formal/hol/EC/formulary_projective.ml

(* ========================================================================= *)
(* Specific formulas for evaluating projective coordinate point operations.  *)
(* ========================================================================= *)

needs "EC/projective.ml";;

(* ------------------------------------------------------------------------- *)
(* Point doubling in projective coordinates.                                 *)
(*                                                                           *)
(* Source: Bernstein-Lange [2007] "Faster addition and doubling..."          *)
(* ------------------------------------------------------------------------- *)

(***
 *** http://hyperelliptic.org/EFD/g1p/auto-code/shortw/projective/doubling/dbl-2007-bl.op3
 ***)

let pr_dbl_2007_bl = new_definition
 `pr_dbl_2007_bl (f:A ring,a:A,b:A) (x1,y1,z1) =
      let xx = ring_pow f x1 2 in
      let zz = ring_pow f z1 2 in
      let t0 = ring_mul f (ring_of_num f 3) xx in
      let t1 = ring_mul f a zz in
      let w = ring_add f t1 t0 in
      let t2 = ring_mul f y1 z1 in
      let s = ring_mul f (ring_of_num f 2) t2 in
      let ss = ring_pow f s 2 in
      let sss = ring_mul f s ss in
      let r = ring_mul f y1 s in
      let rr = ring_pow f r 2 in
      let t3 = ring_add f x1 r in
      let t4 = ring_pow f t3 2 in
      let t5 = ring_sub f t4 xx in
      let b = ring_sub f t5 rr in
      let t6 = ring_pow f w 2 in
      let t7 = ring_mul f (ring_of_num f 2) b in
      let h = ring_sub f t6 t7 in
      let x3 = ring_mul f h s in
      let t8 = ring_sub f b h in
      let t9 = ring_mul f (ring_of_num f 2) rr in
      let t10 = ring_mul f w t8 in
      let y3 = ring_sub f t10 t9 in
      let z3 = sss in
      (x3,y3,z3)`;;

let PR_DBL_2007_BL = prove
 (`!f a b x1 y1 z1:A.
        field f /\
        a IN ring_carrier f /\ b IN ring_carrier f /\
        projective_point f (x1,y1,z1)
        ==> projective_eq f (pr_dbl_2007_bl (f,a,b) (x1,y1,z1))
                          (projective_add (f,a,b) (x1,y1,z1) (x1,y1,z1))`,
  REPEAT GEN_TAC THEN REWRITE_TAC[projective_point] THEN STRIP_TAC THEN
  REWRITE_TAC[pr_dbl_2007_bl; projective_add; projective_eq;
              projective_neg; projective_0] THEN
  ASM_CASES_TAC `z1:A = ring_0 f` THEN
  ASM_REWRITE_TAC[projective_add; projective_eq;
                  projective_neg; projective_0] THEN
  ASM_REWRITE_TAC[LET_DEF; LET_END_DEF; PAIR_EQ; projective_eq] THEN
  FIELD_TAC THEN ASM_SIMP_TAC[FIELD_IMP_INTEGRAL_DOMAIN]);;

(* ------------------------------------------------------------------------- *)
(* Point doubling in projective coordinates assuming a = -3.                 *)
(*                                                                           *)
(* Source: Bernstein-Lange [2007] "Faster addition and doubling..."          *)
(* ------------------------------------------------------------------------- *)

(***
 *** http://hyperelliptic.org/EFD/g1p/auto-code/shortw/projective-3/doubling/dbl-2007-bl-2.op3
 ***)

let p3_dbl_2007_bl_2 = new_definition
 `p3_dbl_2007_bl_2 (f:A ring,a:A,b:A) (x1,y1,z1) =
      let t0 = ring_sub f x1 z1 in
      let t1 = ring_add f x1 z1 in
      let t2 = ring_mul f t0 t1 in
      let w = ring_mul f (ring_of_num f 3) t2 in
      let t3 = ring_mul f y1 z1 in
      let s = ring_mul f (ring_of_num f 2) t3 in
      let ss = ring_pow f s 2 in
      let sss = ring_mul f s ss in
      let r = ring_mul f y1 s in
      let rr = ring_pow f r 2 in
      let t4 = ring_mul f x1 r in
      let b = ring_mul f (ring_of_num f 2) t4 in
      let t5 = ring_pow f w 2 in
      let t6 = ring_mul f (ring_of_num f 2) b in
      let h = ring_sub f t5 t6 in
      let x3 = ring_mul f h s in
      let t7 = ring_sub f b h in
      let t8 = ring_mul f (ring_of_num f 2) rr in
      let t9 = ring_mul f w t7 in
      let y3 = ring_sub f t9 t8 in
      let z3 = sss in
      (x3,y3,z3)`;;

let P3_DBL_2007_BL_2 = prove
 (`!f a b x1 y1 z1:A.
        field f /\
        a = ring_of_int f (-- &3) /\ b IN ring_carrier f /\
        projective_point f (x1,y1,z1)
        ==> projective_eq f (p3_dbl_2007_bl_2 (f,a,b) (x1,y1,z1))
                            (projective_add (f,a,b) (x1,y1,z1) (x1,y1,z1))`,
  REPEAT GEN_TAC THEN REWRITE_TAC[projective_point] THEN STRIP_TAC THEN
  FIRST_X_ASSUM SUBST_ALL_TAC THEN
  REWRITE_TAC[p3_dbl_2007_bl_2; projective_add; projective_eq;
              projective_neg; projective_0] THEN
  ASM_CASES_TAC `z1:A = ring_0 f` THEN
  ASM_REWRITE_TAC[projective_add; projective_eq;
                  projective_neg; projective_0] THEN
  ASM_REWRITE_TAC[LET_DEF; LET_END_DEF; PAIR_EQ; projective_eq] THEN
  FIELD_TAC THEN ASM_SIMP_TAC[FIELD_IMP_INTEGRAL_DOMAIN]);;

(* ------------------------------------------------------------------------- *)
(* Point doubling in projective coordinates assuming a = 0.                  *)
(*                                                                           *)
(* Source: Bernstein-Lange [2007] "Faster addition and doubling..." with     *)
(* trivial constant propagation from a = 0.                                  *)
(* ------------------------------------------------------------------------- *)

(***
 *** http://hyperelliptic.org/EFD/g1p/auto-code/shortw/projective/doubling/dbl-2007-bl.op3
 *** plus trivial constant propagation
 ***)

let p0_dbl_2007_bl = new_definition
 `p0_dbl_2007_bl (f:A ring,a:A,b:A) (x1,y1,z1) =
      let xx = ring_pow f x1 2 in
      let zz = ring_pow f z1 2 in
      let w = ring_mul f (ring_of_num f 3) xx in
      let t2 = ring_mul f y1 z1 in
      let s = ring_mul f (ring_of_num f 2) t2 in
      let ss = ring_pow f s 2 in
      let sss = ring_mul f s ss in
      let r = ring_mul f y1 s in
      let rr = ring_pow f r 2 in
      let t3 = ring_add f x1 r in
      let t4 = ring_pow f t3 2 in
      let t5 = ring_sub f t4 xx in
      let b = ring_sub f t5 rr in
      let t6 = ring_pow f w 2 in
      let t7 = ring_mul f (ring_of_num f 2) b in
      let h = ring_sub f t6 t7 in
      let x3 = ring_mul f h s in
      let t8 = ring_sub f b h in
      let t9 = ring_mul f (ring_of_num f 2) rr in
      let t10 = ring_mul f w t8 in
      let y3 = ring_sub f t10 t9 in
      let z3 = sss in
      (x3,y3,z3)`;;

let P0_DBL_2007_BL = prove
 (`!f a b x1 y1 z1:A.
        field f /\
        a = ring_0 f /\ b IN ring_carrier f /\
        projective_point f (x1,y1,z1)
        ==> projective_eq f (p0_dbl_2007_bl (f,a,b) (x1,y1,z1))
                            (projective_add (f,a,b) (x1,y1,z1) (x1,y1,z1))`,
  REPEAT GEN_TAC THEN REWRITE_TAC[projective_point] THEN STRIP_TAC THEN
  FIRST_X_ASSUM SUBST_ALL_TAC THEN
  REWRITE_TAC[p0_dbl_2007_bl; projective_add; projective_eq;
              projective_neg; projective_0] THEN
  ASM_CASES_TAC `z1:A = ring_0 f` THEN
  ASM_REWRITE_TAC[projective_add; projective_eq;
                  projective_neg; projective_0] THEN
  ASM_REWRITE_TAC[LET_DEF; LET_END_DEF; PAIR_EQ; projective_eq] THEN
  FIELD_TAC THEN ASM_SIMP_TAC[FIELD_IMP_INTEGRAL_DOMAIN]);;

(* ------------------------------------------------------------------------- *)
(* Pure point addition in projective coordinates. This sequence never uses   *)
(* the value of "a" so there's no special optimized version for special "a". *)
(*                                                                           *)
(* Source Cohen-Miyaji-Ono [1998] "Efficient elliptic curve exponentiation"  *)
(*                                                                           *)
(* Note the correctness is not proved in cases where the points are equal    *)
(* (or projectively equivalent), or either input is 0 (point at infinity).   *)
(* ------------------------------------------------------------------------- *)

(***
 *** http://hyperelliptic.org/EFD/g1p/auto-code/shortw/projective/addition/add-1998-cmo-2.op3
 ***)

let pr_add_1998_cmo_2 = new_definition
 `pr_add_1998_cmo_2 (f:A ring,a:A,b:A) (x1,y1,z1) (x2,y2,z2) =
      let y1z2 = ring_mul f y1 z2 in
      let x1z2 = ring_mul f x1 z2 in
      let z1z2 = ring_mul f z1 z2 in
      let t0 = ring_mul f y2 z1 in
      let u = ring_sub f t0 y1z2 in
      let uu = ring_pow f u 2 in
      let t1 = ring_mul f x2 z1 in
      let v = ring_sub f t1 x1z2 in
      let vv = ring_pow f v 2 in
      let vvv = ring_mul f v vv in
      let r = ring_mul f vv x1z2 in
      let t2 = ring_mul f (ring_of_num f 2) r in
      let t3 = ring_mul f uu z1z2 in
      let t4 = ring_sub f t3 vvv in
      let a = ring_sub f t4 t2 in
      let x3 = ring_mul f v a in
      let t5 = ring_sub f r a in
      let t6 = ring_mul f vvv y1z2 in
      let t7 = ring_mul f u t5 in
      let y3 = ring_sub f t7 t6 in
      let z3 = ring_mul f vvv z1z2 in
      (x3,y3,z3)`;;

let PR_ADD_1998_CMO_2 = prove
 (`!f a b x1 y1 z1 x2 y2 z2:A.
        field f /\ ~(ring_char f = 2) /\
        a IN ring_carrier f /\ b IN ring_carrier f /\
        projective_point f (x1,y1,z1) /\ projective_point f (x2,y2,z2) /\
        ~(z1 = ring_0 f) /\ ~(z2 = ring_0 f) /\
        ~(projective_eq f (x1,y1,z1) (x2,y2,z2))
        ==> projective_eq f (pr_add_1998_cmo_2 (f,a,b) (x1,y1,z1) (x2,y2,z2))
                            (projective_add (f,a,b) (x1,y1,z1) (x2,y2,z2))`,
  REPEAT GEN_TAC THEN REWRITE_TAC[projective_point] THEN
  DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
  ASM_SIMP_TAC[GSYM RING_CHAR_DIVIDES_PRIME; PRIME_2] THEN
  REPEAT(DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC)) THEN
  REPEAT(FIRST_X_ASSUM SUBST_ALL_TAC) THEN
  ASM_REWRITE_TAC[projective_eq; pr_add_1998_cmo_2; projective_add] THEN
  REPEAT(COND_CASES_TAC THEN
   ASM_REWRITE_TAC[projective_add; projective_eq;
                   projective_neg; projective_0; LET_DEF; LET_END_DEF]) THEN
  REPEAT(FIRST_X_ASSUM(MP_TAC o check (free_in `(=):A->A->bool` o concl))) THEN
  FIELD_TAC);;

(* ------------------------------------------------------------------------- *)
(* Mixed point addition in projective coordinates. Here "mixed" means        *)
(* assuming z2 = 1, which holds if the second point was directly injected    *)
(* from the Weierstrass coordinates via (x,y) |-> (x,y,1). This never uses   *)
(* the value of "a" so there's no special optimized version for special "a". *)
(*                                                                           *)
(* Source Cohen-Miyaji-Ono [1998] "Efficient elliptic curve exponentiation"  *)
(*                                                                           *)
(* Note the correctness is not proved in the case where the points are equal *)
(* or projectively equivalent, nor where the first is the group identity     *)
(* (i.e. the point at infinity, anything with z1 = 0 in projective coords).  *)
(* ------------------------------------------------------------------------- *)

(***
 *** http://hyperelliptic.org/EFD/g1p/auto-code/shortw/projective/addition/madd-1998-cmo.op3
 ***)

let pr_madd_1998_cmo = new_definition
 `pr_madd_1998_cmo (f:A ring,a:A,b:A) (x1,y1,z1) (x2,y2,z2) =
      let t0 = ring_mul f y2 z1 in
      let u = ring_sub f t0 y1 in
      let uu = ring_pow f u 2 in
      let t1 = ring_mul f x2 z1 in
      let v = ring_sub f t1 x1 in
      let vv = ring_pow f v 2 in
      let vvv = ring_mul f v vv in
      let r = ring_mul f vv x1 in
      let t2 = ring_mul f (ring_of_num f 2) r in
      let t3 = ring_mul f uu z1 in
      let t4 = ring_sub f t3 vvv in
      let a = ring_sub f t4 t2 in
      let x3 = ring_mul f v a in
      let t5 = ring_sub f r a in
      let t6 = ring_mul f vvv y1 in
      let t7 = ring_mul f u t5 in
      let y3 = ring_sub f t7 t6 in
      let z3 = ring_mul f vvv z1 in
      (x3,y3,z3)`;;

let PR_MADD_1998_CMO = prove
 (`!f a b x1 y1 z1 x2 y2 z2:A.
        field f /\ ~(ring_char f = 2) /\
        a IN ring_carrier f /\ b IN ring_carrier f /\
        projective_point f (x1,y1,z1) /\ projective_point f (x2,y2,z2) /\
        z2 = ring_1 f /\
        ~(z1 = ring_0 f) /\ ~(projective_eq f (x1,y1,z1) (x2,y2,z2))
        ==> projective_eq f (pr_madd_1998_cmo (f,a,b) (x1,y1,z1) (x2,y2,z2))
                            (projective_add (f,a,b) (x1,y1,z1) (x2,y2,z2))`,
  REPEAT GEN_TAC THEN REWRITE_TAC[projective_point] THEN
  DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
  ASM_SIMP_TAC[GSYM RING_CHAR_DIVIDES_PRIME; PRIME_2] THEN
  REPEAT(DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC)) THEN
  REPEAT(FIRST_X_ASSUM SUBST_ALL_TAC) THEN
  ASM_REWRITE_TAC[projective_eq; pr_madd_1998_cmo; projective_add] THEN
  REPEAT(COND_CASES_TAC THEN
   ASM_REWRITE_TAC[projective_add; projective_eq;
                   projective_neg; projective_0; LET_DEF; LET_END_DEF]) THEN
  REPEAT(FIRST_X_ASSUM(MP_TAC o check (free_in `(=):A->A->bool` o concl))) THEN
  FIELD_TAC);;