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(* ========================================================================= *)
(* The general notion of an elliptic curve, in the basic Weierstrass form. *)
(* We use the option type to augment it with the "point at infinity" NONE. *)
(* Follows Washington "Elliptic Curves, Number Theory and Cryptography" p14. *)
(* *)
(* y^2 = x^3 + a * x + b over some field F *)
(* *)
(* This isn't general enough for working over characteristics 2 and 3. *)
(* ========================================================================= *)
needs "EC/misc.ml";;
(* ------------------------------------------------------------------------- *)
(* Basic definitions and naive cardinality bounds. *)
(* ------------------------------------------------------------------------- *)
let weierstrass_point = define
`(weierstrass_point f NONE <=> T) /\
(weierstrass_point f (SOME(x:A,y)) <=>
x IN ring_carrier f /\ y IN ring_carrier f)`;;
let weierstrass_curve = define
`(weierstrass_curve(f:A ring,a:A,b:A) NONE <=> T) /\
(weierstrass_curve(f:A ring,a:A,b:A) (SOME(x,y)) <=>
x IN ring_carrier f /\ y IN ring_carrier f /\
ring_pow f y 2 =
ring_add f (ring_pow f x 3) (ring_add f (ring_mul f a x) b))`;;
let weierstrass_neg = define
`(weierstrass_neg (f:A ring,a:A,b:A) NONE = NONE) /\
(weierstrass_neg (f:A ring,a:A,b:A) (SOME(x:A,y)) = SOME(x,ring_neg f y))`;;
let weierstrass_add = define
`(!y. weierstrass_add (f:A ring,a:A,b:A) NONE y = y) /\
(!x. weierstrass_add (f:A ring,a:A,b:A) x NONE = x) /\
(!x1 y1 x2 y2.
weierstrass_add (f:A ring,a:A,b:A) (SOME(x1,y1)) (SOME(x2,y2)) =
if x1 = x2 then
if y1 = y2 /\ ~(y1 = ring_0 f) then
let m = ring_div f
(ring_add f (ring_mul f (ring_of_num f 3) (ring_pow f x1 2)) a)
(ring_mul f (ring_of_num f 2) y1) in
let x3 = ring_sub f (ring_pow f m 2)
(ring_mul f (ring_of_num f 2) x1) in
let y3 = ring_sub f (ring_mul f m (ring_sub f x1 x3)) y1 in
SOME(x3,y3)
else NONE
else
let m = ring_div f (ring_sub f y2 y1) (ring_sub f x2 x1) in
let x3 = ring_sub f (ring_pow f m 2)
(ring_add f x1 x2) in
let y3 = ring_sub f (ring_mul f m (ring_sub f x1 x3)) y1 in
SOME(x3,y3))`;;
let weierstrass_nonsingular = define
`weierstrass_nonsingular (f:A ring,a:A,b:A) <=>
~(ring_add f (ring_mul f (ring_of_num f 4) (ring_pow f a 3))
(ring_mul f (ring_of_num f 27) (ring_pow f b 2)) =
ring_0 f)`;;
let weierstrass_group = define
`weierstrass_group (f:A ring,a:A,b:A) =
group(weierstrass_curve(f,a,b),
NONE,
weierstrass_neg(f,a,b),
weierstrass_add(f,a,b))`;;
let (FINITE_WEIERSTRASS_CURVE,CARD_BOUND_WEIERSTRASS_CURVE) =
(CONJ_PAIR o prove)
(`(!f a b:A. field f /\ FINITE(ring_carrier f)
==> FINITE(weierstrass_curve(f,a,b))) /\
(!f a b:A. field f /\ FINITE(ring_carrier f)
==> CARD(weierstrass_curve(f,a,b))
<= 2 * CARD(ring_carrier f) + 1)`,
REWRITE_TAC[AND_FORALL_THM] THEN REPEAT GEN_TAC THEN
REWRITE_TAC[TAUT `(p ==> q) /\ (p ==> r) <=> p ==> q /\ r`] THEN
STRIP_TAC THEN MATCH_MP_TAC FINITE_CARD_LE_SUBSET THEN EXISTS_TAC
`IMAGE SOME
{(x,y) | (x:A) IN ring_carrier f /\ y IN ring_carrier f /\
ring_pow f y 2 =
ring_add f (ring_pow f x 3) (ring_add f (ring_mul f a x) b)}
UNION {NONE}` THEN
CONJ_TAC THENL
[REWRITE_TAC[weierstrass_curve; SUBSET; FORALL_OPTION_THM;
IN_UNION; IN_SING; IN_IMAGE; option_INJ; FORALL_PAIR_THM;
IN_ELIM_PAIR_THM; UNWIND_THM1] THEN
REWRITE_TAC[option_DISTINCT; IN_CROSS] THEN REWRITE_TAC[IN] THEN
REWRITE_TAC[weierstrass_curve] THEN SIMP_TAC[IN];
MATCH_MP_TAC FINITE_CARD_LE_UNION] THEN
REWRITE_TAC[FINITE_SING; CARD_SING; LE_REFL] THEN
MATCH_MP_TAC FINITE_CARD_LE_IMAGE THEN
MATCH_MP_TAC FINITE_QUADRATIC_CURVE THEN
ASM_SIMP_TAC[FIELD_IMP_INTEGRAL_DOMAIN]);;
(* ------------------------------------------------------------------------- *)
(* Proof of the group properties. This is just done by algebraic brute *)
(* force except for the use of ASSOCIATIVITY_LEMMA to reduce the explosion *)
(* of case distinctions. *)
(* ------------------------------------------------------------------------- *)
let WEIERSTRASS_CURVE_IMP_POINT = prove
(`!f a b p. weierstrass_curve(f,a,b) p ==> weierstrass_point f p`,
REWRITE_TAC[FORALL_OPTION_THM; FORALL_PAIR_THM] THEN
SIMP_TAC[weierstrass_curve; weierstrass_point]);;
let WEIERSTRASS_POINT_NEG = prove
(`!(f:A ring) a b p.
weierstrass_point f p
==> weierstrass_point f (weierstrass_neg (f,a,b) p)`,
REWRITE_TAC[FORALL_OPTION_THM; FORALL_PAIR_THM] THEN
SIMP_TAC[weierstrass_neg; weierstrass_point; RING_NEG]);;
let WEIERSTRASS_POINT_ADD = prove
(`!(f:A ring) a b p q.
a IN ring_carrier f /\ b IN ring_carrier f /\
weierstrass_point f p /\ weierstrass_point f q
==> weierstrass_point f (weierstrass_add (f,a,b) p q)`,
REWRITE_TAC[FORALL_OPTION_THM; FORALL_PAIR_THM] THEN
SIMP_TAC[weierstrass_add; weierstrass_point; LET_DEF; LET_END_DEF] THEN
REPEAT STRIP_TAC THEN
REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[weierstrass_point]) THEN
REPEAT STRIP_TAC THEN RING_CARRIER_TAC);;
let WEIERSTRASS_CURVE_0 = prove
(`!f a b:A. weierstrass_curve(f,a,b) NONE`,
REWRITE_TAC[weierstrass_curve]);;
let WEIERSTRASS_CURVE_NEG = prove
(`!f a (b:A) p.
integral_domain f /\ a IN ring_carrier f /\ b IN ring_carrier f /\
weierstrass_curve(f,a,b) p
==> weierstrass_curve(f,a,b) (weierstrass_neg (f,a,b) p)`,
SIMP_TAC[FORALL_OPTION_THM; FORALL_PAIR_THM; RING_NEG;
weierstrass_curve; weierstrass_neg] THEN
REPEAT GEN_TAC THEN CONV_TAC INTEGRAL_DOMAIN_RULE);;
let WEIERSTRASS_CURVE_ADD = prove
(`!f a (b:A) p q.
field f /\ ~(ring_char f = 2) /\
a IN ring_carrier f /\ b IN ring_carrier f /\
weierstrass_curve(f,a,b) p /\ weierstrass_curve(f,a,b) q
==> weierstrass_curve(f,a,b) (weierstrass_add (f,a,b) p q)`,
REWRITE_TAC[FIELD_CHAR_NOT2] THEN REPLICATE_TAC 3 GEN_TAC THEN
SIMP_TAC[FORALL_OPTION_THM; FORALL_PAIR_THM; RING_ADD;
weierstrass_curve; weierstrass_add] THEN
MAP_EVERY X_GEN_TAC [`x1:A`; `y1:A`; `x2:A`; `y2:A`] THEN
REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[]) THEN
RULE_ASSUM_TAC(REWRITE_RULE[DE_MORGAN_THM]) THEN
REPEAT(FIRST_X_ASSUM(DISJ_CASES_THEN ASSUME_TAC)) THEN
REPEAT(FIRST_X_ASSUM(CONJUNCTS_THEN ASSUME_TAC)) THEN
REPEAT(FIRST_X_ASSUM SUBST_ALL_TAC) THENL
[CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN
REWRITE_TAC[weierstrass_curve] THEN
REPEAT STRIP_TAC THEN TRY RING_CARRIER_TAC THEN
SUBGOAL_THEN `~(ring_of_num f 2:A = ring_0 f)` ASSUME_TAC THENL
[FIELD_TAC; RING_PULL_DIV_TAC THEN RING_TAC];
ALL_TAC; ALL_TAC; ALL_TAC] THEN
REPEAT LET_TAC THEN REWRITE_TAC[weierstrass_curve] THEN
REPEAT STRIP_TAC THEN TRY RING_CARRIER_TAC THEN
FIELD_TAC THEN NOT_RING_CHAR_DIVIDES_TAC);;
let WEIERSTRASS_ADD_LNEG = prove
(`!f a (b:A) p.
field f /\ ~(ring_char f = 2) /\
a IN ring_carrier f /\ b IN ring_carrier f /\
weierstrass_curve(f,a,b) p
==> weierstrass_add(f,a,b) (weierstrass_neg (f,a,b) p) p = NONE`,
REWRITE_TAC[FIELD_CHAR_NOT2] THEN REPLICATE_TAC 3 GEN_TAC THEN
SIMP_TAC[FORALL_OPTION_THM; FORALL_PAIR_THM; RING_ADD;
weierstrass_curve; weierstrass_neg; weierstrass_add] THEN
MAP_EVERY X_GEN_TAC [`x:A`; `y:A`] THEN
REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[]) THEN
RULE_ASSUM_TAC(REWRITE_RULE[DE_MORGAN_THM]) THEN
REPEAT(FIRST_X_ASSUM(DISJ_CASES_THEN ASSUME_TAC)) THEN
REPEAT(FIRST_X_ASSUM(CONJUNCTS_THEN ASSUME_TAC)) THEN
REPEAT LET_TAC THEN REWRITE_TAC[option_DISTINCT] THEN
REPEAT STRIP_TAC THEN TRY RING_CARRIER_TAC THEN
FIELD_TAC);;
let WEIERSTRASS_ADD_SYM = prove
(`!f a (b:A) p q.
field f /\
a IN ring_carrier f /\ b IN ring_carrier f /\
weierstrass_curve(f,a,b) p /\ weierstrass_curve(f,a,b) q
==> weierstrass_add (f,a,b) p q = weierstrass_add (f,a,b) q p`,
REPLICATE_TAC 3 GEN_TAC THEN
SIMP_TAC[FORALL_OPTION_THM; FORALL_PAIR_THM; RING_ADD;
weierstrass_curve; weierstrass_add] THEN
MAP_EVERY X_GEN_TAC [`x1:A`; `y1:A`; `x2:A`; `y2:A`] THEN
REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[]) THEN
RULE_ASSUM_TAC(REWRITE_RULE[DE_MORGAN_THM]) THEN
REPEAT(FIRST_X_ASSUM(DISJ_CASES_THEN ASSUME_TAC)) THEN
REPEAT(FIRST_X_ASSUM(CONJUNCTS_THEN ASSUME_TAC)) THEN
REPEAT(FIRST_X_ASSUM SUBST_ALL_TAC) THEN
REPEAT LET_TAC THEN
REWRITE_TAC[option_DISTINCT; option_INJ; PAIR_EQ] THEN
REPEAT STRIP_TAC THEN TRY RING_CARRIER_TAC THEN
FIELD_TAC);;
let WEIERSTRASS_ADD_ASSOC = prove
(`!f a (b:A) p q r.
field f /\ ~(ring_char f = 2) /\ ~(ring_char f = 3) /\
weierstrass_nonsingular(f,a,b) /\
a IN ring_carrier f /\ b IN ring_carrier f /\
weierstrass_curve(f,a,b) p /\
weierstrass_curve(f,a,b) q /\
weierstrass_curve(f,a,b) r
==> weierstrass_add (f,a,b) p (weierstrass_add (f,a,b) q r) =
weierstrass_add (f,a,b) (weierstrass_add (f,a,b) p q) r`,
let assoclemma = prove
(`!f (a:A) b x1 y1 x2 y2.
field f /\
a IN ring_carrier f /\ b IN ring_carrier f /\
weierstrass_curve(f,a,b) (SOME(x1,y1)) /\
weierstrass_curve(f,a,b) (SOME(x2,y2))
==> (~(SOME(x2,y2) = SOME(x1,y1)) /\
~(SOME(x2,y2) = weierstrass_neg (f,a,b) (SOME(x1,y1))) <=>
~(x1 = x2))`,
REWRITE_TAC[weierstrass_curve; weierstrass_neg; option_INJ; PAIR_EQ] THEN
FIELD_TAC) in
REWRITE_TAC[FIELD_CHAR_NOT23] THEN
REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
X_GEN_TAC `f:A ring` THEN REPEAT DISCH_TAC THEN
MAP_EVERY X_GEN_TAC [`a:A`; `b:A`] THEN REPEAT DISCH_TAC THEN
REWRITE_TAC[IMP_IMP; GSYM CONJ_ASSOC; RIGHT_IMP_FORALL_THM] THEN
MATCH_MP_TAC ASSOCIATIVITY_LEMMA THEN
MAP_EVERY EXISTS_TAC [`weierstrass_neg(f,a:A,b)`; `NONE:(A#A)option`] THEN
REWRITE_TAC[FORALL_OPTION_THM; FORALL_PAIR_THM] THEN
REWRITE_TAC(CONJUNCT1 weierstrass_curve :: CONJUNCT1 weierstrass_neg ::
fst(chop_list 2 (CONJUNCTS weierstrass_add))) THEN
REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THEN REWRITE_TAC[GSYM CONJ_ASSOC] THENL
[REPEAT CONJ_TAC THEN
MAP_EVERY X_GEN_TAC [`x1:A`; `y1:A`] THEN
TRY(MAP_EVERY X_GEN_TAC [`x2:A`; `y2:A`]) THEN
TRY(MAP_EVERY X_GEN_TAC [`x3:A`; `y3:A`]);
MAP_EVERY X_GEN_TAC [`x1:A`; `y1:A`; `x2:A`; `y2:A`; `x3:A`; `y3:A`] THEN
ASM_SIMP_TAC[assoclemma; DE_MORGAN_THM] THEN
REWRITE_TAC[option_DISTINCT] THEN
REPEAT GEN_TAC THEN REWRITE_TAC[weierstrass_curve] THEN
STRIP_TAC THEN STRIP_TAC THEN
ASM_REWRITE_TAC[weierstrass_add]] THEN
REWRITE_TAC[weierstrass_curve; weierstrass_add; weierstrass_neg] THEN
REPEAT(GEN_TAC ORELSE CONJ_TAC) THEN
REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[]) THEN
REPEAT LET_TAC THEN
REWRITE_TAC[option_DISTINCT; option_INJ; PAIR_EQ] THEN
REPEAT STRIP_TAC THEN
REWRITE_TAC[weierstrass_curve; weierstrass_add; weierstrass_neg] THEN
REPEAT(GEN_TAC ORELSE CONJ_TAC) THEN
REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[]) THEN
REPEAT LET_TAC THEN
REWRITE_TAC[option_DISTINCT; option_INJ; PAIR_EQ] THEN
REPEAT STRIP_TAC THEN
REPEAT(FIRST_X_ASSUM(CONJUNCTS_THEN ASSUME_TAC)) THEN
REPEAT(FIRST_X_ASSUM(SUBST_ALL_TAC o SYM o check
(fun th -> is_eq(concl th) && is_var(lhand(concl th)) &&
is_var(rand(concl th))))) THEN
TRY RING_CARRIER_TAC THEN
(FIELD_TAC ORELSE
(RULE_ASSUM_TAC(REWRITE_RULE[weierstrass_nonsingular]) THEN FIELD_TAC)) THEN
NOT_RING_CHAR_DIVIDES_TAC);;
let WEIERSTRASS_GROUP = prove
(`!f a (b:A).
field f /\ ~(ring_char f = 2) /\ ~(ring_char f = 3) /\
a IN ring_carrier f /\ b IN ring_carrier f /\
weierstrass_nonsingular(f,a,b)
==> group_carrier(weierstrass_group(f,a,b)) = weierstrass_curve(f,a,b) /\
group_id(weierstrass_group(f,a,b)) = NONE /\
group_inv(weierstrass_group(f,a,b)) = weierstrass_neg(f,a,b) /\
group_mul(weierstrass_group(f,a,b)) = weierstrass_add(f,a,b)`,
REPEAT GEN_TAC THEN STRIP_TAC THEN
REWRITE_TAC[group_carrier; group_id; group_inv; group_mul; GSYM PAIR_EQ] THEN
REWRITE_TAC[weierstrass_group; GSYM(CONJUNCT2 group_tybij)] THEN
REPEAT CONJ_TAC THENL
[REWRITE_TAC[IN; weierstrass_curve];
REWRITE_TAC[IN] THEN
ASM_SIMP_TAC[WEIERSTRASS_CURVE_NEG; FIELD_IMP_INTEGRAL_DOMAIN];
REWRITE_TAC[IN] THEN ASM_SIMP_TAC[WEIERSTRASS_CURVE_ADD];
REWRITE_TAC[IN] THEN ASM_SIMP_TAC[WEIERSTRASS_ADD_ASSOC];
REWRITE_TAC[FORALL_OPTION_THM; FORALL_PAIR_THM; weierstrass_add];
REWRITE_TAC[IN] THEN GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC(MESON[]
`x = a /\ x = y ==> x = a /\ y = a`) THEN
CONJ_TAC THENL
[ASM_SIMP_TAC[WEIERSTRASS_ADD_LNEG];
MATCH_MP_TAC WEIERSTRASS_ADD_SYM THEN
ASM_SIMP_TAC[WEIERSTRASS_CURVE_NEG; FIELD_IMP_INTEGRAL_DOMAIN]]]);;
(* ------------------------------------------------------------------------- *)
(* Easily computable endomorphisms in some special Weierstrass curves. *)
(* (x,y) |-> (c * x,y) where c^3 = 1 for curves y^2 = x^3 + b. *)
(* (x,y) |-> (-x, c * y) where c^4 = 1 for curves y^2 = x^3 + a * x. *)
(* ------------------------------------------------------------------------- *)
let weierstrass_triplex = define
`weierstrass_triplex f (c:A) NONE = NONE /\
weierstrass_triplex f c (SOME(x:A,y:A)) = SOME(ring_mul f c x,y)`;;
let GROUP_ENDOMORPHISM_TRIPLEX = prove
(`!f a b c:A.
field f /\ ~(ring_char f = 2) /\ ~(ring_char f = 3) /\
a IN ring_carrier f /\ b IN ring_carrier f /\
weierstrass_nonsingular (f,a,b) /\
c IN ring_carrier f /\ ring_pow f c 3 = ring_1 f /\ a = ring_0 f
==> group_endomorphism (weierstrass_group(f,a,b))
(weierstrass_triplex f c)`,
REPEAT GEN_TAC THEN ASM_CASES_TAC `a:A = ring_0 f` THEN
ASM_SIMP_TAC[group_endomorphism; GROUP_HOMOMORPHISM] THEN
SIMP_TAC[SUBSET; FORALL_IN_IMAGE; WEIERSTRASS_GROUP; GROUP_ID] THEN
POP_ASSUM(K ALL_TAC) THEN REWRITE_TAC[FIELD_CHAR_NOT23] THEN STRIP_TAC THEN
REWRITE_TAC[FORALL_OPTION_THM; FORALL_PAIR_THM; IN] THEN
REWRITE_TAC[weierstrass_curve; weierstrass_triplex; weierstrass_add] THEN
REPEAT STRIP_TAC THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[]) THEN
REPEAT LET_TAC THEN REWRITE_TAC[weierstrass_triplex] THEN
TRY RING_CARRIER_TAC THEN
REWRITE_TAC[option_DISTINCT; option_INJ; PAIR_EQ] THEN
RULE_ASSUM_TAC(REWRITE_RULE[weierstrass_nonsingular]) THEN FIELD_TAC);;
let weierstrass_quady = define
`weierstrass_quady f (c:A) NONE = NONE /\
weierstrass_quady f c (SOME(x:A,y:A)) = SOME(ring_neg f x,ring_mul f c y)`;;
let GROUP_ENDOMORPHISM_QUADY = prove
(`!f a b c:A.
field f /\ ~(ring_char f = 2) /\ ~(ring_char f = 3) /\
a IN ring_carrier f /\ b IN ring_carrier f /\
weierstrass_nonsingular (f,a,b) /\
c IN ring_carrier f /\ b = ring_0 f /\
ring_pow f c 4 = ring_1 f /\ ~(ring_pow f c 2 = ring_1 f)
==> group_endomorphism (weierstrass_group(f,a,b))
(weierstrass_quady f c)`,
REPEAT GEN_TAC THEN ASM_CASES_TAC `b:A = ring_0 f` THEN
ASM_SIMP_TAC[group_endomorphism; GROUP_HOMOMORPHISM] THEN
SIMP_TAC[SUBSET; FORALL_IN_IMAGE; WEIERSTRASS_GROUP; GROUP_ID] THEN
POP_ASSUM(K ALL_TAC) THEN REWRITE_TAC[FIELD_CHAR_NOT23] THEN STRIP_TAC THEN
REWRITE_TAC[FORALL_OPTION_THM; FORALL_PAIR_THM; IN] THEN
REWRITE_TAC[weierstrass_curve; weierstrass_quady; weierstrass_add] THEN
REPEAT STRIP_TAC THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[]) THEN
REPEAT LET_TAC THEN REWRITE_TAC[weierstrass_quady] THEN
TRY RING_CARRIER_TAC THEN
REWRITE_TAC[option_DISTINCT; option_INJ; PAIR_EQ] THEN
RULE_ASSUM_TAC(REWRITE_RULE[weierstrass_nonsingular]) THEN FIELD_TAC THEN
NOT_RING_CHAR_DIVIDES_TAC);;
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