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(* ========================================================================= *)
(* Montgomery curves in general, B * y^2 = x^3 + A * x^2 + x. *)
(* ========================================================================= *)
needs "EC/misc.ml";;
(* ------------------------------------------------------------------------- *)
(* Basic definitions and naive cardinality bounds. *)
(* ------------------------------------------------------------------------- *)
let montgomery_point = define
`(montgomery_point f NONE <=> T) /\
(montgomery_point f (SOME(x:A,y)) <=>
x IN ring_carrier f /\ y IN ring_carrier f)`;;
let montgomery_curve = define
`(montgomery_curve(f:A ring,a:A,b:A) NONE <=> T) /\
(montgomery_curve(f:A ring,a:A,b:A) (SOME(x,y)) <=>
x IN ring_carrier f /\ y IN ring_carrier f /\
ring_mul f b (ring_pow f y 2) =
ring_add f (ring_pow f x 3)
(ring_add f (ring_mul f a (ring_pow f x 2)) x))`;;
let montgomery_nonsingular = define
`montgomery_nonsingular (f:A ring,a:A,b:A) <=>
~(b = ring_0 f) /\ ~(ring_pow f a 2 = ring_of_num f 4)`;;
let montgomery_neg = define
`(montgomery_neg (f:A ring,a:A,b:A) NONE = NONE) /\
(montgomery_neg (f:A ring,a:A,b:A) (SOME(x:A,y)) = SOME(x,ring_neg f y))`;;
let montgomery_add = define
`(!y. montgomery_add (f:A ring,a:A,b:A) NONE y = y) /\
(!x. montgomery_add (f:A ring,a:A,b:A) x NONE = x) /\
(!x1 y1 x2 y2.
montgomery_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 l = ring_div f
(ring_add f (ring_mul f (ring_of_num f 3) (ring_pow f x1 2))
(ring_add f (ring_mul f (ring_of_num f 2)
(ring_mul f a x1))
(ring_of_num f 1)))
(ring_mul f (ring_of_num f 2) (ring_mul f b y1)) in
let x3 = ring_sub f (ring_sub f (ring_mul f b (ring_pow f l 2)) a)
(ring_mul f (ring_of_num f 2) x1) in
let y3 = ring_sub f (ring_mul f l (ring_sub f x1 x3)) y1 in
SOME(x3,y3)
else NONE
else
let l = ring_div f (ring_sub f y2 y1) (ring_sub f x2 x1) in
let x3 = ring_sub f (ring_sub f (ring_mul f b (ring_pow f l 2)) a)
(ring_add f x1 x2) in
let y3 = ring_sub f (ring_mul f l (ring_sub f x1 x3)) y1 in
SOME(x3,y3))`;;
let montgomery_group = define
`montgomery_group (f:A ring,a:A,b:A) =
group(montgomery_curve(f,a,b),
NONE,
montgomery_neg(f,a,b),
montgomery_add(f,a,b))`;;
let FINITE_MONTGOMERY_CURVE = prove
(`!f a b:A. field f /\ FINITE(ring_carrier f)
==> FINITE(montgomery_curve(f,a,b))`,
REPEAT STRIP_TAC THEN
MATCH_MP_TAC FINITE_SUBSET THEN EXISTS_TAC
`IMAGE SOME {(x,y) | (x:A) IN ring_carrier f /\ y IN ring_carrier f} UNION
{NONE}` THEN
ASM_SIMP_TAC[FINITE_UNION; FINITE_IMAGE; FINITE_SING; FINITE_PRODUCT] THEN
REWRITE_TAC[montgomery_curve; SUBSET; FORALL_OPTION_THM;
IN_UNION; IN_SING; IN_IMAGE; option_INJ; FORALL_PAIR_THM;
IN_ELIM_PAIR_THM; UNWIND_THM1] THEN
SIMP_TAC[montgomery_curve; IN]);;
let CARD_BOUND_MONTGOMERY_CURVE = prove
(`!f a b:A. field f /\ FINITE(ring_carrier f) /\
a IN ring_carrier f /\ b IN ring_carrier f /\ ~(b = ring_0 f)
==> CARD(montgomery_curve(f,a,b))
<= 2 * CARD(ring_carrier f) + 1`,
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
`IMAGE SOME
{(x,y) | (x:A) IN ring_carrier f /\ y IN ring_carrier f /\
ring_pow f y 2 =
ring_div f (ring_add f (ring_pow f x 3)
(ring_add f (ring_mul f a (ring_pow f x 2)) x)) b}
UNION {NONE}` THEN
CONJ_TAC THENL
[REWRITE_TAC[montgomery_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
REPEAT GEN_TAC THEN GEN_REWRITE_TAC LAND_CONV [IN] THEN
ASM_REWRITE_TAC[montgomery_curve] THEN
STRIP_TAC THEN ASM_REWRITE_TAC[] THEN FIELD_TAC;
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]);;
(* ------------------------------------------------------------------------- *)
(* A stronger form of nonsingularity, implying not only that the core cubic *)
(* doesn't have repeated roots but has no roots at all except for (0,0). *)
(* ------------------------------------------------------------------------- *)
let montgomery_strongly_nonsingular = define
`montgomery_strongly_nonsingular(f,a:A,b:A) <=>
~(b = ring_0 f) /\
~(?z. z IN ring_carrier f /\
ring_pow f z 2 = ring_sub f (ring_pow f a 2) (ring_of_num f 4))`;;
let MONTGOMERY_STRONGLY_NONSINGULAR = prove
(`!f a b:A.
field f /\ ~(ring_char f = 2) /\
a IN ring_carrier f /\ b IN ring_carrier f
==> (montgomery_strongly_nonsingular(f,a,b) <=>
!x y. montgomery_curve(f,a,b) (SOME(x,y))
==> (y = ring_0 f <=> x = ring_0 f))`,
REWRITE_TAC[FIELD_CHAR_NOT2] THEN
REPEAT STRIP_TAC THEN REWRITE_TAC[montgomery_strongly_nonsingular] THEN
REWRITE_TAC[FORALL_AND_THM; TAUT
`p ==> (q <=> r) <=> (r ==> p ==> q) /\ (q ==> p ==> r)`] THEN
REWRITE_TAC[RIGHT_FORALL_IMP_THM; FORALL_UNWIND_THM2] THEN
REWRITE_TAC[montgomery_curve] THEN BINOP_TAC THENL
[MATCH_MP_TAC(TAUT `(~p ==> q) /\ (p ==> ~q) ==> (~p <=> q)`) THEN
CONJ_TAC THENL
[REPEAT STRIP_TAC THEN TRY RING_CARRIER_TAC THEN FIELD_TAC;
DISCH_THEN SUBST1_TAC THEN DISCH_THEN(MP_TAC o SPEC `ring_1 f:A`) THEN
REWRITE_TAC[NOT_IMP] THEN
REPEAT STRIP_TAC THEN TRY RING_CARRIER_TAC THEN FIELD_TAC];
REWRITE_TAC[GSYM NOT_EXISTS_THM; montgomery_curve;
TAUT `p ==> q <=> ~(p /\ ~q)`] THEN
AP_TERM_TAC THEN EQ_TAC THENL
[DISCH_THEN(X_CHOOSE_THEN `z:A` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `ring_div f (ring_sub f z a) (ring_of_num f 2):A`;
DISCH_THEN(X_CHOOSE_THEN `x:A` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `ring_add f (ring_mul f (ring_of_num f 2) x) a:A`] THEN
REPEAT STRIP_TAC THEN TRY RING_CARRIER_TAC THEN FIELD_TAC THEN
NOT_RING_CHAR_DIVIDES_TAC]);;
let MONTGOMERY_STRONGLY_NONSINGULAR_IMP_NONSINGULAR = prove
(`!f a b:A.
a IN ring_carrier f /\ b IN ring_carrier f /\
montgomery_strongly_nonsingular(f,a,b)
==> montgomery_nonsingular(f,a,b)`,
REWRITE_TAC[montgomery_strongly_nonsingular; montgomery_nonsingular] THEN
REPEAT GEN_TAC THEN REWRITE_TAC[NOT_EXISTS_THM] THEN STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `ring_0 f:A`) THEN
ASM_REWRITE_TAC[RING_0] THEN RING_TAC);;
(* ------------------------------------------------------------------------- *)
(* 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 MONTGOMERY_CURVE_IMP_POINT = prove
(`!f a b p. montgomery_curve(f,a,b) p ==> montgomery_point f p`,
REWRITE_TAC[FORALL_OPTION_THM; FORALL_PAIR_THM] THEN
SIMP_TAC[montgomery_curve; montgomery_point]);;
let MONTGOMERY_POINT_NEG = prove
(`!(f:A ring) a b p.
montgomery_point f p
==> montgomery_point f (montgomery_neg (f,a,b) p)`,
REWRITE_TAC[FORALL_OPTION_THM; FORALL_PAIR_THM] THEN
SIMP_TAC[montgomery_neg; montgomery_point; RING_NEG]);;
let MONTGOMERY_POINT_ADD = prove
(`!(f:A ring) a b p q.
a IN ring_carrier f /\ b IN ring_carrier f /\
montgomery_point f p /\ montgomery_point f q
==> montgomery_point f (montgomery_add (f,a,b) p q)`,
REWRITE_TAC[FORALL_OPTION_THM; FORALL_PAIR_THM] THEN
SIMP_TAC[montgomery_add; montgomery_point; LET_DEF; LET_END_DEF] THEN
REPEAT STRIP_TAC THEN
REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[montgomery_point]) THEN
REPEAT STRIP_TAC THEN RING_CARRIER_TAC);;
let MONTGOMERY_CURVE_0 = prove
(`!f a b:A. montgomery_curve(f,a,b) NONE`,
REWRITE_TAC[montgomery_curve]);;
let MONTGOMERY_CURVE_NEG = prove
(`!f a (b:A) p.
integral_domain f /\ a IN ring_carrier f /\ b IN ring_carrier f /\
montgomery_curve(f,a,b) p
==> montgomery_curve(f,a,b) (montgomery_neg (f,a,b) p)`,
SIMP_TAC[FORALL_OPTION_THM; FORALL_PAIR_THM; RING_NEG;
montgomery_curve; montgomery_neg] THEN
REPEAT GEN_TAC THEN CONV_TAC INTEGRAL_DOMAIN_RULE);;
let MONTGOMERY_CURVE_ADD = prove
(`!f a (b:A) p q.
field f /\ ~(ring_char f = 2) /\
montgomery_nonsingular(f,a,b) /\
a IN ring_carrier f /\ b IN ring_carrier f /\
montgomery_curve(f,a,b) p /\ montgomery_curve(f,a,b) q
==> montgomery_curve(f,a,b) (montgomery_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;
montgomery_nonsingular; montgomery_curve; montgomery_add] THEN
REWRITE_TAC[GSYM DE_MORGAN_THM] 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[montgomery_curve] THEN
REPEAT STRIP_TAC THEN TRY RING_CARRIER_TAC THEN
FIELD_TAC THEN NOT_RING_CHAR_DIVIDES_TAC);;
let MONTGOMERY_ADD_LNEG = prove
(`!f a (b:A) p.
field f /\ ~(ring_char f = 2) /\
a IN ring_carrier f /\ b IN ring_carrier f /\
montgomery_curve(f,a,b) p
==> montgomery_add(f,a,b) (montgomery_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;
montgomery_curve; montgomery_neg; montgomery_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 MONTGOMERY_ADD_SYM = prove
(`!f a (b:A) p q.
field f /\
a IN ring_carrier f /\ b IN ring_carrier f /\
montgomery_curve(f,a,b) p /\ montgomery_curve(f,a,b) q
==> montgomery_add (f,a,b) p q = montgomery_add (f,a,b) q p`,
REPLICATE_TAC 3 GEN_TAC THEN
SIMP_TAC[FORALL_OPTION_THM; FORALL_PAIR_THM; RING_ADD;
montgomery_curve; montgomery_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 MONTGOMERY_ADD_ASSOC = prove
(`!f a (b:A) p q r.
field f /\ ~(ring_char f = 2) /\
montgomery_nonsingular(f,a,b) /\
a IN ring_carrier f /\ b IN ring_carrier f /\
montgomery_curve(f,a,b) p /\
montgomery_curve(f,a,b) q /\
montgomery_curve(f,a,b) r
==> montgomery_add (f,a,b) p (montgomery_add (f,a,b) q r) =
montgomery_add (f,a,b) (montgomery_add (f,a,b) p q) r`,
let assoclemma = prove
(`!f (a:A) b x1 y1 x2 y2.
field f /\
montgomery_nonsingular(f,a,b) /\
a IN ring_carrier f /\ b IN ring_carrier f /\
montgomery_curve(f,a,b) (SOME(x1,y1)) /\
montgomery_curve(f,a,b) (SOME(x2,y2))
==> (~(SOME(x2,y2) = SOME(x1,y1)) /\
~(SOME(x2,y2) = montgomery_neg (f,a,b) (SOME(x1,y1))) <=>
~(x1 = x2))`,
REWRITE_TAC[montgomery_nonsingular] THEN
REWRITE_TAC[montgomery_curve; montgomery_neg; option_INJ; PAIR_EQ] THEN
FIELD_TAC) in
REWRITE_TAC[FIELD_CHAR_NOT2] 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 [`montgomery_neg(f,a:A,b)`; `NONE:(A#A)option`] THEN
REWRITE_TAC[FORALL_OPTION_THM; FORALL_PAIR_THM] THEN
REWRITE_TAC(CONJUNCT1 montgomery_curve :: CONJUNCT1 montgomery_neg ::
fst(chop_list 2 (CONJUNCTS montgomery_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 REPEAT CONJ_TAC THEN
TRY(MAP_EVERY X_GEN_TAC [`x2:A`; `y2:A`]) THEN REPEAT CONJ_TAC THEN
TRY(MAP_EVERY X_GEN_TAC [`x3:A`; `y3:A`]) THEN REPEAT CONJ_TAC;
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[montgomery_curve] THEN
STRIP_TAC THEN STRIP_TAC THEN
ASM_REWRITE_TAC[montgomery_add]] THEN
REWRITE_TAC[montgomery_curve; montgomery_add; montgomery_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[montgomery_curve; montgomery_add; montgomery_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
RULE_ASSUM_TAC(REWRITE_RULE[montgomery_nonsingular]) THEN FIELD_TAC THEN
NOT_RING_CHAR_DIVIDES_TAC);;
let MONTGOMERY_GROUP = prove
(`!f a (b:A).
field f /\ ~(ring_char f = 2) /\
a IN ring_carrier f /\ b IN ring_carrier f /\
montgomery_nonsingular(f,a,b)
==> group_carrier(montgomery_group(f,a,b)) = montgomery_curve(f,a,b) /\
group_id(montgomery_group(f,a,b)) = NONE /\
group_inv(montgomery_group(f,a,b)) = montgomery_neg(f,a,b) /\
group_mul(montgomery_group(f,a,b)) = montgomery_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[montgomery_group; GSYM(CONJUNCT2 group_tybij)] THEN
REPEAT CONJ_TAC THENL
[REWRITE_TAC[IN; montgomery_curve];
REWRITE_TAC[IN] THEN
ASM_SIMP_TAC[MONTGOMERY_CURVE_NEG; FIELD_IMP_INTEGRAL_DOMAIN];
REWRITE_TAC[IN] THEN ASM_SIMP_TAC[MONTGOMERY_CURVE_ADD];
REWRITE_TAC[IN] THEN ASM_SIMP_TAC[MONTGOMERY_ADD_ASSOC];
REWRITE_TAC[FORALL_OPTION_THM; FORALL_PAIR_THM; montgomery_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[MONTGOMERY_ADD_LNEG];
MATCH_MP_TAC MONTGOMERY_ADD_SYM THEN
ASM_SIMP_TAC[MONTGOMERY_CURVE_NEG; FIELD_IMP_INTEGRAL_DOMAIN]]]);;
let ABELIAN_MONTGOMERY_GROUP = prove
(`!f a (b:A).
field f /\ ~(ring_char f = 2) /\
a IN ring_carrier f /\ b IN ring_carrier f /\
montgomery_nonsingular(f,a,b)
==> abelian_group(montgomery_group(f,a,b))`,
REPEAT STRIP_TAC THEN ASM_SIMP_TAC[abelian_group; MONTGOMERY_GROUP] THEN
REWRITE_TAC[IN] THEN ASM_MESON_TAC[MONTGOMERY_ADD_SYM]);;
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