aboutsummaryrefslogtreecommitdiff
path: root/frozen_deps/ecdsa/keys.py
blob: f74252c7989cba2f5d82eba4e666747f8daf8d22 (plain) (blame)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
1138
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
1171
1172
1173
1174
1175
1176
1177
1178
1179
1180
1181
1182
1183
1184
1185
1186
1187
1188
1189
1190
1191
1192
1193
1194
1195
1196
1197
1198
1199
1200
1201
1202
1203
1204
1205
1206
1207
1208
1209
1210
1211
1212
1213
1214
1215
1216
1217
1218
1219
1220
1221
1222
1223
1224
1225
1226
1227
1228
1229
1230
1231
1232
1233
1234
1235
1236
1237
1238
1239
1240
1241
1242
1243
1244
1245
1246
1247
1248
1249
1250
1251
1252
1253
1254
1255
1256
1257
1258
1259
1260
1261
1262
1263
1264
1265
1266
1267
1268
1269
1270
1271
1272
1273
1274
1275
1276
1277
1278
1279
1280
1281
1282
1283
1284
1285
1286
1287
1288
1289
1290
1291
1292
1293
1294
1295
1296
1297
1298
1299
1300
1301
1302
1303
1304
1305
1306
1307
1308
1309
1310
1311
1312
1313
1314
1315
1316
1317
1318
1319
1320
1321
1322
1323
1324
1325
1326
1327
1328
1329
1330
1331
1332
1333
1334
1335
1336
1337
1338
1339
1340
1341
1342
1343
1344
1345
1346
1347
1348
1349
1350
1351
1352
1353
1354
1355
1356
1357
1358
1359
1360
1361
1362
1363
1364
1365
1366
1367
1368
1369
1370
1371
1372
1373
1374
1375
1376
1377
1378
1379
1380
1381
1382
1383
1384
1385
1386
1387
1388
1389
1390
1391
1392
1393
1394
1395
1396
1397
1398
1399
1400
1401
1402
1403
1404
1405
1406
1407
1408
1409
1410
1411
1412
1413
1414
1415
1416
1417
1418
1419
1420
1421
1422
1423
1424
1425
1426
1427
1428
1429
1430
1431
1432
1433
1434
1435
1436
1437
1438
1439
1440
1441
1442
1443
1444
1445
1446
1447
1448
1449
1450
1451
1452
1453
1454
1455
1456
1457
1458
1459
1460
1461
1462
1463
1464
1465
1466
1467
1468
1469
1470
1471
1472
1473
1474
1475
1476
1477
1478
1479
1480
1481
1482
1483
1484
1485
1486
1487
1488
1489
1490
1491
1492
1493
1494
1495
1496
1497
1498
1499
1500
1501
1502
1503
1504
1505
1506
1507
1508
1509
1510
1511
1512
1513
1514
1515
1516
1517
1518
1519
1520
1521
1522
1523
1524
1525
1526
1527
1528
1529
1530
1531
1532
1533
1534
1535
1536
1537
1538
1539
1540
1541
1542
1543
1544
1545
1546
1547
1548
1549
1550
1551
1552
1553
1554
1555
1556
1557
1558
1559
1560
1561
1562
1563
1564
1565
1566
1567
1568
1569
1570
1571
1572
1573
1574
1575
1576
1577
1578
1579
1580
1581
1582
1583
1584
1585
1586
1587
1588
1589
1590
1591
1592
1593
1594
1595
1596
1597
1598
1599
1600
1601
1602
1603
1604
1605
1606
1607
1608
1609
1610
1611
1612
1613
1614
1615
1616
1617
1618
1619
1620
1621
1622
1623
1624
1625
1626
1627
1628
1629
1630
1631
"""
Primary classes for performing signing and verification operations.
"""

import binascii
from hashlib import sha1
import os
from six import PY2
from . import ecdsa, eddsa
from . import der, ssh
from . import rfc6979
from . import ellipticcurve
from .curves import NIST192p, Curve, Ed25519, Ed448
from .ecdsa import RSZeroError
from .util import string_to_number, number_to_string, randrange
from .util import sigencode_string, sigdecode_string, bit_length
from .util import (
    oid_ecPublicKey,
    encoded_oid_ecPublicKey,
    oid_ecDH,
    oid_ecMQV,
    MalformedSignature,
)
from ._compat import normalise_bytes
from .errors import MalformedPointError
from .ellipticcurve import PointJacobi, CurveEdTw


__all__ = [
    "BadSignatureError",
    "BadDigestError",
    "VerifyingKey",
    "SigningKey",
    "MalformedPointError",
]


class BadSignatureError(Exception):
    """
    Raised when verification of signature failed.

    Will be raised irrespective of reason of the failure:

    * the calculated or provided hash does not match the signature
    * the signature does not match the curve/public key
    * the encoding of the signature is malformed
    * the size of the signature does not match the curve of the VerifyingKey
    """

    pass


class BadDigestError(Exception):
    """Raised in case the selected hash is too large for the curve."""

    pass


def _truncate_and_convert_digest(digest, curve, allow_truncate):
    """Truncates and converts digest to an integer."""
    if not allow_truncate:
        if len(digest) > curve.baselen:
            raise BadDigestError(
                "this curve ({0}) is too short "
                "for the length of your digest ({1})".format(
                    curve.name, 8 * len(digest)
                )
            )
    else:
        digest = digest[: curve.baselen]
    number = string_to_number(digest)
    if allow_truncate:
        max_length = bit_length(curve.order)
        # we don't use bit_length(number) as that truncates leading zeros
        length = len(digest) * 8

        # See NIST FIPS 186-4:
        #
        # When the length of the output of the hash function is greater
        # than N (i.e., the bit length of q), then the leftmost N bits of
        # the hash function output block shall be used in any calculation
        # using the hash function output during the generation or
        # verification of a digital signature.
        #
        # as such, we need to shift-out the low-order bits:
        number >>= max(0, length - max_length)

    return number


class VerifyingKey(object):
    """
    Class for handling keys that can verify signatures (public keys).

    :ivar `~ecdsa.curves.Curve` ~.curve: The Curve over which all the
        cryptographic operations will take place
    :ivar default_hashfunc: the function that will be used for hashing the
        data. Should implement the same API as hashlib.sha1
    :vartype default_hashfunc: callable
    :ivar pubkey: the actual public key
    :vartype pubkey: ~ecdsa.ecdsa.Public_key
    """

    def __init__(self, _error__please_use_generate=None):
        """Unsupported, please use one of the classmethods to initialise."""
        if not _error__please_use_generate:
            raise TypeError(
                "Please use VerifyingKey.generate() to construct me"
            )
        self.curve = None
        self.default_hashfunc = None
        self.pubkey = None

    def __repr__(self):
        pub_key = self.to_string("compressed")
        if self.default_hashfunc:
            hash_name = self.default_hashfunc().name
        else:
            hash_name = "None"
        return "VerifyingKey.from_string({0!r}, {1!r}, {2})".format(
            pub_key, self.curve, hash_name
        )

    def __eq__(self, other):
        """Return True if the points are identical, False otherwise."""
        if isinstance(other, VerifyingKey):
            return self.curve == other.curve and self.pubkey == other.pubkey
        return NotImplemented

    def __ne__(self, other):
        """Return False if the points are identical, True otherwise."""
        return not self == other

    @classmethod
    def from_public_point(
        cls, point, curve=NIST192p, hashfunc=sha1, validate_point=True
    ):
        """
        Initialise the object from a Point object.

        This is a low-level method, generally you will not want to use it.

        :param point: The point to wrap around, the actual public key
        :type point: ~ecdsa.ellipticcurve.AbstractPoint
        :param curve: The curve on which the point needs to reside, defaults
            to NIST192p
        :type curve: ~ecdsa.curves.Curve
        :param hashfunc: The default hash function that will be used for
            verification, needs to implement the same interface
            as :py:class:`hashlib.sha1`
        :type hashfunc: callable
        :type bool validate_point: whether to check if the point lays on curve
            should always be used if the public point is not a result
            of our own calculation

        :raises MalformedPointError: if the public point does not lay on the
            curve

        :return: Initialised VerifyingKey object
        :rtype: VerifyingKey
        """
        self = cls(_error__please_use_generate=True)
        if isinstance(curve.curve, CurveEdTw):
            raise ValueError("Method incompatible with Edwards curves")
        if not isinstance(point, ellipticcurve.PointJacobi):
            point = ellipticcurve.PointJacobi.from_affine(point)
        self.curve = curve
        self.default_hashfunc = hashfunc
        try:
            self.pubkey = ecdsa.Public_key(
                curve.generator, point, validate_point
            )
        except ecdsa.InvalidPointError:
            raise MalformedPointError("Point does not lay on the curve")
        self.pubkey.order = curve.order
        return self

    def precompute(self, lazy=False):
        """
        Precompute multiplication tables for faster signature verification.

        Calling this method will cause the library to precompute the
        scalar multiplication tables, used in signature verification.
        While it's an expensive operation (comparable to performing
        as many signatures as the bit size of the curve, i.e. 256 for NIST256p)
        it speeds up verification 2 times. You should call this method
        if you expect to verify hundreds of signatures (or more) using the same
        VerifyingKey object.

        Note: You should call this method only once, this method generates a
        new precomputation table every time it's called.

        :param bool lazy: whether to calculate the precomputation table now
           (if set to False) or if it should be delayed to the time of first
           use (when set to True)
        """
        if isinstance(self.curve.curve, CurveEdTw):
            pt = self.pubkey.point
            self.pubkey.point = ellipticcurve.PointEdwards(
                pt.curve(),
                pt.x(),
                pt.y(),
                1,
                pt.x() * pt.y(),
                self.curve.order,
                generator=True,
            )
        else:
            self.pubkey.point = ellipticcurve.PointJacobi.from_affine(
                self.pubkey.point, True
            )
        # as precomputation in now delayed to the time of first use of the
        # point and we were asked specifically to precompute now, make
        # sure the precomputation is performed now to preserve the behaviour
        if not lazy:
            self.pubkey.point * 2

    @classmethod
    def from_string(
        cls,
        string,
        curve=NIST192p,
        hashfunc=sha1,
        validate_point=True,
        valid_encodings=None,
    ):
        """
        Initialise the object from byte encoding of public key.

        The method does accept and automatically detect the type of point
        encoding used. It supports the :term:`raw encoding`,
        :term:`uncompressed`, :term:`compressed`, and :term:`hybrid` encodings.
        It also works with the native encoding of Ed25519 and Ed448 public
        keys (technically those are compressed, but encoded differently than
        in other signature systems).

        Note, while the method is named "from_string" it's a misnomer from
        Python 2 days when there were no binary strings. In Python 3 the
        input needs to be a bytes-like object.

        :param string: single point encoding of the public key
        :type string: :term:`bytes-like object`
        :param curve: the curve on which the public key is expected to lay
        :type curve: ~ecdsa.curves.Curve
        :param hashfunc: The default hash function that will be used for
            verification, needs to implement the same interface as
            hashlib.sha1. Ignored for EdDSA.
        :type hashfunc: callable
        :param validate_point: whether to verify that the point lays on the
            provided curve or not, defaults to True. Ignored for EdDSA.
        :type validate_point: bool
        :param valid_encodings: list of acceptable point encoding formats,
            supported ones are: :term:`uncompressed`, :term:`compressed`,
            :term:`hybrid`, and :term:`raw encoding` (specified with ``raw``
            name). All formats by default (specified with ``None``).
            Ignored for EdDSA.
        :type valid_encodings: :term:`set-like object`

        :raises MalformedPointError: if the public point does not lay on the
            curve or the encoding is invalid

        :return: Initialised VerifyingKey object
        :rtype: VerifyingKey
        """
        if isinstance(curve.curve, CurveEdTw):
            self = cls(_error__please_use_generate=True)
            self.curve = curve
            self.default_hashfunc = None  # ignored for EdDSA
            try:
                self.pubkey = eddsa.PublicKey(curve.generator, string)
            except ValueError:
                raise MalformedPointError("Malformed point for the curve")
            return self

        point = PointJacobi.from_bytes(
            curve.curve,
            string,
            validate_encoding=validate_point,
            valid_encodings=valid_encodings,
        )
        return cls.from_public_point(point, curve, hashfunc, validate_point)

    @classmethod
    def from_pem(
        cls,
        string,
        hashfunc=sha1,
        valid_encodings=None,
        valid_curve_encodings=None,
    ):
        """
        Initialise from public key stored in :term:`PEM` format.

        The PEM header of the key should be ``BEGIN PUBLIC KEY``.

        See the :func:`~VerifyingKey.from_der()` method for details of the
        format supported.

        Note: only a single PEM object decoding is supported in provided
        string.

        :param string: text with PEM-encoded public ECDSA key
        :type string: str
        :param valid_encodings: list of allowed point encodings.
            By default :term:`uncompressed`, :term:`compressed`, and
            :term:`hybrid`. To read malformed files, include
            :term:`raw encoding` with ``raw`` in the list.
        :type valid_encodings: :term:`set-like object`
        :param valid_curve_encodings: list of allowed encoding formats
            for curve parameters. By default (``None``) all are supported:
            ``named_curve`` and ``explicit``.
        :type valid_curve_encodings: :term:`set-like object`


        :return: Initialised VerifyingKey object
        :rtype: VerifyingKey
        """
        return cls.from_der(
            der.unpem(string),
            hashfunc=hashfunc,
            valid_encodings=valid_encodings,
            valid_curve_encodings=valid_curve_encodings,
        )

    @classmethod
    def from_der(
        cls,
        string,
        hashfunc=sha1,
        valid_encodings=None,
        valid_curve_encodings=None,
    ):
        """
        Initialise the key stored in :term:`DER` format.

        The expected format of the key is the SubjectPublicKeyInfo structure
        from RFC5912 (for RSA keys, it's known as the PKCS#1 format)::

           SubjectPublicKeyInfo {PUBLIC-KEY: IOSet} ::= SEQUENCE {
               algorithm        AlgorithmIdentifier {PUBLIC-KEY, {IOSet}},
               subjectPublicKey BIT STRING
           }

        Note: only public EC keys are supported by this method. The
        SubjectPublicKeyInfo.algorithm.algorithm field must specify
        id-ecPublicKey (see RFC3279).

        Only the named curve encoding is supported, thus the
        SubjectPublicKeyInfo.algorithm.parameters field needs to be an
        object identifier. A sequence in that field indicates an explicit
        parameter curve encoding, this format is not supported. A NULL object
        in that field indicates an "implicitlyCA" encoding, where the curve
        parameters come from CA certificate, those, again, are not supported.

        :param string: binary string with the DER encoding of public ECDSA key
        :type string: bytes-like object
        :param valid_encodings: list of allowed point encodings.
            By default :term:`uncompressed`, :term:`compressed`, and
            :term:`hybrid`. To read malformed files, include
            :term:`raw encoding` with ``raw`` in the list.
        :type valid_encodings: :term:`set-like object`
        :param valid_curve_encodings: list of allowed encoding formats
            for curve parameters. By default (``None``) all are supported:
            ``named_curve`` and ``explicit``.
        :type valid_curve_encodings: :term:`set-like object`

        :return: Initialised VerifyingKey object
        :rtype: VerifyingKey
        """
        if valid_encodings is None:
            valid_encodings = set(["uncompressed", "compressed", "hybrid"])
        string = normalise_bytes(string)
        # [[oid_ecPublicKey,oid_curve], point_str_bitstring]
        s1, empty = der.remove_sequence(string)
        if empty != b"":
            raise der.UnexpectedDER(
                "trailing junk after DER pubkey: %s" % binascii.hexlify(empty)
            )
        s2, point_str_bitstring = der.remove_sequence(s1)
        # s2 = oid_ecPublicKey,oid_curve
        oid_pk, rest = der.remove_object(s2)
        if oid_pk in (Ed25519.oid, Ed448.oid):
            if oid_pk == Ed25519.oid:
                curve = Ed25519
            else:
                assert oid_pk == Ed448.oid
                curve = Ed448
            point_str, empty = der.remove_bitstring(point_str_bitstring, 0)
            if empty:
                raise der.UnexpectedDER("trailing junk after public key")
            return cls.from_string(point_str, curve, None)
        if not oid_pk == oid_ecPublicKey:
            raise der.UnexpectedDER(
                "Unexpected object identifier in DER "
                "encoding: {0!r}".format(oid_pk)
            )
        curve = Curve.from_der(rest, valid_curve_encodings)
        point_str, empty = der.remove_bitstring(point_str_bitstring, 0)
        if empty != b"":
            raise der.UnexpectedDER(
                "trailing junk after pubkey pointstring: %s"
                % binascii.hexlify(empty)
            )
        # raw encoding of point is invalid in DER files
        if len(point_str) == curve.verifying_key_length:
            raise der.UnexpectedDER("Malformed encoding of public point")
        return cls.from_string(
            point_str,
            curve,
            hashfunc=hashfunc,
            valid_encodings=valid_encodings,
        )

    @classmethod
    def from_public_key_recovery(
        cls,
        signature,
        data,
        curve,
        hashfunc=sha1,
        sigdecode=sigdecode_string,
        allow_truncate=True,
    ):
        """
        Return keys that can be used as verifiers of the provided signature.

        Tries to recover the public key that can be used to verify the
        signature, usually returns two keys like that.

        :param signature: the byte string with the encoded signature
        :type signature: bytes-like object
        :param data: the data to be hashed for signature verification
        :type data: bytes-like object
        :param curve: the curve over which the signature was performed
        :type curve: ~ecdsa.curves.Curve
        :param hashfunc: The default hash function that will be used for
            verification, needs to implement the same interface as hashlib.sha1
        :type hashfunc: callable
        :param sigdecode: Callable to define the way the signature needs to
            be decoded to an object, needs to handle `signature` as the
            first parameter, the curve order (an int) as the second and return
            a tuple with two integers, "r" as the first one and "s" as the
            second one. See :func:`ecdsa.util.sigdecode_string` and
            :func:`ecdsa.util.sigdecode_der` for examples.
        :param bool allow_truncate: if True, the provided hashfunc can generate
            values larger than the bit size of the order of the curve, the
            extra bits (at the end of the digest) will be truncated.
        :type sigdecode: callable

        :return: Initialised VerifyingKey objects
        :rtype: list of VerifyingKey
        """
        if isinstance(curve.curve, CurveEdTw):
            raise ValueError("Method unsupported for Edwards curves")
        data = normalise_bytes(data)
        digest = hashfunc(data).digest()
        return cls.from_public_key_recovery_with_digest(
            signature,
            digest,
            curve,
            hashfunc=hashfunc,
            sigdecode=sigdecode,
            allow_truncate=allow_truncate,
        )

    @classmethod
    def from_public_key_recovery_with_digest(
        cls,
        signature,
        digest,
        curve,
        hashfunc=sha1,
        sigdecode=sigdecode_string,
        allow_truncate=False,
    ):
        """
        Return keys that can be used as verifiers of the provided signature.

        Tries to recover the public key that can be used to verify the
        signature, usually returns two keys like that.

        :param signature: the byte string with the encoded signature
        :type signature: bytes-like object
        :param digest: the hash value of the message signed by the signature
        :type digest: bytes-like object
        :param curve: the curve over which the signature was performed
        :type curve: ~ecdsa.curves.Curve
        :param hashfunc: The default hash function that will be used for
            verification, needs to implement the same interface as hashlib.sha1
        :type hashfunc: callable
        :param sigdecode: Callable to define the way the signature needs to
            be decoded to an object, needs to handle `signature` as the
            first parameter, the curve order (an int) as the second and return
            a tuple with two integers, "r" as the first one and "s" as the
            second one. See :func:`ecdsa.util.sigdecode_string` and
            :func:`ecdsa.util.sigdecode_der` for examples.
        :type sigdecode: callable
        :param bool allow_truncate: if True, the provided hashfunc can generate
            values larger than the bit size of the order of the curve (and
            the length of provided `digest`), the extra bits (at the end of the
            digest) will be truncated.

        :return: Initialised VerifyingKey object
        :rtype: VerifyingKey
        """
        if isinstance(curve.curve, CurveEdTw):
            raise ValueError("Method unsupported for Edwards curves")
        generator = curve.generator
        r, s = sigdecode(signature, generator.order())
        sig = ecdsa.Signature(r, s)

        digest = normalise_bytes(digest)
        digest_as_number = _truncate_and_convert_digest(
            digest, curve, allow_truncate
        )
        pks = sig.recover_public_keys(digest_as_number, generator)

        # Transforms the ecdsa.Public_key object into a VerifyingKey
        verifying_keys = [
            cls.from_public_point(pk.point, curve, hashfunc) for pk in pks
        ]
        return verifying_keys

    def to_string(self, encoding="raw"):
        """
        Convert the public key to a byte string.

        The method by default uses the :term:`raw encoding` (specified
        by `encoding="raw"`. It can also output keys in :term:`uncompressed`,
        :term:`compressed` and :term:`hybrid` formats.

        Remember that the curve identification is not part of the encoding
        so to decode the point using :func:`~VerifyingKey.from_string`, curve
        needs to be specified.

        Note: while the method is called "to_string", it's a misnomer from
        Python 2 days when character strings and byte strings shared type.
        On Python 3 the returned type will be `bytes`.

        :return: :term:`raw encoding` of the public key (public point) on the
            curve
        :rtype: bytes
        """
        assert encoding in ("raw", "uncompressed", "compressed", "hybrid")
        return self.pubkey.point.to_bytes(encoding)

    def to_pem(
        self, point_encoding="uncompressed", curve_parameters_encoding=None
    ):
        """
        Convert the public key to the :term:`PEM` format.

        The PEM header of the key will be ``BEGIN PUBLIC KEY``.

        The format of the key is described in the
        :func:`~VerifyingKey.from_der()` method.
        This method supports only "named curve" encoding of keys.

        :param str point_encoding: specification of the encoding format
            of public keys. "uncompressed" is most portable, "compressed" is
            smallest. "hybrid" is uncommon and unsupported by most
            implementations, it is as big as "uncompressed".
        :param str curve_parameters_encoding: the encoding for curve parameters
            to use, by default tries to use ``named_curve`` encoding,
            if that is not possible, falls back to ``explicit`` encoding.

        :return: portable encoding of the public key
        :rtype: bytes

        .. warning:: The PEM is encoded to US-ASCII, it needs to be
            re-encoded if the system is incompatible (e.g. uses UTF-16)
        """
        return der.topem(
            self.to_der(point_encoding, curve_parameters_encoding),
            "PUBLIC KEY",
        )

    def to_der(
        self, point_encoding="uncompressed", curve_parameters_encoding=None
    ):
        """
        Convert the public key to the :term:`DER` format.

        The format of the key is described in the
        :func:`~VerifyingKey.from_der()` method.
        This method supports only "named curve" encoding of keys.

        :param str point_encoding: specification of the encoding format
            of public keys. "uncompressed" is most portable, "compressed" is
            smallest. "hybrid" is uncommon and unsupported by most
            implementations, it is as big as "uncompressed".
        :param str curve_parameters_encoding: the encoding for curve parameters
            to use, by default tries to use ``named_curve`` encoding,
            if that is not possible, falls back to ``explicit`` encoding.

        :return: DER encoding of the public key
        :rtype: bytes
        """
        if point_encoding == "raw":
            raise ValueError("raw point_encoding not allowed in DER")
        point_str = self.to_string(point_encoding)
        if isinstance(self.curve.curve, CurveEdTw):
            return der.encode_sequence(
                der.encode_sequence(der.encode_oid(*self.curve.oid)),
                der.encode_bitstring(bytes(point_str), 0),
            )
        return der.encode_sequence(
            der.encode_sequence(
                encoded_oid_ecPublicKey,
                self.curve.to_der(curve_parameters_encoding, point_encoding),
            ),
            # 0 is the number of unused bits in the
            # bit string
            der.encode_bitstring(point_str, 0),
        )

    def to_ssh(self):
        """
        Convert the public key to the SSH format.

        :return: SSH encoding of the public key
        :rtype: bytes
        """
        return ssh.serialize_public(
            self.curve.name,
            self.to_string(),
        )

    def verify(
        self,
        signature,
        data,
        hashfunc=None,
        sigdecode=sigdecode_string,
        allow_truncate=True,
    ):
        """
        Verify a signature made over provided data.

        Will hash `data` to verify the signature.

        By default expects signature in :term:`raw encoding`. Can also be used
        to verify signatures in ASN.1 DER encoding by using
        :func:`ecdsa.util.sigdecode_der`
        as the `sigdecode` parameter.

        :param signature: encoding of the signature
        :type signature: sigdecode method dependent
        :param data: data signed by the `signature`, will be hashed using
            `hashfunc`, if specified, or default hash function
        :type data: :term:`bytes-like object`
        :param hashfunc: The default hash function that will be used for
            verification, needs to implement the same interface as hashlib.sha1
        :type hashfunc: callable
        :param sigdecode: Callable to define the way the signature needs to
            be decoded to an object, needs to handle `signature` as the
            first parameter, the curve order (an int) as the second and return
            a tuple with two integers, "r" as the first one and "s" as the
            second one. See :func:`ecdsa.util.sigdecode_string` and
            :func:`ecdsa.util.sigdecode_der` for examples.
        :type sigdecode: callable
        :param bool allow_truncate: if True, the provided digest can have
            bigger bit-size than the order of the curve, the extra bits (at
            the end of the digest) will be truncated. Use it when verifying
            SHA-384 output using NIST256p or in similar situations. Defaults to
            True.

        :raises BadSignatureError: if the signature is invalid or malformed

        :return: True if the verification was successful
        :rtype: bool
        """
        # signature doesn't have to be a bytes-like-object so don't normalise
        # it, the decoders will do that
        data = normalise_bytes(data)
        if isinstance(self.curve.curve, CurveEdTw):
            signature = normalise_bytes(signature)
            try:
                return self.pubkey.verify(data, signature)
            except (ValueError, MalformedPointError) as e:
                raise BadSignatureError("Signature verification failed", e)

        hashfunc = hashfunc or self.default_hashfunc
        digest = hashfunc(data).digest()
        return self.verify_digest(signature, digest, sigdecode, allow_truncate)

    def verify_digest(
        self,
        signature,
        digest,
        sigdecode=sigdecode_string,
        allow_truncate=False,
    ):
        """
        Verify a signature made over provided hash value.

        By default expects signature in :term:`raw encoding`. Can also be used
        to verify signatures in ASN.1 DER encoding by using
        :func:`ecdsa.util.sigdecode_der`
        as the `sigdecode` parameter.

        :param signature: encoding of the signature
        :type signature: sigdecode method dependent
        :param digest: raw hash value that the signature authenticates.
        :type digest: :term:`bytes-like object`
        :param sigdecode: Callable to define the way the signature needs to
            be decoded to an object, needs to handle `signature` as the
            first parameter, the curve order (an int) as the second and return
            a tuple with two integers, "r" as the first one and "s" as the
            second one. See :func:`ecdsa.util.sigdecode_string` and
            :func:`ecdsa.util.sigdecode_der` for examples.
        :type sigdecode: callable
        :param bool allow_truncate: if True, the provided digest can have
            bigger bit-size than the order of the curve, the extra bits (at
            the end of the digest) will be truncated. Use it when verifying
            SHA-384 output using NIST256p or in similar situations.

        :raises BadSignatureError: if the signature is invalid or malformed
        :raises BadDigestError: if the provided digest is too big for the curve
            associated with this VerifyingKey and allow_truncate was not set

        :return: True if the verification was successful
        :rtype: bool
        """
        # signature doesn't have to be a bytes-like-object so don't normalise
        # it, the decoders will do that
        digest = normalise_bytes(digest)
        number = _truncate_and_convert_digest(
            digest,
            self.curve,
            allow_truncate,
        )

        try:
            r, s = sigdecode(signature, self.pubkey.order)
        except (der.UnexpectedDER, MalformedSignature) as e:
            raise BadSignatureError("Malformed formatting of signature", e)
        sig = ecdsa.Signature(r, s)
        if self.pubkey.verifies(number, sig):
            return True
        raise BadSignatureError("Signature verification failed")


class SigningKey(object):
    """
    Class for handling keys that can create signatures (private keys).

    :ivar `~ecdsa.curves.Curve` curve: The Curve over which all the
        cryptographic operations will take place
    :ivar default_hashfunc: the function that will be used for hashing the
        data. Should implement the same API as :py:class:`hashlib.sha1`
    :ivar int baselen: the length of a :term:`raw encoding` of private key
    :ivar `~ecdsa.keys.VerifyingKey` verifying_key: the public key
        associated with this private key
    :ivar `~ecdsa.ecdsa.Private_key` privkey: the actual private key
    """

    def __init__(self, _error__please_use_generate=None):
        """Unsupported, please use one of the classmethods to initialise."""
        if not _error__please_use_generate:
            raise TypeError("Please use SigningKey.generate() to construct me")
        self.curve = None
        self.default_hashfunc = None
        self.baselen = None
        self.verifying_key = None
        self.privkey = None

    def __eq__(self, other):
        """Return True if the points are identical, False otherwise."""
        if isinstance(other, SigningKey):
            return (
                self.curve == other.curve
                and self.verifying_key == other.verifying_key
                and self.privkey == other.privkey
            )
        return NotImplemented

    def __ne__(self, other):
        """Return False if the points are identical, True otherwise."""
        return not self == other

    @classmethod
    def _twisted_edwards_keygen(cls, curve, entropy):
        """Generate a private key on a Twisted Edwards curve."""
        if not entropy:
            entropy = os.urandom
        random = entropy(curve.baselen)
        private_key = eddsa.PrivateKey(curve.generator, random)
        public_key = private_key.public_key()

        verifying_key = VerifyingKey.from_string(
            public_key.public_key(), curve
        )

        self = cls(_error__please_use_generate=True)
        self.curve = curve
        self.default_hashfunc = None
        self.baselen = curve.baselen
        self.privkey = private_key
        self.verifying_key = verifying_key
        return self

    @classmethod
    def _weierstrass_keygen(cls, curve, entropy, hashfunc):
        """Generate a private key on a Weierstrass curve."""
        secexp = randrange(curve.order, entropy)
        return cls.from_secret_exponent(secexp, curve, hashfunc)

    @classmethod
    def generate(cls, curve=NIST192p, entropy=None, hashfunc=sha1):
        """
        Generate a random private key.

        :param curve: The curve on which the point needs to reside, defaults
            to NIST192p
        :type curve: ~ecdsa.curves.Curve
        :param entropy: Source of randomness for generating the private keys,
            should provide cryptographically secure random numbers if the keys
            need to be secure. Uses os.urandom() by default.
        :type entropy: callable
        :param hashfunc: The default hash function that will be used for
            signing, needs to implement the same interface
            as hashlib.sha1
        :type hashfunc: callable

        :return: Initialised SigningKey object
        :rtype: SigningKey
        """
        if isinstance(curve.curve, CurveEdTw):
            return cls._twisted_edwards_keygen(curve, entropy)
        return cls._weierstrass_keygen(curve, entropy, hashfunc)

    @classmethod
    def from_secret_exponent(cls, secexp, curve=NIST192p, hashfunc=sha1):
        """
        Create a private key from a random integer.

        Note: it's a low level method, it's recommended to use the
        :func:`~SigningKey.generate` method to create private keys.

        :param int secexp: secret multiplier (the actual private key in ECDSA).
            Needs to be an integer between 1 and the curve order.
        :param curve: The curve on which the point needs to reside
        :type curve: ~ecdsa.curves.Curve
        :param hashfunc: The default hash function that will be used for
            signing, needs to implement the same interface
            as hashlib.sha1
        :type hashfunc: callable

        :raises MalformedPointError: when the provided secexp is too large
            or too small for the curve selected
        :raises RuntimeError: if the generation of public key from private
            key failed

        :return: Initialised SigningKey object
        :rtype: SigningKey
        """
        if isinstance(curve.curve, CurveEdTw):
            raise ValueError(
                "Edwards keys don't support setting the secret scalar "
                "(exponent) directly"
            )
        self = cls(_error__please_use_generate=True)
        self.curve = curve
        self.default_hashfunc = hashfunc
        self.baselen = curve.baselen
        n = curve.order
        if not 1 <= secexp < n:
            raise MalformedPointError(
                "Invalid value for secexp, expected integer "
                "between 1 and {0}".format(n)
            )
        pubkey_point = curve.generator * secexp
        if hasattr(pubkey_point, "scale"):
            pubkey_point = pubkey_point.scale()
        self.verifying_key = VerifyingKey.from_public_point(
            pubkey_point, curve, hashfunc, False
        )
        pubkey = self.verifying_key.pubkey
        self.privkey = ecdsa.Private_key(pubkey, secexp)
        self.privkey.order = n
        return self

    @classmethod
    def from_string(cls, string, curve=NIST192p, hashfunc=sha1):
        """
        Decode the private key from :term:`raw encoding`.

        Note: the name of this method is a misnomer coming from days of
        Python 2, when binary strings and character strings shared a type.
        In Python 3, the expected type is `bytes`.

        :param string: the raw encoding of the private key
        :type string: :term:`bytes-like object`
        :param curve: The curve on which the point needs to reside
        :type curve: ~ecdsa.curves.Curve
        :param hashfunc: The default hash function that will be used for
            signing, needs to implement the same interface
            as hashlib.sha1
        :type hashfunc: callable

        :raises MalformedPointError: if the length of encoding doesn't match
            the provided curve or the encoded values is too large
        :raises RuntimeError: if the generation of public key from private
            key failed

        :return: Initialised SigningKey object
        :rtype: SigningKey
        """
        string = normalise_bytes(string)

        if len(string) != curve.baselen:
            raise MalformedPointError(
                "Invalid length of private key, received {0}, "
                "expected {1}".format(len(string), curve.baselen)
            )
        if isinstance(curve.curve, CurveEdTw):
            self = cls(_error__please_use_generate=True)
            self.curve = curve
            self.default_hashfunc = None  # Ignored for EdDSA
            self.baselen = curve.baselen
            self.privkey = eddsa.PrivateKey(curve.generator, string)
            self.verifying_key = VerifyingKey.from_string(
                self.privkey.public_key().public_key(), curve
            )
            return self
        secexp = string_to_number(string)
        return cls.from_secret_exponent(secexp, curve, hashfunc)

    @classmethod
    def from_pem(cls, string, hashfunc=sha1, valid_curve_encodings=None):
        """
        Initialise from key stored in :term:`PEM` format.

        The PEM formats supported are the un-encrypted RFC5915
        (the ssleay format) supported by OpenSSL, and the more common
        un-encrypted RFC5958 (the PKCS #8 format).

        The legacy format files have the header with the string
        ``BEGIN EC PRIVATE KEY``.
        PKCS#8 files have the header ``BEGIN PRIVATE KEY``.
        Encrypted files (ones that include the string
        ``Proc-Type: 4,ENCRYPTED``
        right after the PEM header) are not supported.

        See :func:`~SigningKey.from_der` for ASN.1 syntax of the objects in
        this files.

        :param string: text with PEM-encoded private ECDSA key
        :type string: str
        :param valid_curve_encodings: list of allowed encoding formats
            for curve parameters. By default (``None``) all are supported:
            ``named_curve`` and ``explicit``.
        :type valid_curve_encodings: :term:`set-like object`


        :raises MalformedPointError: if the length of encoding doesn't match
            the provided curve or the encoded values is too large
        :raises RuntimeError: if the generation of public key from private
            key failed
        :raises UnexpectedDER: if the encoding of the PEM file is incorrect

        :return: Initialised SigningKey object
        :rtype: SigningKey
        """
        if not PY2 and isinstance(string, str):  # pragma: no branch
            string = string.encode()

        # The privkey pem may have multiple sections, commonly it also has
        # "EC PARAMETERS", we need just "EC PRIVATE KEY". PKCS#8 should not
        # have the "EC PARAMETERS" section; it's just "PRIVATE KEY".
        private_key_index = string.find(b"-----BEGIN EC PRIVATE KEY-----")
        if private_key_index == -1:
            private_key_index = string.index(b"-----BEGIN PRIVATE KEY-----")

        return cls.from_der(
            der.unpem(string[private_key_index:]),
            hashfunc,
            valid_curve_encodings,
        )

    @classmethod
    def from_der(cls, string, hashfunc=sha1, valid_curve_encodings=None):
        """
        Initialise from key stored in :term:`DER` format.

        The DER formats supported are the un-encrypted RFC5915
        (the ssleay format) supported by OpenSSL, and the more common
        un-encrypted RFC5958 (the PKCS #8 format).

        Both formats contain an ASN.1 object following the syntax specified
        in RFC5915::

            ECPrivateKey ::= SEQUENCE {
              version        INTEGER { ecPrivkeyVer1(1) }} (ecPrivkeyVer1),
              privateKey     OCTET STRING,
              parameters [0] ECParameters {{ NamedCurve }} OPTIONAL,
              publicKey  [1] BIT STRING OPTIONAL
            }

        `publicKey` field is ignored completely (errors, if any, in it will
        be undetected).

        Two formats are supported for the `parameters` field: the named
        curve and the explicit encoding of curve parameters.
        In the legacy ssleay format, this implementation requires the optional
        `parameters` field to get the curve name. In PKCS #8 format, the curve
        is part of the PrivateKeyAlgorithmIdentifier.

        The PKCS #8 format includes an ECPrivateKey object as the `privateKey`
        field within a larger structure::

            OneAsymmetricKey ::= SEQUENCE {
                version                   Version,
                privateKeyAlgorithm       PrivateKeyAlgorithmIdentifier,
                privateKey                PrivateKey,
                attributes            [0] Attributes OPTIONAL,
                ...,
                [[2: publicKey        [1] PublicKey OPTIONAL ]],
                ...
            }

        The `attributes` and `publicKey` fields are completely ignored; errors
        in them will not be detected.

        :param string: binary string with DER-encoded private ECDSA key
        :type string: :term:`bytes-like object`
        :param valid_curve_encodings: list of allowed encoding formats
            for curve parameters. By default (``None``) all are supported:
            ``named_curve`` and ``explicit``.
            Ignored for EdDSA.
        :type valid_curve_encodings: :term:`set-like object`

        :raises MalformedPointError: if the length of encoding doesn't match
            the provided curve or the encoded values is too large
        :raises RuntimeError: if the generation of public key from private
            key failed
        :raises UnexpectedDER: if the encoding of the DER file is incorrect

        :return: Initialised SigningKey object
        :rtype: SigningKey
        """
        s = normalise_bytes(string)
        curve = None

        s, empty = der.remove_sequence(s)
        if empty != b"":
            raise der.UnexpectedDER(
                "trailing junk after DER privkey: %s" % binascii.hexlify(empty)
            )

        version, s = der.remove_integer(s)

        # At this point, PKCS #8 has a sequence containing the algorithm
        # identifier and the curve identifier. The ssleay format instead has
        # an octet string containing the key data, so this is how we can
        # distinguish the two formats.
        if der.is_sequence(s):
            if version not in (0, 1):
                raise der.UnexpectedDER(
                    "expected version '0' or '1' at start of privkey, got %d"
                    % version
                )

            sequence, s = der.remove_sequence(s)
            algorithm_oid, algorithm_identifier = der.remove_object(sequence)

            if algorithm_oid in (Ed25519.oid, Ed448.oid):
                if algorithm_identifier:
                    raise der.UnexpectedDER(
                        "Non NULL parameters for a EdDSA key"
                    )
                key_str_der, s = der.remove_octet_string(s)

                # As RFC5958 describe, there are may be optional Attributes
                # and Publickey. Don't raise error if something after
                # Privatekey

                # TODO parse attributes or validate publickey
                # if s:
                #     raise der.UnexpectedDER(
                #         "trailing junk inside the privateKey"
                #     )
                key_str, s = der.remove_octet_string(key_str_der)
                if s:
                    raise der.UnexpectedDER(
                        "trailing junk after the encoded private key"
                    )

                if algorithm_oid == Ed25519.oid:
                    curve = Ed25519
                else:
                    assert algorithm_oid == Ed448.oid
                    curve = Ed448

                return cls.from_string(key_str, curve, None)

            if algorithm_oid not in (oid_ecPublicKey, oid_ecDH, oid_ecMQV):
                raise der.UnexpectedDER(
                    "unexpected algorithm identifier '%s'" % (algorithm_oid,)
                )

            curve = Curve.from_der(algorithm_identifier, valid_curve_encodings)

            # Up next is an octet string containing an ECPrivateKey. Ignore
            # the optional "attributes" and "publicKey" fields that come after.
            s, _ = der.remove_octet_string(s)

            # Unpack the ECPrivateKey to get to the key data octet string,
            # and rejoin the ssleay parsing path.
            s, empty = der.remove_sequence(s)
            if empty != b"":
                raise der.UnexpectedDER(
                    "trailing junk after DER privkey: %s"
                    % binascii.hexlify(empty)
                )

            version, s = der.remove_integer(s)

        # The version of the ECPrivateKey must be 1.
        if version != 1:
            raise der.UnexpectedDER(
                "expected version '1' at start of DER privkey, got %d"
                % version
            )

        privkey_str, s = der.remove_octet_string(s)

        if not curve:
            tag, curve_oid_str, s = der.remove_constructed(s)
            if tag != 0:
                raise der.UnexpectedDER(
                    "expected tag 0 in DER privkey, got %d" % tag
                )
            curve = Curve.from_der(curve_oid_str, valid_curve_encodings)

        # we don't actually care about the following fields
        #
        # tag, pubkey_bitstring, s = der.remove_constructed(s)
        # if tag != 1:
        #     raise der.UnexpectedDER("expected tag 1 in DER privkey, got %d"
        #                             % tag)
        # pubkey_str = der.remove_bitstring(pubkey_bitstring, 0)
        # if empty != "":
        #     raise der.UnexpectedDER("trailing junk after DER privkey "
        #                             "pubkeystr: %s"
        #                             % binascii.hexlify(empty))

        # our from_string method likes fixed-length privkey strings
        if len(privkey_str) < curve.baselen:
            privkey_str = (
                b"\x00" * (curve.baselen - len(privkey_str)) + privkey_str
            )
        return cls.from_string(privkey_str, curve, hashfunc)

    def to_string(self):
        """
        Convert the private key to :term:`raw encoding`.

        Note: while the method is named "to_string", its name comes from
        Python 2 days, when binary and character strings used the same type.
        The type used in Python 3 is `bytes`.

        :return: raw encoding of private key
        :rtype: bytes
        """
        if isinstance(self.curve.curve, CurveEdTw):
            return bytes(self.privkey.private_key)
        secexp = self.privkey.secret_multiplier
        s = number_to_string(secexp, self.privkey.order)
        return s

    def to_pem(
        self,
        point_encoding="uncompressed",
        format="ssleay",
        curve_parameters_encoding=None,
    ):
        """
        Convert the private key to the :term:`PEM` format.

        See :func:`~SigningKey.from_pem` method for format description.

        Only the named curve format is supported.
        The public key will be included in generated string.

        The PEM header will specify ``BEGIN EC PRIVATE KEY`` or
        ``BEGIN PRIVATE KEY``, depending on the desired format.

        :param str point_encoding: format to use for encoding public point
        :param str format: either ``ssleay`` (default) or ``pkcs8``
        :param str curve_parameters_encoding: format of encoded curve
            parameters, default depends on the curve, if the curve has
            an associated OID, ``named_curve`` format will be used,
            if no OID is associated with the curve, the fallback of
            ``explicit`` parameters will be used.

        :return: PEM encoded private key
        :rtype: bytes

        .. warning:: The PEM is encoded to US-ASCII, it needs to be
            re-encoded if the system is incompatible (e.g. uses UTF-16)
        """
        # TODO: "BEGIN ECPARAMETERS"
        assert format in ("ssleay", "pkcs8")
        header = "EC PRIVATE KEY" if format == "ssleay" else "PRIVATE KEY"
        return der.topem(
            self.to_der(point_encoding, format, curve_parameters_encoding),
            header,
        )

    def _encode_eddsa(self):
        """Create a PKCS#8 encoding of EdDSA keys."""
        ec_private_key = der.encode_octet_string(self.to_string())
        return der.encode_sequence(
            der.encode_integer(0),
            der.encode_sequence(der.encode_oid(*self.curve.oid)),
            der.encode_octet_string(ec_private_key),
        )

    def to_der(
        self,
        point_encoding="uncompressed",
        format="ssleay",
        curve_parameters_encoding=None,
    ):
        """
        Convert the private key to the :term:`DER` format.

        See :func:`~SigningKey.from_der` method for format specification.

        Only the named curve format is supported.
        The public key will be included in the generated string.

        :param str point_encoding: format to use for encoding public point
            Ignored for EdDSA
        :param str format: either ``ssleay`` (default) or ``pkcs8``.
            EdDSA keys require ``pkcs8``.
        :param str curve_parameters_encoding: format of encoded curve
            parameters, default depends on the curve, if the curve has
            an associated OID, ``named_curve`` format will be used,
            if no OID is associated with the curve, the fallback of
            ``explicit`` parameters will be used.
            Ignored for EdDSA.

        :return: DER encoded private key
        :rtype: bytes
        """
        # SEQ([int(1), octetstring(privkey),cont[0], oid(secp224r1),
        #      cont[1],bitstring])
        if point_encoding == "raw":
            raise ValueError("raw encoding not allowed in DER")
        assert format in ("ssleay", "pkcs8")
        if isinstance(self.curve.curve, CurveEdTw):
            if format != "pkcs8":
                raise ValueError("Only PKCS#8 format supported for EdDSA keys")
            return self._encode_eddsa()
        encoded_vk = self.get_verifying_key().to_string(point_encoding)
        priv_key_elems = [
            der.encode_integer(1),
            der.encode_octet_string(self.to_string()),
        ]
        if format == "ssleay":
            priv_key_elems.append(
                der.encode_constructed(
                    0, self.curve.to_der(curve_parameters_encoding)
                )
            )
        # the 0 in encode_bitstring specifies the number of unused bits
        # in the `encoded_vk` string
        priv_key_elems.append(
            der.encode_constructed(1, der.encode_bitstring(encoded_vk, 0))
        )
        ec_private_key = der.encode_sequence(*priv_key_elems)

        if format == "ssleay":
            return ec_private_key
        else:
            return der.encode_sequence(
                # version = 1 means the public key is not present in the
                # top-level structure.
                der.encode_integer(1),
                der.encode_sequence(
                    der.encode_oid(*oid_ecPublicKey),
                    self.curve.to_der(curve_parameters_encoding),
                ),
                der.encode_octet_string(ec_private_key),
            )

    def to_ssh(self):
        """
        Convert the private key to the SSH format.

        :return: SSH encoded private key
        :rtype: bytes
        """
        return ssh.serialize_private(
            self.curve.name,
            self.verifying_key.to_string(),
            self.to_string(),
        )

    def get_verifying_key(self):
        """
        Return the VerifyingKey associated with this private key.

        Equivalent to reading the `verifying_key` field of an instance.

        :return: a public key that can be used to verify the signatures made
            with this SigningKey
        :rtype: VerifyingKey
        """
        return self.verifying_key

    def sign_deterministic(
        self,
        data,
        hashfunc=None,
        sigencode=sigencode_string,
        extra_entropy=b"",
    ):
        """
        Create signature over data.

        For Weierstrass curves it uses the deterministic RFC6979 algorithm.
        For Edwards curves it uses the standard EdDSA algorithm.

        For ECDSA the data will be hashed using the `hashfunc` function before
        signing.
        For EdDSA the data will be hashed with the hash associated with the
        curve (SHA-512 for Ed25519 and SHAKE-256 for Ed448).

        This is the recommended method for performing signatures when hashing
        of data is necessary.

        :param data: data to be hashed and computed signature over
        :type data: :term:`bytes-like object`
        :param hashfunc: hash function to use for computing the signature,
            if unspecified, the default hash function selected during
            object initialisation will be used (see
            `VerifyingKey.default_hashfunc`). The object needs to implement
            the same interface as hashlib.sha1.
            Ignored with EdDSA.
        :type hashfunc: callable
        :param sigencode: function used to encode the signature.
            The function needs to accept three parameters: the two integers
            that are the signature and the order of the curve over which the
            signature was computed. It needs to return an encoded signature.
            See `ecdsa.util.sigencode_string` and `ecdsa.util.sigencode_der`
            as examples of such functions.
            Ignored with EdDSA.
        :type sigencode: callable
        :param extra_entropy: additional data that will be fed into the random
            number generator used in the RFC6979 process. Entirely optional.
            Ignored with EdDSA.
        :type extra_entropy: :term:`bytes-like object`

        :return: encoded signature over `data`
        :rtype: bytes or sigencode function dependent type
        """
        hashfunc = hashfunc or self.default_hashfunc
        data = normalise_bytes(data)

        if isinstance(self.curve.curve, CurveEdTw):
            return self.privkey.sign(data)

        extra_entropy = normalise_bytes(extra_entropy)
        digest = hashfunc(data).digest()

        return self.sign_digest_deterministic(
            digest,
            hashfunc=hashfunc,
            sigencode=sigencode,
            extra_entropy=extra_entropy,
            allow_truncate=True,
        )

    def sign_digest_deterministic(
        self,
        digest,
        hashfunc=None,
        sigencode=sigencode_string,
        extra_entropy=b"",
        allow_truncate=False,
    ):
        """
        Create signature for digest using the deterministic RFC6979 algorithm.

        `digest` should be the output of cryptographically secure hash function
        like SHA256 or SHA-3-256.

        This is the recommended method for performing signatures when no
        hashing of data is necessary.

        :param digest: hash of data that will be signed
        :type digest: :term:`bytes-like object`
        :param hashfunc: hash function to use for computing the random "k"
            value from RFC6979 process,
            if unspecified, the default hash function selected during
            object initialisation will be used (see
            :attr:`.VerifyingKey.default_hashfunc`). The object needs to
            implement
            the same interface as :func:`~hashlib.sha1` from :py:mod:`hashlib`.
        :type hashfunc: callable
        :param sigencode: function used to encode the signature.
            The function needs to accept three parameters: the two integers
            that are the signature and the order of the curve over which the
            signature was computed. It needs to return an encoded signature.
            See :func:`~ecdsa.util.sigencode_string` and
            :func:`~ecdsa.util.sigencode_der`
            as examples of such functions.
        :type sigencode: callable
        :param extra_entropy: additional data that will be fed into the random
            number generator used in the RFC6979 process. Entirely optional.
        :type extra_entropy: :term:`bytes-like object`
        :param bool allow_truncate: if True, the provided digest can have
            bigger bit-size than the order of the curve, the extra bits (at
            the end of the digest) will be truncated. Use it when signing
            SHA-384 output using NIST256p or in similar situations.

        :return: encoded signature for the `digest` hash
        :rtype: bytes or sigencode function dependent type
        """
        if isinstance(self.curve.curve, CurveEdTw):
            raise ValueError("Method unsupported for Edwards curves")
        secexp = self.privkey.secret_multiplier
        hashfunc = hashfunc or self.default_hashfunc
        digest = normalise_bytes(digest)
        extra_entropy = normalise_bytes(extra_entropy)

        def simple_r_s(r, s, order):
            return r, s, order

        retry_gen = 0
        while True:
            k = rfc6979.generate_k(
                self.curve.generator.order(),
                secexp,
                hashfunc,
                digest,
                retry_gen=retry_gen,
                extra_entropy=extra_entropy,
            )
            try:
                r, s, order = self.sign_digest(
                    digest,
                    sigencode=simple_r_s,
                    k=k,
                    allow_truncate=allow_truncate,
                )
                break
            except RSZeroError:
                retry_gen += 1

        return sigencode(r, s, order)

    def sign(
        self,
        data,
        entropy=None,
        hashfunc=None,
        sigencode=sigencode_string,
        k=None,
        allow_truncate=True,
    ):
        """
        Create signature over data.

        Uses the probabilistic ECDSA algorithm for Weierstrass curves
        (NIST256p, etc.) and the deterministic EdDSA algorithm for the
        Edwards curves (Ed25519, Ed448).

        This method uses the standard ECDSA algorithm that requires a
        cryptographically secure random number generator.

        It's recommended to use the :func:`~SigningKey.sign_deterministic`
        method instead of this one.

        :param data: data that will be hashed for signing
        :type data: :term:`bytes-like object`
        :param callable entropy: randomness source, :func:`os.urandom` by
            default. Ignored with EdDSA.
        :param hashfunc: hash function to use for hashing the provided
            ``data``.
            If unspecified the default hash function selected during
            object initialisation will be used (see
            :attr:`.VerifyingKey.default_hashfunc`).
            Should behave like :func:`~hashlib.sha1` from :py:mod:`hashlib`.
            The output length of the
            hash (in bytes) must not be longer than the length of the curve
            order (rounded up to the nearest byte), so using SHA256 with
            NIST256p is ok, but SHA256 with NIST192p is not. (In the 2**-96ish
            unlikely event of a hash output larger than the curve order, the
            hash will effectively be wrapped mod n).
            If you want to explicitly allow use of large hashes with small
            curves set the ``allow_truncate`` to ``True``.
            Use ``hashfunc=hashlib.sha1`` to match openssl's
            ``-ecdsa-with-SHA1`` mode,
            or ``hashfunc=hashlib.sha256`` for openssl-1.0.0's
            ``-ecdsa-with-SHA256``.
            Ignored for EdDSA
        :type hashfunc: callable
        :param sigencode: function used to encode the signature.
            The function needs to accept three parameters: the two integers
            that are the signature and the order of the curve over which the
            signature was computed. It needs to return an encoded signature.
            See :func:`~ecdsa.util.sigencode_string` and
            :func:`~ecdsa.util.sigencode_der`
            as examples of such functions.
            Ignored for EdDSA
        :type sigencode: callable
        :param int k: a pre-selected nonce for calculating the signature.
            In typical use cases, it should be set to None (the default) to
            allow its generation from an entropy source.
            Ignored for EdDSA.
        :param bool allow_truncate: if ``True``, the provided digest can have
            bigger bit-size than the order of the curve, the extra bits (at
            the end of the digest) will be truncated. Use it when signing
            SHA-384 output using NIST256p or in similar situations. True by
            default.
            Ignored for EdDSA.

        :raises RSZeroError: in the unlikely event when *r* parameter or
            *s* parameter of the created signature is equal 0, as that would
            leak the key. Caller should try a better entropy source, retry with
            different ``k``, or use the
            :func:`~SigningKey.sign_deterministic` in such case.

        :return: encoded signature of the hash of `data`
        :rtype: bytes or sigencode function dependent type
        """
        hashfunc = hashfunc or self.default_hashfunc
        data = normalise_bytes(data)
        if isinstance(self.curve.curve, CurveEdTw):
            return self.sign_deterministic(data)
        h = hashfunc(data).digest()
        return self.sign_digest(h, entropy, sigencode, k, allow_truncate)

    def sign_digest(
        self,
        digest,
        entropy=None,
        sigencode=sigencode_string,
        k=None,
        allow_truncate=False,
    ):
        """
        Create signature over digest using the probabilistic ECDSA algorithm.

        This method uses the standard ECDSA algorithm that requires a
        cryptographically secure random number generator.

        This method does not hash the input.

        It's recommended to use the
        :func:`~SigningKey.sign_digest_deterministic` method
        instead of this one.

        :param digest: hash value that will be signed
        :type digest: :term:`bytes-like object`
        :param callable entropy: randomness source, os.urandom by default
        :param sigencode: function used to encode the signature.
            The function needs to accept three parameters: the two integers
            that are the signature and the order of the curve over which the
            signature was computed. It needs to return an encoded signature.
            See `ecdsa.util.sigencode_string` and `ecdsa.util.sigencode_der`
            as examples of such functions.
        :type sigencode: callable
        :param int k: a pre-selected nonce for calculating the signature.
            In typical use cases, it should be set to None (the default) to
            allow its generation from an entropy source.
        :param bool allow_truncate: if True, the provided digest can have
            bigger bit-size than the order of the curve, the extra bits (at
            the end of the digest) will be truncated. Use it when signing
            SHA-384 output using NIST256p or in similar situations.

        :raises RSZeroError: in the unlikely event when "r" parameter or
            "s" parameter of the created signature is equal 0, as that would
            leak the key. Caller should try a better entropy source, retry with
            different 'k', or use the
            :func:`~SigningKey.sign_digest_deterministic` in such case.

        :return: encoded signature for the `digest` hash
        :rtype: bytes or sigencode function dependent type
        """
        if isinstance(self.curve.curve, CurveEdTw):
            raise ValueError("Method unsupported for Edwards curves")
        digest = normalise_bytes(digest)
        number = _truncate_and_convert_digest(
            digest,
            self.curve,
            allow_truncate,
        )
        r, s = self.sign_number(number, entropy, k)
        return sigencode(r, s, self.privkey.order)

    def sign_number(self, number, entropy=None, k=None):
        """
        Sign an integer directly.

        Note, this is a low level method, usually you will want to use
        :func:`~SigningKey.sign_deterministic` or
        :func:`~SigningKey.sign_digest_deterministic`.

        :param int number: number to sign using the probabilistic ECDSA
            algorithm.
        :param callable entropy: entropy source, os.urandom by default
        :param int k: pre-selected nonce for signature operation. If unset
            it will be selected at random using the entropy source.

        :raises RSZeroError: in the unlikely event when "r" parameter or
            "s" parameter of the created signature is equal 0, as that would
            leak the key. Caller should try a better entropy source, retry with
            different 'k', or use the
            :func:`~SigningKey.sign_digest_deterministic` in such case.

        :return: the "r" and "s" parameters of the signature
        :rtype: tuple of ints
        """
        if isinstance(self.curve.curve, CurveEdTw):
            raise ValueError("Method unsupported for Edwards curves")
        order = self.privkey.order

        if k is not None:
            _k = k
        else:
            _k = randrange(order, entropy)

        assert 1 <= _k < order
        sig = self.privkey.sign(number, _k)
        return sig.r, sig.s