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(* ========================================================================= *)
(* Projective coordinates, x = X / Z without y for Montgomery curves. *)
(* ========================================================================= *)
needs "EC/montgomery.ml";;
(* ------------------------------------------------------------------------- *)
(* The representation is only relational as we throw away the y coordinate. *)
(* ------------------------------------------------------------------------- *)
let montgomery_xz = define
`(montgomery_xz (f:A ring) NONE (X,Z) <=>
X IN ring_carrier f /\ Z IN ring_carrier f /\
~(X = ring_0 f) /\ Z = ring_0 f) /\
(montgomery_xz f (SOME(x,y:A)) (X,Z) <=>
X IN ring_carrier f /\ Z IN ring_carrier f /\
~(Z = ring_0 f) /\ ring_div f X Z = x)`;;
(* ------------------------------------------------------------------------- *)
(* However, doubling and *differential* addition are calculable. *)
(* ------------------------------------------------------------------------- *)
let montgomery_xzdouble = define
`montgomery_xzdouble (f,a:A,b:A) (X,Z) =
ring_pow f (ring_sub f (ring_pow f X 2) (ring_pow f Z 2)) 2,
ring_mul f (ring_mul f (ring_of_num f 4) (ring_mul f X Z))
(ring_add f (ring_pow f X 2)
(ring_add f (ring_mul f a
(ring_mul f X Z)) (ring_pow f Z 2)))`;;
let montgomery_xzdiffadd = define
`montgomery_xzdiffadd (f:A ring,a:A,b:A) (X,Z) (Xm,Zm) (Xn,Zn) =
ring_mul f (ring_mul f (ring_of_num f 4) Z)
(ring_pow f (ring_sub f (ring_mul f Xm Xn)
(ring_mul f Zm Zn)) 2),
ring_mul f (ring_mul f (ring_of_num f 4) X)
(ring_pow f (ring_sub f (ring_mul f Xm Zn)
(ring_mul f Xn Zm)) 2)`;;
let MONTGOMERY_XZDOUBLE = prove
(`!f (a:A) b p q.
field f /\ ~(ring_char f = 2) /\
a IN ring_carrier f /\ b IN ring_carrier f /\
montgomery_nonsingular(f,a,b) /\
montgomery_curve(f,a,b) p /\
montgomery_xz f p q
==> montgomery_xz f (montgomery_add(f,a,b) p p)
(montgomery_xzdouble(f,a,b) q)`,
REWRITE_TAC[FIELD_CHAR_NOT2] THEN
REWRITE_TAC[FORALL_OPTION_THM; FORALL_PAIR_THM] THEN
REWRITE_TAC[montgomery_curve; montgomery_add; montgomery_xz;
GSYM DE_MORGAN_THM; montgomery_xzdouble; montgomery_nonsingular] THEN
REPEAT STRIP_TAC THEN REWRITE_TAC[COND_SWAP] THEN
TRY(COND_CASES_TAC THEN ASM_REWRITE_TAC[]) THEN
REWRITE_TAC[LET_DEF; LET_END_DEF; montgomery_xz] THEN
REPEAT STRIP_TAC THEN TRY RING_CARRIER_TAC THEN
FIELD_TAC THEN NOT_RING_CHAR_DIVIDES_TAC);;
let MONTGOMERY_XZDIFFADD = prove
(`!f (a:A) b p q pm qm pn qn.
field f /\ ~(ring_char f = 2) /\ ~(ring_char f = 3) /\
a IN ring_carrier f /\ b IN ring_carrier f /\
montgomery_nonsingular(f,a,b) /\
montgomery_curve(f,a,b) p /\
montgomery_curve(f,a,b) pm /\
montgomery_curve(f,a,b) pn /\
montgomery_xz f p q /\
montgomery_xz f pm qm /\
montgomery_xz f pn qn /\
~(FST q = ring_0 f) /\ ~(SND q = ring_0 f) /\
montgomery_add (f,a,b) pm (montgomery_neg (f,a,b) pn) = p
==> montgomery_xz f (montgomery_add(f,a,b) pm pn)
(montgomery_xzdiffadd(f,a,b) q qm qn)`,
REWRITE_TAC[FIELD_CHAR_NOT23] THEN REPEAT GEN_TAC THEN ASM_CASES_TAC
`p = montgomery_add (f,a:A,b) pm (montgomery_neg (f,a,b) pn)` THEN
ASM_REWRITE_TAC[] THEN POP_ASSUM(K ALL_TAC) THEN
W(fun (asl,w) -> MAP_EVERY (fun t -> SPEC_TAC(t,t)) (frees w)) THEN
REWRITE_TAC[FORALL_OPTION_THM; FORALL_PAIR_THM] THEN
REWRITE_TAC[montgomery_curve; montgomery_add; montgomery_xz; montgomery_neg;
LET_DEF; LET_END_DEF; montgomery_xzdiffadd; montgomery_nonsingular] THEN
REPEAT(GEN_TAC ORELSE CONJ_TAC) THEN
REWRITE_TAC[COND_SWAP; option_DISTINCT; option_INJ; PAIR_EQ] THEN
REPEAT(COND_CASES_TAC THEN
ASM_REWRITE_TAC[option_DISTINCT; option_INJ; PAIR_EQ; montgomery_add;
LET_DEF; LET_END_DEF; montgomery_xz]) THEN
REPEAT STRIP_TAC THEN TRY RING_CARRIER_TAC THEN
FIELD_TAC THEN NOT_RING_CHAR_DIVIDES_TAC);;
(* ------------------------------------------------------------------------- *)
(* The y coordinate can be recovered from any nondegenerate addition (e.g. *)
(* differential) where we know the y coordinate of one of the addends. This *)
(* formula is from Okeya and Sakurai's paper in CHES 2001 (LNCS 2162, p129). *)
(* *)
(* Suppose (x1,y1) + (x,y) = (x2,y2). Then *)
(* y1 = ((x1 * x + 1) * (x1 + x + 2 * A) - 2 * A - (x1 - x)^2 * x2) / *)
(* (2 * B * y) *)
(* ------------------------------------------------------------------------- *)
let MONTGOMERY_ADD_YRECOVERY = prove
(`!f a (b:A) x y x1 y1 x2 y2.
field f /\ ~(ring_char f = 2) /\
a IN ring_carrier f /\ b IN ring_carrier f /\ ~(b = ring_0 f) /\
montgomery_curve (f,a,b) (SOME(x,y)) /\ ~(y = ring_0 f) /\
montgomery_curve (f,a,b) (SOME(x1,y1)) /\
montgomery_add(f,a,b) (SOME(x,y)) (SOME(x1,y1)) = SOME(x2,y2)
==> y1 = ring_div f
(ring_sub f
(ring_sub f
(ring_mul f (ring_add f (ring_mul f x1 x) (ring_1 f))
(ring_add f x1
(ring_add f x (ring_mul f (ring_of_num f 2) a))))
(ring_mul f (ring_of_num f 2) a))
(ring_mul f (ring_pow f (ring_sub f x1 x) 2) x2))
(ring_mul f (ring_of_num f 2) (ring_mul f b y))`,
REWRITE_TAC[FIELD_CHAR_NOT2] THEN
REWRITE_TAC[montgomery_nonsingular] THEN REPEAT GEN_TAC THEN
REWRITE_TAC[montgomery_curve; montgomery_add; LET_DEF; LET_END_DEF] THEN
REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[option_DISTINCT]) THEN
REWRITE_TAC[option_INJ; PAIR_EQ] THEN REPEAT STRIP_TAC THEN
REPLICATE_TAC 2 (FIRST_X_ASSUM(SUBST1_TAC o SYM)) THEN
FIELD_TAC THEN NOT_RING_CHAR_DIVIDES_TAC);;
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