (L4 b0 b1 b2 b3)) = 146 C.eq_wider# (# a0, a1, a2, a3 #) (# b0, b1, b2, b3 #) 147 {-# INLINE eq #-} 148 149 -- | Variable-time equality comparison. 150 eq_vartime :: Montgomery -> Montgomery -> Bool 151 eq_vartime (Montgomery (Wider -> a)) (Montgomery (Wider -> b)) = 152 WW.eq_vartime a b 153 154 -- innards -------------------------------------------------------------------- 155 156 redc_inner# 157 :: Limb4 -- ^ upper limbs 158 -> Limb4 -- ^ lower limbs 159 -> (# Limb4, Limb #) -- ^ upper limbs, meta-carry 160 redc_inner# (# u0, u1, u2, u3 #) (# l0, l1, l2, l3 #) = 161 let !(# m0, m1, m2, m3 #) = 162 (# Limb 0xFFFFFFFEFFFFFC2F##, Limb 0xFFFFFFFFFFFFFFFF## 163 , Limb 0xFFFFFFFFFFFFFFFF##, Limb 0xFFFFFFFFFFFFFFFF## #) 164 !n = Limb 0xD838091DD2253531## 165 !w_0 = L.mul_w# l0 n 166 !(# _, c_00 #) = L.mac# w_0 m0 l0 (Limb 0##) 167 !(# l0_1, c_01 #) = L.mac# w_0 m1 l1 c_00 168 !(# l0_2, c_02 #) = L.mac# w_0 m2 l2 c_01 169 !(# l0_3, c_03 #) = L.mac# w_0 m3 l3 c_02 170 !(# u_0, mc_0 #) = L.add_c# u0 c_03 (Limb 0##) 171 !w_1 = L.mul_w# l0_1 n 172 !(# _, c_10 #) = L.mac# w_1 m0 l0_1 (Limb 0##) 173 !(# l1_1, c_11 #) = L.mac# w_1 m1 l0_2 c_10 174 !(# l1_2, c_12 #) = L.mac# w_1 m2 l0_3 c_11 175 !(# u1_3, c_13 #) = L.mac# w_1 m3 u_0 c_12 176 !(# u_1, mc_1 #) = L.add_c# u1 c_13 mc_0 177 !w_2 = L.mul_w# l1_1 n 178 !(# _, c_20 #) = L.mac# w_2 m0 l1_1 (Limb 0##) 179 !(# l2_1, c_21 #) = L.mac# w_2 m1 l1_2 c_20 180 !(# u2_2, c_22 #) = L.mac# w_2 m2 u1_3 c_21 181 !(# u2_3, c_23 #) = L.mac# w_2 m3 u_1 c_22 182 !(# u_2, mc_2 #) = L.add_c# u2 c_23 mc_1 183 !w_3 = L.mul_w# l2_1 n 184 !(# _, c_30 #) = L.mac# w_3 m0 l2_1 (Limb 0##) 185 !(# u3_1, c_31 #) = L.mac# w_3 m1 u2_2 c_30 186 !(# u3_2, c_32 #) = L.mac# w_3 m2 u2_3 c_31 187 !(# u3_3, c_33 #) = L.mac# w_3 m3 u_2 c_32 188 !(# u_3, mc_3 #) = L.add_c# u3 c_33 mc_2 189 in (# (# u3_1, u3_2, u3_3, u_3 #), mc_3 #) 190 {-# INLINE redc_inner# #-} 191 192 -- | Montgomery reduction. 193 redc# 194 :: Limb4 -- ^ lower limbs 195 -> Limb4 -- ^ upper limbs 196 -> Limb4 -- ^ result 197 redc# l u = 198 let -- field prime 199 !m = L4 0xFFFFFFFEFFFFFC2F## 0xFFFFFFFFFFFFFFFF## 200 0xFFFFFFFFFFFFFFFF## 0xFFFFFFFFFFFFFFFF## 201 !(# nu, mc #) = redc_inner# u l 202 in WW.sub_mod_c# nu mc m m 203 {-# INLINE redc# #-} 204 205 -- | Montgomery reduction. 206 -- 207 -- The first argument represents the low words, and the second the 208 -- high words, of an extra-large eight-limb word in Montgomery form. 209 redc 210 :: Montgomery -- ^ low wider-word, Montgomery form 211 -> Montgomery -- ^ high wider-word, Montgomery form 212 -> Montgomery -- ^ reduced value 213 redc (Montgomery l) (Montgomery u) = 214 let !res = redc# l u 215 in Montgomery res 216 217 retr_inner# 218 :: Limb4 -- ^ value in montgomery form 219 -> Limb4 -- ^ retrieved value 220 retr_inner# (# x0, x1, x2, x3 #) = 221 let !(# m0, m1, m2, m3 #) = 222 L4 0xFFFFFFFEFFFFFC2F## 0xFFFFFFFFFFFFFFFF## 223 0xFFFFFFFFFFFFFFFF## 0xFFFFFFFFFFFFFFFF## 224 !n = Limb 0xD838091DD2253531## 225 !u_0 = L.mul_w# x0 n 226 !(# _, o0 #) = L.mac# u_0 m0 x0 (Limb 0##) 227 !(# o0_1, p0_1 #) = L.mac# u_0 m1 (Limb 0##) o0 228 !(# p0_2, q0_2 #) = L.mac# u_0 m2 (Limb 0##) p0_1 229 !(# q0_3, r0_3 #) = L.mac# u_0 m3 (Limb 0##) q0_2 230 !u_1 = L.mul_w# (L.add_w# o0_1 x1) n 231 !(# _, o1 #) = L.mac# u_1 m0 x1 o0_1 232 !(# o1_1, p1_1 #) = L.mac# u_1 m1 p0_2 o1 233 !(# p1_2, q1_2 #) = L.mac# u_1 m2 q0_3 p1_1 234 !(# q1_3, r1_3 #) = L.mac# u_1 m3 r0_3 q1_2 235 !u_2 = L.mul_w# (L.add_w# o1_1 x2) n 236 !(# _, o2 #) = L.mac# u_2 m0 x2 o1_1 237 !(# o2_1, p2_1 #) = L.mac# u_2 m1 p1_2 o2 238 !(# p2_2, q2_2 #) = L.mac# u_2 m2 q1_3 p2_1 239 !(# q2_3, r2_3 #) = L.mac# u_2 m3 r1_3 q2_2 240 !u_3 = L.mul_w# (L.add_w# o2_1 x3) n 241 !(# _, o3 #) = L.mac# u_3 m0 x3 o2_1 242 !(# o3_1, p3_1 #) = L.mac# u_3 m1 p2_2 o3 243 !(# p3_2, q3_2 #) = L.mac# u_3 m2 q2_3 p3_1 244 !(# q3_3, r3_3 #) = L.mac# u_3 m3 r2_3 q3_2 245 in (# o3_1, p3_2, q3_3, r3_3 #) 246 {-# INLINE retr_inner# #-} 247 248 retr# 249 :: Limb4 -- montgomery form 250 -> Limb4 251 retr# f = retr_inner# f 252 {-# INLINE retr# #-} 253 254 -- | Retrieve a 'Montgomery' value from the Montgomery domain, producing 255 -- a 'Wider' word. 256 retr 257 :: Montgomery -- ^ value in montgomery form 258 -> Wider -- ^ retrieved value 259 retr (Montgomery f) = 260 let !res = retr# f 261 in (Wider res) 262 263 -- | Montgomery multiplication (FIOS), without conditional subtract. 264 mul_inner# 265 :: Limb4 -- ^ x 266 -> Limb4 -- ^ y 267 -> (# Limb4, Limb #) -- ^ product, meta-carry 268 mul_inner# (# x0, x1, x2, x3 #) (# y0, y1, y2, y3 #) = 269 let !(# m0, m1, m2, m3 #) = 270 L4 0xFFFFFFFEFFFFFC2F## 0xFFFFFFFFFFFFFFFF## 271 0xFFFFFFFFFFFFFFFF## 0xFFFFFFFFFFFFFFFF## 272 !n = Limb 0xD838091DD2253531## 273 !axy0 = L.mul_c# x0 y0 274 !u0 = L.mul_w# (lo axy0) n 275 !(# (# _, a0 #), c0 #) = W.add_o# (L.mul_c# u0 m0) axy0 276 !carry0 = (# a0, c0 #) 277 !axy0_1 = L.mul_c# x0 y1 278 !umc0_1 = W.add_w# (L.mul_c# u0 m1) carry0 279 !(# (# o0, ab0_1 #), c0_1 #) = W.add_o# axy0_1 umc0_1 280 !carry0_1 = (# ab0_1, c0_1 #) 281 !axy0_2 = L.mul_c# x0 y2 282 !umc0_2 = W.add_w# (L.mul_c# u0 m2) carry0_1 283 !(# (# p0, ab0_2 #), c0_2 #) = W.add_o# axy0_2 umc0_2 284 !carry0_2 = (# ab0_2, c0_2 #) 285 !axy0_3 = L.mul_c# x0 y3 286 !umc0_3 = W.add_w# (L.mul_c# u0 m3) carry0_2 287 !(# (# q0, ab0_3 #), c0_3 #) = W.add_o# axy0_3 umc0_3 288 !carry0_3 = (# ab0_3, c0_3 #) 289 !(# r0, mc0 #) = carry0_3 290 !axy1 = wadd_w# (L.mul_c# x1 y0) o0 291 !u1 = L.mul_w# (lo axy1) n 292 !(# (# _, a1 #), c1 #) = W.add_o# (L.mul_c# u1 m0) axy1 293 !carry1 = (# a1, c1 #) 294 !axy1_1 = wadd_w# (L.mul_c# x1 y1) p0 295 !umc1_1 = W.add_w# (L.mul_c# u1 m1) carry1 296 !(# (# o1, ab1_1 #), c1_1 #) = W.add_o# axy1_1 umc1_1 297 !carry1_1 = (# ab1_1, c1_1 #) 298 !axy1_2 = wadd_w# (L.mul_c# x1 y2) q0 299 !umc1_2 = W.add_w# (L.mul_c# u1 m2) carry1_1 300 !(# (# p1, ab1_2 #), c1_2 #) = W.add_o# axy1_2 umc1_2 301 !carry1_2 = (# ab1_2, c1_2 #) 302 !axy1_3 = wadd_w# (L.mul_c# x1 y3) r0 303 !umc1_3 = W.add_w# (L.mul_c# u1 m3) carry1_2 304 !(# (# q1, ab1_3 #), c1_3 #) = W.add_o# axy1_3 umc1_3 305 !carry1_3 = (# ab1_3, c1_3 #) 306 !(# r1, mc1 #) = wadd_w# carry1_3 mc0 307 !axy2 = wadd_w# (L.mul_c# x2 y0) o1 308 !u2 = L.mul_w# (lo axy2) n 309 !(# (# _, a2 #), c2 #) = W.add_o# (L.mul_c# u2 m0) axy2 310 !carry2 = (# a2, c2 #) 311 !axy2_1 = wadd_w# (L.mul_c# x2 y1) p1 312 !umc2_1 = W.add_w# (L.mul_c# u2 m1) carry2 313 !(# (# o2, ab2_1 #), c2_1 #) = W.add_o# axy2_1 umc2_1 314 !carry2_1 = (# ab2_1, c2_1 #) 315 !axy2_2 = wadd_w# (L.mul_c# x2 y2) q1 316 !umc2_2 = W.add_w# (L.mul_c# u2 m2) carry2_1 317 !(# (# p2, ab2_2 #), c2_2 #) = W.add_o# axy2_2 umc2_2 318 !carry2_2 = (# ab2_2, c2_2 #) 319 !axy2_3 = wadd_w# (L.mul_c# x2 y3) r1 320 !umc2_3 = W.add_w# (L.mul_c# u2 m3) carry2_2 321 !(# (# q2, ab2_3 #), c2_3 #) = W.add_o# axy2_3 umc2_3 322 !carry2_3 = (# ab2_3, c2_3 #) 323 !(# r2, mc2 #) = wadd_w# carry2_3 mc1 324 !axy3 = wadd_w# (L.mul_c# x3 y0) o2 325 !u3 = L.mul_w# (lo axy3) n 326 !(# (# _, a3 #), c3 #) = W.add_o# (L.mul_c# u3 m0) axy3 327 !carry3 = (# a3, c3 #) 328 !axy3_1 = wadd_w# (L.mul_c# x3 y1) p2 329 !umc3_1 = W.add_w# (L.mul_c# u3 m1) carry3 330 !(# (# o3, ab3_1 #), c3_1 #) = W.add_o# axy3_1 umc3_1 331 !carry3_1 = (# ab3_1, c3_1 #) 332 !axy3_2 = wadd_w# (L.mul_c# x3 y2) q2 333 !umc3_2 = W.add_w# (L.mul_c# u3 m2) carry3_1 334 !(# (# p3, ab3_2 #), c3_2 #) = W.add_o# axy3_2 umc3_2 335 !carry3_2 = (# ab3_2, c3_2 #) 336 !axy3_3 = wadd_w# (L.mul_c# x3 y3) r2 337 !umc3_3 = W.add_w# (L.mul_c# u3 m3) carry3_2 338 !(# (# q3, ab3_3 #), c3_3 #) = W.add_o# axy3_3 umc3_3 339 !carry3_3 = (# ab3_3, c3_3 #) 340 !(# r3, mc3 #) = wadd_w# carry3_3 mc2 341 in (# (# o3, p3, q3, r3 #), mc3 #) 342 {-# INLINE mul_inner# #-} 343 344 mul# 345 :: Limb4 346 -> Limb4 347 -> Limb4 348 mul# a b = 349 let -- field prime 350 !m = L4 0xFFFFFFFEFFFFFC2F## 0xFFFFFFFFFFFFFFFF## 351 0xFFFFFFFFFFFFFFFF## 0xFFFFFFFFFFFFFFFF## 352 !(# nu, mc #) = mul_inner# a b 353 in WW.sub_mod_c# nu mc m m 354 {-# NOINLINE mul# #-} -- cannot be inlined without exploding comp time 355 356 -- | Multiplication in the Montgomery domain. 357 -- 358 -- Note that 'Montgomery' is an instance of 'Num', so you can use '*' 359 -- to apply this function. 360 -- 361 -- >>> 1 * 1 :: Montgomery 362 -- 1 363 mul 364 :: Montgomery -- ^ multiplicand in montgomery form 365 -> Montgomery -- ^ multiplier in montgomery form 366 -> Montgomery -- ^ montgomery product 367 mul (Montgomery a) (Montgomery b) = Montgomery (mul# a b) 368 369 to# 370 :: Limb4 -- ^ integer 371 -> Limb4 372 to# x = 373 let !r2 = L4 0x000007A2000E90A1## 0x1## 0## 0## -- r^2 mod m 374 in mul# x r2 375 {-# INLINE to# #-} 376 377 -- | Convert a 'Wider' word to the Montgomery domain. 378 to :: Wider -> Montgomery 379 to (Wider x) = Montgomery (to# x) 380 381 -- | Retrieve a 'Montgomery' word from the Montgomery domain. 382 -- 383 -- This function is a synonym for 'retr'. 384 from :: Montgomery -> Wider 385 from = retr 386 387 add# 388 :: Limb4 -- ^ augend 389 -> Limb4 -- ^ addend 390 -> Limb4 -- ^ sum 391 add# a b = 392 let -- field prime 393 !m = L4 0xFFFFFFFEFFFFFC2F## 0xFFFFFFFFFFFFFFFF## 394 0xFFFFFFFFFFFFFFFF## 0xFFFFFFFFFFFFFFFF## 395 in WW.add_mod# a b m 396 {-# INLINE add# #-} 397 398 -- | Addition in the Montgomery domain. 399 -- 400 -- Note that 'Montgomery' is an instance of 'Num', so you can use '+' 401 -- to apply this function. 402 -- 403 -- >>> 1 + 1 :: Montgomery 404 -- 2 405 add :: Montgomery -> Montgomery -> Montgomery 406 add (Montgomery a) (Montgomery b) = Montgomery (add# a b) 407 408 sub# 409 :: Limb4 -- ^ minuend 410 -> Limb4 -- ^ subtrahend 411 -> Limb4 -- ^ difference 412 sub# a b = 413 let -- field prime 414 !m = L4 0xFFFFFFFEFFFFFC2F## 0xFFFFFFFFFFFFFFFF## 415 0xFFFFFFFFFFFFFFFF## 0xFFFFFFFFFFFFFFFF## 416 in WW.sub_mod# a b m 417 {-# INLINE sub# #-} 418 419 -- | Subtraction in the Montgomery domain. 420 -- 421 -- Note that 'Montgomery' is an instance of 'Num', so you can use '-' 422 -- to apply this function. 423 -- 424 -- >>> 1 - 1 :: Montgomery 425 -- 0 426 sub :: Montgomery -> Montgomery -> Montgomery 427 sub (Montgomery a) (Montgomery b) = Montgomery (sub# a b) 428 429 neg# 430 :: Limb4 -- ^ argument 431 -> Limb4 -- ^ modular negation 432 neg# a = sub# (L4 0## 0## 0## 0##) a 433 {-# INLINE neg# #-} 434 435 -- | Additive inverse in the Montgomery domain. 436 -- 437 -- Note that 'Montgomery' is an instance of 'Num', so you can use 'negate' 438 -- to apply this function. 439 -- 440 -- >>> negate 1 :: Montgomery 441 -- 115792089237316195423570985008687907853269984665640564039457584007908834671662 442 -- >>> (negate 1 :: Montgomery) + 1 443 -- 0 444 neg :: Montgomery -> Montgomery 445 neg (Montgomery a) = Montgomery (neg# a) 446 447 sqr# :: Limb4 -> Limb4 448 sqr# a = 449 let !(# l, h #) = WW.sqr# a 450 in redc# l h 451 {-# NOINLINE sqr# #-} -- cannot be inlined without exploding comp time 452 453 -- | Squaring in the Montgomery domain. 454 -- 455 -- >>> sqr 1 456 -- 1 457 -- >>> sqr 2 458 -- 4 459 -- >>> sqr (negate 2) 460 -- 4 461 sqr :: Montgomery -> Montgomery 462 sqr (Montgomery a) = Montgomery (mul# a a) 463 464 -- | Zero (the additive unit) in the Montgomery domain. 465 zero :: Montgomery 466 zero = Montgomery (L4 0## 0## 0## 0##) 467 468 -- | One (the multiplicative unit) in the Montgomery domain. 469 one :: Montgomery 470 one = Montgomery (L4 0x1000003D1## 0## 0## 0##) 471 472 -- generated by etc/generate_inv.sh 473 inv# 474 :: Limb4 475 -> Limb4 476 inv# a = 477 let -- montgomery 'one' 478 !t0 = L4 0x1000003D1## 0## 0## 0## 479 !t1 = sqr# t0 480 !t2 = mul# a t1 481 !t3 = sqr# t2 482 !t4 = mul# a t3 483 !t5 = sqr# t4 484 !t6 = mul# a t5 485 !t7 = sqr# t6 486 !t8 = mul# a t7 487 !t9 = sqr# t8 488 !t10 = mul# a t9 489 !t11 = sqr# t10 490 !t12 = mul# a t11 491 !t13 = sqr# t12 492 !t14 = mul# a t13 493 !t15 = sqr# t14 494 !t16 = mul# a t15 495 !t17 = sqr# t16 496 !t18 = mul# a t17 497 !t19 = sqr# t18 498 !t20 = mul# a t19 499 !t21 = sqr# t20 500 !t22 = mul# a t21 501 !t23 = sqr# t22 502 !t24 = mul# a t23 503 !t25 = sqr# t24 504 !t26 = mul# a t25 505 !t27 = sqr# t26 506 !t28 = mul# a t27 507 !t29 = sqr# t28 508 !t30 = mul# a t29 509 !t31 = sqr# t30 510 !t32 = mul# a t31 511 !t33 = sqr# t32 512 !t34 = mul# a t33 513 !t35 = sqr# t34 514 !t36 = mul# a t35 515 !t37 = sqr# t36 516 !t38 = mul# a t37 517 !t39 = sqr# t38 518 !t40 = mul# a t39 519 !t41 = sqr# t40 520 !t42 = mul# a t41 521 !t43 = sqr# t42 522 !t44 = mul# a t43 523 !t45 = sqr# t44 524 !t46 = mul# a t45 525 !t47 = sqr# t46 526 !t48 = mul# a t47 527 !t49 = sqr# t48 528 !t50 = mul# a t49 529 !t51 = sqr# t50 530 !t52 = mul# a t51 531 !t53 = sqr# t52 532 !t54 = mul# a t53 533 !t55 = sqr# t54 534 !t56 = mul# a t55 535 !t57 = sqr# t56 536 !t58 = mul# a t57 537 !t59 = sqr# t58 538 !t60 = mul# a t59 539 !t61 = sqr# t60 540 !t62 = mul# a t61 541 !t63 = sqr# t62 542 !t64 = mul# a t63 543 !t65 = sqr# t64 544 !t66 = mul# a t65 545 !t67 = sqr# t66 546 !t68 = mul# a t67 547 !t69 = sqr# t68 548 !t70 = mul# a t69 549 !t71 = sqr# t70 550 !t72 = mul# a t71 551 !t73 = sqr# t72 552 !t74 = mul# a t73 553 !t75 = sqr# t74 554 !t76 = mul# a t75 555 !t77 = sqr# t76 556 !t78 = mul# a t77 557 !t79 = sqr# t78 558 !t80 = mul# a t79 559 !t81 = sqr# t80 560 !t82 = mul# a t81 561 !t83 = sqr# t82 562 !t84 = mul# a t83 563 !t85 = sqr# t84 564 !t86 = mul# a t85 565 !t87 = sqr# t86 566 !t88 = mul# a t87 567 !t89 = sqr# t88 568 !t90 = mul# a t89 569 !t91 = sqr# t90 570 !t92 = mul# a t91 571 !t93 = sqr# t92 572 !t94 = mul# a t93 573 !t95 = sqr# t94 574 !t96 = mul# a t95 575 !t97 = sqr# t96 576 !t98 = mul# a t97 577 !t99 = sqr# t98 578 !t100 = mul# a t99 579 !t101 = sqr# t100 580 !t102 = mul# a t101 581 !t103 = sqr# t102 582 !t104 = mul# a t103 583 !t105 = sqr# t104 584 !t106 = mul# a t105 585 !t107 = sqr# t106 586 !t108 = mul# a t107 587 !t109 = sqr# t108 588 !t110 = mul# a t109 589 !t111 = sqr# t110 590 !t112 = mul# a t111 591 !t113 = sqr# t112 592 !t114 = mul# a t113 593 !t115 = sqr# t114 594 !t116 = mul# a t115 595 !t117 = sqr# t116 596 !t118 = mul# a t117 597 !t119 = sqr# t118 598 !t120 = mul# a t119 599 !t121 = sqr# t120 600 !t122 = mul# a t121 601 !t123 = sqr# t122 602 !t124 = mul# a t123 603 !t125 = sqr# t124 604 !t126 = mul# a t125 605 !t127 = sqr# t126 606 !t128 = mul# a t127 607 !t129 = sqr# t128 608 !t130 = mul# a t129 609 !t131 = sqr# t130 610 !t132 = mul# a t131 611 !t133 = sqr# t132 612 !t134 = mul# a t133 613 !t135 = sqr# t134 614 !t136 = mul# a t135 615 !t137 = sqr# t136 616 !t138 = mul# a t137 617 !t139 = sqr# t138 618 !t140 = mul# a t139 619 !t141 = sqr# t140 620 !t142 = mul# a t141 621 !t143 = sqr# t142 622 !t144 = mul# a t143 623 !t145 = sqr# t144 624 !t146 = mul# a t145 625 !t147 = sqr# t146 626 !t148 = mul# a t147 627 !t149 = sqr# t148 628 !t150 = mul# a t149 629 !t151 = sqr# t150 630 !t152 = mul# a t151 631 !t153 = sqr# t152 632 !t154 = mul# a t153 633 !t155 = sqr# t154 634 !t156 = mul# a t155 635 !t157 = sqr# t156 636 !t158 = mul# a t157 637 !t159 = sqr# t158 638 !t160 = mul# a t159 639 !t161 = sqr# t160 640 !t162 = mul# a t161 641 !t163 = sqr# t162 642 !t164 = mul# a t163 643 !t165 = sqr# t164 644 !t166 = mul# a t165 645 !t167 = sqr# t166 646 !t168 = mul# a t167 647 !t169 = sqr# t168 648 !t170 = mul# a t169 649 !t171 = sqr# t170 650 !t172 = mul# a t171 651 !t173 = sqr# t172 652 !t174 = mul# a t173 653 !t175 = sqr# t174 654 !t176 = mul# a t175 655 !t177 = sqr# t176 656 !t178 = mul# a t177 657 !t179 = sqr# t178 658 !t180 = mul# a t179 659 !t181 = sqr# t180 660 !t182 = mul# a t181 661 !t183 = sqr# t182 662 !t184 = mul# a t183 663 !t185 = sqr# t184 664 !t186 = mul# a t185 665 !t187 = sqr# t186 666 !t188 = mul# a t187 667 !t189 = sqr# t188 668 !t190 = mul# a t189 669 !t191 = sqr# t190 670 !t192 = mul# a t191 671 !t193 = sqr# t192 672 !t194 = mul# a t193 673 !t195 = sqr# t194 674 !t196 = mul# a t195 675 !t197 = sqr# t196 676 !t198 = mul# a t197 677 !t199 = sqr# t198 678 !t200 = mul# a t199 679 !t201 = sqr# t200 680 !t202 = mul# a t201 681 !t203 = sqr# t202 682 !t204 = mul# a t203 683 !t205 = sqr# t204 684 !t206 = mul# a t205 685 !t207 = sqr# t206 686 !t208 = mul# a t207 687 !t209 = sqr# t208 688 !t210 = mul# a t209 689 !t211 = sqr# t210 690 !t212 = mul# a t211 691 !t213 = sqr# t212 692 !t214 = mul# a t213 693 !t215 = sqr# t214 694 !t216 = mul# a t215 695 !t217 = sqr# t216 696 !t218 = mul# a t217 697 !t219 = sqr# t218 698 !t220 = mul# a t219 699 !t221 = sqr# t220 700 !t222 = mul# a t221 701 !t223 = sqr# t222 702 !t224 = mul# a t223 703 !t225 = sqr# t224 704 !t226 = mul# a t225 705 !t227 = sqr# t226 706 !t228 = mul# a t227 707 !t229 = sqr# t228 708 !t230 = mul# a t229 709 !t231 = sqr# t230 710 !t232 = mul# a t231 711 !t233 = sqr# t232 712 !t234 = mul# a t233 713 !t235 = sqr# t234 714 !t236 = mul# a t235 715 !t237 = sqr# t236 716 !t238 = mul# a t237 717 !t239 = sqr# t238 718 !t240 = mul# a t239 719 !t241 = sqr# t240 720 !t242 = mul# a t241 721 !t243 = sqr# t242 722 !t244 = mul# a t243 723 !t245 = sqr# t244 724 !t246 = mul# a t245 725 !t247 = sqr# t246 726 !t248 = mul# a t247 727 !t249 = sqr# t248 728 !t250 = mul# a t249 729 !t251 = sqr# t250 730 !t252 = mul# a t251 731 !t253 = sqr# t252 732 !t254 = mul# a t253 733 !t255 = sqr# t254 734 !t256 = mul# a t255 735 !t257 = sqr# t256 736 !t258 = mul# a t257 737 !t259 = sqr# t258 738 !t260 = mul# a t259 739 !t261 = sqr# t260 740 !t262 = mul# a t261 741 !t263 = sqr# t262 742 !t264 = mul# a t263 743 !t265 = sqr# t264 744 !t266 = mul# a t265 745 !t267 = sqr# t266 746 !t268 = mul# a t267 747 !t269 = sqr# t268 748 !t270 = mul# a t269 749 !t271 = sqr# t270 750 !t272 = mul# a t271 751 !t273 = sqr# t272 752 !t274 = mul# a t273 753 !t275 = sqr# t274 754 !t276 = mul# a t275 755 !t277 = sqr# t276 756 !t278 = mul# a t277 757 !t279 = sqr# t278 758 !t280 = mul# a t279 759 !t281 = sqr# t280 760 !t282 = mul# a t281 761 !t283 = sqr# t282 762 !t284 = mul# a t283 763 !t285 = sqr# t284 764 !t286 = mul# a t285 765 !t287 = sqr# t286 766 !t288 = mul# a t287 767 !t289 = sqr# t288 768 !t290 = mul# a t289 769 !t291 = sqr# t290 770 !t292 = mul# a t291 771 !t293 = sqr# t292 772 !t294 = mul# a t293 773 !t295 = sqr# t294 774 !t296 = mul# a t295 775 !t297 = sqr# t296 776 !t298 = mul# a t297 777 !t299 = sqr# t298 778 !t300 = mul# a t299 779 !t301 = sqr# t300 780 !t302 = mul# a t301 781 !t303 = sqr# t302 782 !t304 = mul# a t303 783 !t305 = sqr# t304 784 !t306 = mul# a t305 785 !t307 = sqr# t306 786 !t308 = mul# a t307 787 !t309 = sqr# t308 788 !t310 = mul# a t309 789 !t311 = sqr# t310 790 !t312 = mul# a t311 791 !t313 = sqr# t312 792 !t314 = mul# a t313 793 !t315 = sqr# t314 794 !t316 = mul# a t315 795 !t317 = sqr# t316 796 !t318 = mul# a t317 797 !t319 = sqr# t318 798 !t320 = mul# a t319 799 !t321 = sqr# t320 800 !t322 = mul# a t321 801 !t323 = sqr# t322 802 !t324 = mul# a t323 803 !t325 = sqr# t324 804 !t326 = mul# a t325 805 !t327 = sqr# t326 806 !t328 = mul# a t327 807 !t329 = sqr# t328 808 !t330 = mul# a t329 809 !t331 = sqr# t330 810 !t332 = mul# a t331 811 !t333 = sqr# t332 812 !t334 = mul# a t333 813 !t335 = sqr# t334 814 !t336 = mul# a t335 815 !t337 = sqr# t336 816 !t338 = mul# a t337 817 !t339 = sqr# t338 818 !t340 = mul# a t339 819 !t341 = sqr# t340 820 !t342 = mul# a t341 821 !t343 = sqr# t342 822 !t344 = mul# a t343 823 !t345 = sqr# t344 824 !t346 = mul# a t345 825 !t347 = sqr# t346 826 !t348 = mul# a t347 827 !t349 = sqr# t348 828 !t350 = mul# a t349 829 !t351 = sqr# t350 830 !t352 = mul# a t351 831 !t353 = sqr# t352 832 !t354 = mul# a t353 833 !t355 = sqr# t354 834 !t356 = mul# a t355 835 !t357 = sqr# t356 836 !t358 = mul# a t357 837 !t359 = sqr# t358 838 !t360 = mul# a t359 839 !t361 = sqr# t360 840 !t362 = mul# a t361 841 !t363 = sqr# t362 842 !t364 = mul# a t363 843 !t365 = sqr# t364 844 !t366 = mul# a t365 845 !t367 = sqr# t366 846 !t368 = mul# a t367 847 !t369 = sqr# t368 848 !t370 = mul# a t369 849 !t371 = sqr# t370 850 !t372 = mul# a t371 851 !t373 = sqr# t372 852 !t374 = mul# a t373 853 !t375 = sqr# t374 854 !t376 = mul# a t375 855 !t377 = sqr# t376 856 !t378 = mul# a t377 857 !t379 = sqr# t378 858 !t380 = mul# a t379 859 !t381 = sqr# t380 860 !t382 = mul# a t381 861 !t383 = sqr# t382 862 !t384 = mul# a t383 863 !t385 = sqr# t384 864 !t386 = mul# a t385 865 !t387 = sqr# t386 866 !t388 = mul# a t387 867 !t389 = sqr# t388 868 !t390 = mul# a t389 869 !t391 = sqr# t390 870 !t392 = mul# a t391 871 !t393 = sqr# t392 872 !t394 = mul# a t393 873 !t395 = sqr# t394 874 !t396 = mul# a t395 875 !t397 = sqr# t396 876 !t398 = mul# a t397 877 !t399 = sqr# t398 878 !t400 = mul# a t399 879 !t401 = sqr# t400 880 !t402 = mul# a t401 881 !t403 = sqr# t402 882 !t404 = mul# a t403 883 !t405 = sqr# t404 884 !t406 = mul# a t405 885 !t407 = sqr# t406 886 !t408 = mul# a t407 887 !t409 = sqr# t408 888 !t410 = mul# a t409 889 !t411 = sqr# t410 890 !t412 = mul# a t411 891 !t413 = sqr# t412 892 !t414 = mul# a t413 893 !t415 = sqr# t414 894 !t416 = mul# a t415 895 !t417 = sqr# t416 896 !t418 = mul# a t417 897 !t419 = sqr# t418 898 !t420 = mul# a t419 899 !t421 = sqr# t420 900 !t422 = mul# a t421 901 !t423 = sqr# t422 902 !t424 = mul# a t423 903 !t425 = sqr# t424 904 !t426 = mul# a t425 905 !t427 = sqr# t426 906 !t428 = mul# a t427 907 !t429 = sqr# t428 908 !t430 = mul# a t429 909 !t431 = sqr# t430 910 !t432 = mul# a t431 911 !t433 = sqr# t432 912 !t434 = mul# a t433 913 !t435 = sqr# t434 914 !t436 = mul# a t435 915 !t437 = sqr# t436 916 !t438 = mul# a t437 917 !t439 = sqr# t438 918 !t440 = mul# a t439 919 !t441 = sqr# t440 920 !t442 = mul# a t441 921 !t443 = sqr# t442 922 !t444 = mul# a t443 923 !t445 = sqr# t444 924 !t446 = mul# a t445 925 !t447 = sqr# t446 926 !t448 = sqr# t447 927 !t449 = mul# a t448 928 !t450 = sqr# t449 929 !t451 = mul# a t450 930 !t452 = sqr# t451 931 !t453 = mul# a t452 932 !t454 = sqr# t453 933 !t455 = mul# a t454 934 !t456 = sqr# t455 935 !t457 = mul# a t456 936 !t458 = sqr# t457 937 !t459 = mul# a t458 938 !t460 = sqr# t459 939 !t461 = mul# a t460 940 !t462 = sqr# t461 941 !t463 = mul# a t462 942 !t464 = sqr# t463 943 !t465 = mul# a t464 944 !t466 = sqr# t465 945 !t467 = mul# a t466 946 !t468 = sqr# t467 947 !t469 = mul# a t468 948 !t470 = sqr# t469 949 !t471 = mul# a t470 950 !t472 = sqr# t471 951 !t473 = mul# a t472 952 !t474 = sqr# t473 953 !t475 = mul# a t474 954 !t476 = sqr# t475 955 !t477 = mul# a t476 956 !t478 = sqr# t477 957 !t479 = mul# a t478 958 !t480 = sqr# t479 959 !t481 = mul# a t480 960 !t482 = sqr# t481 961 !t483 = mul# a t482 962 !t484 = sqr# t483 963 !t485 = mul# a t484 964 !t486 = sqr# t485 965 !t487 = mul# a t486 966 !t488 = sqr# t487 967 !t489 = mul# a t488 968 !t490 = sqr# t489 969 !t491 = mul# a t490 970 !t492 = sqr# t491 971 !t493 = sqr# t492 972 !t494 = sqr# t493 973 !t495 = sqr# t494 974 !t496 = sqr# t495 975 !t497 = mul# a t496 976 !t498 = sqr# t497 977 !t499 = sqr# t498 978 !t500 = mul# a t499 979 !t501 = sqr# t500 980 !t502 = mul# a t501 981 !t503 = sqr# t502 982 !t504 = sqr# t503 983 !t505 = mul# a t504 984 !r = t505 985 in r 986 {-# INLINE inv# #-} 987 988 -- | Multiplicative inverse in the Montgomery domain. 989 -- 990 -- >> inv 2 991 -- 57896044618658097711785492504343953926634992332820282019728792003954417335832 992 -- >> inv 2 * 2 993 -- 1 994 inv :: Montgomery -> Montgomery 995 inv (Montgomery w) = Montgomery (inv# w) 996 997 -- | Square root (Tonelli-Shanks) in the Montgomery domain. 998 -- 999 -- Returns 'Nothing' if the square root doesn't exist. 1000 -- 1001 -- Note that the square root calculation itself is performed in 1002 -- constant time; we branch only when casting to 'Maybe' at the end. 1003 -- 1004 -- >>> sqrt_vartime 4 1005 -- Just 2 1006 -- >>> sqrt_vartime 15 1007 -- Just 69211104694897500952317515077652022726490027694212560352756646854116994689233 1008 -- >>> (*) <$> sqrt_vartime 15 <*> sqrt_vartime 15 1009 -- Just 15 1010 sqrt_vartime :: Montgomery -> Maybe Montgomery 1011 sqrt_vartime (Montgomery n) = case sqrt# n of 1012 (# a, c #) 1013 | C.decide c -> Just $! Montgomery a 1014 | otherwise -> Nothing 1015 1016 -- generated by etc/generate_sqrt.sh 1017 sqrt# 1018 :: Limb4 1019 -> (# Limb4, C.Choice #) 1020 sqrt# a = 1021 let !t0 = L4 0x1000003D1## 0## 0## 0## 1022 !t1 = sqr# t0 1023 !t2 = sqr# t1 1024 !t3 = sqr# t2 1025 !t4 = mul# a t3 1026 !t5 = sqr# t4 1027 !t6 = mul# a t5 1028 !t7 = sqr# t6 1029 !t8 = mul# a t7 1030 !t9 = sqr# t8 1031 !t10 = mul# a t9 1032 !t11 = sqr# t10 1033 !t12 = mul# a t11 1034 !t13 = sqr# t12 1035 !t14 = mul# a t13 1036 !t15 = sqr# t14 1037 !t16 = mul# a t15 1038 !t17 = sqr# t16 1039 !t18 = mul# a t17 1040 !t19 = sqr# t18 1041 !t20 = mul# a t19 1042 !t21 = sqr# t20 1043 !t22 = mul# a t21 1044 !t23 = sqr# t22 1045 !t24 = mul# a t23 1046 !t25 = sqr# t24 1047 !t26 = mul# a t25 1048 !t27 = sqr# t26 1049 !t28 = mul# a t27 1050 !t29 = sqr# t28 1051 !t30 = mul# a t29 1052 !t31 = sqr# t30 1053 !t32 = mul# a t31 1054 !t33 = sqr# t32 1055 !t34 = mul# a t33 1056 !t35 = sqr# t34 1057 !t36 = mul# a t35 1058 !t37 = sqr# t36 1059 !t38 = mul# a t37 1060 !t39 = sqr# t38 1061 !t40 = mul# a t39 1062 !t41 = sqr# t40 1063 !t42 = mul# a t41 1064 !t43 = sqr# t42 1065 !t44 = mul# a t43 1066 !t45 = sqr# t44 1067 !t46 = mul# a t45 1068 !t47 = sqr# t46 1069 !t48 = mul# a t47 1070 !t49 = sqr# t48 1071 !t50 = mul# a t49 1072 !t51 = sqr# t50 1073 !t52 = mul# a t51 1074 !t53 = sqr# t52 1075 !t54 = mul# a t53 1076 !t55 = sqr# t54 1077 !t56 = mul# a t55 1078 !t57 = sqr# t56 1079 !t58 = mul# a t57 1080 !t59 = sqr# t58 1081 !t60 = mul# a t59 1082 !t61 = sqr# t60 1083 !t62 = mul# a t61 1084 !t63 = sqr# t62 1085 !t64 = mul# a t63 1086 !t65 = sqr# t64 1087 !t66 = mul# a t65 1088 !t67 = sqr# t66 1089 !t68 = mul# a t67 1090 !t69 = sqr# t68 1091 !t70 = mul# a t69 1092 !t71 = sqr# t70 1093 !t72 = mul# a t71 1094 !t73 = sqr# t72 1095 !t74 = mul# a t73 1096 !t75 = sqr# t74 1097 !t76 = mul# a t75 1098 !t77 = sqr# t76 1099 !t78 = mul# a t77 1100 !t79 = sqr# t78 1101 !t80 = mul# a t79 1102 !t81 = sqr# t80 1103 !t82 = mul# a t81 1104 !t83 = sqr# t82 1105 !t84 = mul# a t83 1106 !t85 = sqr# t84 1107 !t86 = mul# a t85 1108 !t87 = sqr# t86 1109 !t88 = mul# a t87 1110 !t89 = sqr# t88 1111 !t90 = mul# a t89 1112 !t91 = sqr# t90 1113 !t92 = mul# a t91 1114 !t93 = sqr# t92 1115 !t94 = mul# a t93 1116 !t95 = sqr# t94 1117 !t96 = mul# a t95 1118 !t97 = sqr# t96 1119 !t98 = mul# a t97 1120 !t99 = sqr# t98 1121 !t100 = mul# a t99 1122 !t101 = sqr# t100 1123 !t102 = mul# a t101 1124 !t103 = sqr# t102 1125 !t104 = mul# a t103 1126 !t105 = sqr# t104 1127 !t106 = mul# a t105 1128 !t107 = sqr# t106 1129 !t108 = mul# a t107 1130 !t109 = sqr# t108 1131 !t110 = mul# a t109 1132 !t111 = sqr# t110 1133 !t112 = mul# a t111 1134 !t113 = sqr# t112 1135 !t114 = mul# a t113 1136 !t115 = sqr# t114 1137 !t116 = mul# a t115 1138 !t117 = sqr# t116 1139 !t118 = mul# a t117 1140 !t119 = sqr# t118 1141 !t120 = mul# a t119 1142 !t121 = sqr# t120 1143 !t122 = mul# a t121 1144 !t123 = sqr# t122 1145 !t124 = mul# a t123 1146 !t125 = sqr# t124 1147 !t126 = mul# a t125 1148 !t127 = sqr# t126 1149 !t128 = mul# a t127 1150 !t129 = sqr# t128 1151 !t130 = mul# a t129 1152 !t131 = sqr# t130 1153 !t132 = mul# a t131 1154 !t133 = sqr# t132 1155 !t134 = mul# a t133 1156 !t135 = sqr# t134 1157 !t136 = mul# a t135 1158 !t137 = sqr# t136 1159 !t138 = mul# a t137 1160 !t139 = sqr# t138 1161 !t140 = mul# a t139 1162 !t141 = sqr# t140 1163 !t142 = mul# a t141 1164 !t143 = sqr# t142 1165 !t144 = mul# a t143 1166 !t145 = sqr# t144 1167 !t146 = mul# a t145 1168 !t147 = sqr# t146 1169 !t148 = mul# a t147 1170 !t149 = sqr# t148 1171 !t150 = mul# a t149 1172 !t151 = sqr# t150 1173 !t152 = mul# a t151 1174 !t153 = sqr# t152 1175 !t154 = mul# a t153 1176 !t155 = sqr# t154 1177 !t156 = mul# a t155 1178 !t157 = sqr# t156 1179 !t158 = mul# a t157 1180 !t159 = sqr# t158 1181 !t160 = mul# a t159 1182 !t161 = sqr# t160 1183 !t162 = mul# a t161 1184 !t163 = sqr# t162 1185 !t164 = mul# a t163 1186 !t165 = sqr# t164 1187 !t166 = mul# a t165 1188 !t167 = sqr# t166 1189 !t168 = mul# a t167 1190 !t169 = sqr# t168 1191 !t170 = mul# a t169 1192 !t171 = sqr# t170 1193 !t172 = mul# a t171 1194 !t173 = sqr# t172 1195 !t174 = mul# a t173 1196 !t175 = sqr# t174 1197 !t176 = mul# a t175 1198 !t177 = sqr# t176 1199 !t178 = mul# a t177 1200 !t179 = sqr# t178 1201 !t180 = mul# a t179 1202 !t181 = sqr# t180 1203 !t182 = mul# a t181 1204 !t183 = sqr# t182 1205 !t184 = mul# a t183 1206 !t185 = sqr# t184 1207 !t186 = mul# a t185 1208 !t187 = sqr# t186 1209 !t188 = mul# a t187 1210 !t189 = sqr# t188 1211 !t190 = mul# a t189 1212 !t191 = sqr# t190 1213 !t192 = mul# a t191 1214 !t193 = sqr# t192 1215 !t194 = mul# a t193 1216 !t195 = sqr# t194 1217 !t196 = mul# a t195 1218 !t197 = sqr# t196 1219 !t198 = mul# a t197 1220 !t199 = sqr# t198 1221 !t200 = mul# a t199 1222 !t201 = sqr# t200 1223 !t202 = mul# a t201 1224 !t203 = sqr# t202 1225 !t204 = mul# a t203 1226 !t205 = sqr# t204 1227 !t206 = mul# a t205 1228 !t207 = sqr# t206 1229 !t208 = mul# a t207 1230 !t209 = sqr# t208 1231 !t210 = mul# a t209 1232 !t211 = sqr# t210 1233 !t212 = mul# a t211 1234 !t213 = sqr# t212 1235 !t214 = mul# a t213 1236 !t215 = sqr# t214 1237 !t216 = mul# a t215 1238 !t217 = sqr# t216 1239 !t218 = mul# a t217 1240 !t219 = sqr# t218 1241 !t220 = mul# a t219 1242 !t221 = sqr# t220 1243 !t222 = mul# a t221 1244 !t223 = sqr# t222 1245 !t224 = mul# a t223 1246 !t225 = sqr# t224 1247 !t226 = mul# a t225 1248 !t227 = sqr# t226 1249 !t228 = mul# a t227 1250 !t229 = sqr# t228 1251 !t230 = mul# a t229 1252 !t231 = sqr# t230 1253 !t232 = mul# a t231 1254 !t233 = sqr# t232 1255 !t234 = mul# a t233 1256 !t235 = sqr# t234 1257 !t236 = mul# a t235 1258 !t237 = sqr# t236 1259 !t238 = mul# a t237 1260 !t239 = sqr# t238 1261 !t240 = mul# a t239 1262 !t241 = sqr# t240 1263 !t242 = mul# a t241 1264 !t243 = sqr# t242 1265 !t244 = mul# a t243 1266 !t245 = sqr# t244 1267 !t246 = mul# a t245 1268 !t247 = sqr# t246 1269 !t248 = mul# a t247 1270 !t249 = sqr# t248 1271 !t250 = mul# a t249 1272 !t251 = sqr# t250 1273 !t252 = mul# a t251 1274 !t253 = sqr# t252 1275 !t254 = mul# a t253 1276 !t255 = sqr# t254 1277 !t256 = mul# a t255 1278 !t257 = sqr# t256 1279 !t258 = mul# a t257 1280 !t259 = sqr# t258 1281 !t260 = mul# a t259 1282 !t261 = sqr# t260 1283 !t262 = mul# a t261 1284 !t263 = sqr# t262 1285 !t264 = mul# a t263 1286 !t265 = sqr# t264 1287 !t266 = mul# a t265 1288 !t267 = sqr# t266 1289 !t268 = mul# a t267 1290 !t269 = sqr# t268 1291 !t270 = mul# a t269 1292 !t271 = sqr# t270 1293 !t272 = mul# a t271 1294 !t273 = sqr# t272 1295 !t274 = mul# a t273 1296 !t275 = sqr# t274 1297 !t276 = mul# a t275 1298 !t277 = sqr# t276 1299 !t278 = mul# a t277 1300 !t279 = sqr# t278 1301 !t280 = mul# a t279 1302 !t281 = sqr# t280 1303 !t282 = mul# a t281 1304 !t283 = sqr# t282 1305 !t284 = mul# a t283 1306 !t285 = sqr# t284 1307 !t286 = mul# a t285 1308 !t287 = sqr# t286 1309 !t288 = mul# a t287 1310 !t289 = sqr# t288 1311 !t290 = mul# a t289 1312 !t291 = sqr# t290 1313 !t292 = mul# a t291 1314 !t293 = sqr# t292 1315 !t294 = mul# a t293 1316 !t295 = sqr# t294 1317 !t296 = mul# a t295 1318 !t297 = sqr# t296 1319 !t298 = mul# a t297 1320 !t299 = sqr# t298 1321 !t300 = mul# a t299 1322 !t301 = sqr# t300 1323 !t302 = mul# a t301 1324 !t303 = sqr# t302 1325 !t304 = mul# a t303 1326 !t305 = sqr# t304 1327 !t306 = mul# a t305 1328 !t307 = sqr# t306 1329 !t308 = mul# a t307 1330 !t309 = sqr# t308 1331 !t310 = mul# a t309 1332 !t311 = sqr# t310 1333 !t312 = mul# a t311 1334 !t313 = sqr# t312 1335 !t314 = mul# a t313 1336 !t315 = sqr# t314 1337 !t316 = mul# a t315 1338 !t317 = sqr# t316 1339 !t318 = mul# a t317 1340 !t319 = sqr# t318 1341 !t320 = mul# a t319 1342 !t321 = sqr# t320 1343 !t322 = mul# a t321 1344 !t323 = sqr# t322 1345 !t324 = mul# a t323 1346 !t325 = sqr# t324 1347 !t326 = mul# a t325 1348 !t327 = sqr# t326 1349 !t328 = mul# a t327 1350 !t329 = sqr# t328 1351 !t330 = mul# a t329 1352 !t331 = sqr# t330 1353 !t332 = mul# a t331 1354 !t333 = sqr# t332 1355 !t334 = mul# a t333 1356 !t335 = sqr# t334 1357 !t336 = mul# a t335 1358 !t337 = sqr# t336 1359 !t338 = mul# a t337 1360 !t339 = sqr# t338 1361 !t340 = mul# a t339 1362 !t341 = sqr# t340 1363 !t342 = mul# a t341 1364 !t343 = sqr# t342 1365 !t344 = mul# a t343 1366 !t345 = sqr# t344 1367 !t346 = mul# a t345 1368 !t347 = sqr# t346 1369 !t348 = mul# a t347 1370 !t349 = sqr# t348 1371 !t350 = mul# a t349 1372 !t351 = sqr# t350 1373 !t352 = mul# a t351 1374 !t353 = sqr# t352 1375 !t354 = mul# a t353 1376 !t355 = sqr# t354 1377 !t356 = mul# a t355 1378 !t357 = sqr# t356 1379 !t358 = mul# a t357 1380 !t359 = sqr# t358 1381 !t360 = mul# a t359 1382 !t361 = sqr# t360 1383 !t362 = mul# a t361 1384 !t363 = sqr# t362 1385 !t364 = mul# a t363 1386 !t365 = sqr# t364 1387 !t366 = mul# a t365 1388 !t367 = sqr# t366 1389 !t368 = mul# a t367 1390 !t369 = sqr# t368 1391 !t370 = mul# a t369 1392 !t371 = sqr# t370 1393 !t372 = mul# a t371 1394 !t373 = sqr# t372 1395 !t374 = mul# a t373 1396 !t375 = sqr# t374 1397 !t376 = mul# a t375 1398 !t377 = sqr# t376 1399 !t378 = mul# a t377 1400 !t379 = sqr# t378 1401 !t380 = mul# a t379 1402 !t381 = sqr# t380 1403 !t382 = mul# a t381 1404 !t383 = sqr# t382 1405 !t384 = mul# a t383 1406 !t385 = sqr# t384 1407 !t386 = mul# a t385 1408 !t387 = sqr# t386 1409 !t388 = mul# a t387 1410 !t389 = sqr# t388 1411 !t390 = mul# a t389 1412 !t391 = sqr# t390 1413 !t392 = mul# a t391 1414 !t393 = sqr# t392 1415 !t394 = mul# a t393 1416 !t395 = sqr# t394 1417 !t396 = mul# a t395 1418 !t397 = sqr# t396 1419 !t398 = mul# a t397 1420 !t399 = sqr# t398 1421 !t400 = mul# a t399 1422 !t401 = sqr# t400 1423 !t402 = mul# a t401 1424 !t403 = sqr# t402 1425 !t404 = mul# a t403 1426 !t405 = sqr# t404 1427 !t406 = mul# a t405 1428 !t407 = sqr# t406 1429 !t408 = mul# a t407 1430 !t409 = sqr# t408 1431 !t410 = mul# a t409 1432 !t411 = sqr# t410 1433 !t412 = mul# a t411 1434 !t413 = sqr# t412 1435 !t414 = mul# a t413 1436 !t415 = sqr# t414 1437 !t416 = mul# a t415 1438 !t417 = sqr# t416 1439 !t418 = mul# a t417 1440 !t419 = sqr# t418 1441 !t420 = mul# a t419 1442 !t421 = sqr# t420 1443 !t422 = mul# a t421 1444 !t423 = sqr# t422 1445 !t424 = mul# a t423 1446 !t425 = sqr# t424 1447 !t426 = mul# a t425 1448 !t427 = sqr# t426 1449 !t428 = mul# a t427 1450 !t429 = sqr# t428 1451 !t430 = mul# a t429 1452 !t431 = sqr# t430 1453 !t432 = mul# a t431 1454 !t433 = sqr# t432 1455 !t434 = mul# a t433 1456 !t435 = sqr# t434 1457 !t436 = mul# a t435 1458 !t437 = sqr# t436 1459 !t438 = mul# a t437 1460 !t439 = sqr# t438 1461 !t440 = mul# a t439 1462 !t441 = sqr# t440 1463 !t442 = mul# a t441 1464 !t443 = sqr# t442 1465 !t444 = mul# a t443 1466 !t445 = sqr# t444 1467 !t446 = mul# a t445 1468 !t447 = sqr# t446 1469 !t448 = mul# a t447 1470 !t449 = sqr# t448 1471 !t450 = sqr# t449 1472 !t451 = mul# a t450 1473 !t452 = sqr# t451 1474 !t453 = mul# a t452 1475 !t454 = sqr# t453 1476 !t455 = mul# a t454 1477 !t456 = sqr# t455 1478 !t457 = mul# a t456 1479 !t458 = sqr# t457 1480 !t459 = mul# a t458 1481 !t460 = sqr# t459 1482 !t461 = mul# a t460 1483 !t462 = sqr# t461 1484 !t463 = mul# a t462 1485 !t464 = sqr# t463 1486 !t465 = mul# a t464 1487 !t466 = sqr# t465 1488 !t467 = mul# a t466 1489 !t468 = sqr# t467 1490 !t469 = mul# a t468 1491 !t470 = sqr# t469 1492 !t471 = mul# a t470 1493 !t472 = sqr# t471 1494 !t473 = mul# a t472 1495 !t474 = sqr# t473 1496 !t475 = mul# a t474 1497 !t476 = sqr# t475 1498 !t477 = mul# a t476 1499 !t478 = sqr# t477 1500 !t479 = mul# a t478 1501 !t480 = sqr# t479 1502 !t481 = mul# a t480 1503 !t482 = sqr# t481 1504 !t483 = mul# a t482 1505 !t484 = sqr# t483 1506 !t485 = mul# a t484 1507 !t486 = sqr# t485 1508 !t487 = mul# a t486 1509 !t488 = sqr# t487 1510 !t489 = mul# a t488 1511 !t490 = sqr# t489 1512 !t491 = mul# a t490 1513 !t492 = sqr# t491 1514 !t493 = mul# a t492 1515 !t494 = sqr# t493 1516 !t495 = sqr# t494 1517 !t496 = sqr# t495 1518 !t497 = sqr# t496 1519 !t498 = sqr# t497 1520 !t499 = mul# a t498 1521 !t500 = sqr# t499 1522 !t501 = mul# a t500 1523 !t502 = sqr# t501 1524 !t503 = sqr# t502 1525 !r = t503 1526 in (# r, WW.eq# (sqr# r) a #) 1527 {-# INLINE sqrt# #-} 1528 1529 -- | Exponentiation in the Montgomery domain. 1530 -- 1531 -- >>> exp 2 3 1532 -- 8 1533 -- >>> exp 2 10 1534 -- 1024 1535 exp :: Montgomery -> Wider -> Montgomery 1536 exp (Montgomery b) (Wider e) = Montgomery (exp# b e) 1537 1538 exp# 1539 :: Limb4 1540 -> Limb4 1541 -> Limb4 1542 exp# b e = 1543 let !o = L4 0x1000003D1## 0## 0## 0## 1544 loop !r !m !ex n = case n of 1545 0 -> r 1546 _ -> 1547 let !(# ne, bit #) = WW.shr1_c# ex 1548 !candidate = mul# r m 1549 !nr = select# r candidate bit 1550 !nm = sqr# m 1551 in loop nr nm ne (n - 1) 1552 in loop o b e (256 :: Word) 1553 {-# INLINE exp# #-} 1554 1555 odd# :: Limb4 -> C.Choice 1556 odd# = WW.odd# 1557 {-# INLINE odd# #-} 1558 1559 -- | Check if a 'Montgomery' value is odd. 1560 -- 1561 -- Note that the comparison is performed in constant time, but we 1562 -- branch when converting to 'Bool'. 1563 -- 1564 -- >>> odd 1 1565 -- True 1566 -- >>> odd 2 1567 -- False 1568 -- >>> Data.Word.Wider.odd (retr 3) -- parity is preserved 1569 -- True 1570 odd_vartime :: Montgomery -> Bool 1571 odd_vartime (Montgomery m) = C.decide (odd# m) 1572 1573 -- constant-time selection ---------------------------------------------------- 1574 1575 select# 1576 :: Limb4 -- ^ a 1577 -> Limb4 -- ^ b 1578 -> C.Choice -- ^ c 1579 -> Limb4 -- ^ result 1580 select# = WW.select# 1581 {-# INLINE select# #-} 1582 1583 -- | Return a if c is truthy, otherwise return b. 1584 -- 1585 -- >>> import qualified Data.Choice as C 1586 -- >>> select 0 1 (C.true# ()) 1587 -- 1 1588 select 1589 :: Montgomery -- ^ a 1590 -> Montgomery -- ^ b 1591 -> C.Choice -- ^ c 1592 -> Montgomery -- ^ result 1593 select (Montgomery a) (Montgomery b) c = Montgomery (select# a b c) 1594