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
|
Require Import BinInt BinNat ZArith Znumtheory.
Require Export Coq.Classes.Morphisms Setoid.
Require Export Ring_theory Field_theory Field_tac.
Require Export Crypto.Galois.GaloisField.
Set Implicit Arguments.
Unset Strict Implicits.
Module GaloisFieldTheory (M: Modulus).
Module Field := GaloisField M.
Import M Field.
(* Notations *)
Notation "x {+} y" := (GFplus x y) (at level 60, right associativity).
Notation "x {-} y" := (GFminus x y) (at level 60, right associativity).
Notation "x {*} y" := (GFmult x y) (at level 50, left associativity).
Notation "x {/} y" := (GFdiv x y) (at level 50, left associativity).
Notation "x {^} y" := (GFexp x y) (at level 40, left associativity).
Notation "{1}" := GFone.
Notation "{0}" := GFzero.
(* Basic Properties *)
Theorem GFplus_0_l: forall x : GF, {0}{+}x = x.
intro x. apply gf_eq. unfold GFzero, GFplus, GFToZ; simpl.
destruct x. destruct e. simpl. rewrite e.
rewrite Zmod_mod. trivial.
Qed.
Theorem GFplus_commutative: forall x y : GF, x{+}y = y{+}x.
intros x y. apply gf_eq. unfold GFzero, GFplus, GFToZ; simpl.
destruct x. destruct e.
destruct y. destruct e0.
simpl. rewrite Zplus_comm. rewrite e, e0. trivial.
Qed.
Theorem GFplus_associative: forall x y z : GF, x{+}y{+}z = (x{+}y){+}z.
intros x y z. apply gf_eq. unfold GFzero, GFplus, GFToZ; simpl.
destruct x. destruct e.
destruct y. destruct e0.
destruct z. destruct e1.
simpl. rewrite Zplus_mod_idemp_l, Zplus_mod_idemp_r, Zplus_assoc. trivial.
Qed.
Theorem GFmult_1_l: forall x : GF, {1}{*}x = x.
intro x. apply gf_eq. unfold GFzero, GFmult, GFToZ; simpl.
destruct x. destruct e. simpl. rewrite e.
(* TODO: how do we make this automatic?
* I'd like to case, but we need to keep the mod.*)
assert (forall z,
match z with
| 0 => 0
| Z.pos y' => Z.pos y'
| Z.neg y' => Z.neg y'
end = z).
intros; induction z; simpl; intuition.
rewrite H, Zmod_mod. trivial.
Qed.
Theorem GFmult_commutative: forall x y : GF, x{*}y = y{*}x.
intros x y. apply gf_eq. unfold GFzero, GFmult, GFToZ; simpl.
destruct x. destruct e.
destruct y. destruct e0.
simpl. rewrite Zmult_comm. rewrite e, e0. trivial.
Qed.
Theorem GFmult_associative: forall x y z : GF, x{*}(y{*}z) = x{*}y{*}z.
intros x y z. apply gf_eq. unfold GFzero, GFmult, GFToZ; simpl.
destruct x. destruct e.
destruct y. destruct e0.
destruct z. destruct e1.
simpl. rewrite Zmult_mod_idemp_l, Zmult_mod_idemp_r, Zmult_assoc. trivial.
Qed.
Theorem GFdistributive: forall x y z : GF, (x{+}y){*}z = x{*}z{+}y{*}z.
intros x y z. apply gf_eq. unfold GFzero, GFmult, GFplus, GFToZ; simpl.
destruct x. destruct e.
destruct y. destruct e0.
destruct z. destruct e1.
simpl.
rewrite Zmult_mod_idemp_l.
rewrite <- Zplus_mod.
rewrite Z.mul_add_distr_r.
trivial.
Qed.
Theorem GFminus_opp: forall x y : GF, x{-}y = x{+}GFopp y.
intros x y. apply gf_eq. unfold GFzero, GFmult, GFToZ; simpl.
destruct x. destruct e.
destruct y. destruct e0.
simpl.
rewrite Zplus_mod_idemp_r.
trivial.
Qed.
Theorem GFopp_inverse: forall x : GF, x{+}GFopp x = {0}.
intro x. apply gf_eq. unfold GFzero, GFplus, GFToZ; simpl.
destruct x. destruct e. simpl. rewrite e.
rewrite <- Zplus_mod.
rewrite Z.add_opp_r.
rewrite Zminus_mod_idemp_r.
rewrite Z.sub_diag.
trivial.
Qed.
(* Ring Theory*)
Instance GFplus_compat : Proper (eq==>eq==>eq) GFplus.
intros (p1, p2) (q1, q2) H (r1, r2) (s1, s2) H0; intuition; simpl in *.
assert (p1 + r1 = q1 + s1).
apply exist_eq in H; apply exist_eq in H0. rewrite H, H0. trivial.
unfold GFplus; simpl; rewrite H1; trivial.
Qed.
Instance GFminus_compat : Proper (eq==>eq==>eq) GFminus.
intros (p1, p2) (q1, q2) H (r1, r2) (s1, s2) H0; intuition; simpl in *.
assert (p1 - r1 = q1 - s1).
apply exist_eq in H; apply exist_eq in H0. rewrite H, H0. trivial.
unfold GFminus; simpl; rewrite H1; trivial.
Qed.
Instance GFmult_compat : Proper (eq==>eq==>eq) GFmult.
intros (p1, p2) (q1, q2) H (r1, r2) (s1, s2) H0; intuition; simpl in *.
assert (p1 * r1 = q1 * s1).
apply exist_eq in H; apply exist_eq in H0. rewrite H, H0. trivial.
unfold GFmult; simpl; rewrite H1; trivial.
Qed.
Instance GFopp_compat : Proper (eq==>eq) GFopp.
intros x y H; intuition; simpl in *.
unfold GFopp, GFzero, GFminus; simpl. apply exist_eq.
destruct x. destruct e. rewrite e in *.
destruct y. destruct e0. rewrite e0 in *.
simpl in *. apply exist_eq in H.
assert (HX := (Z.eq_dec (x0 mod modulus) 0)); destruct HX.
rewrite e1; rewrite H in e1; rewrite e1; simpl; intuition.
repeat rewrite Z_mod_nz_opp_full;
try repeat rewrite Zmod_mod;
simpl; intuition.
Qed.
Definition GFring_theory : ring_theory GFzero GFone GFplus GFmult GFminus GFopp eq.
constructor.
exact GFplus_0_l.
exact GFplus_commutative.
exact GFplus_associative.
exact GFmult_1_l.
exact GFmult_commutative.
exact GFmult_associative.
exact GFdistributive.
exact GFminus_opp.
exact GFopp_inverse.
Defined.
Add Ring GFring : GFring_theory.
(* Power theory *)
Lemma GFpower_theory : power_theory GFone GFmult eq id GFexp.
constructor. intros.
induction n; simpl; intuition.
(* Prove the doubling case, which we'll use alot *)
assert (forall q r0, GFexp' r0 q{*}GFexp' r0 q = GFexp' (r0{*}r0) q).
induction q.
intros. assert (Q := (IHq (r0{*}r0))).
unfold GFexp'; simpl; fold GFexp'.
rewrite <- Q. ring.
intros. assert (Q := (IHq (r0{*}r0))).
unfold GFexp'; simpl; fold GFexp'.
rewrite <- Q. ring.
intros; simpl; ring.
(* Take it case-by-case *)
unfold GFexp; induction p.
(* Odd case *)
unfold GFexp'; simpl; fold GFexp'; fold pow_pos.
rewrite <- (H p r).
rewrite <- IHp. trivial.
(* Even case *)
unfold GFexp'; simpl; fold GFexp'; fold pow_pos.
rewrite <- (H p r).
rewrite <- IHp. trivial.
(* Base case *)
simpl; intuition.
Qed.
(* Field Theory*)
Instance GFinv_compat : Proper (eq==>eq) GFinv.
unfold GFinv; simpl; intuition.
Qed.
Instance GFdiv_compat : Proper (eq==>eq==>eq) GFdiv.
unfold GFdiv; simpl; intuition.
Qed.
Definition GFfield_theory : field_theory GFzero GFone GFplus GFmult GFminus GFopp GFdiv GFinv eq.
constructor; simpl; intuition.
exact GFring_theory.
unfold GFone, GFzero in H; apply exist_eq in H; simpl in H; intuition.
(* Prove multiplicative inverses *)
unfold GFinv.
destruct totient; simpl. destruct i.
replace (p{^}(N.pred x){*}p) with (p{^}x); simpl; intuition.
apply N.gt_lt in H0; apply N.lt_neq in H0.
replace x with (N.succ (N.pred x)).
induction (N.pred x).
simpl. ring.
rewrite N.pred_succ.
(* Just the N.succ case *)
generalize p. induction p0; simpl in *; intuition.
rewrite (IHp0 (p1{*}p1)). ring.
ring.
(* cleanup *)
rewrite N.succ_pred; intuition.
Defined.
Add Field GFfield : GFfield_theory.
End GaloisFieldTheory.
|
