Scalar.hs (29474B)
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.Scalar 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 scalar group order. 18 19 module Numeric.Montgomery.Secp256k1.Scalar ( 20 -- * Montgomery form, secp256k1 scalar group order 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 , exp 56 , exp# 57 , odd_vartime 58 , odd# 59 ) where 60 61 import Control.DeepSeq 62 import qualified Data.Choice as C 63 import Data.Word.Limb (Limb(..)) 64 import qualified Data.Word.Limb as L 65 import qualified Data.Word.Wide as W 66 import Data.Word.Wider (Wider(..)) 67 import qualified Data.Word.Wider as WW 68 import GHC.Exts (Word(..), Word#) 69 import Prelude hiding (or, and, not, exp) 70 71 -- montgomery arithmetic, specialized to the secp256k1 scalar group order 72 -- 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141 73 74 -- | Montgomery-form 'Wider' words, on the Montgomery domain defined by 75 -- the secp256k1 scalar group order. 76 -- 77 -- >>> let one = 1 :: Montgomery 78 -- >>> one 79 -- 1 80 -- >>> putStrLn (render one) 81 -- (4624529908474429119, 4994812053365940164, 1, 0) 82 data Montgomery = Montgomery !Limb4 83 84 instance Show Montgomery where 85 show = show . from 86 87 -- | Render a 'Montgomery' value as a 'String', showing its individual 88 -- 'Limb's. 89 -- 90 -- >>> putStrLn (render 1) 91 -- (4624529908474429119, 4994812053365940164, 1, 0) 92 render :: Montgomery -> String 93 render (Montgomery (L4 a b c d)) = 94 "(" <> show (W# a) <> ", " <> show (W# b) <> ", " 95 <> show (W# c) <> ", " <> show (W# d) <> ")" 96 97 -- | Note that 'fromInteger' necessarily runs in variable time due 98 -- to conversion from the variable-size, potentially heap-allocated 99 -- 'Integer' type. 100 instance Num Montgomery where 101 a + b = add a b 102 a - b = sub a b 103 a * b = mul a b 104 negate a = neg a 105 abs = id 106 fromInteger = to . WW.to_vartime 107 signum (Montgomery (# l0, l1, l2, l3 #)) = 108 let !(Limb l) = l0 `L.or#` l1 `L.or#` l2 `L.or#` l3 109 !n = C.from_word_nonzero# l 110 !b = C.to_word# n 111 in Montgomery (L4 b 0## 0## 0##) 112 113 instance NFData Montgomery where 114 rnf (Montgomery a) = case a of (# _, _, _, _ #) -> () 115 116 -- utilities ------------------------------------------------------------------ 117 118 type Limb2 = (# Limb, Limb #) 119 120 type Limb4 = (# Limb, Limb, Limb, Limb #) 121 122 pattern L4 :: Word# -> Word# -> Word# -> Word# -> Limb4 123 pattern L4 w0 w1 w2 w3 = (# Limb w0, Limb w1, Limb w2, Limb w3 #) 124 {-# COMPLETE L4 #-} 125 126 -- Wide wrapping addition, when addend is only a limb. 127 wadd_w# :: Limb2 -> Limb -> Limb2 128 wadd_w# (# x_lo, x_hi #) y_lo = 129 let !(# s0, c0 #) = L.add_o# x_lo y_lo 130 !(# s1, _ #) = L.add_o# x_hi c0 131 in (# s0, s1 #) 132 {-# INLINE wadd_w# #-} 133 134 -- Truncate a wide word to a 'Limb'. 135 lo :: Limb2 -> Limb 136 lo (# l, _ #) = l 137 {-# INLINE lo #-} 138 139 -- comparison ----------------------------------------------------------------- 140 141 -- | Constant-time equality comparison. 142 eq :: Montgomery -> Montgomery -> C.Choice 143 eq (Montgomery (L4 a0 a1 a2 a3)) (Montgomery (L4 b0 b1 b2 b3)) = 144 C.eq_wider# (# a0, a1, a2, a3 #) (# b0, b1, b2, b3 #) 145 {-# INLINE eq #-} 146 147 -- | Variable-time equality comparison. 148 eq_vartime :: Montgomery -> Montgomery -> Bool 149 eq_vartime (Montgomery (Wider -> a)) (Montgomery (Wider -> b)) = 150 WW.eq_vartime a b 151 152 -- innards -------------------------------------------------------------------- 153 154 redc_inner# 155 :: Limb4 -- ^ upper limbs 156 -> Limb4 -- ^ lower limbs 157 -> (# Limb4, Limb #) -- ^ upper limbs, meta-carry 158 redc_inner# (# u0, u1, u2, u3 #) (# l0, l1, l2, l3 #) = 159 let !(# m0, m1, m2, m3 #) = 160 L4 0xBFD25E8CD0364141## 0xBAAEDCE6AF48A03B## 161 0xFFFFFFFFFFFFFFFE## 0xFFFFFFFFFFFFFFFF## 162 !n = Limb 0x4B0DFF665588B13F## 163 !w_0 = L.mul_w# l0 n 164 !(# _, c_00 #) = L.mac# w_0 m0 l0 (Limb 0##) 165 !(# l0_1, c_01 #) = L.mac# w_0 m1 l1 c_00 166 !(# l0_2, c_02 #) = L.mac# w_0 m2 l2 c_01 167 !(# l0_3, c_03 #) = L.mac# w_0 m3 l3 c_02 168 !(# u_0, mc_0 #) = L.add_c# u0 c_03 (Limb 0##) 169 !w_1 = L.mul_w# l0_1 n 170 !(# _, c_10 #) = L.mac# w_1 m0 l0_1 (Limb 0##) 171 !(# l1_1, c_11 #) = L.mac# w_1 m1 l0_2 c_10 172 !(# l1_2, c_12 #) = L.mac# w_1 m2 l0_3 c_11 173 !(# u1_3, c_13 #) = L.mac# w_1 m3 u_0 c_12 174 !(# u_1, mc_1 #) = L.add_c# u1 c_13 mc_0 175 !w_2 = L.mul_w# l1_1 n 176 !(# _, c_20 #) = L.mac# w_2 m0 l1_1 (Limb 0##) 177 !(# l2_1, c_21 #) = L.mac# w_2 m1 l1_2 c_20 178 !(# u2_2, c_22 #) = L.mac# w_2 m2 u1_3 c_21 179 !(# u2_3, c_23 #) = L.mac# w_2 m3 u_1 c_22 180 !(# u_2, mc_2 #) = L.add_c# u2 c_23 mc_1 181 !w_3 = L.mul_w# l2_1 n 182 !(# _, c_30 #) = L.mac# w_3 m0 l2_1 (Limb 0##) 183 !(# u3_1, c_31 #) = L.mac# w_3 m1 u2_2 c_30 184 !(# u3_2, c_32 #) = L.mac# w_3 m2 u2_3 c_31 185 !(# u3_3, c_33 #) = L.mac# w_3 m3 u_2 c_32 186 !(# u_3, mc_3 #) = L.add_c# u3 c_33 mc_2 187 in (# (# u3_1, u3_2, u3_3, u_3 #), mc_3 #) 188 {-# INLINE redc_inner# #-} 189 190 redc# 191 :: Limb4 -- ^ lower limbs 192 -> Limb4 -- ^ upper limbs 193 -> Limb4 -- ^ result 194 redc# l u = 195 let -- group order 196 !m = L4 0xBFD25E8CD0364141## 0xBAAEDCE6AF48A03B## 197 0xFFFFFFFFFFFFFFFE## 0xFFFFFFFFFFFFFFFF## 198 !(# nu, mc #) = redc_inner# u l 199 in WW.sub_mod_c# nu mc m m 200 {-# INLINE redc# #-} 201 202 -- | Montgomery reduction. 203 -- 204 -- The first argument represents the low words, and the second the 205 -- high words, of an extra-large eight-limb word in Montgomery form. 206 redc 207 :: Montgomery -- ^ low wider-word, Montgomery form 208 -> Montgomery -- ^ high wider-word, Montgomery form 209 -> Montgomery -- ^ reduced value 210 redc (Montgomery l) (Montgomery u) = 211 let !res = redc# l u 212 in (Montgomery res) 213 214 retr_inner# 215 :: Limb4 -- ^ value in montgomery form 216 -> Limb4 -- ^ retrieved value 217 retr_inner# (# x0, x1, x2, x3 #) = 218 let !(# m0, m1, m2, m3 #) = 219 L4 0xBFD25E8CD0364141## 0xBAAEDCE6AF48A03B## 220 0xFFFFFFFFFFFFFFFE## 0xFFFFFFFFFFFFFFFF## 221 !n = Limb 0x4B0DFF665588B13F## 222 !u_0 = L.mul_w# x0 n 223 !(# _, o0 #) = L.mac# u_0 m0 x0 (Limb 0##) 224 !(# o0_1, p0_1 #) = L.mac# u_0 m1 (Limb 0##) o0 225 !(# p0_2, q0_2 #) = L.mac# u_0 m2 (Limb 0##) p0_1 226 !(# q0_3, r0_3 #) = L.mac# u_0 m3 (Limb 0##) q0_2 227 !u_1 = L.mul_w# (L.add_w# o0_1 x1) n 228 !(# _, o1 #) = L.mac# u_1 m0 x1 o0_1 229 !(# o1_1, p1_1 #) = L.mac# u_1 m1 p0_2 o1 230 !(# p1_2, q1_2 #) = L.mac# u_1 m2 q0_3 p1_1 231 !(# q1_3, r1_3 #) = L.mac# u_1 m3 r0_3 q1_2 232 !u_2 = L.mul_w# (L.add_w# o1_1 x2) n 233 !(# _, o2 #) = L.mac# u_2 m0 x2 o1_1 234 !(# o2_1, p2_1 #) = L.mac# u_2 m1 p1_2 o2 235 !(# p2_2, q2_2 #) = L.mac# u_2 m2 q1_3 p2_1 236 !(# q2_3, r2_3 #) = L.mac# u_2 m3 r1_3 q2_2 237 !u_3 = L.mul_w# (L.add_w# o2_1 x3) n 238 !(# _, o3 #) = L.mac# u_3 m0 x3 o2_1 239 !(# o3_1, p3_1 #) = L.mac# u_3 m1 p2_2 o3 240 !(# p3_2, q3_2 #) = L.mac# u_3 m2 q2_3 p3_1 241 !(# q3_3, r3_3 #) = L.mac# u_3 m3 r2_3 q3_2 242 in (# o3_1, p3_2, q3_3, r3_3 #) 243 {-# INLINE retr_inner# #-} 244 245 retr# 246 :: Limb4 247 -> Limb4 248 retr# f = retr_inner# f 249 {-# INLINE retr# #-} 250 251 -- | Retrieve a 'Montgomery' value from the Montgomery domain, producing 252 -- a 'Wider' word. 253 retr 254 :: Montgomery -- ^ value in Montgomery form 255 -> Wider -- ^ retrieved value 256 retr (Montgomery f) = 257 let !res = retr# f 258 in (Wider res) 259 260 -- | Montgomery multiplication (FIOS), without conditional subtract. 261 mul_inner# 262 :: Limb4 -- ^ x 263 -> Limb4 -- ^ y 264 -> (# Limb4, Limb #) -- ^ product, meta-carry 265 mul_inner# (# x0, x1, x2, x3 #) (# y0, y1, y2, y3 #) = 266 let !(# m0, m1, m2, m3 #) = 267 L4 0xBFD25E8CD0364141## 0xBAAEDCE6AF48A03B## 268 0xFFFFFFFFFFFFFFFE## 0xFFFFFFFFFFFFFFFF## 269 !n = Limb 0x4B0DFF665588B13F## 270 !axy0 = L.mul_c# x0 y0 271 !u0 = L.mul_w# (lo axy0) n 272 !(# (# _, a0 #), c0 #) = W.add_o# (L.mul_c# u0 m0) axy0 273 !carry0 = (# a0, c0 #) 274 !axy0_1 = L.mul_c# x0 y1 275 !umc0_1 = W.add_w# (L.mul_c# u0 m1) carry0 276 !(# (# o0, ab0_1 #), c0_1 #) = W.add_o# axy0_1 umc0_1 277 !carry0_1 = (# ab0_1, c0_1 #) 278 !axy0_2 = L.mul_c# x0 y2 279 !umc0_2 = W.add_w# (L.mul_c# u0 m2) carry0_1 280 !(# (# p0, ab0_2 #), c0_2 #) = W.add_o# axy0_2 umc0_2 281 !carry0_2 = (# ab0_2, c0_2 #) 282 !axy0_3 = L.mul_c# x0 y3 283 !umc0_3 = W.add_w# (L.mul_c# u0 m3) carry0_2 284 !(# (# q0, ab0_3 #), c0_3 #) = W.add_o# axy0_3 umc0_3 285 !carry0_3 = (# ab0_3, c0_3 #) 286 !(# r0, mc0 #) = carry0_3 287 !axy1 = wadd_w# (L.mul_c# x1 y0) o0 288 !u1 = L.mul_w# (lo axy1) n 289 !(# (# _, a1 #), c1 #) = W.add_o# (L.mul_c# u1 m0) axy1 290 !carry1 = (# a1, c1 #) 291 !axy1_1 = wadd_w# (L.mul_c# x1 y1) p0 292 !umc1_1 = W.add_w# (L.mul_c# u1 m1) carry1 293 !(# (# o1, ab1_1 #), c1_1 #) = W.add_o# axy1_1 umc1_1 294 !carry1_1 = (# ab1_1, c1_1 #) 295 !axy1_2 = wadd_w# (L.mul_c# x1 y2) q0 296 !umc1_2 = W.add_w# (L.mul_c# u1 m2) carry1_1 297 !(# (# p1, ab1_2 #), c1_2 #) = W.add_o# axy1_2 umc1_2 298 !carry1_2 = (# ab1_2, c1_2 #) 299 !axy1_3 = wadd_w# (L.mul_c# x1 y3) r0 300 !umc1_3 = W.add_w# (L.mul_c# u1 m3) carry1_2 301 !(# (# q1, ab1_3 #), c1_3 #) = W.add_o# axy1_3 umc1_3 302 !carry1_3 = (# ab1_3, c1_3 #) 303 !(# r1, mc1 #) = wadd_w# carry1_3 mc0 304 !axy2 = wadd_w# (L.mul_c# x2 y0) o1 305 !u2 = L.mul_w# (lo axy2) n 306 !(# (# _, a2 #), c2 #) = W.add_o# (L.mul_c# u2 m0) axy2 307 !carry2 = (# a2, c2 #) 308 !axy2_1 = wadd_w# (L.mul_c# x2 y1) p1 309 !umc2_1 = W.add_w# (L.mul_c# u2 m1) carry2 310 !(# (# o2, ab2_1 #), c2_1 #) = W.add_o# axy2_1 umc2_1 311 !carry2_1 = (# ab2_1, c2_1 #) 312 !axy2_2 = wadd_w# (L.mul_c# x2 y2) q1 313 !umc2_2 = W.add_w# (L.mul_c# u2 m2) carry2_1 314 !(# (# p2, ab2_2 #), c2_2 #) = W.add_o# axy2_2 umc2_2 315 !carry2_2 = (# ab2_2, c2_2 #) 316 !axy2_3 = wadd_w# (L.mul_c# x2 y3) r1 317 !umc2_3 = W.add_w# (L.mul_c# u2 m3) carry2_2 318 !(# (# q2, ab2_3 #), c2_3 #) = W.add_o# axy2_3 umc2_3 319 !carry2_3 = (# ab2_3, c2_3 #) 320 !(# r2, mc2 #) = wadd_w# carry2_3 mc1 321 !axy3 = wadd_w# (L.mul_c# x3 y0) o2 322 !u3 = L.mul_w# (lo axy3) n 323 !(# (# _, a3 #), c3 #) = W.add_o# (L.mul_c# u3 m0) axy3 324 !carry3 = (# a3, c3 #) 325 !axy3_1 = wadd_w# (L.mul_c# x3 y1) p2 326 !umc3_1 = W.add_w# (L.mul_c# u3 m1) carry3 327 !(# (# o3, ab3_1 #), c3_1 #) = W.add_o# axy3_1 umc3_1 328 !carry3_1 = (# ab3_1, c3_1 #) 329 !axy3_2 = wadd_w# (L.mul_c# x3 y2) q2 330 !umc3_2 = W.add_w# (L.mul_c# u3 m2) carry3_1 331 !(# (# p3, ab3_2 #), c3_2 #) = W.add_o# axy3_2 umc3_2 332 !carry3_2 = (# ab3_2, c3_2 #) 333 !axy3_3 = wadd_w# (L.mul_c# x3 y3) r2 334 !umc3_3 = W.add_w# (L.mul_c# u3 m3) carry3_2 335 !(# (# q3, ab3_3 #), c3_3 #) = W.add_o# axy3_3 umc3_3 336 !carry3_3 = (# ab3_3, c3_3 #) 337 !(# r3, mc3 #) = wadd_w# carry3_3 mc2 338 in (# (# o3, p3, q3, r3 #), mc3 #) 339 {-# INLINE mul_inner# #-} 340 341 mul# 342 :: Limb4 343 -> Limb4 344 -> Limb4 345 mul# a b = 346 let -- group order 347 !m = L4 0xBFD25E8CD0364141## 0xBAAEDCE6AF48A03B## 348 0xFFFFFFFFFFFFFFFE## 0xFFFFFFFFFFFFFFFF## 349 !(# nu, mc #) = mul_inner# a b 350 in WW.sub_mod_c# nu mc m m 351 {-# NOINLINE mul# #-} -- cannot be inlined without exploding comp time 352 353 -- | Multiplication in the Montgomery domain. 354 -- 355 -- Note that 'Montgomery' is an instance of 'Num', so you can use '*' 356 -- to apply this function. 357 -- 358 -- >>> 1 * 1 :: Montgomery 359 -- 1 360 mul 361 :: Montgomery -- ^ multiplicand in montgomery form 362 -> Montgomery -- ^ multiplier in montgomery form 363 -> Montgomery -- ^ montgomery product 364 mul (Montgomery a) (Montgomery b) = Montgomery (mul# a b) 365 366 to# 367 :: Limb4 -- ^ integer 368 -> Limb4 369 to# x = 370 let !r2 = L4 0x896CF21467D7D140## 0x741496C20E7CF878## -- r^2 mod m 371 0xE697F5E45BCD07C6## 0x9D671CD581C69BC5## 372 in mul# x r2 373 {-# INLINE to# #-} 374 375 -- | Convert a 'Wider' word to the Montgomery domain. 376 to :: Wider -> Montgomery 377 to (Wider x) = Montgomery (to# x) 378 379 -- | Retrieve a 'Montgomery' word from the Montgomery domain. 380 -- 381 -- This function is a synonym for 'retr'. 382 from :: Montgomery -> Wider 383 from = retr 384 385 add# 386 :: Limb4 -- ^ augend 387 -> Limb4 -- ^ addend 388 -> Limb4 -- ^ sum 389 add# a b = 390 let -- group order 391 !m = L4 0xBFD25E8CD0364141## 0xBAAEDCE6AF48A03B## 392 0xFFFFFFFFFFFFFFFE## 0xFFFFFFFFFFFFFFFF## 393 in WW.add_mod# a b m 394 {-# INLINE add# #-} 395 396 -- | Addition in the Montgomery domain. 397 -- 398 -- Note that 'Montgomery' is an instance of 'Num', so you can use '+' 399 -- to apply this function. 400 -- 401 -- >>> 1 + 1 :: Montgomery 402 -- 2 403 add 404 :: Montgomery -- ^ augend 405 -> Montgomery -- ^ addend 406 -> Montgomery -- ^ sum 407 add (Montgomery a) (Montgomery b) = Montgomery (add# a b) 408 409 sub# 410 :: Limb4 -- ^ minuend 411 -> Limb4 -- ^ subtrahend 412 -> Limb4 -- ^ difference 413 sub# a b = 414 let !m = L4 0xBFD25E8CD0364141## 0xBAAEDCE6AF48A03B## 415 0xFFFFFFFFFFFFFFFE## 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 427 :: Montgomery -- ^ minuend 428 -> Montgomery -- ^ subtrahend 429 -> Montgomery -- ^ difference 430 sub (Montgomery a) (Montgomery b) = Montgomery (sub# a b) 431 432 neg# 433 :: Limb4 -- ^ argument 434 -> Limb4 -- ^ modular negation 435 neg# a = sub# (L4 0## 0## 0## 0##) a 436 {-# INLINE neg# #-} 437 438 -- | Additive inverse in the Montgomery domain. 439 -- 440 -- Note that 'Montgomery' is an instance of 'Num', so you can use 'negate' 441 -- to apply this function. 442 -- 443 -- >>> negate 1 :: Montgomery 444 -- 115792089237316195423570985008687907852837564279074904382605163141518161494336 445 -- >>> (negate 1 :: Montgomery) + 1 446 -- 0 447 neg :: Montgomery -> Montgomery 448 neg (Montgomery a) = Montgomery (neg# a) 449 450 sqr# :: Limb4 -> Limb4 451 sqr# a = 452 let !(# l, h #) = WW.sqr# a 453 in redc# l h 454 {-# NOINLINE sqr# #-} -- cannot be inlined without exploding comp time 455 456 -- | Squaring in the Montgomery domain. 457 -- 458 -- >>> sqr 1 459 -- 1 460 -- >>> sqr 2 461 -- 4 462 -- >>> sqr (negate 2) 463 -- 4 464 sqr 465 :: Montgomery -- ^ argument 466 -> Montgomery -- ^ square 467 sqr (Montgomery a) = Montgomery (mul# a a) 468 469 -- | Zero (the additive unit) in the Montgomery domain. 470 zero :: Montgomery 471 zero = Montgomery (L4 0## 0## 0## 0##) 472 473 -- | One (the multiplicative unit) in the Montgomery domain. 474 one :: Montgomery 475 one = Montgomery (L4 0x402DA1732FC9BEBF## 0x4551231950B75FC4## 476 0x0000000000000001## 0x0000000000000000##) 477 478 -- generated by etc/generate_inv.sh 479 inv# 480 :: Limb4 481 -> Limb4 482 inv# a = 483 let !t0 = L4 0x402DA1732FC9BEBF## 0x4551231950B75FC4## 484 0x0000000000000001## 0x0000000000000000## 485 !t1 = sqr# t0 486 !t2 = mul# a t1 487 !t3 = sqr# t2 488 !t4 = mul# a t3 489 !t5 = sqr# t4 490 !t6 = mul# a t5 491 !t7 = sqr# t6 492 !t8 = mul# a t7 493 !t9 = sqr# t8 494 !t10 = mul# a t9 495 !t11 = sqr# t10 496 !t12 = mul# a t11 497 !t13 = sqr# t12 498 !t14 = mul# a t13 499 !t15 = sqr# t14 500 !t16 = mul# a t15 501 !t17 = sqr# t16 502 !t18 = mul# a t17 503 !t19 = sqr# t18 504 !t20 = mul# a t19 505 !t21 = sqr# t20 506 !t22 = mul# a t21 507 !t23 = sqr# t22 508 !t24 = mul# a t23 509 !t25 = sqr# t24 510 !t26 = mul# a t25 511 !t27 = sqr# t26 512 !t28 = mul# a t27 513 !t29 = sqr# t28 514 !t30 = mul# a t29 515 !t31 = sqr# t30 516 !t32 = mul# a t31 517 !t33 = sqr# t32 518 !t34 = mul# a t33 519 !t35 = sqr# t34 520 !t36 = mul# a t35 521 !t37 = sqr# t36 522 !t38 = mul# a t37 523 !t39 = sqr# t38 524 !t40 = mul# a t39 525 !t41 = sqr# t40 526 !t42 = mul# a t41 527 !t43 = sqr# t42 528 !t44 = mul# a t43 529 !t45 = sqr# t44 530 !t46 = mul# a t45 531 !t47 = sqr# t46 532 !t48 = mul# a t47 533 !t49 = sqr# t48 534 !t50 = mul# a t49 535 !t51 = sqr# t50 536 !t52 = mul# a t51 537 !t53 = sqr# t52 538 !t54 = mul# a t53 539 !t55 = sqr# t54 540 !t56 = mul# a t55 541 !t57 = sqr# t56 542 !t58 = mul# a t57 543 !t59 = sqr# t58 544 !t60 = mul# a t59 545 !t61 = sqr# t60 546 !t62 = mul# a t61 547 !t63 = sqr# t62 548 !t64 = mul# a t63 549 !t65 = sqr# t64 550 !t66 = mul# a t65 551 !t67 = sqr# t66 552 !t68 = mul# a t67 553 !t69 = sqr# t68 554 !t70 = mul# a t69 555 !t71 = sqr# t70 556 !t72 = mul# a t71 557 !t73 = sqr# t72 558 !t74 = mul# a t73 559 !t75 = sqr# t74 560 !t76 = mul# a t75 561 !t77 = sqr# t76 562 !t78 = mul# a t77 563 !t79 = sqr# t78 564 !t80 = mul# a t79 565 !t81 = sqr# t80 566 !t82 = mul# a t81 567 !t83 = sqr# t82 568 !t84 = mul# a t83 569 !t85 = sqr# t84 570 !t86 = mul# a t85 571 !t87 = sqr# t86 572 !t88 = mul# a t87 573 !t89 = sqr# t88 574 !t90 = mul# a t89 575 !t91 = sqr# t90 576 !t92 = mul# a t91 577 !t93 = sqr# t92 578 !t94 = mul# a t93 579 !t95 = sqr# t94 580 !t96 = mul# a t95 581 !t97 = sqr# t96 582 !t98 = mul# a t97 583 !t99 = sqr# t98 584 !t100 = mul# a t99 585 !t101 = sqr# t100 586 !t102 = mul# a t101 587 !t103 = sqr# t102 588 !t104 = mul# a t103 589 !t105 = sqr# t104 590 !t106 = mul# a t105 591 !t107 = sqr# t106 592 !t108 = mul# a t107 593 !t109 = sqr# t108 594 !t110 = mul# a t109 595 !t111 = sqr# t110 596 !t112 = mul# a t111 597 !t113 = sqr# t112 598 !t114 = mul# a t113 599 !t115 = sqr# t114 600 !t116 = mul# a t115 601 !t117 = sqr# t116 602 !t118 = mul# a t117 603 !t119 = sqr# t118 604 !t120 = mul# a t119 605 !t121 = sqr# t120 606 !t122 = mul# a t121 607 !t123 = sqr# t122 608 !t124 = mul# a t123 609 !t125 = sqr# t124 610 !t126 = mul# a t125 611 !t127 = sqr# t126 612 !t128 = mul# a t127 613 !t129 = sqr# t128 614 !t130 = mul# a t129 615 !t131 = sqr# t130 616 !t132 = mul# a t131 617 !t133 = sqr# t132 618 !t134 = mul# a t133 619 !t135 = sqr# t134 620 !t136 = mul# a t135 621 !t137 = sqr# t136 622 !t138 = mul# a t137 623 !t139 = sqr# t138 624 !t140 = mul# a t139 625 !t141 = sqr# t140 626 !t142 = mul# a t141 627 !t143 = sqr# t142 628 !t144 = mul# a t143 629 !t145 = sqr# t144 630 !t146 = mul# a t145 631 !t147 = sqr# t146 632 !t148 = mul# a t147 633 !t149 = sqr# t148 634 !t150 = mul# a t149 635 !t151 = sqr# t150 636 !t152 = mul# a t151 637 !t153 = sqr# t152 638 !t154 = mul# a t153 639 !t155 = sqr# t154 640 !t156 = mul# a t155 641 !t157 = sqr# t156 642 !t158 = mul# a t157 643 !t159 = sqr# t158 644 !t160 = mul# a t159 645 !t161 = sqr# t160 646 !t162 = mul# a t161 647 !t163 = sqr# t162 648 !t164 = mul# a t163 649 !t165 = sqr# t164 650 !t166 = mul# a t165 651 !t167 = sqr# t166 652 !t168 = mul# a t167 653 !t169 = sqr# t168 654 !t170 = mul# a t169 655 !t171 = sqr# t170 656 !t172 = mul# a t171 657 !t173 = sqr# t172 658 !t174 = mul# a t173 659 !t175 = sqr# t174 660 !t176 = mul# a t175 661 !t177 = sqr# t176 662 !t178 = mul# a t177 663 !t179 = sqr# t178 664 !t180 = mul# a t179 665 !t181 = sqr# t180 666 !t182 = mul# a t181 667 !t183 = sqr# t182 668 !t184 = mul# a t183 669 !t185 = sqr# t184 670 !t186 = mul# a t185 671 !t187 = sqr# t186 672 !t188 = mul# a t187 673 !t189 = sqr# t188 674 !t190 = mul# a t189 675 !t191 = sqr# t190 676 !t192 = mul# a t191 677 !t193 = sqr# t192 678 !t194 = mul# a t193 679 !t195 = sqr# t194 680 !t196 = mul# a t195 681 !t197 = sqr# t196 682 !t198 = mul# a t197 683 !t199 = sqr# t198 684 !t200 = mul# a t199 685 !t201 = sqr# t200 686 !t202 = mul# a t201 687 !t203 = sqr# t202 688 !t204 = mul# a t203 689 !t205 = sqr# t204 690 !t206 = mul# a t205 691 !t207 = sqr# t206 692 !t208 = mul# a t207 693 !t209 = sqr# t208 694 !t210 = mul# a t209 695 !t211 = sqr# t210 696 !t212 = mul# a t211 697 !t213 = sqr# t212 698 !t214 = mul# a t213 699 !t215 = sqr# t214 700 !t216 = mul# a t215 701 !t217 = sqr# t216 702 !t218 = mul# a t217 703 !t219 = sqr# t218 704 !t220 = mul# a t219 705 !t221 = sqr# t220 706 !t222 = mul# a t221 707 !t223 = sqr# t222 708 !t224 = mul# a t223 709 !t225 = sqr# t224 710 !t226 = mul# a t225 711 !t227 = sqr# t226 712 !t228 = mul# a t227 713 !t229 = sqr# t228 714 !t230 = mul# a t229 715 !t231 = sqr# t230 716 !t232 = mul# a t231 717 !t233 = sqr# t232 718 !t234 = mul# a t233 719 !t235 = sqr# t234 720 !t236 = mul# a t235 721 !t237 = sqr# t236 722 !t238 = mul# a t237 723 !t239 = sqr# t238 724 !t240 = mul# a t239 725 !t241 = sqr# t240 726 !t242 = mul# a t241 727 !t243 = sqr# t242 728 !t244 = mul# a t243 729 !t245 = sqr# t244 730 !t246 = mul# a t245 731 !t247 = sqr# t246 732 !t248 = mul# a t247 733 !t249 = sqr# t248 734 !t250 = mul# a t249 735 !t251 = sqr# t250 736 !t252 = mul# a t251 737 !t253 = sqr# t252 738 !t254 = mul# a t253 739 !t255 = sqr# t254 740 !t256 = sqr# t255 741 !t257 = mul# a t256 742 !t258 = sqr# t257 743 !t259 = sqr# t258 744 !t260 = mul# a t259 745 !t261 = sqr# t260 746 !t262 = mul# a t261 747 !t263 = sqr# t262 748 !t264 = mul# a t263 749 !t265 = sqr# t264 750 !t266 = sqr# t265 751 !t267 = mul# a t266 752 !t268 = sqr# t267 753 !t269 = sqr# t268 754 !t270 = mul# a t269 755 !t271 = sqr# t270 756 !t272 = sqr# t271 757 !t273 = mul# a t272 758 !t274 = sqr# t273 759 !t275 = sqr# t274 760 !t276 = mul# a t275 761 !t277 = sqr# t276 762 !t278 = mul# a t277 763 !t279 = sqr# t278 764 !t280 = mul# a t279 765 !t281 = sqr# t280 766 !t282 = sqr# t281 767 !t283 = mul# a t282 768 !t284 = sqr# t283 769 !t285 = mul# a t284 770 !t286 = sqr# t285 771 !t287 = sqr# t286 772 !t288 = mul# a t287 773 !t289 = sqr# t288 774 !t290 = mul# a t289 775 !t291 = sqr# t290 776 !t292 = mul# a t291 777 !t293 = sqr# t292 778 !t294 = sqr# t293 779 !t295 = sqr# t294 780 !t296 = mul# a t295 781 !t297 = sqr# t296 782 !t298 = mul# a t297 783 !t299 = sqr# t298 784 !t300 = mul# a t299 785 !t301 = sqr# t300 786 !t302 = sqr# t301 787 !t303 = sqr# t302 788 !t304 = mul# a t303 789 !t305 = sqr# t304 790 !t306 = mul# a t305 791 !t307 = sqr# t306 792 !t308 = sqr# t307 793 !t309 = mul# a t308 794 !t310 = sqr# t309 795 !t311 = sqr# t310 796 !t312 = mul# a t311 797 !t313 = sqr# t312 798 !t314 = sqr# t313 799 !t315 = mul# a t314 800 !t316 = sqr# t315 801 !t317 = mul# a t316 802 !t318 = sqr# t317 803 !t319 = mul# a t318 804 !t320 = sqr# t319 805 !t321 = mul# a t320 806 !t322 = sqr# t321 807 !t323 = sqr# t322 808 !t324 = mul# a t323 809 !t325 = sqr# t324 810 !t326 = sqr# t325 811 !t327 = sqr# t326 812 !t328 = mul# a t327 813 !t329 = sqr# t328 814 !t330 = sqr# t329 815 !t331 = sqr# t330 816 !t332 = sqr# t331 817 !t333 = mul# a t332 818 !t334 = sqr# t333 819 !t335 = sqr# t334 820 !t336 = mul# a t335 821 !t337 = sqr# t336 822 !t338 = sqr# t337 823 !t339 = sqr# t338 824 !t340 = sqr# t339 825 !t341 = sqr# t340 826 !t342 = sqr# t341 827 !t343 = sqr# t342 828 !t344 = sqr# t343 829 !t345 = mul# a t344 830 !t346 = sqr# t345 831 !t347 = mul# a t346 832 !t348 = sqr# t347 833 !t349 = mul# a t348 834 !t350 = sqr# t349 835 !t351 = sqr# t350 836 !t352 = mul# a t351 837 !t353 = sqr# t352 838 !t354 = mul# a t353 839 !t355 = sqr# t354 840 !t356 = mul# a t355 841 !t357 = sqr# t356 842 !t358 = sqr# t357 843 !t359 = mul# a t358 844 !t360 = sqr# t359 845 !t361 = mul# a t360 846 !t362 = sqr# t361 847 !t363 = mul# a t362 848 !t364 = sqr# t363 849 !t365 = mul# a t364 850 !t366 = sqr# t365 851 !t367 = mul# a t366 852 !t368 = sqr# t367 853 !t369 = mul# a t368 854 !t370 = sqr# t369 855 !t371 = mul# a t370 856 !t372 = sqr# t371 857 !t373 = mul# a t372 858 !t374 = sqr# t373 859 !t375 = sqr# t374 860 !t376 = mul# a t375 861 !t377 = sqr# t376 862 !t378 = sqr# t377 863 !t379 = sqr# t378 864 !t380 = mul# a t379 865 !t381 = sqr# t380 866 !t382 = sqr# t381 867 !t383 = sqr# t382 868 !t384 = mul# a t383 869 !t385 = sqr# t384 870 !t386 = sqr# t385 871 !t387 = mul# a t386 872 !t388 = sqr# t387 873 !t389 = mul# a t388 874 !t390 = sqr# t389 875 !t391 = mul# a t390 876 !t392 = sqr# t391 877 !t393 = mul# a t392 878 !t394 = sqr# t393 879 !t395 = sqr# t394 880 !t396 = mul# a t395 881 !t397 = sqr# t396 882 !t398 = sqr# t397 883 !t399 = sqr# t398 884 !t400 = sqr# t399 885 !t401 = mul# a t400 886 !t402 = sqr# t401 887 !t403 = mul# a t402 888 !t404 = sqr# t403 889 !t405 = sqr# t404 890 !t406 = sqr# t405 891 !t407 = mul# a t406 892 !t408 = sqr# t407 893 !t409 = mul# a t408 894 !t410 = sqr# t409 895 !t411 = sqr# t410 896 !t412 = mul# a t411 897 !t413 = sqr# t412 898 !t414 = sqr# t413 899 !t415 = sqr# t414 900 !t416 = sqr# t415 901 !t417 = sqr# t416 902 !t418 = sqr# t417 903 !t419 = sqr# t418 904 !t420 = mul# a t419 905 !t421 = sqr# t420 906 !t422 = mul# a t421 907 !t423 = sqr# t422 908 !t424 = sqr# t423 909 !t425 = mul# a t424 910 !t426 = sqr# t425 911 !t427 = mul# a t426 912 !t428 = sqr# t427 913 !t429 = sqr# t428 914 !t430 = sqr# t429 915 !t431 = mul# a t430 916 !t432 = sqr# t431 917 !t433 = sqr# t432 918 !t434 = sqr# t433 919 !t435 = sqr# t434 920 !t436 = sqr# t435 921 !t437 = sqr# t436 922 !t438 = mul# a t437 923 !t439 = sqr# t438 924 !t440 = sqr# t439 925 !t441 = sqr# t440 926 !t442 = mul# a t441 927 !t443 = sqr# t442 928 !t444 = mul# a t443 929 !t445 = sqr# t444 930 !t446 = mul# a t445 931 !t447 = sqr# t446 932 !t448 = mul# a t447 933 !t449 = sqr# t448 934 !t450 = mul# a t449 935 !t451 = sqr# t450 936 !t452 = mul# a t451 937 !r = t452 938 in r 939 {-# INLINE inv# #-} 940 941 -- | Multiplicative inverse in the Montgomery domain. 942 -- 943 -- >> inv 2 944 -- 57896044618658097711785492504343953926418782139537452191302581570759080747169 945 -- >> inv 2 * 2 946 -- 1 947 inv 948 :: Montgomery -- ^ argument 949 -> Montgomery -- ^ inverse 950 inv (Montgomery w) = Montgomery (inv# w) 951 952 -- | Exponentiation in the Montgomery domain. 953 -- 954 -- >>> exp 2 3 955 -- 8 956 -- >>> exp 2 10 957 -- 1024 958 exp :: Montgomery -> Wider -> Montgomery 959 exp (Montgomery b) (Wider e) = Montgomery (exp# b e) 960 961 exp# 962 :: Limb4 963 -> Limb4 964 -> Limb4 965 exp# b e = 966 let !o = L4 0x402DA1732FC9BEBF## 0x4551231950B75FC4## 967 0x0000000000000001## 0x0000000000000000## 968 loop !r !m !ex n = case n of 969 0 -> r 970 _ -> 971 let !(# ne, bit #) = WW.shr1_c# ex 972 !candidate = mul# r m 973 !nr = select# r candidate bit 974 !nm = sqr# m 975 in loop nr nm ne (n - 1) 976 in loop o b e (256 :: Word) 977 {-# INLINE exp# #-} 978 979 odd# :: Limb4 -> C.Choice 980 odd# = WW.odd# 981 {-# INLINE odd# #-} 982 983 -- | Check if a 'Montgomery' value is odd. 984 -- 985 -- Note that the comparison is performed in constant time, but we 986 -- branch when converting to 'Bool'. 987 -- 988 -- >>> odd 1 989 -- True 990 -- >>> odd 2 991 -- False 992 -- >>> Data.Word.Wider.odd (retr 3) -- parity is preserved 993 -- True 994 odd_vartime :: Montgomery -> Bool 995 odd_vartime (Montgomery m) = C.decide (odd# m) 996 997 -- constant-time selection ---------------------------------------------------- 998 999 select# 1000 :: Limb4 -- ^ a 1001 -> Limb4 -- ^ b 1002 -> C.Choice -- ^ c 1003 -> Limb4 -- ^ result 1004 select# = WW.select# 1005 {-# INLINE select# #-} 1006 1007 -- | Return a if c is truthy, otherwise return b. 1008 -- 1009 -- >>> import qualified Data.Choice as C 1010 -- >>> select 0 1 (C.true# ()) 1011 -- 1 1012 select 1013 :: Montgomery -- ^ a 1014 -> Montgomery -- ^ b 1015 -> C.Choice -- ^ c 1016 -> Montgomery -- ^ result 1017 select (Montgomery a) (Montgomery b) c = Montgomery (select# a b c) 1018