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
// -*- mode: rust; -*-
//
// This file is part of schnorrkel.
// Copyright (c) 2019 Web 3 Foundation
// See LICENSE for licensing information.
//
// Authors:
// - Jeffrey Burdges <jeff@web3.foundation>

//! ### Implementation of a Verifiable Random Function (VRF) using Ristretto points and Schnorr DLEQ proofs.
//!
//! *Warning*  We warn that our VRF construction supports malleable
//! outputs via the `*malleable*` methods.  These are insecure when
//! used in  conjunction with our HDKD provided in dervie.rs.
//! Attackers could translate malleable VRF outputs from one soft subkey 
//! to another soft subkey, gaining early knowledge of the VRF output.
//! We suggest using either non-malleable VRFs or using implicit
//! certificates instead of HDKD when using VRFs.
//!
//! We model the VRF on "Making NSEC5 Practical for DNSSEC" by
//! Dimitrios Papadopoulos, Duane Wessels, Shumon Huque, Moni Naor,
//! Jan Včelák, Leonid Rezyin, andd Sharon Goldberg.
//! https://eprint.iacr.org/2017/099.pdf
//! We note the V(X)EdDSA signature scheme by Trevor Perrin at
//! https://www.signal.org/docs/specifications/xeddsa/#vxeddsa
//! is almost identical to the NSEC5 construction, except that
//! V(X)Ed25519 fails to be a VRF by giving signers multiple
//! outputs per input.  There is another even later variant at
//! https://datatracker.ietf.org/doc/draft-irtf-cfrg-vrf/
//!
//! We support individual signers merging numerous VRF outputs created
//! with the same keypair, which follows the "DLEQ Proofs" and "Batching
//! the Proofs" sections of "Privacy Pass - The Math" by Alex Davidson,
//! https://new.blog.cloudflare.com/privacy-pass-the-math/#dleqproofs
//! and "Privacy Pass: Bypassing Internet Challenges Anonymously"
//! by Alex Davidson, Ian Goldberg, Nick Sullivan, George Tankersley,
//! and Filippo Valsorda.
//! https://www.petsymposium.org/2018/files/papers/issue3/popets-2018-0026.pdf
//!
//! As noted there, our merging technique's soundness appeals to
//! Theorem 3.17 on page 74 of Ryan Henry's PhD thesis
//! "Efficient Zero-Knowledge Proofs and Applications"
//! https://uwspace.uwaterloo.ca/bitstream/handle/10012/8621/Henry_Ryan.pdf
//! See also the attack on Peng and Bao’s batch proof protocol in
//! "Batch Proofs of Partial Knowledge" by Ryan Henry and Ian Goldberg
//! https://www.cypherpunks.ca/~iang/pubs/batchzkp-acns.pdf
//!
//! We might reasonably ask if the VRF signer's public key should
//! really be hashed when creating the scalars in `vrfs_merge*`.
//! After all, there is no similar requirement when the values being
//! hashed are BLS public keys in say
//! https://crypto.stanford.edu/~dabo/pubs/papers/BLSmultisig.html
//! In fact, we expect the public key could be dropped both in
//! Privacy Pass' case, due to using randomness in the messages,
//! and in the VRF case, provided the message depends upon shared
//! randomness created after the public key.  Yet, there are VRF
//! applications outside these two cases, and DLEQ proof applications
//! where the points are not even hashes.  At minimum, we expect
//! hashing the public key prevents malicious signers from choosing
//! their key to cancel out the blinding of a particular point,
//! which might become important in a some anonymity applications.
//! In any case, there is no cost to hashing the public key for VRF
//! applications, but important such an approach cannot yield a
//! verifiable shuffle.
//! TODO: Explain better!
//!
//! We also implement verifier side batching analogous to batched
//! verification of Schnorr signatures, but note this requires an
//! extra curve point, which enlarges the VRF proofs from 64 bytes
//! to 96 bytes.  We provide `shorten_*` methods to produce the
//! non-batchable proof from the batchable proof because doing so
//! is an inherent part of the batch verification anyways.
//! TODO: Security arguments!
//!
//! We do not provide DLEQ proofs optimized for the same signer using
//! multiple public keys because such constructions sound more the
//! domain of zero-knowledge proof libraries.

use core::borrow::Borrow;

#[cfg(any(feature = "alloc", feature = "std"))]
use core::iter::once;

#[cfg(feature = "alloc")]
use alloc::{boxed::Box, vec::Vec};
#[cfg(feature = "std")]
use std::{boxed::Box, vec::Vec};

use curve25519_dalek::constants;
use curve25519_dalek::ristretto::{CompressedRistretto, RistrettoPoint};
use curve25519_dalek::scalar::Scalar;
use curve25519_dalek::traits::{IsIdentity,MultiscalarMul,VartimeMultiscalarMul}; // Identity

use merlin::Transcript;

use super::*;
use crate::context::SigningTranscript;
use crate::points::RistrettoBoth;
// use crate::errors::SignatureError;

/// Value for `kusama` paramater to `*dleq*` methods that yields the VRF for kusama.
/// 
/// Greg Maxwell argue that nonce generation should hash all parameters
/// that challenge generation does in https://moderncrypto.org/mail-archive/curves/2020/001012.html
/// We support this position in prionciple as a defense in depth against
/// attacks that cause missalignment between the public and secret keys.
///
/// We did this for signatures but not for the VRF deployed in Kusama.
/// We cannot justify add this defense to the deployed VRF because
/// several layers already address this attack, including merlin's
/// witnesses and that signers normally only sign VRF outputs once.
/// 
/// We suggest using Greg Maxwell's trick if you use a stand alone DLEQ
/// proof though, meaning call `*dleq*` methods with `kusama: false`.
///
/// see: https://github.com/w3f/schnorrkel/issues/53
// We currently lack tests for the case when this is false, but you can
// rerun cargo test with this set to false for that.
pub const KUSAMA_VRF : bool = true;

/// Length of VRF output.
pub const VRF_OUTPUT_LENGTH : usize = 32;

/// Length of the short VRF proof which lacks support for batch verification.
pub const VRF_PROOF_LENGTH : usize = 64;

/// Length of the longer VRF proof which supports batch verification.
pub const VRF_PROOF_BATCHABLE_LENGTH : usize = 96;

/// `SigningTranscript` helper trait that manages VRF output malleability.
///
/// In short, `VRFSigningTranscript` acts like a default argument
/// `malleabe : bool = false` for every mathod that uses it instead of
/// `SigningTranscript`.
pub trait VRFSigningTranscript {
    /// Real underlying `SigningTranscript`
    type T: SigningTranscript;
    /// Return the underlying `SigningTranscript` after addressing
    /// VRF output malleability, usually by making it non-malleable,
    fn transcript_with_malleability_addressed(self, publickey: &PublicKey) -> Self::T;
}

impl<T> VRFSigningTranscript for T where T: SigningTranscript {
    type T = T;
    #[inline(always)]
    fn transcript_with_malleability_addressed(mut self, publickey: &PublicKey) -> T {
        self.commit_point(b"vrf-nm-pk", publickey.as_compressed());        
        // publickey.make_transcript_nonmalleable(&mut self);
        self
    }
}

/// VRF SigningTranscript for malleable VRF ouputs.
///
/// *Warning*  We caution that malleable VRF outputs are insecure when
/// used in conjunction with HDKD, as provided in dervie.rs. 
/// Attackers could translate malleable VRF outputs from one soft subkey 
/// to another soft subkey, gaining early knowledge of the VRF output.
/// We think most VRF applicaitons for which HDKH soudns suitable
/// benefit from using implicit certificates insead of HDKD anyways,
/// which should also be secure in combination with HDKD.
/// We always use non-malleable VRF inputs in our convenience methods.
#[derive(Clone)]
pub struct Malleable<T: SigningTranscript>(pub T);
impl<T> VRFSigningTranscript for Malleable<T> where T: SigningTranscript {
    type T = T;
    #[inline(always)]
    fn transcript_with_malleability_addressed(self, _publickey: &PublicKey) -> T { self.0 }
}


/// Create a malleable VRF input point by hashing a transcript to a point.
///
/// *Warning*  We caution that malleable VRF inputs are insecure when
/// used in conjunction with HDKD, as provided in dervie.rs. 
/// Attackers could translate malleable VRF outputs from one soft subkey 
/// to another soft subkey, gaining early knowledge of the VRF output.
/// We think most VRF applicaitons for which HDKH soudns suitable
/// benefit from using implicit certificates insead of HDKD anyways,
/// which should also be secure in combination with HDKH.
/// We always use non-malleable VRF inputs in our convenience methods.
pub fn vrf_malleable_hash<T: SigningTranscript>(mut t: T) -> RistrettoBoth {
    let mut b = [0u8; 64];
    t.challenge_bytes(b"VRFHash", &mut b);
    RistrettoBoth::from_point(RistrettoPoint::from_uniform_bytes(&b))
}

impl PublicKey {
    /// Create a non-malleable VRF input point by hashing a transcript to a point.
    pub fn vrf_hash<T>(&self, t: T) -> RistrettoBoth
    where T: VRFSigningTranscript {
        vrf_malleable_hash(t.transcript_with_malleability_addressed(self))
    }

    /// Pair a non-malleable VRF output with the hash of the given transcript.
    pub fn vrf_attach_hash<T>(&self, output: VRFOutput, t: T) -> SignatureResult<VRFInOut>
    where T: VRFSigningTranscript {
        output.attach_input_hash(self,t)
    }
}

/// VRF output, possibly unverified.
///
/// Internally, we keep both `RistrettoPoint` and `CompressedRistretto`
/// forms using `RistrettoBoth`.
///
/// We'd actually love to statically distinguish here between inputs
/// and outputs, as well as whether outputs were verified, but doing
/// so would disrupt our general purpose DLEQ proof mechanism, so
/// users must be responcible for this themselves.  We do however
/// consume by value in actual output methods, and do not implement
/// `Copy`, as a reminder that VRF outputs should only be used once
/// and should be checked before usage.
#[derive(Debug, Copy, Clone, Default, PartialEq, Eq, PartialOrd, Ord, Hash)]
pub struct VRFOutput(pub [u8; PUBLIC_KEY_LENGTH]);

impl VRFOutput {
    const DESCRIPTION: &'static str =
        "A Ristretto Schnorr VRF output represented as a 32-byte Ristretto compressed point";

    /// Convert this VRF output to a byte array.
    #[inline]
    pub fn to_bytes(&self) -> [u8; VRF_OUTPUT_LENGTH] {
        self.0
    }

    /// View this secret key as a byte array.
    #[inline]
    pub fn as_bytes(&self) -> &[u8; VRF_OUTPUT_LENGTH] {
        &self.0
    }

    /// Construct a `VRFOutput` from a slice of bytes.
    #[inline]
    pub fn from_bytes(bytes: &[u8]) -> SignatureResult<VRFOutput> {
        if bytes.len() != VRF_OUTPUT_LENGTH {
            return Err(SignatureError::BytesLengthError {
                name: "VRFOutput",
                description: VRFOutput::DESCRIPTION,
                length: VRF_OUTPUT_LENGTH
            });
        }
        let mut bits: [u8; 32] = [0u8; 32];
        bits.copy_from_slice(&bytes[..32]);
        Ok(VRFOutput(bits))
    }

    /// Pair a non-malleable VRF output with the hash of the given transcript.
    pub fn attach_input_hash<T>(&self, public: &PublicKey, t: T) -> SignatureResult<VRFInOut>
    where T: VRFSigningTranscript {
        let input = public.vrf_hash(t);
        let output = RistrettoBoth::from_bytes_ser("VRFOutput", VRFOutput::DESCRIPTION, &self.0) ?;
        if output.as_point().is_identity() { return Err(SignatureError::PointDecompressionError); }
        Ok(VRFInOut { input, output })
    }
}

serde_boilerplate!(VRFOutput);

/// VRF input and output paired together, possibly unverified.
///
/// Internally, we keep both `RistrettoPoint` and `CompressedRistretto`
/// forms using `RistrettoBoth`.
#[derive(Debug, Clone, PartialEq, Eq, PartialOrd, Ord, Hash)]
pub struct VRFInOut {
    /// VRF input point
    pub input: RistrettoBoth,
    /// VRF output point
    pub output: RistrettoBoth,
}

impl SecretKey {
    /// Evaluate the VRF-like multiplication on an uncompressed point,
    /// probably not useful in this form.
    pub fn vrf_create_from_point(&self, input: RistrettoBoth) -> VRFInOut {
        let output = RistrettoBoth::from_point(&self.key * input.as_point());
        VRFInOut { input, output }
    }

    /// Evaluate the VRF-like multiplication on a compressed point,
    /// useful for proving key exchanges, OPRFs, or sequential VRFs.
    ///
    /// We caution that such protocols could provide signing oracles
    /// and note that `vrf_create_from_point` cannot check for
    /// problematic inputs like `attach_input_hash` does.
    pub fn vrf_create_from_compressed_point(&self, input: &VRFOutput) -> SignatureResult<VRFInOut> {
        let input = RistrettoBoth::from_compressed(CompressedRistretto(input.0)) ?;
        Ok(self.vrf_create_from_point(input))
    }
}

impl Keypair {
    /// Evaluate the VRF on the given transcript.
    pub fn vrf_create_hash<T: VRFSigningTranscript>(&self, t: T) -> VRFInOut {
        self.secret.vrf_create_from_point(self.public.vrf_hash(t))
    }
}

impl VRFInOut {
    /// VRF output point bytes for serialization.
    pub fn as_output_bytes(&self) -> &[u8; 32] {
        self.output.as_compressed().as_bytes()
    }

    /// VRF output point bytes for serialization.
    pub fn to_output(&self) -> VRFOutput {
        VRFOutput(self.output.as_compressed().to_bytes())
    }

    /// Commit VRF input and output to a transcript.
    ///
    /// We commit both the input and output to provide the 2Hash-DH
    /// construction from Theorem 2 on page 32 in appendix C of
    /// ["Ouroboros Praos: An adaptively-secure, semi-synchronous proof-of-stake blockchain"](https://eprint.iacr.org/2017/573.pdf)
    /// by Bernardo David, Peter Gazi, Aggelos Kiayias, and Alexander Russell.
    ///
    /// We use this construction both for the VRF usage methods
    /// `VRFInOut::make_*` as well as for signer side batching.
    pub fn commit<T: SigningTranscript>(&self, t: &mut T) {
        t.commit_point(b"vrf-in", self.input.as_compressed());
        t.commit_point(b"vrf-out", self.output.as_compressed());
    }

    /// Raw bytes output from the VRF.
    ///
    /// If you are not the signer then you must verify the VRF before calling this method.
    ///
    /// If called with distinct contexts then outputs should be independent.
    ///
    /// We incorporate both the input and output to provide the 2Hash-DH
    /// construction from Theorem 2 on page 32 in appendex C of
    /// ["Ouroboros Praos: An adaptively-secure, semi-synchronous proof-of-stake blockchain"](https://eprint.iacr.org/2017/573.pdf)
    /// by Bernardo David, Peter Gazi, Aggelos Kiayias, and Alexander Russell.
    pub fn make_bytes<B: Default + AsMut<[u8]>>(&self, context: &[u8]) -> B {
        let mut t = Transcript::new(b"VRFResult");
        t.append_message(b"",context);
        self.commit(&mut t);
        let mut seed = B::default();
        t.challenge_bytes(b"", seed.as_mut());
        seed
    }

    /// VRF output converted into any `SeedableRng`.
    ///
    /// If you are not the signer then you must verify the VRF before calling this method.
    ///
    /// We expect most users would prefer the less generic `VRFInOut::make_chacharng` method.
    pub fn make_rng<R: ::rand_core::SeedableRng>(&self, context: &[u8]) -> R {
        R::from_seed(self.make_bytes::<R::Seed>(context))
    }

    /// VRF output converted into a `ChaChaRng`.
    ///
    /// If you are not the signer then you must verify the VRF before calling this method.
    ///
    /// If called with distinct contexts then outputs should be independent.
    /// Independent output streams are available via `ChaChaRng::set_stream` too.
    ///
    /// We incorporate both the input and output to provide the 2Hash-DH
    /// construction from Theorem 2 on page 32 in appendex C of
    /// ["Ouroboros Praos: An adaptively-secure, semi-synchronous proof-of-stake blockchain"](https://eprint.iacr.org/2017/573.pdf)
    /// by Bernardo David, Peter Gazi, Aggelos Kiayias, and Alexander Russell.
    #[cfg(feature = "rand_chacha")]
    pub fn make_chacharng(&self, context: &[u8]) -> ::rand_chacha::ChaChaRng {
        self.make_rng::<::rand_chacha::ChaChaRng>(context)
    }

    /// VRF output converted into Merlin's Keccek based `Rng`.
    ///
    /// If you are not the signer then you must verify the VRF before calling this method.
    ///
    /// We think this might be marginally slower than `ChaChaRng`
    /// when considerable output is required, but it should reduce
    /// the final linked binary size slightly, and improves domain
    /// separation.
    #[inline(always)]
    pub fn make_merlin_rng(&self, context: &[u8]) -> merlin::TranscriptRng {
        // Very insecure hack except for our commit_witness_bytes below
        struct ZeroFakeRng;
        impl ::rand_core::RngCore for ZeroFakeRng {
            fn next_u32(&mut self) -> u32 {  panic!()  }
            fn next_u64(&mut self) -> u64 {  panic!()  }
            fn fill_bytes(&mut self, dest: &mut [u8]) {
                for i in dest.iter_mut() {  *i = 0;  }
            }
            fn try_fill_bytes(&mut self, dest: &mut [u8]) -> Result<(), ::rand_core::Error> {
                self.fill_bytes(dest);
                Ok(())
            }
        }
        impl ::rand_core::CryptoRng for ZeroFakeRng {}

        let mut t = Transcript::new(b"VRFResult");
        t.append_message(b"",context);
        self.commit(&mut t);
        t.build_rng().finalize(&mut ZeroFakeRng)
    }
}

fn challenge_scalar_128<T: SigningTranscript>(mut t: T) -> Scalar {
    let mut s = [0u8; 16];
    t.challenge_bytes(b"", &mut s);
    Scalar::from(u128::from_le_bytes(s))
}

impl PublicKey {
    /// Merge VRF input and output pairs from the same signer,
    /// using variable time arithmetic
    ///
    /// You should use `vartime=true` when verifying VRF proofs batched
    /// by the singer.  You could usually use `vartime=true` even when
    /// producing proofs, provided the set being signed is not secret.
    ///
    /// There is sadly no constant time 128 bit multiplication in dalek,
    /// making `vartime=false` somewhat slower than necessary.  It should
    /// only impact signers in niche scenarios however, so the slower
    /// variant should normally be unnecessary.
    ///
    /// Panics if given an empty points list.
    ///
    /// TODO: Add constant time 128 bit batched multiplication to dalek.
    /// TODO: Is rand_chacha's `gen::<u128>()` standardizable enough to
    /// prefer it over merlin for the output?  
    pub fn vrfs_merge<B>(&self, ps: &[B], vartime: bool) -> VRFInOut
    where
        B: Borrow<VRFInOut>,
    {
        assert!( ps.len() > 0);
        let mut t = ::merlin::Transcript::new(b"MergeVRFs");
        t.commit_point(b"vrf:pk", self.as_compressed());
        for p in ps.iter() {
            p.borrow().commit(&mut t);
        }

        let zf = || ps.iter().map(|p| {
            let mut t0 = t.clone();
            p.borrow().commit(&mut t0);
            challenge_scalar_128(t0)
        });
        #[cfg(any(feature = "alloc", feature = "std"))]
        let zs: Vec<Scalar> = zf().collect();
        #[cfg(any(feature = "alloc", feature = "std"))]
        let zf = || zs.iter();

        // We need actual fns here because closures cannot easily take
        // closures as arguments, due to Rust lacking polymorphic
        // closures but giving all closures unique types.
        fn get_input(p: &VRFInOut) -> &RistrettoPoint { p.input.as_point() }
        fn get_output(p: &VRFInOut) -> &RistrettoPoint { p.output.as_point() }
        #[cfg(any(feature = "alloc", feature = "std"))]
        let go = |io: fn(p: &VRFInOut) -> &RistrettoPoint| {
            let ps = ps.iter().map( |p| io(p.borrow()) );
            RistrettoBoth::from_point(if vartime {
                RistrettoPoint::vartime_multiscalar_mul(zf(), ps)
            } else {
                RistrettoPoint::multiscalar_mul(zf(), ps)
            })
        };
        #[cfg(not(any(feature = "alloc", feature = "std")))]
        let go = |io: fn(p: &VRFInOut) -> &RistrettoPoint| {
            use curve25519_dalek::traits::Identity;
            let mut acc = RistrettoPoint::identity();
            for (z,p) in zf().zip(ps) {
                acc += z * io(p.borrow());
            }
            RistrettoBoth::from_point(acc)
        };

        let input = go( get_input );
        let output = go( get_output );
        VRFInOut { input, output }
    }
}

/// Short proof of correctness for associated VRF output,
/// for which no batched verification works.
#[derive(Debug, Clone, PartialEq, Eq)] // PartialOrd, Ord, Hash
pub struct VRFProof {
    /// Challenge
    c: Scalar,
    /// Schnorr proof
    s: Scalar,
}

impl VRFProof {
    const DESCRIPTION : &'static str = "A Ristretto Schnorr VRF proof without batch verification support, which consists of two scalars, making it 64 bytes.";

    /// Convert this `VRFProof` to a byte array.
    #[inline]
    pub fn to_bytes(&self) -> [u8; VRF_PROOF_LENGTH] {
        let mut bytes = [0u8; VRF_PROOF_LENGTH];

        bytes[..32].copy_from_slice(&self.c.as_bytes()[..]);
        bytes[32..].copy_from_slice(&self.s.as_bytes()[..]);
        bytes
    }

    /// Construct a `VRFProof` from a slice of bytes.
    #[inline]
    pub fn from_bytes(bytes: &[u8]) -> SignatureResult<VRFProof> {
        if bytes.len() != VRF_PROOF_LENGTH {
            return Err(SignatureError::BytesLengthError {
                name: "VRFProof",
                description: VRFProof::DESCRIPTION,
                length: VRF_PROOF_LENGTH
            });
        }
        let mut c: [u8; 32] = [0u8; 32];
        let mut s: [u8; 32] = [0u8; 32];

        c.copy_from_slice(&bytes[..32]);
        s.copy_from_slice(&bytes[32..]);

        let c = Scalar::from_canonical_bytes(c).ok_or(SignatureError::ScalarFormatError) ?;
        let s = Scalar::from_canonical_bytes(s).ok_or(SignatureError::ScalarFormatError) ?;
        Ok(VRFProof { c, s })
    }
}

serde_boilerplate!(VRFProof);

/// Longer proof of correctness for associated VRF output,
/// which supports batching.
#[derive(Debug, Clone, PartialEq, Eq)] // PartialOrd, Ord, Hash
#[allow(non_snake_case)]
pub struct VRFProofBatchable {
    /// Our nonce R = r G to permit batching the first verification equation
    R: CompressedRistretto,
    /// Our input hashed and raised to r to permit batching the second verification equation
    Hr: CompressedRistretto,
    /// Schnorr proof
    s: Scalar,
}

impl VRFProofBatchable {
    const DESCRIPTION : &'static str = "A Ristretto Schnorr VRF proof that supports batch verification, which consists of two Ristretto compressed points and one scalar, making it 96 bytes.";

    /// Convert this `VRFProofBatchable` to a byte array.
    #[allow(non_snake_case)]
    #[inline]
    pub fn to_bytes(&self) -> [u8; VRF_PROOF_BATCHABLE_LENGTH] {
        let mut bytes = [0u8; VRF_PROOF_BATCHABLE_LENGTH];

        bytes[0..32].copy_from_slice(&self.R.as_bytes()[..]);
        bytes[32..64].copy_from_slice(&self.Hr.as_bytes()[..]);
        bytes[64..96].copy_from_slice(&self.s.as_bytes()[..]);
        bytes
    }

    /// Construct a `VRFProofBatchable` from a slice of bytes.
    #[allow(non_snake_case)]
    #[inline]
    pub fn from_bytes(bytes: &[u8]) -> SignatureResult<VRFProofBatchable> {
        if bytes.len() != VRF_PROOF_BATCHABLE_LENGTH {
            return Err(SignatureError::BytesLengthError {
                name: "VRFProofBatchable",
                description: VRFProofBatchable::DESCRIPTION,
                length: VRF_PROOF_BATCHABLE_LENGTH,
            });
        }
        let mut R: [u8; 32] = [0u8; 32];
        let mut Hr: [u8; 32] = [0u8; 32];
        let mut s: [u8; 32] = [0u8; 32];

        R.copy_from_slice(&bytes[0..32]);
        Hr.copy_from_slice(&bytes[32..64]);
        s.copy_from_slice(&bytes[64..96]);

        let s = Scalar::from_canonical_bytes(s).ok_or(SignatureError::ScalarFormatError) ?;
        Ok(VRFProofBatchable { R: CompressedRistretto(R), Hr: CompressedRistretto(Hr), s })
    }

    /// Return the shortened `VRFProof` for retransmitting in not batched situations
    #[allow(non_snake_case)]
    pub fn shorten_dleq<T>(&self, mut t: T, public: &PublicKey, p: &VRFInOut, kusama: bool) -> VRFProof
    where T: SigningTranscript,
    {
        t.proto_name(b"DLEQProof");
        // t.commit_point(b"vrf:g",constants::RISTRETTO_BASEPOINT_TABLE.basepoint().compress());
        t.commit_point(b"vrf:h", p.input.as_compressed());
        if !kusama {  t.commit_point(b"vrf:pk", public.as_compressed());  }

        t.commit_point(b"vrf:R=g^r", &self.R);
        t.commit_point(b"vrf:h^r", &self.Hr);

        if kusama {  t.commit_point(b"vrf:pk", public.as_compressed());  }
        t.commit_point(b"vrf:h^sk", p.output.as_compressed());

        VRFProof {
            c: t.challenge_scalar(b"prove"), // context, message, A/public_key, R=rG
            s: self.s,
        }
    }

    /// Return the shortened `VRFProof` for retransmitting in non-batched situations
    ///
    /// TODO: Avoid the error path here by avoiding decompressing,
    /// either locally here, or more likely by decompressing
    /// `VRFOutput` in deserialization.
    pub fn shorten_vrf<T>( &self, public: &PublicKey, t: T, out: &VRFOutput)
     -> SignatureResult<VRFProof>
    where T: VRFSigningTranscript,
    {
        let p = out.attach_input_hash(public,t) ?; // Avoidable errors if decompressed earlier
        let t0 = Transcript::new(b"VRF");  // We have context in t and another hear confuses batching
        Ok(self.shorten_dleq(t0, public, &p, KUSAMA_VRF))
    }
}

serde_boilerplate!(VRFProofBatchable);

impl Keypair {
    /// Produce DLEQ proof.
    ///
    /// We assume the `VRFInOut` paramater has been computed correctly
    /// by multiplying every input point by `self.secret`, like by
    /// using one of the `vrf_create_*` methods on `SecretKey`.
    /// If so, we produce a proof that this multiplication was done correctly.
    #[allow(non_snake_case)]
    pub fn dleq_proove<T>(&self, mut t: T, p: &VRFInOut, kusama: bool) -> (VRFProof, VRFProofBatchable)
    where
        T: SigningTranscript,
    {
        t.proto_name(b"DLEQProof");
        // t.commit_point(b"vrf:g",constants::RISTRETTO_BASEPOINT_TABLE.basepoint().compress());
        t.commit_point(b"vrf:h", p.input.as_compressed());
        if !kusama {  t.commit_point(b"vrf:pk", self.public.as_compressed());  }

        // We compute R after adding pk and all h.
        let mut r = t.witness_scalar(b"proving\00",&[&self.secret.nonce]);
        let R = (&r * &constants::RISTRETTO_BASEPOINT_TABLE).compress();
        t.commit_point(b"vrf:R=g^r", &R);

        let Hr = (&r * p.input.as_point()).compress();
        t.commit_point(b"vrf:h^r", &Hr);

        if kusama {  t.commit_point(b"vrf:pk", self.public.as_compressed());  }
        // We add h^sk last to save an allocation if we ever need to hash multiple h together.
        t.commit_point(b"vrf:h^sk", p.output.as_compressed());

        let c = t.challenge_scalar(b"prove"); // context, message, A/public_key, R=rG
        let s = &r - &(&c * &self.secret.key);

        ::zeroize::Zeroize::zeroize(&mut r);

        (VRFProof { c, s }, VRFProofBatchable { R, Hr, s })
    }

    /// Run VRF on one single input transcript, producing the outpus
    /// and correspodning short proof.
    ///
    /// There are schemes like Ouroboros Praos in which nodes evaluate
    /// VRFs repeatedly until they win some contest.  In these case,
    /// you should probably use vrf_sign_n_check to gain access to the
    /// `VRFInOut` from `vrf_create_hash` first, and then avoid computing
    /// the proof whenever you do not win. 
    pub fn vrf_sign<T>(&self, t: T) -> (VRFInOut, VRFProof, VRFProofBatchable)
    where T: VRFSigningTranscript,
    {
        self.vrf_sign_extra(t,Transcript::new(b"VRF"))
        // We have context in t and another hear confuses batching
    }

    /// Run VRF on one single input transcript and an extra message transcript, 
    /// producing the outpus and correspodning short proof.
    pub fn vrf_sign_extra<T,E>(&self, t: T, extra: E) -> (VRFInOut, VRFProof, VRFProofBatchable)
    where T: VRFSigningTranscript,
          E: SigningTranscript,
    {
        let p = self.vrf_create_hash(t);
        let (proof, proof_batchable) = self.dleq_proove(extra, &p, KUSAMA_VRF);
        (p, proof, proof_batchable)
    }


    /// Run VRF on one single input transcript, producing the outpus
    /// and correspodning short proof only if the result first passes
    /// some check.
    ///
    /// There are schemes like Ouroboros Praos in which nodes evaluate
    /// VRFs repeatedly until they win some contest.  In these case,
    /// you might use this function to short circuit computing the full
    /// proof.
    pub fn vrf_sign_after_check<T,F>(&self, t: T, mut check: F)
     -> Option<(VRFInOut, VRFProof, VRFProofBatchable)>
    where T: VRFSigningTranscript,
          F: FnMut(&VRFInOut) -> bool,
    {
        self.vrf_sign_extra_after_check(t,
            |io| if check(io) { Some(Transcript::new(b"VRF")) } else { None }
        )
    }

    /// Run VRF on one single input transcript, producing the outpus
    /// and correspodning short proof only if the result first passes
    /// some check, which itself returns an extra message transcript.
    pub fn vrf_sign_extra_after_check<T,E,F>(&self, t: T, mut check: F)
     -> Option<(VRFInOut, VRFProof, VRFProofBatchable)>
    where T: VRFSigningTranscript,
          E: SigningTranscript,
          F: FnMut(&VRFInOut) -> Option<E>,
    {
        let p = self.vrf_create_hash(t);
        let extra = check(&p) ?;
        let (proof, proof_batchable) = self.dleq_proove(extra, &p, KUSAMA_VRF);
        Some((p, proof, proof_batchable))
    }

    /// Run VRF on several input transcripts, producing their outputs
    /// and a common short proof.
    ///
    /// We merge the VRF outputs using variable time arithmetic, so
    /// if even the hash of the message being signed is sensitive then
    /// you might reimplement some constant time variant.
    #[cfg(any(feature = "alloc", feature = "std"))]
    pub fn vrfs_sign<T, I>(&self, ts: I) -> (Box<[VRFInOut]>, VRFProof, VRFProofBatchable)
    where
        T: VRFSigningTranscript,
        I: IntoIterator<Item = T>,
    {
        self.vrfs_sign_extra(ts, Transcript::new(b"VRF"))
    }

    /// Run VRF on several input transcripts and an extra message transcript,
    /// producing their outputs and a common short proof.
    ///
    /// We merge the VRF outputs using variable time arithmetic, so
    /// if even the hash of the message being signed is sensitive then
    /// you might reimplement some constant time variant.
    #[cfg(any(feature = "alloc", feature = "std"))]
    pub fn vrfs_sign_extra<T,E,I>(&self, ts: I, extra: E) -> (Box<[VRFInOut]>, VRFProof, VRFProofBatchable)
    where
        T: VRFSigningTranscript,
        E: SigningTranscript,
        I: IntoIterator<Item = T>,
    {
        let ps = ts.into_iter()
            .map(|t| self.vrf_create_hash(t))
            .collect::<Vec<VRFInOut>>();
        let p = self.public.vrfs_merge(&ps,true);
        let (proof, proof_batchable) = self.dleq_proove(extra, &p, KUSAMA_VRF);
        (ps.into_boxed_slice(), proof, proof_batchable)
    }
}

impl PublicKey {
    /// Verify DLEQ proof that `p.output = s * p.input` where `self`
    /// `s` times the basepoint.
    ///
    /// We return an enlarged `VRFProofBatchable` instead of just true,
    /// so that verifiers can forward batchable proofs.
    ///
    /// In principle, one might provide "blindly verifiable" VRFs that
    /// avoid requiring `self` here, but naively such constructions
    /// risk the same flaws as DLEQ based blind signatures, and this
    /// version exploits the slightly faster basepoint arithmetic.
    #[allow(non_snake_case)]
    pub fn dleq_verify<T>(
        &self,
        mut t: T,
        p: &VRFInOut,
        proof: &VRFProof,
        kusama: bool,
    ) -> SignatureResult<VRFProofBatchable>
    where
        T: SigningTranscript,
    {
        t.proto_name(b"DLEQProof");
        // t.commit_point(b"vrf:g",constants::RISTRETTO_BASEPOINT_TABLE.basepoint().compress());
        t.commit_point(b"vrf:h", p.input.as_compressed());
        if !kusama {  t.commit_point(b"vrf:pk", self.as_compressed());  }

        // We recompute R aka u from the proof
        // let R = (&proof.c * self.as_point()) + (&proof.s * &constants::RISTRETTO_BASEPOINT_TABLE);
        let R = RistrettoPoint::vartime_double_scalar_mul_basepoint(
            &proof.c,
            self.as_point(),
            &proof.s,
        ).compress();
        t.commit_point(b"vrf:R=g^r", &R);

        // We also recompute h^r aka u using the proof
        #[cfg(not(any(feature = "alloc", feature = "std")))]
        let Hr = (&proof.c * p.output.as_point()) + (&proof.s * p.input.as_point());

        // TODO: Verify if this is actually faster using benchmarks
        #[cfg(any(feature = "alloc", feature = "std"))]
        let Hr = RistrettoPoint::vartime_multiscalar_mul(
            &[proof.c, proof.s],
            &[*p.output.as_point(), *p.input.as_point()],
        );

        let Hr = Hr.compress();
        t.commit_point(b"vrf:h^r", &Hr);

        if kusama {  t.commit_point(b"vrf:pk", self.as_compressed());  }
        // We add h^sk last to save an allocation if we ever need to hash multiple h together.
        t.commit_point(b"vrf:h^sk", p.output.as_compressed());

        // We need not check that h^pk lies on the curve because Ristretto ensures this.
        let VRFProof { c, s } = *proof;
        if c == t.challenge_scalar(b"prove") {
            Ok(VRFProofBatchable { R, Hr, s }) // Scalar: Copy ?!?
        } else {
            Err(SignatureError::EquationFalse)
        }
    }

    /// Verify VRF proof for one single input transcript and corresponding output.
    pub fn vrf_verify<T: VRFSigningTranscript>(
        &self,
        t: T,
        out: &VRFOutput,
        proof: &VRFProof,
    ) -> SignatureResult<(VRFInOut, VRFProofBatchable)> {
        self.vrf_verify_extra(t,out,proof,Transcript::new(b"VRF"))
    }

    /// Verify VRF proof for one single input transcript and corresponding output.
    pub fn vrf_verify_extra<T,E>(
        &self,
        t: T,
        out: &VRFOutput,
        proof: &VRFProof,
        extra: E,
    ) -> SignatureResult<(VRFInOut, VRFProofBatchable)> 
    where T: VRFSigningTranscript,
          E: SigningTranscript,
    {
        let p = out.attach_input_hash(self,t)?;
        let proof_batchable = self.dleq_verify(extra, &p, proof, KUSAMA_VRF)?;
        Ok((p, proof_batchable))
    }

    /// Verify a common VRF short proof for several input transcripts and corresponding outputs.
    #[cfg(any(feature = "alloc", feature = "std"))]
    pub fn vrfs_verify<T,I,O>(
        &self,
        transcripts: I,
        outs: &[O],
        proof: &VRFProof,
    ) -> SignatureResult<(Box<[VRFInOut]>, VRFProofBatchable)>
    where
        T: VRFSigningTranscript,
        I: IntoIterator<Item = T>,
        O: Borrow<VRFOutput>,
    {
        self.vrfs_verify_extra(transcripts,outs,proof,Transcript::new(b"VRF"))
    }

    /// Verify a common VRF short proof for several input transcripts and corresponding outputs.
    #[cfg(any(feature = "alloc", feature = "std"))]
    pub fn vrfs_verify_extra<T,E,I,O>(
        &self,
        transcripts: I,
        outs: &[O],
        proof: &VRFProof,
        extra: E,
    ) -> SignatureResult<(Box<[VRFInOut]>, VRFProofBatchable)>
    where
        T: VRFSigningTranscript,
        E: SigningTranscript,
        I: IntoIterator<Item = T>,
        O: Borrow<VRFOutput>,
    {
        let mut ts = transcripts.into_iter();
        let ps = ts.by_ref().zip(outs)
            .map(|(t, out)| out.borrow().attach_input_hash(self,t))
            .collect::<SignatureResult<Vec<VRFInOut>>>()?;
        assert!(ts.next().is_none(), "Too few VRF outputs for VRF inputs.");
        assert!(
            ps.len() == outs.len(),
            "Too few VRF inputs for VRF outputs."
        );
        let p = self.vrfs_merge(&ps[..],true);
        let proof_batchable = self.dleq_verify(extra, &p, proof, KUSAMA_VRF)?;
        Ok((ps.into_boxed_slice(), proof_batchable))
    }
}

/// Batch verify DLEQ proofs where the public keys were held by
/// different parties.
///
/// We first reconstruct the `c`s present in the `VRFProof`s but absent
/// in the `VRFProofBatchable`s, using `shorten_dleq`.  We then verify
/// the `R` and `Hr` components of the `VRFProofBatchable`s using the
/// two equations a normal verification uses to discover them.
/// We do this by delinearizing both verification equations with
/// random numbers.
///
/// TODO: Assess when the two verification equations should be
/// combined, presumably by benchmarking both forms.  At smaller batch
/// sizes then we should clearly benefit form the combined form, but
/// bany combination doubles the scalar by scalar multiplicications
/// and hashing, so large enough batch verifications should favor two
/// seperate calls.
#[cfg(any(feature = "alloc", feature = "std"))]
#[allow(non_snake_case)]
pub fn dleq_verify_batch(
    ps: &[VRFInOut],
    proofs: &[VRFProofBatchable],
    public_keys: &[PublicKey],
    kusama: bool,
) -> SignatureResult<()> {
    const ASSERT_MESSAGE: &'static str = "The number of messages/transcripts / input points, output points, proofs, and public keys must be equal.";
    assert!(ps.len() == proofs.len(), ASSERT_MESSAGE);
    assert!(proofs.len() == public_keys.len(), ASSERT_MESSAGE);

    // Use a random number generator keyed by the publidc keys, the
    // inout and putput points, and the system randomn number gnerator.
    let mut csprng = {
        let mut t = Transcript::new(b"VB-RNG");
        for (pk,p) in public_keys.iter().zip(ps) {
            t.commit_point(b"",pk.as_compressed());
            p.commit(&mut t);
        }
        t.build_rng().finalize(&mut rand_hack())
    };

    // Select a random 128-bit scalar for each signature.
    // We may represent these as scalars because we use
    // variable time 256 bit multiplication below.
    let rnd_128bit_scalar = |_| {
        let mut r = [0u8; 16];
        csprng.fill_bytes(&mut r);
        Scalar::from(u128::from_le_bytes(r))
    };
    let zz: Vec<Scalar> = proofs.iter().map(rnd_128bit_scalar).collect();

    let z_s: Vec<Scalar> = zz.iter().zip(proofs)
        .map(|(z, proof)| z * proof.s)
        .collect();

    // Compute the basepoint coefficient, ∑ s[i] z[i] (mod l)
    let B_coefficient: Scalar = z_s.iter().sum();

    let t0 = Transcript::new(b"VRF");
    let z_c: Vec<Scalar> = zz.iter().enumerate()
        .map( |(i, z)| z * proofs[i].shorten_dleq(t0.clone(), &public_keys[i], &ps[i], kusama).c )
        .collect();

    // Compute (∑ z[i] s[i] (mod l)) B + ∑ (z[i] c[i] (mod l)) A[i] - ∑ z[i] R[i] = 0
    let mut b = RistrettoPoint::optional_multiscalar_mul(
        zz.iter().map(|z| -z)
            .chain(z_c.iter().cloned())
            .chain(once(B_coefficient)),
        proofs.iter().map(|proof| proof.R.decompress())
            .chain(public_keys.iter().map(|pk| Some(*pk.as_point())))
            .chain(once(Some(constants::RISTRETTO_BASEPOINT_POINT))),
    ).map(|id| id.is_identity()).unwrap_or(false);

    // Compute (∑ z[i] s[i] (mod l)) Input[i] + ∑ (z[i] c[i] (mod l)) Output[i] - ∑ z[i] Hr[i] = 0
    b &= RistrettoPoint::optional_multiscalar_mul(
        zz.iter().map(|z| -z)
            .chain(z_c)
            .chain(z_s),
        proofs.iter().map(|proof| proof.Hr.decompress())
            .chain(ps.iter().map(|p| Some(*p.output.as_point())))
            .chain(ps.iter().map(|p| Some(*p.input.as_point()))),
    ).map(|id| id.is_identity()).unwrap_or(false);

    if b { Ok(()) } else { Err(SignatureError::EquationFalse) }
}

/// Batch verify VRFs by different signers
///
///
#[cfg(any(feature = "alloc", feature = "std"))]
pub fn vrf_verify_batch<T, I>(
    transcripts: I,
    outs: &[VRFOutput],
    proofs: &[VRFProofBatchable],
    publickeys: &[PublicKey],
) -> SignatureResult<Box<[VRFInOut]>>
where
    T: VRFSigningTranscript,
    I: IntoIterator<Item = T>,
{
    let mut ts = transcripts.into_iter();
    let ps = ts.by_ref()
        .zip(publickeys)
        .zip(outs)
        .map(|((t, pk), out)| out.attach_input_hash(pk,t))
        .collect::<SignatureResult<Vec<VRFInOut>>>()?;
    assert!(ts.next().is_none(), "Too few VRF outputs for VRF inputs.");
    assert!(
        ps.len() == outs.len(),
        "Too few VRF inputs for VRF outputs."
    );
    if dleq_verify_batch(&ps[..], proofs, publickeys, KUSAMA_VRF).is_ok() {
        Ok(ps.into_boxed_slice())
    } else {
        Err(SignatureError::EquationFalse)
    }
}

#[cfg(test)]
mod tests {
    #[cfg(feature = "alloc")]
    use alloc::vec::Vec;
    #[cfg(feature = "std")]
    use std::vec::Vec;

    use super::*;

    #[test]
    fn vrf_single() {
        // #[cfg(feature = "getrandom")]
        let mut csprng = ::rand_core::OsRng;

        let keypair1 = Keypair::generate_with(&mut csprng);

        let ctx = signing_context(b"yo!");
        let msg = b"meow";
        let (io1, proof1, proof1batchable) = keypair1.vrf_sign(ctx.bytes(msg));
        let out1 = &io1.to_output();
        assert_eq!(
            proof1,
            proof1batchable
                .shorten_vrf(&keypair1.public, ctx.bytes(msg), &out1)
                .unwrap(),
            "Oops `shorten_vrf` failed"
        );
        let (io1too, proof1too) = keypair1.public.vrf_verify(ctx.bytes(msg), &out1, &proof1)
            .expect("Correct VRF verification failed!");
        assert_eq!(
            io1too, io1,
            "Output differs between signing and verification!"
        );
        assert_eq!(
            proof1batchable, proof1too,
            "VRF verification yielded incorrect batchable proof"
        );
        assert_eq!(
            keypair1.vrf_sign(ctx.bytes(msg)).0,
            io1,
            "Rerunning VRF gave different output"
        );

        assert!(
            keypair1.public.vrf_verify(ctx.bytes(b"not meow"), &out1, &proof1).is_err(),
            "VRF verification with incorrect message passed!"
        );

        let keypair2 = Keypair::generate_with(&mut csprng);
        assert!(
            keypair2.public.vrf_verify(ctx.bytes(msg), &out1, &proof1).is_err(),
            "VRF verification with incorrect signer passed!"
        );
    }

    #[test]
    fn vrf_malleable() {
        // #[cfg(feature = "getrandom")]
        let mut csprng = ::rand_core::OsRng;

        let keypair1 = Keypair::generate_with(&mut csprng);

        let ctx = signing_context(b"yo!");
        let msg = b"meow";
        let (io1, proof1, proof1batchable) = keypair1.vrf_sign(Malleable(ctx.bytes(msg)));
        let out1 = &io1.to_output();
        assert_eq!(
            proof1,
            proof1batchable.shorten_vrf(&keypair1.public, Malleable(ctx.bytes(msg)), &out1).unwrap(),
            "Oops `shorten_vrf` failed"
        );
        let (io1too, proof1too) = keypair1
            .public.vrf_verify(Malleable(ctx.bytes(msg)), &out1, &proof1)
            .expect("Correct VRF verification failed!");
        assert_eq!(
            io1too, io1,
            "Output differs between signing and verification!"
        );
        assert_eq!(
            proof1batchable, proof1too,
            "VRF verification yielded incorrect batchable proof"
        );
        assert_eq!(
            keypair1.vrf_sign(Malleable(ctx.bytes(msg))).0,
            io1,
            "Rerunning VRF gave different output"
        );
        assert!(
            keypair1.public.vrf_verify(Malleable(ctx.bytes(b"not meow")), &out1, &proof1).is_err(),
            "VRF verification with incorrect message passed!"
        );

        let keypair2 = Keypair::generate_with(&mut csprng);
        assert!(
            keypair2.public.vrf_verify(Malleable(ctx.bytes(msg)), &out1, &proof1).is_err(),
            "VRF verification with incorrect signer passed!"
        );
        let (io2, _proof2, _proof2batchable) = keypair2.vrf_sign(Malleable(ctx.bytes(msg)));
        let out2 = &io2.to_output();

        // Verified key exchange, aka sequential two party VRF.
        let t0 = Transcript::new(b"VRF");
        let io21 = keypair2.secret.vrf_create_from_compressed_point(out1).unwrap();
        let proofs21 = keypair2.dleq_proove(t0.clone(), &io21, KUSAMA_VRF);
        let io12 = keypair1.secret.vrf_create_from_compressed_point(out2).unwrap();
        let proofs12 = keypair1.dleq_proove(t0.clone(), &io12, KUSAMA_VRF);
        assert_eq!(io12.output, io21.output, "Sequential two-party VRF failed");
        assert_eq!(
            proofs21.0,
            proofs21.1.shorten_dleq(t0.clone(), &keypair2.public, &io21, KUSAMA_VRF),
            "Oops `shorten_dleq` failed"
        );
        assert_eq!(
            proofs12.0,
            proofs12.1.shorten_dleq(t0.clone(), &keypair1.public, &io12, KUSAMA_VRF),
            "Oops `shorten_dleq` failed"
        );
        assert!(keypair1
            .public
            .dleq_verify(t0.clone(), &io12, &proofs12.0, KUSAMA_VRF)
            .is_ok());
        assert!(keypair2
            .public
            .dleq_verify(t0.clone(), &io21, &proofs21.0, KUSAMA_VRF)
            .is_ok());
    }

    #[cfg(any(feature = "alloc", feature = "std"))]
    #[test]
    fn vrfs_merged_and_batched() {
        let mut csprng = ::rand_core::OsRng;
        let keypairs: Vec<Keypair> = (0..4)
            .map(|_| Keypair::generate_with(&mut csprng))
            .collect();

        let ctx = signing_context(b"yo!");
        let messages: [&[u8; 4]; 2] = [b"meow", b"woof"];
        let ts = || messages.iter().map(|m| ctx.bytes(*m));

        let ios_n_proofs = keypairs.iter().map(|k| k.vrfs_sign(ts())).collect::<Vec<(
            Box<[VRFInOut]>,
            VRFProof,
            VRFProofBatchable,
        )>>();

        for (k, (ios, proof, proof_batchable)) in keypairs.iter().zip(&ios_n_proofs) {
            let outs = ios
                .iter()
                .map(|io| io.to_output())
                .collect::<Vec<VRFOutput>>();
            let (ios_too, proof_too) = k
                .public
                .vrfs_verify(ts(), &outs, &proof)
                .expect("Valid VRF output verification failed!");
            assert_eq!(
                ios_too, *ios,
                "Output differs between signing and verification!"
            );
            assert_eq!(
                proof_too, *proof_batchable,
                "Returning batchable proof failed!"
            );
        }
        for (k, (ios, proof, _proof_batchable)) in keypairs.iter().zip(&ios_n_proofs) {
            let outs = ios.iter()
                .rev()
                .map(|io| io.to_output())
                .collect::<Vec<VRFOutput>>();
            assert!(
                k.public.vrfs_verify(ts(), &outs, &proof).is_err(),
                "Incorrect VRF output verification passed!"
            );
        }
        for (k, (ios, proof, _proof_batchable)) in keypairs.iter().rev().zip(&ios_n_proofs) {
            let outs = ios.iter()
                .map(|io| io.to_output())
                .collect::<Vec<VRFOutput>>();
            assert!(
                k.public.vrfs_verify(ts(), &outs, &proof).is_err(),
                "VRF output verification by a different signer passed!"
            );
        }

        let mut ios = keypairs.iter().enumerate()
            .map(|(i, keypair)| keypair.public.vrfs_merge(&ios_n_proofs[i].0,true))
            .collect::<Vec<VRFInOut>>();

        let mut proofs = ios_n_proofs.iter()
            .map(|(_ios, _proof, proof_batchable)| proof_batchable.clone())
            .collect::<Vec<VRFProofBatchable>>();

        let mut public_keys = keypairs.iter()
            .map(|keypair| keypair.public.clone())
            .collect::<Vec<PublicKey>>();

        assert!(
            dleq_verify_batch(&ios, &proofs, &public_keys, KUSAMA_VRF).is_ok(),
            "Batch verification failed!"
        );
        proofs.reverse();
        assert!(
            dleq_verify_batch(&ios, &proofs, &public_keys, KUSAMA_VRF).is_err(),
            "Batch verification with incorrect proofs passed!"
        );
        proofs.reverse();
        public_keys.reverse();
        assert!(
            dleq_verify_batch(&ios, &proofs, &public_keys, KUSAMA_VRF).is_err(),
            "Batch verification with incorrect public keys passed!"
        );
        public_keys.reverse();
        ios.reverse();
        assert!(
            dleq_verify_batch(&ios, &proofs, &public_keys, KUSAMA_VRF).is_err(),
            "Batch verification with incorrect points passed!"
        );
    }
}