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(* ========================================================================= *)
(* The NIST-recommended elliptic curve P-256, aka secp256r1. *)
(* ========================================================================= *)
needs "EC/weierstrass.ml";;
needs "EC/excluderoots.ml";;
needs "EC/computegroup.ml";;
add_curve weierstrass_curve;;
add_curveneg weierstrass_neg;;
add_curveadd weierstrass_add;;
(* ------------------------------------------------------------------------- *)
(* The NIST curve parameters, copied from the NIST FIPS 186-4 document. *)
(* See https://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.186-4.pdf *)
(* ------------------------------------------------------------------------- *)
let p_256 = new_definition `p_256 = 115792089210356248762697446949407573530086143415290314195533631308867097853951`;;
let n_256 = new_definition `n_256 = 115792089210356248762697446949407573529996955224135760342422259061068512044369`;;
let SEED_256 = new_definition `SEED_256 = 0xc49d360886e704936a6678e1139d26b7819f7e90`;;
let c_256 = new_definition `c_256 = 0x7efba1662985be9403cb055c75d4f7e0ce8d84a9c5114abcaf3177680104fa0d`;;
let b_256 = new_definition `b_256 = 0x5ac635d8aa3a93e7b3ebbd55769886bc651d06b0cc53b0f63bce3c3e27d2604b`;;
let G_256 = new_definition `G_256 = SOME(&0x6b17d1f2e12c4247f8bce6e563a440f277037d812deb33a0f4a13945d898c296:int,&0x4fe342e2fe1a7f9b8ee7eb4a7c0f9e162bce33576b315ececbb6406837bf51f5:int)`;;
(* ------------------------------------------------------------------------- *)
(* Primality of the field characteristic and group order. *)
(* ------------------------------------------------------------------------- *)
let P_256 = prove
(`p_256 = 2 EXP 256 - 2 EXP 224 + 2 EXP 192 + 2 EXP 96 - 1`,
REWRITE_TAC[p_256] THEN CONV_TAC NUM_REDUCE_CONV);;
let PRIME_P256 = time prove
(`prime p_256`,
REWRITE_TAC[p_256] THEN CONV_TAC NUM_REDUCE_CONV THEN
(CONV_TAC o PRIME_RULE)
["2"; "3"; "5"; "7"; "11"; "13"; "17"; "23"; "43"; "53"; "107"; "157";
"173"; "181"; "197"; "241"; "257"; "313"; "641"; "661"; "727"; "757";
"919"; "1087"; "1531"; "2411"; "3677"; "3769"; "4349"; "17449"; "18169";
"65537"; "78283"; "490463"; "704251"; "6700417"; "204061199";
"34282281433"; "66417393611"; "11290956913871"; "46076956964474543";
"774023187263532362759620327192479577272145303";
"835945042244614951780389953367877943453916927241"]);;
let PRIME_N256 = time prove
(`prime n_256`,
REWRITE_TAC[n_256] THEN CONV_TAC NUM_REDUCE_CONV THEN
(CONV_TAC o PRIME_RULE)
["2"; "3"; "5"; "7"; "11"; "13"; "17"; "19"; "29"; "31"; "37"; "41"; "43";
"71"; "97"; "127"; "131"; "151"; "229"; "263"; "311"; "337"; "373"; "727";
"1201"; "1297"; "1511"; "3023"; "3407"; "9547"; "16879"; "17449"; "38189";
"104471"; "126241"; "155317"; "3969899"; "9350987"; "187019741";
"191039911"; "311245691"; "622491383"; "1002328039319";
"208150935158385979"; "2624747550333869278416773953"]);;
(* ------------------------------------------------------------------------- *)
(* Basic sanity check on the (otherwise unused) c parameter. *)
(* ------------------------------------------------------------------------- *)
let SANITY_CHECK_256 = prove
(`(&b_256 pow 2 * &c_256:int == -- &27) (mod &p_256)`,
REWRITE_TAC[G_256; p_256; b_256; c_256] THEN
REWRITE_TAC[GSYM INT_REM_EQ] THEN CONV_TAC INT_REDUCE_CONV);;
(* ------------------------------------------------------------------------- *)
(* Definition of the curve group and proof of its key properties. *)
(* ------------------------------------------------------------------------- *)
let p256_group = define
`p256_group =
weierstrass_group
(integer_mod_ring p_256,
ring_neg (integer_mod_ring p_256) (&3),
&b_256)`;;
let P256_GROUP = prove
(`group_carrier p256_group =
weierstrass_curve
(integer_mod_ring p_256,ring_neg (integer_mod_ring p_256) (&3),&b_256) /\
group_id p256_group =
NONE /\
group_inv p256_group =
weierstrass_neg
(integer_mod_ring p_256,ring_neg (integer_mod_ring p_256) (&3),&b_256) /\
group_mul p256_group =
weierstrass_add
(integer_mod_ring p_256,ring_neg (integer_mod_ring p_256) (&3),&b_256)`,
REWRITE_TAC[p256_group] THEN
MATCH_MP_TAC WEIERSTRASS_GROUP THEN
REWRITE_TAC[FIELD_INTEGER_MOD_RING; INTEGER_MOD_RING_CHAR; PRIME_P256] THEN
REWRITE_TAC[p_256; b_256; weierstrass_nonsingular] THEN
SIMP_TAC[INTEGER_MOD_RING_CLAUSES; ARITH; IN_ELIM_THM] THEN
CONV_TAC INT_REDUCE_CONV);;
add_ecgroup [p_256; b_256] P256_GROUP;;
let NO_ROOTS_P256 = prove
(`!x:int. ~((x pow 3 - &3 * x + &b_256 == &0) (mod &p_256))`,
EXCLUDE_MODULAR_CUBIC_ROOTS_TAC PRIME_P256 [p_256;b_256]);;
let GENERATOR_IN_GROUP_CARRIER_256 = prove
(`G_256 IN group_carrier p256_group`,
REWRITE_TAC[G_256] THEN CONV_TAC ECGROUP_CARRIER_CONV);;
let GROUP_ELEMENT_ORDER_G256 = prove
(`group_element_order p256_group G_256 = n_256`,
SIMP_TAC[GROUP_ELEMENT_ORDER_UNIQUE_PRIME; GENERATOR_IN_GROUP_CARRIER_256;
PRIME_N256] THEN
REWRITE_TAC[G_256; el 1 (CONJUNCTS P256_GROUP); option_DISTINCT] THEN
REWRITE_TAC[n_256] THEN CONV_TAC(LAND_CONV ECGROUP_POW_CONV) THEN
REFL_TAC);;
let FINITE_GROUP_CARRIER_256 = prove
(`FINITE(group_carrier p256_group)`,
REWRITE_TAC[P256_GROUP] THEN MATCH_MP_TAC FINITE_WEIERSTRASS_CURVE THEN
REWRITE_TAC[FINITE_INTEGER_MOD_RING; FIELD_INTEGER_MOD_RING; PRIME_P256] THEN
REWRITE_TAC[p_256] THEN CONV_TAC NUM_REDUCE_CONV);;
let SIZE_P256_GROUP = prove
(`group_carrier p256_group HAS_SIZE n_256`,
MATCH_MP_TAC GROUP_ADHOC_ORDER_UNIQUE_LEMMA THEN
EXISTS_TAC `G_256:(int#int)option` THEN
REWRITE_TAC[GENERATOR_IN_GROUP_CARRIER_256; GROUP_ELEMENT_ORDER_G256;
FINITE_GROUP_CARRIER_256] THEN
REWRITE_TAC[P256_GROUP] THEN CONJ_TAC THENL
[W(MP_TAC o PART_MATCH (lhand o rand)
CARD_BOUND_WEIERSTRASS_CURVE o lhand o snd) THEN
REWRITE_TAC[FINITE_INTEGER_MOD_RING; FIELD_INTEGER_MOD_RING] THEN
REWRITE_TAC[PRIME_P256] THEN ANTS_TAC THENL
[REWRITE_TAC[p_256] THEN CONV_TAC NUM_REDUCE_CONV;
MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] LET_TRANS)] THEN
SIMP_TAC[CARD_INTEGER_MOD_RING; p_256; ARITH] THEN
REWRITE_TAC[n_256] THEN CONV_TAC NUM_REDUCE_CONV;
REWRITE_TAC[FORALL_OPTION_THM; IN; FORALL_PAIR_THM] THEN
REWRITE_TAC[weierstrass_curve; weierstrass_neg; option_DISTINCT] THEN
MAP_EVERY X_GEN_TAC [`x:int`; `y:int`] THEN REWRITE_TAC[option_INJ] THEN
REWRITE_TAC[IN_INTEGER_MOD_RING_CARRIER; INTEGER_MOD_RING_CLAUSES] THEN
CONV_TAC INT_REM_DOWN_CONV THEN REWRITE_TAC[p_256; PAIR_EQ] THEN
CONV_TAC INT_REDUCE_CONV] THEN
ASM_CASES_TAC `y:int = &0` THENL
[ASM_REWRITE_TAC[] THEN CONV_TAC INT_REDUCE_CONV THEN
DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC (MP_TAC o SYM)) THEN
CONV_TAC INT_REM_DOWN_CONV THEN MP_TAC(SPEC `x:int` NO_ROOTS_P256) THEN
REWRITE_TAC[INT_ARITH `y - &3 * x + b:int = y + (-- &3) * x + b`] THEN
REWRITE_TAC[GSYM INT_REM_EQ; p_256; INT_REM_ZERO];
STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP (INT_ARITH
`--y rem p = y ==> y rem p = y ==> (--y rem p = y rem p)`)) THEN
ANTS_TAC THENL [ASM_SIMP_TAC[INT_REM_LT]; ALL_TAC] THEN
REWRITE_TAC[INT_REM_EQ; INTEGER_RULE
`(--y:int == y) (mod p) <=> p divides (&2 * y)`] THEN
DISCH_THEN(MP_TAC o MATCH_MP (INTEGER_RULE
`p divides (a * b:int) ==> coprime(a,p) ==> p divides b`)) THEN
REWRITE_TAC[GSYM num_coprime; ARITH; COPRIME_2] THEN
DISCH_THEN(MP_TAC o MATCH_MP INT_DIVIDES_LE) THEN ASM_INT_ARITH_TAC]);;
let GENERATED_P256_GROUP = prove
(`subgroup_generated p256_group {G_256} = p256_group`,
SIMP_TAC[SUBGROUP_GENERATED_ELEMENT_ORDER;
GENERATOR_IN_GROUP_CARRIER_256;
FINITE_GROUP_CARRIER_256] THEN
REWRITE_TAC[GROUP_ELEMENT_ORDER_G256;
REWRITE_RULE[HAS_SIZE] SIZE_P256_GROUP]);;
let CYCLIC_P256_GROUP = prove
(`cyclic_group p256_group`,
MESON_TAC[CYCLIC_GROUP_ALT; GENERATED_P256_GROUP]);;
let ABELIAN_P256_GROUP = prove
(`abelian_group p256_group`,
MESON_TAC[CYCLIC_P256_GROUP; CYCLIC_IMP_ABELIAN_GROUP]);;
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