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(* ========================================================================= *)
(* The NIST-recommended elliptic curve P-224, aka secp224r1. *)
(* ========================================================================= *)
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_224 = new_definition `p_224 = 26959946667150639794667015087019630673557916260026308143510066298881`;;
let n_224 = new_definition `n_224 = 26959946667150639794667015087019625940457807714424391721682722368061`;;
let SEED_224 = new_definition `SEED_224 = 0xbd71344799d5c7fcdc45b59fa3b9ab8f6a948bc5`;;
let c_224 = new_definition `c_224 = 0x5b056c7e11dd68f40469ee7f3c7a7d74f7d121116506d031218291fb`;;
let b_224 = new_definition `b_224 = 0xb4050a850c04b3abf54132565044b0b7d7bfd8ba270b39432355ffb4`;;
let G_224 = new_definition `G_224 = SOME(&0xb70e0cbd6bb4bf7f321390b94a03c1d356c21122343280d6115c1d21:int,&0xbd376388b5f723fb4c22dfe6cd4375a05a07476444d5819985007e34:int)`;;
(* ------------------------------------------------------------------------- *)
(* Primality of the field characteristic and group order. *)
(* ------------------------------------------------------------------------- *)
let P_224 = prove
(`p_224 = 2 EXP 224 - 2 EXP 96 + 1`,
REWRITE_TAC[p_224] THEN CONV_TAC NUM_REDUCE_CONV);;
let PRIME_P224 = time prove
(`prime p_224`,
REWRITE_TAC[p_224] THEN CONV_TAC NUM_REDUCE_CONV THEN
(CONV_TAC o PRIME_RULE)
["2"; "3"; "5"; "7"; "11"; "17"; "23"; "31"; "43"; "47"; "173"; "257";
"347"; "373"; "641"; "727"; "16657"; "17449"; "65537"; "166571"; "274177";
"2998279"; "6700417"; "67280421310721"]);;
let PRIME_N224 = time prove
(`prime n_224`,
REWRITE_TAC[n_224] THEN CONV_TAC NUM_REDUCE_CONV THEN
(CONV_TAC o PRIME_RULE)
["2"; "3"; "5"; "7"; "11"; "13"; "17"; "23"; "29"; "31"; "37"; "41"; "43";
"47"; "61"; "67"; "89"; "101"; "127"; "139"; "173"; "239"; "269"; "347";
"349"; "509"; "631"; "659"; "1303"; "1319"; "2089"; "2153"; "2707";
"3433"; "10909"; "20599"; "30859"; "85999"; "87739"; "145091"; "166823";
"11105363"; "13928737"; "821796863"; "432621809776543";
"136401162692544977256234449"; "34646440928557194402992574983797";
"375503554633724504423937478103159147573209";
"50520606258875818707470860153287666700917696099933389351507"]);;
(* ------------------------------------------------------------------------- *)
(* Basic sanity check on the (otherwise unused) c parameter. *)
(* ------------------------------------------------------------------------- *)
let SANITY_CHECK_224 = prove
(`(&b_224 pow 2 * &c_224:int == -- &27) (mod &p_224)`,
REWRITE_TAC[G_224; p_224; b_224; c_224] 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 p224_group = define
`p224_group =
weierstrass_group
(integer_mod_ring p_224,
ring_neg (integer_mod_ring p_224) (&3),
&b_224)`;;
let P224_GROUP = prove
(`group_carrier p224_group =
weierstrass_curve
(integer_mod_ring p_224,ring_neg (integer_mod_ring p_224) (&3),&b_224) /\
group_id p224_group =
NONE /\
group_inv p224_group =
weierstrass_neg
(integer_mod_ring p_224,ring_neg (integer_mod_ring p_224) (&3),&b_224) /\
group_mul p224_group =
weierstrass_add
(integer_mod_ring p_224,ring_neg (integer_mod_ring p_224) (&3),&b_224)`,
REWRITE_TAC[p224_group] THEN
MATCH_MP_TAC WEIERSTRASS_GROUP THEN
REWRITE_TAC[FIELD_INTEGER_MOD_RING; INTEGER_MOD_RING_CHAR; PRIME_P224] THEN
REWRITE_TAC[p_224; b_224; weierstrass_nonsingular] THEN
SIMP_TAC[INTEGER_MOD_RING_CLAUSES; ARITH; IN_ELIM_THM] THEN
CONV_TAC INT_REDUCE_CONV);;
add_ecgroup [p_224; b_224] P224_GROUP;;
let NO_ROOTS_P224 = prove
(`!x:int. ~((x pow 3 - &3 * x + &b_224 == &0) (mod &p_224))`,
EXCLUDE_MODULAR_CUBIC_ROOTS_TAC PRIME_P224 [p_224;b_224]);;
let GENERATOR_IN_GROUP_CARRIER_224 = prove
(`G_224 IN group_carrier p224_group`,
REWRITE_TAC[G_224] THEN CONV_TAC ECGROUP_CARRIER_CONV);;
let GROUP_ELEMENT_ORDER_G224 = prove
(`group_element_order p224_group G_224 = n_224`,
SIMP_TAC[GROUP_ELEMENT_ORDER_UNIQUE_PRIME; GENERATOR_IN_GROUP_CARRIER_224;
PRIME_N224] THEN
REWRITE_TAC[G_224; el 1 (CONJUNCTS P224_GROUP); option_DISTINCT] THEN
REWRITE_TAC[n_224] THEN CONV_TAC(LAND_CONV ECGROUP_POW_CONV) THEN
REFL_TAC);;
let FINITE_GROUP_CARRIER_224 = prove
(`FINITE(group_carrier p224_group)`,
REWRITE_TAC[P224_GROUP] THEN MATCH_MP_TAC FINITE_WEIERSTRASS_CURVE THEN
REWRITE_TAC[FINITE_INTEGER_MOD_RING; FIELD_INTEGER_MOD_RING; PRIME_P224] THEN
REWRITE_TAC[p_224] THEN CONV_TAC NUM_REDUCE_CONV);;
let SIZE_P224_GROUP = prove
(`group_carrier p224_group HAS_SIZE n_224`,
MATCH_MP_TAC GROUP_ADHOC_ORDER_UNIQUE_LEMMA THEN
EXISTS_TAC `G_224:(int#int)option` THEN
REWRITE_TAC[GENERATOR_IN_GROUP_CARRIER_224; GROUP_ELEMENT_ORDER_G224;
FINITE_GROUP_CARRIER_224] THEN
REWRITE_TAC[P224_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_P224] THEN ANTS_TAC THENL
[REWRITE_TAC[p_224] THEN CONV_TAC NUM_REDUCE_CONV;
MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] LET_TRANS)] THEN
SIMP_TAC[CARD_INTEGER_MOD_RING; p_224; ARITH] THEN
REWRITE_TAC[n_224] 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_224; 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_P224) THEN
REWRITE_TAC[INT_ARITH `y - &3 * x + b:int = y + (-- &3) * x + b`] THEN
REWRITE_TAC[GSYM INT_REM_EQ; p_224; 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_P224_GROUP = prove
(`subgroup_generated p224_group {G_224} = p224_group`,
SIMP_TAC[SUBGROUP_GENERATED_ELEMENT_ORDER;
GENERATOR_IN_GROUP_CARRIER_224;
FINITE_GROUP_CARRIER_224] THEN
REWRITE_TAC[GROUP_ELEMENT_ORDER_G224;
REWRITE_RULE[HAS_SIZE] SIZE_P224_GROUP]);;
let CYCLIC_P224_GROUP = prove
(`cyclic_group p224_group`,
MESON_TAC[CYCLIC_GROUP_ALT; GENERATED_P224_GROUP]);;
let ABELIAN_P224_GROUP = prove
(`abelian_group p224_group`,
MESON_TAC[CYCLIC_P224_GROUP; CYCLIC_IMP_ABELIAN_GROUP]);;
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