forked from SSProve/ssprove
-
Notifications
You must be signed in to change notification settings - Fork 0
/
SigmaProtocol.v
1033 lines (945 loc) · 30.7 KB
/
SigmaProtocol.v
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
From Relational Require Import OrderEnrichedCategory GenericRulesSimple.
Set Warnings "-notation-overridden,-ambiguous-paths".
From mathcomp Require Import all_ssreflect all_algebra reals distr realsum
fingroup.fingroup solvable.cyclic prime ssrnat ssreflect ssrfun ssrbool ssrnum
eqtype choice seq.
Set Warnings "notation-overridden,ambiguous-paths".
From Crypt Require Import Axioms ChoiceAsOrd SubDistr Couplings
UniformDistrLemmas FreeProbProg Theta_dens RulesStateProb UniformStateProb
pkg_core_definition choice_type pkg_composition pkg_rhl
Package Prelude RandomOracle.
From Coq Require Import Utf8.
From extructures Require Import ord fset fmap.
From Equations Require Import Equations.
Require Equations.Prop.DepElim.
Set Equations With UIP.
Set Bullet Behavior "Strict Subproofs".
Set Default Goal Selector "!".
Set Primitive Projections.
Import Num.Def.
Import Num.Theory.
Import Order.POrderTheory.
Import PackageNotation.
#[local] Open Scope ring_scope.
Module Type SigmaProtocolParams.
Parameter Witness Statement Message Challenge Response : finType.
Parameter w0 : Witness.
Parameter e0 : Challenge.
Parameter z0 : Response.
Parameter R : Statement → Witness → bool.
Parameter Statement_pos : Positive #|Statement|.
Parameter Witness_pos : Positive #|Witness|.
Parameter Message_pos : Positive #|Message|.
Parameter Challenge_pos : Positive #|Challenge|.
Parameter Response_pos : Positive #|Response|.
Parameter Bool_pos : Positive #|bool_choiceType|.
End SigmaProtocolParams.
Module Type SigmaProtocolAlgorithms (π : SigmaProtocolParams).
Import π.
#[local] Open Scope package_scope.
#[local] Existing Instance Bool_pos.
#[local] Existing Instance Statement_pos.
#[local] Existing Instance Witness_pos.
#[local] Existing Instance Message_pos.
#[local] Existing Instance Challenge_pos.
#[local] Existing Instance Response_pos.
Definition choiceWitness := 'fin #|Witness|.
Definition choiceStatement := 'fin #|Statement|.
Definition choiceMessage := 'fin #|Message|.
Definition choiceChallenge := 'fin #|Challenge|.
Definition choiceResponse := 'fin #|Response|.
Definition choiceTranscript :=
chProd (chProd (chProd choiceStatement choiceMessage) choiceChallenge) choiceResponse.
Definition choiceBool := 'fin #|bool_choiceType|.
Parameter Sigma_locs : {fset Location}.
Parameter Simulator_locs : {fset Location}.
Parameter Commit :
∀ (h : choiceStatement) (w : choiceWitness),
code Sigma_locs [interface] choiceMessage.
Parameter Response :
∀ (h : choiceStatement) (w : choiceWitness)
(a : choiceMessage) (e : choiceChallenge),
code Sigma_locs [interface] choiceResponse.
Parameter Verify :
∀ (h : choiceStatement) (a : choiceMessage) (e : choiceChallenge)
(z : choiceResponse),
choiceBool.
Parameter Simulate :
∀ (h : choiceStatement) (e : choiceChallenge),
code Simulator_locs [interface] choiceTranscript.
Parameter Extractor :
∀ (h : choiceStatement) (a : choiceMessage)
(e : choiceChallenge) (e' : choiceChallenge)
(z : choiceResponse) (z' : choiceResponse),
'option choiceWitness.
Parameter KeyGen : ∀ (w : choiceWitness), choiceStatement.
End SigmaProtocolAlgorithms.
Module SigmaProtocol (π : SigmaProtocolParams)
(Alg : SigmaProtocolAlgorithms π).
Import π.
Import Alg.
Notation " 'chStatement' " :=
choiceStatement (in custom pack_type at level 2).
Notation " 'chWitness' " :=
choiceWitness (in custom pack_type at level 2).
Notation " 'chChallenge' " :=
choiceChallenge (in custom pack_type at level 2).
Notation " 'chRelation' " :=
(chProd choiceStatement choiceWitness) (in custom pack_type at level 2).
Definition choiceInput := (chProd (chProd choiceStatement choiceWitness) choiceChallenge).
Notation " 'chInput' " :=
choiceInput
(in custom pack_type at level 2).
Notation " 'chMessage' " := choiceMessage (in custom pack_type at level 2).
Notation " 'chTranscript' " :=
choiceTranscript (in custom pack_type at level 2).
Definition Opening := chProd choiceChallenge choiceResponse.
Notation " 'chSoundness' " :=
(chProd choiceStatement (chProd choiceMessage (chProd Opening Opening)))
(in custom pack_type at level 2).
Definition i_challenge := #|Challenge|.
Definition i_witness := #|Witness|.
Definition TRANSCRIPT : nat := 0.
Definition COM : nat := 1.
Definition VER : nat := 2.
Definition ADV : nat := 3.
Definition SOUNDNESS : nat := 4.
Definition i_challenge_pos : Positive i_challenge.
Proof.
unfold i_challenge.
apply Challenge_pos.
Qed.
Definition i_witness_pos : Positive i_witness.
Proof.
unfold i_witness.
apply Witness_pos.
Qed.
#[local] Existing Instance i_challenge_pos.
#[local] Existing Instance i_witness_pos.
#[local] Open Scope package_scope.
Definition SHVZK_real:
package Sigma_locs
[interface]
[interface #val #[ TRANSCRIPT ] : chInput → chTranscript]
:=
[package
#def #[ TRANSCRIPT ] (hwe : chInput) : chTranscript
{
let '(h,w,e) := hwe in
#assert (R (otf h) (otf w)) ;;
a ← Commit h w ;;
z ← Response h w a e ;;
@ret choiceTranscript (h,a,e,z)
}
].
Definition SHVZK_ideal:
package Simulator_locs
[interface]
[interface #val #[ TRANSCRIPT ] : chInput → chTranscript]
:=
[package
#def #[ TRANSCRIPT ] (hwe : chInput) : chTranscript
{
let '(h, w, e) := hwe in
#assert (R (otf h) (otf w)) ;;
t ← Simulate h e ;;
ret t
}
].
(* Main security statement for Special Honest-Verifier Zero-Knowledge. *)
Definition ɛ_SHVZK A := AdvantageE SHVZK_real SHVZK_ideal A.
Definition Special_Soundness_f :
package fset0
[interface]
[interface #val #[ SOUNDNESS ] : chSoundness → 'bool ]
:=
[package
#def #[ SOUNDNESS ] (t : chSoundness) : 'bool
{
let '(h, (a, ((e, z), (e', z')))) := t in
let v1 := Verify h a e z in
let v2 := Verify h a e' z' in
if [&& (e != e') , (otf v1) & (otf v2) ] then
match Extractor h a e e' z z' with
| Some w => ret (R (otf h) (otf w))
| None => ret false
end
else ret false
}
].
Definition Special_Soundness_t :
package fset0
[interface]
[interface #val #[ SOUNDNESS ] : chSoundness → 'bool ]
:=
[package
#def #[ SOUNDNESS ] (t : chSoundness) : 'bool
{
let '(h, (a, ((e, z), (e', z')))) := t in
let v1 := Verify h a e z in
let v2 := Verify h a e' z' in
ret [&& (e != e') , (otf v1) & (otf v2) ]
}
].
(* Main security statement for 2-special soundness. *)
Definition ɛ_soundness A :=
AdvantageE Special_Soundness_t Special_Soundness_f A.
(**************************************)
(* Start of Commitment Scheme Section *)
(**************************************)
Section Commitments.
Definition HIDING : nat := 5.
Definition OPEN : nat := 6.
Definition INIT : nat := 7.
Definition GET : nat := 8.
Definition challenge_loc : Location := ('option choiceChallenge; 7%N).
Definition response_loc : Location := ('option choiceResponse; 8%N).
Definition Com_locs : {fset Location} :=
fset [:: challenge_loc ; response_loc ].
Definition setup_loc : Location := ('bool; 10%N).
Definition statement_loc : Location := (choiceStatement; 11%N).
Definition witness_loc : Location := (choiceWitness; 12%N).
Definition KEY_locs : {fset Location} := fset [:: setup_loc; witness_loc ; statement_loc].
Definition choiceOpen := (chProd choiceChallenge choiceResponse).
Notation " 'chOpen' " := choiceOpen (in custom pack_type at level 2).
Notation " 'chKeys' " := (chProd choiceStatement choiceWitness) (in custom pack_type at level 2).
Lemma in_fset_left l (L1 L2 : {fset Location}) :
is_true (l \in L1) →
is_true (l \in (L1 :|: L2)).
Proof.
intros H.
apply /fsetUP.
left. assumption.
Qed.
Hint Extern 20 (is_true (_ \in (_ :|: _))) =>
apply in_fset_left; solve [auto_in_fset]
: typeclass_instances ssprove_valid_db.
Definition KEY:
package KEY_locs
[interface]
[interface
#val #[ INIT ] : 'unit → 'unit ;
#val #[ GET ] : 'unit → chStatement
]
:=
[package
#def #[ INIT ] (_ : 'unit) : 'unit
{
b ← get setup_loc ;;
#assert (negb b) ;;
w ← sample uniform i_witness ;;
let h := KeyGen w in
#assert (R (otf h) (otf w)) ;;
#put setup_loc := true ;;
#put statement_loc := h ;;
#put witness_loc := w ;;
@ret 'unit Datatypes.tt
}
;
#def #[ GET ] (_ : 'unit) : chStatement
{
b ← get setup_loc ;;
if b then
h ← get statement_loc ;;
w ← get witness_loc ;;
ret h
else
fail
}
].
Definition Sigma_to_Com_locs := (Com_locs :|: Simulator_locs).
#[tactic=notac] Equations? Sigma_to_Com:
package Sigma_to_Com_locs
[interface
#val #[ INIT ] : 'unit → 'unit ;
#val #[ GET ] : 'unit → chStatement
]
[interface
#val #[ COM ] : chChallenge → chMessage ;
#val #[ OPEN ] : 'unit → chOpen ;
#val #[ VER ] : chTranscript → 'bool
]
:= Sigma_to_Com :=
[package
#def #[ COM ] (e : chChallenge) : chMessage
{
#import {sig #[ INIT ] : 'unit → 'unit } as key_gen_init ;;
#import {sig #[ GET ] : 'unit → chStatement } as key_gen_get ;;
_ ← key_gen_init Datatypes.tt ;;
h ← key_gen_get Datatypes.tt ;;
'(h,a,e,z) ← Simulate h e ;;
#put challenge_loc := Some e ;;
#put response_loc := Some z ;;
ret a
}
;
#def #[ OPEN ] (_ : 'unit) : chOpen
{
o_e ← get challenge_loc ;;
o_z ← get response_loc ;;
match (o_e, o_z) with
| (Some e, Some z) => @ret choiceOpen (e, z)
| _ => fail
end
}
;
#def #[ VER ] (t : chTranscript) : 'bool
{
let '(h,a,e,z) := t in
ret (otf (Verify h a e z))
}
].
Proof.
unfold Sigma_to_Com_locs.
ssprove_valid.
eapply valid_injectLocations.
1: apply fsubsetUr.
eapply valid_injectMap.
2: apply (Simulate x1 x).
rewrite -fset0E.
apply fsub0set.
Qed.
#[tactic=notac] Equations? Sigma_to_Com_Aux:
package (setup_loc |: Sigma_to_Com_locs)
[interface
#val #[ TRANSCRIPT ] : chInput → chTranscript
]
[interface
#val #[ COM ] : chChallenge → chMessage ;
#val #[ OPEN ] : 'unit → chOpen ;
#val #[ VER ] : chTranscript → 'bool
]
:= Sigma_to_Com_Aux :=
[package
#def #[ COM ] (e : chChallenge) : chMessage
{
#import {sig #[ TRANSCRIPT ] : chInput → chTranscript } as RUN ;;
b ← get setup_loc ;;
#assert (negb b) ;;
w ← sample uniform i_witness ;;
let h := KeyGen w in
#assert (R (otf h) (otf w)) ;;
#put setup_loc := true ;;
'(h, a, e, z) ← RUN (h, w, e) ;;
#put challenge_loc := Some e ;;
#put response_loc := Some z ;;
@ret choiceMessage a
}
;
#def #[ OPEN ] (_ : 'unit) : chOpen
{
o_e ← get challenge_loc ;;
o_z ← get response_loc ;;
match (o_e, o_z) with
| (Some e, Some z) => @ret choiceOpen (e, z)
| _ => fail
end
}
;
#def #[ VER ] (t : chTranscript) : 'bool
{
let '(h,a,e,z) := t in
ret (otf (Verify h a e z))
}
].
Proof.
unfold Sigma_to_Com_locs, Com_locs.
ssprove_valid.
all: rewrite in_fsetU ; apply /orP ; right.
all: rewrite in_fsetU ; apply /orP ; left.
all: rewrite !fset_cons.
1,3 : rewrite in_fsetU ; apply /orP ; left ; rewrite in_fset1 ; done.
1,2 : rewrite in_fsetU ; apply /orP ; right ;
rewrite in_fsetU ; apply /orP ; left ;
rewrite in_fset1 ; done.
Qed.
Notation " 'chHiding' " := (chProd choiceChallenge choiceChallenge) (in custom pack_type at level 2).
Definition Hiding_E := [interface #val #[ HIDING ] : chHiding → chMessage ].
(* Commitment to input value*)
Definition Hiding_real:
package fset0
[interface
#val #[ COM ] : chChallenge → chMessage ;
#val #[ OPEN ] : 'unit → chOpen ;
#val #[ VER ] : chTranscript → 'bool
]
Hiding_E
:=
[package
#def #[ HIDING ] (ms : chHiding) : chMessage
{
#import {sig #[ COM ] : chChallenge → chMessage } as com ;;
let '(m1, m2) := ms in
b ← sample uniform 1 ;;
if Nat.even b then
a ← com m1 ;;
ret a
else
a ← com m2 ;;
ret a
}
].
(* Commitment to random value *)
Definition Hiding_ideal :
package fset0
[interface
#val #[ COM ] : chChallenge → chMessage ;
#val #[ OPEN ] : 'unit → chOpen ;
#val #[ VER ] : chTranscript → 'bool
]
Hiding_E
:=
[package
#def #[ HIDING ] (_ : chHiding) : chMessage
{
#import {sig #[ COM ] : chChallenge → chMessage } as com ;;
e ← sample uniform i_challenge ;;
t ← com e ;;
ret t
}
].
Definition ɛ_hiding A :=
AdvantageE
(Hiding_real ∘ Sigma_to_Com ∘ KEY)
(Hiding_ideal ∘ Sigma_to_Com ∘ KEY) (A ∘ (par KEY (ID Hiding_E))).
Notation inv := (
heap_ignore (fset [:: statement_loc ; witness_loc])
).
Instance Invariant_inv : Invariant (Sigma_to_Com_locs :|: KEY_locs) (setup_loc |: Sigma_to_Com_locs) inv.
Proof.
ssprove_invariant.
unfold KEY_locs.
apply fsubsetU ; apply /orP ; left.
apply fsubsetU ; apply /orP ; right.
rewrite !fset_cons.
apply fsubsetU ; apply /orP ; right.
rewrite fsubUset ; apply /andP ; split.
- apply fsubsetU ; apply /orP ; right.
apply fsubsetU ; apply /orP ; left.
apply fsubsetxx.
- apply fsubsetU ; apply /orP ; left.
rewrite fsubUset ; apply /andP ; split.
+ apply fsubsetxx.
+ rewrite -fset0E. apply fsub0set.
Qed.
Hint Extern 50 (_ = code_link _ _) =>
rewrite code_link_scheme
: ssprove_code_simpl.
Theorem commitment_hiding :
∀ LA A,
ValidPackage LA [interface
#val #[ HIDING ] : chHiding → chMessage
] A_export (A ∘ (par KEY (ID Hiding_E))) →
fdisjoint LA KEY_locs ->
fdisjoint LA Sigma_to_Com_locs ->
fdisjoint LA (fset [:: setup_loc]) ->
fdisjoint LA Sigma_locs ->
fdisjoint LA Simulator_locs ->
fdisjoint Simulator_locs (fset [:: statement_loc ; witness_loc]) ->
fdisjoint Sigma_locs (fset [:: statement_loc ; witness_loc]) ->
(ɛ_hiding A) <= 0 +
AdvantageE SHVZK_ideal SHVZK_real (((A ∘ par KEY (ID Hiding_E)) ∘ Hiding_real) ∘ Sigma_to_Com_Aux) +
AdvantageE (Hiding_real ∘ Sigma_to_Com_Aux ∘ SHVZK_real)
(Hiding_ideal ∘ Sigma_to_Com_Aux ∘ SHVZK_real) (A ∘ par KEY (ID Hiding_E)) +
AdvantageE SHVZK_real SHVZK_ideal (((A ∘ par KEY (ID Hiding_E)) ∘ Hiding_ideal) ∘ Sigma_to_Com_Aux) +
0.
Proof.
unfold ɛ_hiding, ɛ_SHVZK.
intros LA A VA Hd1 Hd2 Hd3 HdSigma HdSimulator Hd4 Hd5.
ssprove triangle (Hiding_real ∘ Sigma_to_Com ∘ KEY) [::
(Hiding_real ∘ Sigma_to_Com_Aux ∘ SHVZK_ideal) ;
(Hiding_real ∘ Sigma_to_Com_Aux ∘ SHVZK_real) ;
(Hiding_ideal ∘ Sigma_to_Com_Aux ∘ SHVZK_real) ;
(Hiding_ideal ∘ Sigma_to_Com_Aux ∘ SHVZK_ideal)
] (Hiding_ideal ∘ Sigma_to_Com ∘ KEY) (A ∘ (par KEY (ID Hiding_E)))
as ineq.
eapply le_trans. 1: exact ineq.
clear ineq.
repeat eapply ler_add.
- apply eq_ler.
eapply eq_rel_perf_ind with (inv := inv).
5: apply VA.
1:{
ssprove_valid.
3: apply fsub0set.
3: apply fsubsetxx.
1: instantiate (1 := (Sigma_to_Com_locs :|: KEY_locs)).
1: apply fsubsetUl.
1: apply fsubsetUr.
}
1:{
ssprove_valid.
1: apply fsubsetxx.
2: apply fsub0set.
2: apply fsubsetxx.
unfold Sigma_to_Com_locs.
apply fsubsetU ; apply /orP ; right.
apply fsubsetUr.
}
3,4: rewrite fdisjointUr ; apply /andP ; split.
3-4,6: assumption.
3: rewrite fset1E ; assumption.
1: exact _.
rewrite Sigma_to_Com_equation_1.
rewrite Sigma_to_Com_Aux_equation_1.
simplify_eq_rel h.
ssprove_code_simpl.
destruct h.
ssprove_code_simpl.
ssprove_code_simpl_more.
ssprove_sync=>b.
case (Nat.even b) eqn:Hb ; rewrite Hb.
+ ssprove_sync=> setup.
ssprove_code_simpl.
ssprove_code_simpl_more.
apply r_assertD.
1: done.
intros _ _.
ssprove_sync=> w.
apply r_assertD.
1: done.
intros _ Rel.
ssprove_swap_seq_lhs [:: 2 ; 1]%N.
ssprove_contract_put_get_lhs.
rewrite !cast_fun_K.
rewrite Rel.
ssprove_code_simpl.
ssprove_code_simpl_more.
ssprove_sync.
ssprove_swap_lhs 1%N.
ssprove_contract_put_get_lhs.
ssprove_swap_seq_lhs [:: 0 ; 1]%N.
ssprove_contract_put_get_lhs.
apply r_put_lhs.
apply r_put_lhs.
ssprove_restore_pre.
1: ssprove_invariant.
eapply rsame_head_alt.
1: exact _.
{
unfold inv.
intros l lin h1 s' h2.
apply h2.
move: Hd4 => /fdisjointP Hd4.
apply Hd4.
apply lin.
}
{
unfold inv.
intros l v lin.
apply put_pre_cond_heap_ignore.
}
intros t.
destruct t.
destruct s1.
destruct s1.
ssprove_sync.
ssprove_sync.
apply r_ret.
done.
+ ssprove_sync=>setup.
ssprove_code_simpl.
ssprove_code_simpl_more.
apply r_assertD.
1: done.
intros _ _.
ssprove_sync=>w.
apply r_assertD.
1: done.
intros _ Rel.
ssprove_swap_seq_lhs [:: 2 ; 1]%N.
ssprove_contract_put_get_lhs.
rewrite !cast_fun_K.
rewrite Rel.
ssprove_code_simpl.
ssprove_code_simpl_more.
ssprove_sync.
ssprove_swap_lhs 1%N.
ssprove_contract_put_get_lhs.
ssprove_swap_seq_lhs [:: 0 ; 1]%N.
ssprove_contract_put_get_lhs.
apply r_put_lhs.
apply r_put_lhs.
ssprove_restore_pre.
1: ssprove_invariant.
eapply rsame_head_alt.
1: exact _.
{
unfold inv.
intros l lin h1 s' h2.
apply h2.
move: Hd4 => /fdisjointP Hd4.
apply Hd4.
apply lin.
}
{
unfold inv.
intros l v lin.
apply put_pre_cond_heap_ignore.
}
intros t.
destruct t.
destruct s1.
destruct s1.
ssprove_sync.
ssprove_sync.
apply r_ret.
done.
- rewrite -!Advantage_link.
1: apply eq_ler ; done.
- done.
- rewrite -!Advantage_link.
1: apply eq_ler ; done.
- apply eq_ler.
eapply eq_rel_perf_ind with (inv := inv).
5: apply VA.
1:{
ssprove_valid.
4: apply fsubsetxx.
3: apply fsub0set.
2: instantiate (1 := (Simulator_locs :|: (setup_loc |: Sigma_to_Com_locs))).
- apply fsubsetUr.
- apply fsubsetUl.
}
1:{
ssprove_valid.
3: apply fsub0set.
3: apply fsubsetxx.
1: instantiate (1 := (Sigma_to_Com_locs :|: KEY_locs)).
- apply fsubsetUl.
- apply fsubsetUr.
}
3,4: rewrite fdisjointUr ; apply /andP ; split.
4: rewrite fdisjointUr ; apply /andP ; split.
3,5-7: assumption.
3: rewrite fset1E ; assumption.
{
ssprove_invariant.
unfold KEY_locs.
apply fsubsetU ; apply /orP ; right.
apply fsubsetU ; apply /orP ; right.
rewrite !fset_cons.
apply fsubsetU ; apply /orP ; right.
rewrite fsubUset ; apply /andP ; split.
- apply fsubsetU ; apply /orP ; right.
apply fsubsetU ; apply /orP ; left.
apply fsubsetxx.
- apply fsubsetU ; apply /orP ; left.
rewrite fsubUset ; apply /andP ; split.
+ apply fsubsetxx.
+ rewrite -fset0E. apply fsub0set.
}
rewrite Sigma_to_Com_equation_1.
rewrite Sigma_to_Com_Aux_equation_1.
simplify_eq_rel h.
ssprove_code_simpl.
destruct h.
ssprove_code_simpl.
ssprove_code_simpl_more.
ssprove_sync=>e.
ssprove_sync=> setup.
ssprove_code_simpl.
ssprove_code_simpl_more.
apply r_assertD.
1: done.
intros _ _.
ssprove_sync=> w.
apply r_assertD.
1: done.
intros _ Rel.
ssprove_swap_seq_rhs [:: 2 ; 1]%N.
ssprove_contract_put_get_rhs.
rewrite !cast_fun_K.
rewrite Rel.
ssprove_code_simpl.
ssprove_code_simpl_more.
ssprove_sync.
ssprove_swap_rhs 1%N.
ssprove_contract_put_get_rhs.
ssprove_swap_seq_rhs [:: 0 ; 1]%N.
ssprove_contract_put_get_rhs.
apply r_put_rhs.
apply r_put_rhs.
ssprove_restore_pre.
1: ssprove_invariant.
eapply rsame_head_alt.
1: exact _.
{
unfold inv.
intros l lin h1 s' h2.
apply h2.
move: Hd4 => /fdisjointP Hd4.
apply Hd4.
apply lin.
}
{
unfold inv.
intros l v lin.
apply put_pre_cond_heap_ignore.
}
intros t.
destruct t.
destruct s1.
destruct s1.
ssprove_sync.
ssprove_sync.
apply r_ret.
done.
Qed.
Definition Com_Binding:
package fset0
[interface
#val #[ COM ] : chChallenge → chMessage ;
#val #[ OPEN ] : 'unit → chOpen ;
#val #[ VER ] : chTranscript → 'bool
]
[interface #val #[ SOUNDNESS ] : chSoundness → 'bool ]
:=
[package
#def #[ SOUNDNESS ] (t : chSoundness) : 'bool
{
#import {sig #[ VER ] : chTranscript → 'bool } as Ver ;;
let '(h, (a, ((e, z), (e', z')))) := t in
v1 ← Ver (h, a, e, z) ;;
v2 ← Ver (h, a, e', z') ;;
ret [&& (e != e'), v1 & v2]
}
].
Lemma commitment_binding :
∀ LA A,
ValidPackage LA [interface
#val #[ SOUNDNESS ] : chSoundness → 'bool
] A_export A →
fdisjoint LA (Sigma_to_Com_locs :|: KEY_locs) →
AdvantageE (Com_Binding ∘ Sigma_to_Com ∘ KEY) (Special_Soundness_t) A = 0.
Proof.
intros LA A VA Hdisj.
eapply eq_rel_perf_ind_eq.
4: apply VA.
1:{
ssprove_valid.
3: apply fsub0set.
1: instantiate (1 := (Sigma_to_Com_locs :|: KEY_locs)).
2: apply fsubsetUr.
1: apply fsubsetUl.
apply fsubsetxx.
}
1: ssprove_valid.
2: assumption.
2: apply fdisjoints0.
rewrite Sigma_to_Com_equation_1.
simplify_eq_rel h.
ssprove_code_simpl.
simpl.
destruct h, s0, s1, s1, s2.
apply r_ret. auto.
Qed.
End Commitments.
(* This section aim to prove an automatic conversation between the sampling of the random challenge and a random oracle. *)
(* The main difference is that the random oracle is a query parametrized by the context of the execution. *)
Import RandomOracle.
Module OracleParams <: ROParams.
Definition Query := prod_finType Statement Message.
Definition Random := Challenge.
Definition Query_pos : Positive #|Query|.
Proof.
unfold Query. rewrite !card_prod.
apply Positive_prod.
- apply Statement_pos.
- apply Message_pos.
Qed.
Definition Random_pos : Positive #|Random| := Challenge_pos.
End OracleParams.
Module Oracle := RO OracleParams.
Import Oracle OracleParams.
Section FiatShamir.
Definition RUN : nat := 6.
Definition VERIFY : nat := 7.
Definition prod_assoc : chProd choiceStatement choiceMessage → chQuery.
Proof.
cbn. intros [statement message].
rewrite !card_prod.
apply mxvec_index. all: assumption.
Qed.
(* TW: I moved it here because it might induce back-tracking and we want to
avoid it because of time-consumption.
*)
Hint Extern 20 (ValidCode ?L ?I ?c.(prog)) =>
eapply valid_injectMap ; [| eapply c.(prog_valid) ]
: typeclass_instances ssprove_valid_db.
Definition Fiat_Shamir :
package Sigma_locs
[interface
#val #[ INIT ] : 'unit → 'unit ;
#val #[ QUERY ] : 'query → 'random
]
[interface
#val #[ VERIFY ] : chTranscript → 'bool ;
#val #[ RUN ] : chRelation → chTranscript
]
:=
[package
#def #[ VERIFY ] (t : chTranscript) : 'bool
{
#import {sig #[ QUERY ] : 'query → 'random } as RO_query ;;
let '(h,a,e,z) := t in
e ← RO_query (prod_assoc (h, a)) ;;
ret (otf (Verify h a e z))
} ;
#def #[ RUN ] (hw : chRelation) : chTranscript
{
#import {sig #[ INIT ] : 'unit → 'unit } as RO_init ;;
#import {sig #[ QUERY ] : 'query → 'random } as RO_query ;;
let '(h,w) := hw in
#assert (R (otf h) (otf w)) ;;
a ← Commit h w ;;
RO_init Datatypes.tt ;;
e ← RO_query (prod_assoc (h, a)) ;;
z ← Response h w a e ;;
@ret choiceTranscript (h,a,e,z)
}
].
Definition RUN_interactive :
package Sigma_locs
[interface]
[interface
#val #[ VERIFY ] : chTranscript → 'bool ;
#val #[ RUN ] : chRelation → chTranscript
]
:=
[package
#def #[ VERIFY ] (t : chTranscript) : 'bool
{
let '(h,a,e,z) := t in
ret (otf (Verify h a e z))
} ;
#def #[ RUN ] (hw : chRelation) : chTranscript
{
let '(h,w) := hw in
#assert (R (otf h) (otf w)) ;;
a ← Commit h w ;;
e ← sample uniform i_random ;;
z ← Response h w a e ;;
@ret choiceTranscript (h,a,e,z)
}
].
Definition SHVZK_real_aux :
package Sigma_locs
[interface #val #[ TRANSCRIPT ] : chInput → chTranscript ]
[interface #val #[ RUN ] : chRelation → chTranscript ]
:=
[package
#def #[ RUN ] (hw : chRelation) : chTranscript
{
#import {sig #[ TRANSCRIPT ] : chInput → chTranscript } as SHVZK ;;
e ← sample uniform i_random ;;
t ← SHVZK (hw, e) ;;
ret t
}
].
Lemma run_interactive_shvzk :
∀ LA A,
ValidPackage LA [interface
#val #[ RUN ] : chRelation → chTranscript
] A_export A →
fdisjoint LA Sigma_locs →
AdvantageE RUN_interactive (SHVZK_real_aux ∘ SHVZK_real) A = 0.
Proof.
intros LA A Va Hdisj.
eapply eq_rel_perf_ind_eq.
5,6: apply Hdisj.
4: apply Va.
2:{
rewrite <- fsetUid.
eapply valid_link.
- apply SHVZK_real_aux.
- apply SHVZK_real.
}
1:{
eapply valid_package_inject_export.
2: apply RUN_interactive.
apply fsubset_ext. intros ? ?.
rewrite fset_cons. apply /fsetUP. right. assumption.
}
simplify_eq_rel hw.
ssprove_code_simpl.
rewrite cast_fun_K.
ssprove_code_simpl.
destruct hw as [h w].
ssprove_code_simpl_more. ssprove_code_simpl.
ssprove_swap_rhs 0%N.
ssprove_sync_eq. intro rel.
ssprove_swap_rhs 0%N.
apply rsame_head. intros [a st].
ssprove_sync_eq. intro e.
apply rsame_head. intro z.
apply r_ret. intuition auto.
Qed.
Hint Extern 50 (_ = code_link _ _) =>
rewrite code_link_scheme
: ssprove_code_simpl.
Theorem fiat_shamir_correct :
∀ LA A ,
ValidPackage LA [interface
#val #[ RUN ] : chRelation → chTranscript
] A_export A →
fdisjoint LA (Sigma_locs :|: RO_locs) →
fdisjoint Sigma_locs RO_locs →
AdvantageE (Fiat_Shamir ∘ RO) RUN_interactive A = 0.
Proof.
intros LA A Va Hdisj Hdisj_oracle.
eapply eq_rel_perf_ind_ignore.
6: apply Hdisj.
6:{
rewrite fdisjointUr in Hdisj. move: Hdisj => /andP [h _].
apply h.
}
5: apply Va.
1:{
ssprove_valid.
2: apply fsubsetUl.
2: apply fsubsetUr.
eapply valid_package_inject_export.
2: apply Fiat_Shamir.
apply fsubset_ext. intros.
rewrite fset_cons. apply /fsetUP. right. assumption.
}
1:{
eapply valid_package_inject_export.
2: apply RUN_interactive.
apply fsubset_ext. intros.
rewrite fset_cons. apply /fsetUP. right. assumption.
}
1:{ apply fsubsetU. erewrite fsubsetUr. auto. }
simplify_eq_rel hw.
ssprove_code_simpl.