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(* ========================================================================= *)
(* Twisted Edwards curves in general, a * x^2 + y^2 = 1 + d * x^2 * y^2. *)
(* ========================================================================= *)
needs "EC/misc.ml";;
(* ------------------------------------------------------------------------- *)
(* Basic definitions and naive cardinality bounds. *)
(* ------------------------------------------------------------------------- *)
let edwards_point = define
`edwards_point f (x:A,y) <=>
x IN ring_carrier f /\ y IN ring_carrier f`;;
let edwards_curve = define
`edwards_curve(f:A ring,a:A,d:A) (x,y) <=>
x IN ring_carrier f /\ y IN ring_carrier f /\
ring_add f (ring_mul f a (ring_pow f x 2)) (ring_pow f y 2) =
ring_add f (ring_1 f)
(ring_mul f d (ring_mul f (ring_pow f x 2) (ring_pow f y 2)))`;;
let edwards_nonsingular = define
`edwards_nonsingular (f:A ring,a:A,d:A) <=>
(?b. b IN ring_carrier f /\ ring_pow f b 2 = a) /\
(d = ring_0 f \/ ~(?c. c IN ring_carrier f /\ ring_pow f c 2 = d))`;;
let edwards_0 = define
`edwards_0 (f:A ring,a:A,d:A) = (ring_0 f,ring_1 f)`;;
let edwards_neg = define
`edwards_neg (f:A ring,a:A,d:A) (x,y:A) = (ring_neg f x,y)`;;
let edwards_add = define
`edwards_add (f:A ring,a:A,d:A) (x1,y1) (x2,y2) =
ring_div f (ring_add f (ring_mul f x1 y2) (ring_mul f y1 x2))
(ring_add f (ring_1 f)
(ring_mul f d (ring_mul f x1 (ring_mul f y1 (ring_mul f x2 y2))))),
ring_div f
(ring_sub f (ring_mul f y1 y2) (ring_mul f a (ring_mul f x1 x2)))
(ring_sub f (ring_1 f)
(ring_mul f d (ring_mul f x1 (ring_mul f y1 (ring_mul f x2 y2)))))`;;
let edwards_group = define
`edwards_group (f:A ring,a:A,d:A) =
group(edwards_curve(f,a,d),
edwards_0(f,a,d),
edwards_neg(f,a,d),
edwards_add(f,a,d))`;;
let EDWARD_NONSINGULAR_ALT = prove
(`!f a (d:A).
field f /\ a IN ring_carrier f /\ d IN ring_carrier f
==> (edwards_nonsingular (f,a,d) <=>
(?b. b IN ring_carrier f /\ ring_pow f b 2 = a) /\
~(?c. c IN ring_carrier f /\ ~(c = ring_0 f) /\ ring_pow f c 2 = d))`,
REPEAT STRIP_TAC THEN REWRITE_TAC[edwards_nonsingular] THEN AP_TERM_TAC THEN
ASM_CASES_TAC `d:A = ring_0 f` THEN ASM_REWRITE_TAC[] THENL
[FIELD_TAC; AP_TERM_TAC THEN AP_TERM_TAC THEN ABS_TAC] THEN
EQ_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN FIELD_TAC);;
let FINITE_EDWARDS_CURVE = prove
(`!f a d:A. field f /\ FINITE(ring_carrier f)
==> FINITE(edwards_curve(f,a,d))`,
REPEAT STRIP_TAC THEN
MATCH_MP_TAC FINITE_SUBSET THEN EXISTS_TAC
`{(x,y) | (x:A) IN ring_carrier f /\ y IN ring_carrier f}` THEN
ASM_SIMP_TAC[FINITE_PRODUCT] THEN
REWRITE_TAC[edwards_curve; SUBSET; FORALL_PAIR_THM; IN_ELIM_PAIR_THM] THEN
SIMP_TAC[edwards_curve; IN]);;
let CARD_BOUND_EDWARDS_CURVE = prove
(`!f a d:A. field f /\ FINITE(ring_carrier f) /\
a IN ring_carrier f /\ d IN ring_carrier f /\
edwards_nonsingular(f,a,d)
==> CARD(edwards_curve(f,a,d)) <= 2 * CARD(ring_carrier f)`,
REPEAT STRIP_TAC THEN
MATCH_MP_TAC(MESON[FINITE_SUBSET; CARD_SUBSET; LE_TRANS]
`!s. t SUBSET s /\ FINITE s /\ CARD s <= n ==> CARD t <= n`) THEN
EXISTS_TAC
`{(x,y) | (x:A) IN ring_carrier f /\ y IN ring_carrier f /\
ring_pow f y 2 =
ring_div f (ring_sub f (ring_1 f)
(ring_mul f a (ring_pow f x 2)))
(ring_sub f (ring_1 f)
(ring_mul f d (ring_pow f x 2)))}`THEN
ASM_SIMP_TAC[FINITE_QUADRATIC_CURVE; FIELD_IMP_INTEGRAL_DOMAIN] THEN
REWRITE_TAC[SUBSET; FORALL_PAIR_THM; IN_ELIM_PAIR_THM] THEN
MAP_EVERY X_GEN_TAC [`x:A`; `y:A`] THEN GEN_REWRITE_TAC LAND_CONV [IN] THEN
REWRITE_TAC[edwards_curve] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM(MP_TAC o SYM o SYM) THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [edwards_nonsingular]) THEN
REWRITE_TAC[NOT_EXISTS_THM; RIGHT_OR_FORALL_THM] THEN
DISCH_THEN(MP_TAC o SPEC `ring_inv f x:A` o CONJUNCT2) THEN
ASM_SIMP_TAC[RING_INV] THEN FIELD_TAC);;
(* ------------------------------------------------------------------------- *)
(* Proof of the group properties by algebraic brute force. We do use a bit *)
(* more delicacy than calling FIELD_TAC in order to avoid assuming anything *)
(* about the characteristic of the field. *)
(* ------------------------------------------------------------------------- *)
let EDWARDS_NONSINGULAR_DENOMINATORS = prove
(`!f a (d:A) x1 y1 x2 y2.
field f /\ a IN ring_carrier f /\ d IN ring_carrier f /\
edwards_nonsingular(f,a,d) /\
edwards_curve(f,a,d) (x1,y1) /\ edwards_curve(f,a,d) (x2,y2)
==> ~(ring_add f (ring_1 f)
(ring_mul f d
(ring_mul f x1 (ring_mul f y1 (ring_mul f x2 y2)))) =
ring_0 f) /\
~(ring_sub f (ring_1 f)
(ring_mul f d
(ring_mul f x1 (ring_mul f y1 (ring_mul f x2 y2)))) =
ring_0 f)`,
REPEAT GEN_TAC THEN REWRITE_TAC[edwards_curve; GSYM DE_MORGAN_THM] THEN
DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN DISCH_TAC THEN
UNDISCH_TAC `edwards_nonsingular(f,a:A,d)` THEN
ASM_SIMP_TAC[EDWARD_NONSINGULAR_ALT] THEN
REWRITE_TAC[TAUT `~(p /\ ~q) <=> p ==> q`; LEFT_IMP_EXISTS_THM] THEN
X_GEN_TAC `e:A` THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (SUBST_ALL_TAC o SYM)) THEN
MATCH_MP_TAC(MESON[] `!a b. P a \/ P b ==> ?x. P x`) THEN
EXISTS_TAC `ring_inv f (ring_mul f x1 y2):A` THEN
EXISTS_TAC `ring_inv f (ring_mul f e (ring_mul f x1 x2)):A` THEN
ASM_SIMP_TAC[RING_INV; RING_MUL] THEN FIELD_TAC);;
let EDWARDS_NONSINGULAR_DENOMINATORS_POINTS =
GEN_REWRITE_RULE (funpow 4 BINDER_CONV) [FORALL_UNPAIR_THM]
(GEN_REWRITE_RULE (funpow 3 BINDER_CONV) [FORALL_UNPAIR_THM]
EDWARDS_NONSINGULAR_DENOMINATORS);;
let EDWARDS_CURVE_IMP_POINT = prove
(`!f a d p. edwards_curve(f,a,d) p ==> edwards_point f p`,
REWRITE_TAC[FORALL_PAIR_THM] THEN SIMP_TAC[edwards_curve; edwards_point]);;
let EDWARDS_POINT_NEG = prove
(`!(f:A ring) a d p.
edwards_point f p
==> edwards_point f (edwards_neg (f,a,d) p)`,
REWRITE_TAC[FORALL_PAIR_THM] THEN
SIMP_TAC[edwards_neg; edwards_point; RING_NEG]);;
let EDWARDS_POINT_ADD = prove
(`!(f:A ring) a d p q.
a IN ring_carrier f /\ d IN ring_carrier f /\
edwards_point f p /\ edwards_point f q
==> edwards_point f (edwards_add (f,a,d) p q)`,
REWRITE_TAC[FORALL_PAIR_THM] THEN
SIMP_TAC[edwards_add; edwards_point; LET_DEF; LET_END_DEF] THEN
REPEAT STRIP_TAC THEN
REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[edwards_point]) THEN
REPEAT STRIP_TAC THEN RING_CARRIER_TAC);;
let EDWARDS_CURVE_0 = prove
(`!f a d:A.
a IN ring_carrier f /\ d IN ring_carrier f
==> edwards_curve(f,a,d) (edwards_0(f,a,d))`,
REWRITE_TAC[edwards_curve; edwards_0] THEN
REPEAT STRIP_TAC THEN TRY RING_CARRIER_TAC THEN RING_TAC);;
let EDWARDS_CURVE_NEG = prove
(`!f a (d:A) p.
integral_domain f /\ a IN ring_carrier f /\ d IN ring_carrier f /\
edwards_curve(f,a,d) p
==> edwards_curve(f,a,d) (edwards_neg (f,a,d) p)`,
SIMP_TAC[FORALL_PAIR_THM; RING_NEG; edwards_curve; edwards_neg] THEN
REPEAT GEN_TAC THEN CONV_TAC INTEGRAL_DOMAIN_RULE);;
let EDWARDS_CURVE_ADD = prove
(`!f a (d:A) p q.
field f /\
a IN ring_carrier f /\ d IN ring_carrier f /\
edwards_nonsingular (f,a,d) /\
edwards_curve(f,a,d) p /\ edwards_curve(f,a,d) q
==> edwards_curve(f,a,d) (edwards_add (f,a,d) p q)`,
REWRITE_TAC[FORALL_PAIR_THM; edwards_curve; edwards_add; PAIR_EQ] THEN
MAP_EVERY X_GEN_TAC
[`f:A ring`; `a:A`; `d:A`; `x1:A`; `y1:A`; `x2:A`; `y2:A`] THEN
STRIP_TAC THEN
MP_TAC(SPECL [`f:A ring`; `a:A`; `d:A`; `x1:A`; `y1:A`; `x2:A`; `y2:A`]
EDWARDS_NONSINGULAR_DENOMINATORS) THEN
ASM_REWRITE_TAC[edwards_curve] THEN
FIRST_X_ASSUM(K ALL_TAC o GEN_REWRITE_RULE I [edwards_nonsingular]) THEN
REPEAT STRIP_TAC THEN TRY RING_CARRIER_TAC THEN
RING_PULL_DIV_TAC THEN RING_TAC);;
let EDWARDS_ADD_LID = prove
(`!f a (d:A) p.
field f /\
a IN ring_carrier f /\ d IN ring_carrier f /\
edwards_curve(f,a,d) p
==> edwards_add(f,a,d) (edwards_0 (f,a,d)) p = p`,
REWRITE_TAC[FORALL_PAIR_THM; edwards_curve; edwards_add;
edwards_0; PAIR_EQ] THEN
REPEAT STRIP_TAC THEN TRY RING_CARRIER_TAC THEN FIELD_TAC);;
let EDWARDS_ADD_LNEG = prove
(`!f a (d:A) p.
field f /\
a IN ring_carrier f /\ d IN ring_carrier f /\
edwards_nonsingular (f,a,d) /\
edwards_curve(f,a,d) p
==> edwards_add(f,a,d) (edwards_neg (f,a,d) p) p = edwards_0(f,a,d)`,
REWRITE_TAC[FORALL_PAIR_THM; edwards_curve; edwards_add;
edwards_neg; edwards_0; PAIR_EQ] THEN
MAP_EVERY X_GEN_TAC [`f:A ring`; `a:A`; `d:A`; `x:A`; `y:A`] THEN
STRIP_TAC THEN
MP_TAC(SPECL [`f:A ring`; `a:A`; `d:A`;
`edwards_neg (f,a,d) (x,y):A#A`; `(x,y):A#A`]
EDWARDS_NONSINGULAR_DENOMINATORS_POINTS) THEN
ASM_REWRITE_TAC[edwards_curve; edwards_neg] THEN
FIRST_X_ASSUM(K ALL_TAC o GEN_REWRITE_RULE I [edwards_nonsingular]) THEN
ANTS_TAC THEN REPEAT STRIP_TAC THEN TRY RING_CARRIER_TAC THEN
FIELD_TAC);;
let EDWARDS_ADD_SYM = prove
(`!f a (d:A) p q.
field f /\
a IN ring_carrier f /\ d IN ring_carrier f /\
edwards_nonsingular (f,a,d) /\
edwards_curve(f,a,d) p /\ edwards_curve(f,a,d) q
==> edwards_add (f,a,d) p q = edwards_add (f,a,d) q p`,
REWRITE_TAC[FORALL_PAIR_THM; edwards_curve; edwards_add; PAIR_EQ] THEN
MAP_EVERY X_GEN_TAC
[`f:A ring`; `a:A`; `d:A`; `x1:A`; `y1:A`; `x2:A`; `y2:A`] THEN
STRIP_TAC THEN
MP_TAC(SPECL [`f:A ring`; `a:A`; `d:A`; `x1:A`; `y1:A`; `x2:A`; `y2:A`]
EDWARDS_NONSINGULAR_DENOMINATORS) THEN
MP_TAC(SPECL [`f:A ring`; `a:A`; `d:A`; `x2:A`; `y2:A`; `x1:A`; `y1:A`]
EDWARDS_NONSINGULAR_DENOMINATORS) THEN
ASM_REWRITE_TAC[edwards_curve] THEN
FIRST_X_ASSUM(K ALL_TAC o GEN_REWRITE_RULE I [edwards_nonsingular]) THEN
REPEAT STRIP_TAC THEN TRY RING_CARRIER_TAC THEN
RING_PULL_DIV_TAC THEN RING_TAC);;
let EDWARDS_ADD_ASSOC = prove
(`!f a (d:A) p q r.
field f /\
a IN ring_carrier f /\ d IN ring_carrier f /\
edwards_nonsingular (f,a,d) /\
edwards_curve(f,a,d) p /\ edwards_curve(f,a,d) q /\
edwards_curve(f,a,d) r
==> edwards_add (f,a,d) p (edwards_add (f,a,d) q r) =
edwards_add (f,a,d) (edwards_add (f,a,d) p q) r`,
let lemma = prove
(`field f /\
x1 IN ring_carrier f /\ y1 IN ring_carrier f /\
x2 IN ring_carrier f /\ y2 IN ring_carrier f /\
~(y1 = ring_0 f \/ y2 = ring_0 f) /\
ring_mul f x1 y2 = ring_mul f x2 y1
==> ring_div f x1 y1 = ring_div f x2 y2`,
FIELD_TAC) in
REWRITE_TAC[FORALL_PAIR_THM; edwards_curve; edwards_add; PAIR_EQ] THEN
MAP_EVERY X_GEN_TAC
[`f:A ring`; `a:A`; `d:A`; `x1:A`; `y1:A`;
`x2:A`; `y2:A`; `x3:A`; `y3:A`] THEN
STRIP_TAC THEN CONJ_TAC THEN MATCH_MP_TAC lemma THEN
ASM_REWRITE_TAC[] THEN
REPLICATE_TAC 4 (CONJ_TAC THENL [RING_CARRIER_TAC; ALL_TAC]) THEN
CONJ_TAC THENL
[REWRITE_TAC[DE_MORGAN_THM] THEN CONJ_TAC THEN
W(MP_TAC o PART_MATCH (lhand o rand) EDWARDS_NONSINGULAR_DENOMINATORS o
snd) THEN
(ANTS_TAC THENL [ALL_TAC; DISCH_THEN(ACCEPT_TAC o CONJUNCT1)]) THEN
REWRITE_TAC[GSYM edwards_add] THEN ASM_REWRITE_TAC[] THEN
REPEAT CONJ_TAC THEN TRY(MATCH_MP_TAC EDWARDS_CURVE_ADD) THEN
ASM_REWRITE_TAC[edwards_curve] THEN ASM_MESON_TAC[DIVIDES_REFL];
ALL_TAC;
REWRITE_TAC[DE_MORGAN_THM] THEN CONJ_TAC THEN
W(MP_TAC o PART_MATCH (rand o rand) EDWARDS_NONSINGULAR_DENOMINATORS o
snd) THEN
(ANTS_TAC THENL [ALL_TAC; DISCH_THEN(ACCEPT_TAC o CONJUNCT2)]) THEN
REWRITE_TAC[GSYM edwards_add] THEN ASM_REWRITE_TAC[] THEN
REPEAT CONJ_TAC THEN TRY(MATCH_MP_TAC EDWARDS_CURVE_ADD) THEN
ASM_REWRITE_TAC[edwards_curve] THEN ASM_MESON_TAC[DIVIDES_REFL];
ALL_TAC] THEN
MP_TAC(ISPECL [`f:A ring`; `a:A`; `d:A`; `(x1,y1):A#A`; `(x2,y2):A#A`]
EDWARDS_NONSINGULAR_DENOMINATORS_POINTS) THEN
MP_TAC(ISPECL [`f:A ring`; `a:A`; `d:A`; `(x2,y2):A#A`; `(x3,y3):A#A`]
EDWARDS_NONSINGULAR_DENOMINATORS_POINTS) THEN
ASM_REWRITE_TAC[edwards_curve] THEN STRIP_TAC THEN STRIP_TAC THEN
RING_PULL_DIV_TAC THEN RING_TAC);;
let EDWARDS_GROUP = prove
(`!f a (d:A).
field f /\
a IN ring_carrier f /\ d IN ring_carrier f /\
edwards_nonsingular(f,a,d)
==> group_carrier(edwards_group(f,a,d)) = edwards_curve(f,a,d) /\
group_id(edwards_group(f,a,d)) = edwards_0(f,a,d) /\
group_inv(edwards_group(f,a,d)) = edwards_neg(f,a,d) /\
group_mul(edwards_group(f,a,d)) = edwards_add(f,a,d)`,
REPEAT GEN_TAC THEN STRIP_TAC THEN
REWRITE_TAC[group_carrier; group_id; group_inv; group_mul; GSYM PAIR_EQ] THEN
REWRITE_TAC[edwards_group; GSYM(CONJUNCT2 group_tybij)] THEN
REPEAT CONJ_TAC THENL
[REWRITE_TAC[IN] THEN REWRITE_TAC[edwards_curve; edwards_0] THEN
REWRITE_TAC[RING_0; RING_1] THEN RING_TAC;
REWRITE_TAC[IN] THEN
ASM_SIMP_TAC[EDWARDS_CURVE_NEG; FIELD_IMP_INTEGRAL_DOMAIN];
REWRITE_TAC[IN] THEN ASM_SIMP_TAC[EDWARDS_CURVE_ADD];
REWRITE_TAC[IN] THEN ASM_SIMP_TAC[EDWARDS_ADD_ASSOC];
REWRITE_TAC[IN] THEN
ASM_MESON_TAC[EDWARDS_ADD_LID; EDWARDS_ADD_SYM; EDWARDS_CURVE_0];
REWRITE_TAC[IN] THEN
ASM_MESON_TAC[EDWARDS_ADD_LNEG; EDWARDS_ADD_SYM; EDWARDS_CURVE_NEG;
FIELD_IMP_INTEGRAL_DOMAIN]]);;
let ABELIAN_EDWARDS_GROUP = prove
(`!f a (d:A).
field f /\
a IN ring_carrier f /\ d IN ring_carrier f /\
edwards_nonsingular(f,a,d)
==> abelian_group(edwards_group(f,a,d))`,
REPEAT STRIP_TAC THEN ASM_SIMP_TAC[abelian_group; EDWARDS_GROUP] THEN
REWRITE_TAC[IN] THEN ASM_MESON_TAC[EDWARDS_ADD_SYM]);;
(* ------------------------------------------------------------------------- *)
(* Characterizing low-order points on an Edwards curve. *)
(* ------------------------------------------------------------------------- *)
let EDWARDS_GROUP_ORDER_EQ_2 = prove
(`!f (a:A) d p.
field f /\ ~(ring_char f = 2) /\
a IN ring_carrier f /\ d IN ring_carrier f /\
edwards_nonsingular (f,a,d) /\
p IN group_carrier (edwards_group(f,a,d))
==> (group_element_order (edwards_group(f,a,d)) p = 2 <=>
p = (ring_0 f,ring_neg f (ring_1 f)))`,
REPLICATE_TAC 3 GEN_TAC THEN REWRITE_TAC[FIELD_CHAR_NOT2] THEN
REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REPEAT DISCH_TAC THEN
ASM_SIMP_TAC[GROUP_ELEMENT_ORDER_EQ_2_ALT] THEN
ASM_SIMP_TAC[EDWARDS_GROUP; IMP_CONJ] THEN
REWRITE_TAC[FORALL_PAIR_THM; edwards_0; edwards_neg; PAIR_EQ; IN] THEN
REWRITE_TAC[edwards_curve] THEN FIELD_TAC);;
let EDWARDS_GROUP_ORDER_EQ_4_EQUIV = prove
(`!f (a:A) d x y.
field f /\ ~(ring_char f = 2) /\
a IN ring_carrier f /\ d IN ring_carrier f /\
edwards_nonsingular (f,a,d) /\
(x,y) IN group_carrier (edwards_group(f,a,d))
==> (group_element_order (edwards_group(f,a,d)) (x,y) = 4 <=>
ring_mul f a (ring_pow f x 2) = ring_1 f /\ y = ring_0 f)`,
REWRITE_TAC[ARITH_RULE `4 = 2 * 2`] THEN
SIMP_TAC[GROUP_ELEMENT_ORDER_EQ_MUL; DIVIDES_2; ARITH_EQ; ARITH_EVEN] THEN
SIMP_TAC[EDWARDS_GROUP_ORDER_EQ_2; GROUP_POW] THEN
REPLICATE_TAC 3 GEN_TAC THEN REWRITE_TAC[FIELD_CHAR_NOT2] THEN
REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REPEAT DISCH_TAC THEN
ASM_SIMP_TAC[GROUP_POW_2; FORALL_PAIR_THM] THEN
ASM_SIMP_TAC[EDWARDS_GROUP] THEN REWRITE_TAC[IN] THEN
REWRITE_TAC[edwards_curve; edwards_add; PAIR_EQ] THEN
MAP_EVERY X_GEN_TAC [`x:A`; `y:A`] THEN STRIP_TAC THEN
MP_TAC(ISPECL [`f:A ring`; `a:A`; `d:A`; `x:A`; `y:A`; `x:A`; `y:A`]
EDWARDS_NONSINGULAR_DENOMINATORS) THEN
ASM_REWRITE_TAC[edwards_curve] THEN FIELD_TAC);;
let EDWARDS_GROUP_ORDER_EQ_4 = prove
(`!f (a:A) d a' p.
field f /\ ~(ring_char f = 2) /\
a IN ring_carrier f /\ d IN ring_carrier f /\
a' IN ring_carrier f /\ ring_mul f a (ring_pow f a' 2) = ring_1 f /\
edwards_nonsingular (f,a,d) /\
p IN group_carrier (edwards_group(f,a,d))
==> (group_element_order (edwards_group(f,a,d)) p = 4 <=>
p = (a',ring_0 f) \/ p = (ring_neg f a',ring_0 f))`,
REWRITE_TAC[ARITH_RULE `4 = 2 * 2`] THEN
SIMP_TAC[GROUP_ELEMENT_ORDER_EQ_MUL; DIVIDES_2; ARITH_EQ; ARITH_EVEN] THEN
SIMP_TAC[EDWARDS_GROUP_ORDER_EQ_2; GROUP_POW] THEN
REPLICATE_TAC 4 GEN_TAC THEN REWRITE_TAC[FIELD_CHAR_NOT2] THEN
REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REPEAT DISCH_TAC THEN
ASM_SIMP_TAC[GROUP_POW_2; FORALL_PAIR_THM] THEN
ASM_SIMP_TAC[EDWARDS_GROUP] THEN REWRITE_TAC[IN] THEN
REWRITE_TAC[edwards_curve; edwards_add; PAIR_EQ] THEN
MAP_EVERY X_GEN_TAC [`x:A`; `y:A`] THEN STRIP_TAC THEN
MP_TAC(ISPECL [`f:A ring`; `a:A`; `d:A`; `x:A`; `y:A`; `x:A`; `y:A`]
EDWARDS_NONSINGULAR_DENOMINATORS) THEN
ASM_REWRITE_TAC[edwards_curve] THEN FIELD_TAC);;
let EDWARDS_GROUP_ORDER_EQ_8_EQUIV = prove
(`!f (a:A) d x y.
field f /\ ~(ring_char f = 2) /\
a IN ring_carrier f /\ d IN ring_carrier f /\
edwards_nonsingular (f,a,d) /\
(x,y) IN group_carrier (edwards_group(f,a,d))
==> (group_element_order (edwards_group(f,a,d)) (x,y) = 8 <=>
ring_mul f a (ring_pow f x 2) = ring_pow f y 2 /\
ring_mul f (ring_of_num f 4)
(ring_mul f (ring_pow f a 2) (ring_pow f x 4)) =
ring_pow f (ring_add f (ring_1 f)
(ring_mul f a (ring_mul f d (ring_pow f x 4)))) 2)`,
REWRITE_TAC[ARITH_RULE `8 = 2 * 4`] THEN
SIMP_TAC[GROUP_ELEMENT_ORDER_EQ_MUL; DIVIDES_2; ARITH_EQ; ARITH_EVEN] THEN
SIMP_TAC[GROUP_POW; REWRITE_RULE[PAIR]
(GEN_REWRITE_RULE (funpow 3 BINDER_CONV) [FORALL_UNPAIR_THM]
EDWARDS_GROUP_ORDER_EQ_4_EQUIV)] THEN
REPLICATE_TAC 3 GEN_TAC THEN REWRITE_TAC[FIELD_CHAR_NOT2] THEN
REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REPEAT DISCH_TAC THEN
ASM_SIMP_TAC[GROUP_POW_2; FORALL_PAIR_THM] THEN
ASM_SIMP_TAC[EDWARDS_GROUP] THEN REWRITE_TAC[IN] THEN
REWRITE_TAC[edwards_curve; edwards_add; PAIR_EQ] THEN
MAP_EVERY X_GEN_TAC [`x:A`; `y:A`] THEN STRIP_TAC THEN
MP_TAC(ISPECL [`f:A ring`; `a:A`; `d:A`; `x:A`; `y:A`; `x:A`; `y:A`]
EDWARDS_NONSINGULAR_DENOMINATORS) THEN
ASM_REWRITE_TAC[edwards_curve] THEN FIELD_TAC);;
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