pygost/gost3410.py

   1 # coding: utf-8
   2 # PyGOST -- Pure Python GOST cryptographic functions library
   3 # Copyright (C) 2015-2023 Sergey Matveev <stargrave@stargrave.org>
   4 #
   5 # This program is free software: you can redistribute it and/or modify
   6 # it under the terms of the GNU General Public License as published by
   7 # the Free Software Foundation, version 3 of the License.
   8 #
   9 # This program is distributed in the hope that it will be useful,
  10 # but WITHOUT ANY WARRANTY; without even the implied warranty of
  11 # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  12 # GNU General Public License for more details.
  13 #
  14 # You should have received a copy of the GNU General Public License
  15 # along with this program.  If not, see <http://www.gnu.org/licenses/>.
  16 """GOST R 34.10 public-key signature function.
  17
  18 This is implementation of GOST R 34.10-2001 (:rfc:`5832`), GOST R
  19 34.10-2012 (:rfc:`7091`). The difference between 2001 and 2012 is the
  20 key, digest and signature lengths.
  21 """
  22
  23 from os import urandom
  24
  25 from pygost.utils import bytes2long
  26 from pygost.utils import hexdec
  27 from pygost.utils import long2bytes
  28 from pygost.utils import modinvert
  29
  30
  31 def point_size(point):
  32     """Determine is it either 256 or 512 bit point
  33     """
  34     return (512 // 8) if point.bit_length() > 256 else (256 // 8)
  35
  36
  37 class GOST3410Curve(object):
  38     """GOST 34.10 validated curve
  39
  40     >>> curve = CURVES["id-GostR3410-2001-TestParamSet"]
  41     >>> prv = prv_unmarshal(urandom(32))
  42     >>> signature = sign(curve, prv, GOST341194(data).digest())
  43     >>> pub = public_key(curve, prv)
  44     >>> verify(curve, pub, GOST341194(data).digest(), signature)
  45     True
  46
  47     :param long p: characteristic of the underlying prime field
  48     :param long q: elliptic curve subgroup order
  49     :param long a, b: coefficients of the equation of the elliptic curve in
  50                       the canonical form
  51     :param long x, y: the coordinate of the point P (generator of the
  52                       subgroup of order q) of the elliptic curve in
  53                       the canonical form
  54     :param long e, d: coefficients of the equation of the elliptic curve in
  55                       the twisted Edwards form
  56     :param str name: human-readable curve name
  57     """
  58
  59     def __init__(self, p, q, a, b, x, y, cofactor=1, e=None, d=None, name=None):
  60         self.p = p
  61         self.q = q
  62         self.a = a
  63         self.b = b
  64         self.x = x
  65         self.y = y
  66         self.cofactor = cofactor
  67         self.e = e
  68         self.d = d
  69         if not self.contains((x, y)):
  70             raise ValueError("Invalid parameters")
  71         self._st = None
  72         self.name = name
  73
  74     @property
  75     def point_size(self):
  76         return point_size(self.p)
  77
  78     def __repr__(self):
  79         return "<%s: %s>" % (self.__class__.__name__, self.name)
  80
  81     def pos(self, v):
  82         """Make positive number
  83         """
  84         if v < 0:
  85             return v + self.p
  86         return v
  87
  88     def contains(self, point):
  89         """Is point on the curve?
  90
  91         :type point: (long, long)
  92         """
  93         x, y = point
  94         r1 = y * y % self.p
  95         r2 = ((x * x + self.a) * x + self.b) % self.p
  96         return r1 == self.pos(r2)
  97
  98     def _add(self, p1x, p1y, p2x, p2y):
  99         if p1x == p2x and p1y == p2y:
 100             # double
 101             t = ((3 * p1x * p1x + self.a) * modinvert(2 * p1y, self.p)) % self.p
 102         else:
 103             tx = self.pos(p2x - p1x) % self.p
 104             ty = self.pos(p2y - p1y) % self.p
 105             t = (ty * modinvert(tx, self.p)) % self.p
 106         tx = self.pos(t * t - p1x - p2x) % self.p
 107         ty = self.pos(t * (p1x - tx) - p1y) % self.p
 108         return tx, ty
 109
 110     def exp(self, degree, x=None, y=None):
 111         x = x or self.x
 112         y = y or self.y
 113         tx = x
 114         ty = y
 115         if degree == 0:
 116             raise ValueError("Bad degree value")
 117         degree -= 1
 118         while degree != 0:
 119             if degree & 1 == 1:
 120                 tx, ty = self._add(tx, ty, x, y)
 121             degree = degree >> 1
 122             x, y = self._add(x, y, x, y)
 123         return tx, ty
 124
 125     def st(self):
 126         """Compute s/t parameters for twisted Edwards curve points conversion
 127         """
 128         if self.e is None or self.d is None:
 129             raise ValueError("Non twisted Edwards curve")
 130         if self._st is not None:
 131             return self._st
 132         self._st = (
 133             self.pos(self.e - self.d) * modinvert(4, self.p) % self.p,
 134             (self.e + self.d) * modinvert(6, self.p) % self.p,
 135         )
 136         return self._st
 137
 138
 139 CURVES = {
 140     "GostR3410_2001_ParamSet_cc": GOST3410Curve(
 141         p=bytes2long(hexdec("C0000000000000000000000000000000000000000000000000000000000003C7")),
 142         q=bytes2long(hexdec("5fffffffffffffffffffffffffffffff606117a2f4bde428b7458a54b6e87b85")),
 143         a=bytes2long(hexdec("C0000000000000000000000000000000000000000000000000000000000003c4")),
 144         b=bytes2long(hexdec("2d06B4265ebc749ff7d0f1f1f88232e81632e9088fd44b7787d5e407e955080c")),
 145         x=bytes2long(hexdec("0000000000000000000000000000000000000000000000000000000000000002")),
 146         y=bytes2long(hexdec("a20e034bf8813ef5c18d01105e726a17eb248b264ae9706f440bedc8ccb6b22c")),
 147     ),
 148     "id-GostR3410-2001-TestParamSet": GOST3410Curve(
 149         p=bytes2long(hexdec("8000000000000000000000000000000000000000000000000000000000000431")),
 150         q=bytes2long(hexdec("8000000000000000000000000000000150FE8A1892976154C59CFC193ACCF5B3")),
 151         a=bytes2long(hexdec("0000000000000000000000000000000000000000000000000000000000000007")),
 152         b=bytes2long(hexdec("5FBFF498AA938CE739B8E022FBAFEF40563F6E6A3472FC2A514C0CE9DAE23B7E")),
 153         x=bytes2long(hexdec("0000000000000000000000000000000000000000000000000000000000000002")),
 154         y=bytes2long(hexdec("08E2A8A0E65147D4BD6316030E16D19C85C97F0A9CA267122B96ABBCEA7E8FC8")),
 155     ),
 156     "id-tc26-gost-3410-12-256-paramSetA": GOST3410Curve(
 157         p=bytes2long(hexdec("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFD97")),
 158         q=bytes2long(hexdec("400000000000000000000000000000000FD8CDDFC87B6635C115AF556C360C67")),
 159         a=bytes2long(hexdec("C2173F1513981673AF4892C23035A27CE25E2013BF95AA33B22C656F277E7335")),
 160         b=bytes2long(hexdec("295F9BAE7428ED9CCC20E7C359A9D41A22FCCD9108E17BF7BA9337A6F8AE9513")),
 161         x=bytes2long(hexdec("91E38443A5E82C0D880923425712B2BB658B9196932E02C78B2582FE742DAA28")),
 162         y=bytes2long(hexdec("32879423AB1A0375895786C4BB46E9565FDE0B5344766740AF268ADB32322E5C")),
 163         cofactor=4,
 164         e=0x01,
 165         d=bytes2long(hexdec("0605F6B7C183FA81578BC39CFAD518132B9DF62897009AF7E522C32D6DC7BFFB")),
 166     ),
 167     "id-tc26-gost-3410-12-256-paramSetB": GOST3410Curve(
 168         p=bytes2long(hexdec("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFD97")),
 169         q=bytes2long(hexdec("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF6C611070995AD10045841B09B761B893")),
 170         a=bytes2long(hexdec("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFD94")),
 171         b=bytes2long(hexdec("00000000000000000000000000000000000000000000000000000000000000a6")),
 172         x=bytes2long(hexdec("0000000000000000000000000000000000000000000000000000000000000001")),
 173         y=bytes2long(hexdec("8D91E471E0989CDA27DF505A453F2B7635294F2DDF23E3B122ACC99C9E9F1E14")),
 174     ),
 175     "id-tc26-gost-3410-12-256-paramSetC": GOST3410Curve(
 176         p=bytes2long(hexdec("8000000000000000000000000000000000000000000000000000000000000C99")),
 177         q=bytes2long(hexdec("800000000000000000000000000000015F700CFFF1A624E5E497161BCC8A198F")),
 178         a=bytes2long(hexdec("8000000000000000000000000000000000000000000000000000000000000C96")),
 179         b=bytes2long(hexdec("3E1AF419A269A5F866A7D3C25C3DF80AE979259373FF2B182F49D4CE7E1BBC8B")),
 180         x=bytes2long(hexdec("0000000000000000000000000000000000000000000000000000000000000001")),
 181         y=bytes2long(hexdec("3FA8124359F96680B83D1C3EB2C070E5C545C9858D03ECFB744BF8D717717EFC")),
 182     ),
 183     "id-tc26-gost-3410-12-256-paramSetD": GOST3410Curve(
 184         p=bytes2long(hexdec("9B9F605F5A858107AB1EC85E6B41C8AACF846E86789051D37998F7B9022D759B")),
 185         q=bytes2long(hexdec("9B9F605F5A858107AB1EC85E6B41C8AA582CA3511EDDFB74F02F3A6598980BB9")),
 186         a=bytes2long(hexdec("9B9F605F5A858107AB1EC85E6B41C8AACF846E86789051D37998F7B9022D7598")),
 187         b=bytes2long(hexdec("000000000000000000000000000000000000000000000000000000000000805a")),
 188         x=bytes2long(hexdec("0000000000000000000000000000000000000000000000000000000000000000")),
 189         y=bytes2long(hexdec("41ECE55743711A8C3CBF3783CD08C0EE4D4DC440D4641A8F366E550DFDB3BB67")),
 190     ),
 191     "id-tc26-gost-3410-12-512-paramSetTest": GOST3410Curve(
 192         p=bytes2long(hexdec("4531ACD1FE0023C7550D267B6B2FEE80922B14B2FFB90F04D4EB7C09B5D2D15DF1D852741AF4704A0458047E80E4546D35B8336FAC224DD81664BBF528BE6373")),
 193         q=bytes2long(hexdec("4531ACD1FE0023C7550D267B6B2FEE80922B14B2FFB90F04D4EB7C09B5D2D15DA82F2D7ECB1DBAC719905C5EECC423F1D86E25EDBE23C595D644AAF187E6E6DF")),
 194         a=7,
 195         b=bytes2long(hexdec("1CFF0806A31116DA29D8CFA54E57EB748BC5F377E49400FDD788B649ECA1AC4361834013B2AD7322480A89CA58E0CF74BC9E540C2ADD6897FAD0A3084F302ADC")),
 196         x=bytes2long(hexdec("24D19CC64572EE30F396BF6EBBFD7A6C5213B3B3D7057CC825F91093A68CD762FD60611262CD838DC6B60AA7EEE804E28BC849977FAC33B4B530F1B120248A9A")),
 197         y=bytes2long(hexdec("2BB312A43BD2CE6E0D020613C857ACDDCFBF061E91E5F2C3F32447C259F39B2C83AB156D77F1496BF7EB3351E1EE4E43DC1A18B91B24640B6DBB92CB1ADD371E")),
 198     ),
 199     "id-tc26-gost-3410-12-512-paramSetA": GOST3410Curve(
 200         p=bytes2long(hexdec("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFDC7")),
 201         q=bytes2long(hexdec("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF27E69532F48D89116FF22B8D4E0560609B4B38ABFAD2B85DCACDB1411F10B275")),
 202         a=bytes2long(hexdec("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFDC4")),
 203         b=bytes2long(hexdec("E8C2505DEDFC86DDC1BD0B2B6667F1DA34B82574761CB0E879BD081CFD0B6265EE3CB090F30D27614CB4574010DA90DD862EF9D4EBEE4761503190785A71C760")),
 204         x=bytes2long(hexdec("00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000003")),
 205         y=bytes2long(hexdec("7503CFE87A836AE3A61B8816E25450E6CE5E1C93ACF1ABC1778064FDCBEFA921DF1626BE4FD036E93D75E6A50E3A41E98028FE5FC235F5B889A589CB5215F2A4")),
 206     ),
 207     "id-tc26-gost-3410-12-512-paramSetB": GOST3410Curve(
 208         p=bytes2long(hexdec("8000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000006F")),
 209         q=bytes2long(hexdec("800000000000000000000000000000000000000000000000000000000000000149A1EC142565A545ACFDB77BD9D40CFA8B996712101BEA0EC6346C54374F25BD")),
 210         a=bytes2long(hexdec("8000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000006C")),
 211         b=bytes2long(hexdec("687D1B459DC841457E3E06CF6F5E2517B97C7D614AF138BCBF85DC806C4B289F3E965D2DB1416D217F8B276FAD1AB69C50F78BEE1FA3106EFB8CCBC7C5140116")),
 212         x=bytes2long(hexdec("00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000002")),
 213         y=bytes2long(hexdec("1A8F7EDA389B094C2C071E3647A8940F3C123B697578C213BE6DD9E6C8EC7335DCB228FD1EDF4A39152CBCAAF8C0398828041055F94CEEEC7E21340780FE41BD")),
 214     ),
 215     "id-tc26-gost-3410-12-512-paramSetC": GOST3410Curve(
 216         p=bytes2long(hexdec("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFDC7")),
 217         q=bytes2long(hexdec("3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC98CDBA46506AB004C33A9FF5147502CC8EDA9E7A769A12694623CEF47F023ED")),
 218         a=bytes2long(hexdec("DC9203E514A721875485A529D2C722FB187BC8980EB866644DE41C68E143064546E861C0E2C9EDD92ADE71F46FCF50FF2AD97F951FDA9F2A2EB6546F39689BD3")),
 219         b=bytes2long(hexdec("B4C4EE28CEBC6C2C8AC12952CF37F16AC7EFB6A9F69F4B57FFDA2E4F0DE5ADE038CBC2FFF719D2C18DE0284B8BFEF3B52B8CC7A5F5BF0A3C8D2319A5312557E1")),
 220         x=bytes2long(hexdec("E2E31EDFC23DE7BDEBE241CE593EF5DE2295B7A9CBAEF021D385F7074CEA043AA27272A7AE602BF2A7B9033DB9ED3610C6FB85487EAE97AAC5BC7928C1950148")),
 221         y=bytes2long(hexdec("F5CE40D95B5EB899ABBCCFF5911CB8577939804D6527378B8C108C3D2090FF9BE18E2D33E3021ED2EF32D85822423B6304F726AA854BAE07D0396E9A9ADDC40F")),
 222         cofactor=4,
 223         e=0x01,
 224         d=bytes2long(hexdec("9E4F5D8C017D8D9F13A5CF3CDF5BFE4DAB402D54198E31EBDE28A0621050439CA6B39E0A515C06B304E2CE43E79E369E91A0CFC2BC2A22B4CA302DBB33EE7550")),
 225     ),
 226 }
 227 CURVES["id-GostR3410-2001-CryptoPro-A-ParamSet"] = CURVES["id-tc26-gost-3410-12-256-paramSetB"]
 228 CURVES["id-GostR3410-2001-CryptoPro-B-ParamSet"] = CURVES["id-tc26-gost-3410-12-256-paramSetC"]
 229 CURVES["id-GostR3410-2001-CryptoPro-C-ParamSet"] = CURVES["id-tc26-gost-3410-12-256-paramSetD"]
 230 CURVES["id-GostR3410-2001-CryptoPro-XchA-ParamSet"] = CURVES["id-GostR3410-2001-CryptoPro-A-ParamSet"]
 231 CURVES["id-GostR3410-2001-CryptoPro-XchB-ParamSet"] = CURVES["id-GostR3410-2001-CryptoPro-C-ParamSet"]
 232 CURVES["id-tc26-gost-3410-2012-256-paramSetA"] = CURVES["id-tc26-gost-3410-12-256-paramSetA"]
 233 CURVES["id-tc26-gost-3410-2012-256-paramSetB"] = CURVES["id-tc26-gost-3410-12-256-paramSetB"]
 234 CURVES["id-tc26-gost-3410-2012-256-paramSetC"] = CURVES["id-tc26-gost-3410-12-256-paramSetC"]
 235 CURVES["id-tc26-gost-3410-2012-256-paramSetD"] = CURVES["id-tc26-gost-3410-12-256-paramSetD"]
 236 CURVES["id-tc26-gost-3410-2012-512-paramSetTest"] = CURVES["id-tc26-gost-3410-12-512-paramSetTest"]
 237 CURVES["id-tc26-gost-3410-2012-512-paramSetA"] = CURVES["id-tc26-gost-3410-12-512-paramSetA"]
 238 CURVES["id-tc26-gost-3410-2012-512-paramSetB"] = CURVES["id-tc26-gost-3410-12-512-paramSetB"]
 239 CURVES["id-tc26-gost-3410-2012-512-paramSetC"] = CURVES["id-tc26-gost-3410-12-512-paramSetC"]
 240 for _name, _curve in CURVES.items():
 241     _curve.name = _name
 242 DEFAULT_CURVE = CURVES["id-tc26-gost-3410-12-256-paramSetB"]
 243
 244
 245 def public_key(curve, prv):
 246     """Generate public key from the private one
 247
 248     :param GOST3410Curve curve: curve to use
 249     :param long prv: private key
 250     :returns: public key's parts, X and Y
 251     :rtype: (long, long)
 252     """
 253     return curve.exp(prv)
 254
 255
 256 def sign(curve, prv, digest, rand=None):
 257     """Calculate signature for provided digest
 258
 259     :param GOST3410Curve curve: curve to use
 260     :param long prv: private key
 261     :param digest: digest for signing
 262     :type digest: bytes, 32 or 64 bytes
 263     :param rand: optional predefined random data used for k/r generation
 264     :type rand: bytes, 32 or 64 bytes
 265     :returns: signature, BE(S) || BE(R)
 266     :rtype: bytes, 64 or 128 bytes
 267     """
 268     size = curve.point_size
 269     q = curve.q
 270     e = bytes2long(digest) % q
 271     if e == 0:
 272         e = 1
 273     while True:
 274         if rand is None:
 275             rand = urandom(size)
 276         elif len(rand) != size:
 277             raise ValueError("rand length != %d" % size)
 278         k = bytes2long(rand) % q
 279         if k == 0:
 280             continue
 281         r, _ = curve.exp(k)
 282         r %= q
 283         if r == 0:
 284             continue
 285         d = prv * r
 286         k *= e
 287         s = (d + k) % q
 288         if s == 0:
 289             continue
 290         break
 291     return long2bytes(s, size) + long2bytes(r, size)
 292
 293
 294 def verify(curve, pub, digest, signature):
 295     """Verify provided digest with the signature
 296
 297     :param GOST3410Curve curve: curve to use
 298     :type pub: (long, long)
 299     :param digest: digest needed to check
 300     :type digest: bytes, 32 or 64 bytes
 301     :param signature: signature to verify with
 302     :type signature: bytes, 64 or 128 bytes
 303     :rtype: bool
 304     """
 305     size = curve.point_size
 306     if len(signature) != size * 2:
 307         raise ValueError("Invalid signature length")
 308     q = curve.q
 309     p = curve.p
 310     s = bytes2long(signature[:size])
 311     r = bytes2long(signature[size:])
 312     if r <= 0 or r >= q or s <= 0 or s >= q:
 313         return False
 314     e = bytes2long(digest) % curve.q
 315     if e == 0:
 316         e = 1
 317     v = modinvert(e, q)
 318     z1 = s * v % q
 319     z2 = q - r * v % q
 320     p1x, p1y = curve.exp(z1)
 321     q1x, q1y = curve.exp(z2, pub[0], pub[1])
 322     lm = q1x - p1x
 323     if lm < 0:
 324         lm += p
 325     lm = modinvert(lm, p)
 326     z1 = q1y - p1y
 327     lm = lm * z1 % p
 328     lm = lm * lm % p
 329     lm = lm - p1x - q1x
 330     lm = lm % p
 331     if lm < 0:
 332         lm += p
 333     lm %= q
 334     # This is not constant time comparison!
 335     return lm == r
 336
 337
 338 def prv_unmarshal(prv):
 339     """Unmarshal little-endian private key
 340
 341     :param bytes prv: serialized private key
 342     :rtype: long
 343
 344     It is advisable to use :py:func:`pygost.gost3410.prv_marshal` to
 345     assure that key i in curve's Q field for better compatibility with
 346     some implementations.
 347     """
 348     return bytes2long(prv[::-1])
 349
 350
 351 def prv_marshal(curve, prv):
 352     """Marshal little-endian private key
 353
 354     :param GOST3410Curve curve: curve to use
 355     :param long prv: serialized private key
 356     :rtype: bytes
 357
 358     Key is in curve's Q field.
 359     """
 360     return long2bytes(prv % curve.q, point_size(prv))[::-1]
 361
 362
 363 def pub_marshal(pub):
 364     """Marshal public key
 365
 366     :type pub: (long, long)
 367     :rtype: bytes
 368     :returns: LE(X) || LE(Y)
 369     """
 370     size = point_size(pub[0])
 371     return (long2bytes(pub[1], size) + long2bytes(pub[0], size))[::-1]
 372
 373
 374 def pub_unmarshal(pub):
 375     """Unmarshal public key
 376
 377     :param pub: LE(X) || LE(Y)
 378     :type pub: bytes
 379     :rtype: (long, long)
 380     """
 381     size = len(pub) // 2
 382     pub = pub[::-1]
 383     return (bytes2long(pub[size:]), bytes2long(pub[:size]))
 384
 385
 386 def uv2xy(curve, u, v):
 387     """Convert twisted Edwards curve U,V coordinates to Weierstrass X,Y
 388     """
 389     s, t = curve.st()
 390     k1 = (s * (1 + v)) % curve.p
 391     k2 = curve.pos(1 - v)
 392     x = t + k1 * modinvert(k2, curve.p)
 393     y = k1 * modinvert(u * k2, curve.p)
 394     return x % curve.p, y % curve.p
 395
 396
 397 def xy2uv(curve, x, y):
 398     """Convert Weierstrass X,Y coordinates to twisted Edwards curve U,V
 399     """
 400     s, t = curve.st()
 401     xmt = curve.pos(x - t)
 402     u = xmt * modinvert(y, curve.p)
 403     v = curve.pos(xmt - s) * modinvert(xmt + s, curve.p)
 404     return u % curve.p, v % curve.p