(* ========================================================================= *) (* The SECG-recommended elliptic curve secp256k1. *) (* ========================================================================= *) 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 SECG curve parameters, copied from the SEC 2 document. *) (* See https://www.secg.org/sec2-v2.pdf *) (* ------------------------------------------------------------------------- *) let p_256k1 = define `p_256k1 = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F`;; let n_256k1 = define `n_256k1 = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141`;; let G_256K1 = define `G_256K1 = SOME(&0x79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798:int,&0x483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8:int)`;; (* ------------------------------------------------------------------------- *) (* Also parameters beta and lambda for an endomorphism. *) (* ------------------------------------------------------------------------- *) let p256k1_beta = define `p256k1_beta = 0x7ae96a2b657c07106e64479eac3434e99cf0497512f58995c1396c28719501ee`;; let p256k1_lambda = define `p256k1_lambda = 0x5363ad4cc05c30e0a5261c028812645a122e22ea20816678df02967c1b23bd72`;; (* ------------------------------------------------------------------------- *) (* Primality of the field characteristic and group order. *) (* ------------------------------------------------------------------------- *) let P_256K1 = prove (`p_256k1 = 2 EXP 256 - 2 EXP 32 - 977`, REWRITE_TAC[p_256k1] THEN CONV_TAC NUM_REDUCE_CONV);; let P_256K1_ALT = prove (`p_256k1 = 2 EXP 256 - 2 EXP 32 - 2 EXP 9 - 2 EXP 8 - 2 EXP 7 - 2 EXP 6 - 2 EXP 4 - 1`, REWRITE_TAC[p_256k1] THEN CONV_TAC NUM_REDUCE_CONV);; let PRIME_P256K1 = time prove (`prime p_256k1`, REWRITE_TAC[p_256k1] THEN CONV_TAC NUM_REDUCE_CONV THEN (CONV_TAC o PRIME_RULE) ["2"; "3"; "5"; "7"; "11"; "13"; "17"; "19"; "29"; "31"; "41"; "53"; "67"; "83"; "97"; "101"; "103"; "131"; "239"; "271"; "419"; "443"; "887"; "971"; "1373"; "1627"; "2621"; "2657"; "4423"; "5323"; "7723"; "13441"; "20113"; "24809"; "41201"; "96557"; "1206781"; "7240687"; "13331831"; "107590001"; "173378833005251801"; "22149492674086928081353"; "132896956044521568488119"; "255515944373312847190720520512484175977"; "205115282021455665897114700593932402728804164701536103180137503955397371"]);; let PRIME_N256K1 = time prove (`prime n_256k1`, REWRITE_TAC[n_256k1] THEN CONV_TAC NUM_REDUCE_CONV THEN (CONV_TAC o PRIME_RULE) ["2"; "3"; "5"; "7"; "11"; "13"; "17"; "19"; "23"; "29"; "37"; "41"; "59"; "67"; "73"; "97"; "109"; "113"; "149"; "199"; "293"; "461"; "631"; "797"; "1409"; "1871"; "2011"; "2731"; "2861"; "4051"; "9349"; "16699"; "28181"; "85831"; "120233"; "305873"; "1627771"; "4681609"; "44706919"; "545358713"; "297159362677"; "107361793816595537"; "174723607534414371449"; "29047611873442575647497758179"; "341948486974166000522343609283189"]);; (* ------------------------------------------------------------------------- *) (* Definition of the curve group and proof of its key properties. *) (* ------------------------------------------------------------------------- *) let p256k1_group = define `p256k1_group = weierstrass_group(integer_mod_ring p_256k1,&0,&7)`;; let P256K1_GROUP = prove (`group_carrier p256k1_group = weierstrass_curve(integer_mod_ring p_256k1,&0,&7) /\ group_id p256k1_group = NONE /\ group_inv p256k1_group = weierstrass_neg(integer_mod_ring p_256k1,&0,&7) /\ group_mul p256k1_group = weierstrass_add(integer_mod_ring p_256k1,&0,&7)`, REWRITE_TAC[p256k1_group] THEN MATCH_MP_TAC WEIERSTRASS_GROUP THEN REWRITE_TAC[FIELD_INTEGER_MOD_RING; INTEGER_MOD_RING_CHAR; PRIME_P256K1] THEN REWRITE_TAC[p_256k1; weierstrass_nonsingular] THEN SIMP_TAC[INTEGER_MOD_RING_CLAUSES; ARITH; IN_ELIM_THM] THEN CONV_TAC INT_REDUCE_CONV);; add_ecgroup [p_256k1] P256K1_GROUP;; let NO_ROOTS_256K1 = prove (`!x:int. ~((x pow 3 + &7 == &0) (mod &p_256k1))`, EXCLUDE_MODULAR_CUBIC_ROOTS_TAC PRIME_P256K1 [p_256k1]);; let GENERATOR_IN_GROUP_CARRIER_256K1 = prove (`G_256K1 IN group_carrier p256k1_group`, REWRITE_TAC[G_256K1] THEN CONV_TAC ECGROUP_CARRIER_CONV);; let GROUP_ELEMENT_ORDER_G256K1 = prove (`group_element_order p256k1_group G_256K1 = n_256k1`, SIMP_TAC[GROUP_ELEMENT_ORDER_UNIQUE_PRIME; GENERATOR_IN_GROUP_CARRIER_256K1; PRIME_N256K1] THEN REWRITE_TAC[G_256K1; el 1 (CONJUNCTS P256K1_GROUP); option_DISTINCT] THEN REWRITE_TAC[n_256k1] THEN CONV_TAC(LAND_CONV ECGROUP_POW_CONV) THEN REFL_TAC);; let FINITE_GROUP_CARRIER_256K1 = prove (`FINITE(group_carrier p256k1_group)`, REWRITE_TAC[P256K1_GROUP] THEN MATCH_MP_TAC FINITE_WEIERSTRASS_CURVE THEN REWRITE_TAC[FINITE_INTEGER_MOD_RING; FIELD_INTEGER_MOD_RING; PRIME_P256K1] THEN REWRITE_TAC[p_256k1] THEN CONV_TAC NUM_REDUCE_CONV);; let SIZE_P256K1_GROUP = prove (`group_carrier p256k1_group HAS_SIZE n_256k1`, MATCH_MP_TAC GROUP_ADHOC_ORDER_UNIQUE_LEMMA THEN EXISTS_TAC `G_256K1:(int#int)option` THEN REWRITE_TAC[GENERATOR_IN_GROUP_CARRIER_256K1; GROUP_ELEMENT_ORDER_G256K1; FINITE_GROUP_CARRIER_256K1] THEN REWRITE_TAC[P256K1_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_P256K1] THEN ANTS_TAC THENL [REWRITE_TAC[p_256k1] THEN CONV_TAC NUM_REDUCE_CONV; MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] LET_TRANS)] THEN SIMP_TAC[CARD_INTEGER_MOD_RING; p_256k1; ARITH] THEN REWRITE_TAC[n_256k1] 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_256k1; 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_256K1) THEN REWRITE_TAC[INT_MUL_LZERO; INT_ADD_LID] THEN REWRITE_TAC[GSYM INT_REM_EQ; p_256k1; 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_P256K1_GROUP = prove (`subgroup_generated p256k1_group {G_256K1} = p256k1_group`, SIMP_TAC[SUBGROUP_GENERATED_ELEMENT_ORDER; GENERATOR_IN_GROUP_CARRIER_256K1; FINITE_GROUP_CARRIER_256K1] THEN REWRITE_TAC[GROUP_ELEMENT_ORDER_G256K1; REWRITE_RULE[HAS_SIZE] SIZE_P256K1_GROUP]);; let CYCLIC_P256K1_GROUP = prove (`cyclic_group p256k1_group`, MESON_TAC[CYCLIC_GROUP_ALT; GENERATED_P256K1_GROUP]);; let ABELIAN_P256K1_GROUP = prove (`abelian_group p256k1_group`, MESON_TAC[CYCLIC_P256K1_GROUP; CYCLIC_IMP_ABELIAN_GROUP]);; (* ------------------------------------------------------------------------- *) (* Easily computable endomorphism of secp256k1 curve. *) (* ------------------------------------------------------------------------- *) let GROUP_ENDOMORPHISM_TRIPLEX_BETA = prove (`group_endomorphism p256k1_group (weierstrass_triplex (integer_mod_ring p_256k1) (&p256k1_beta))`, REWRITE_TAC[p256k1_group] THEN MATCH_MP_TAC GROUP_ENDOMORPHISM_TRIPLEX THEN SIMP_TAC[INTEGER_MOD_RING; INTEGER_MOD_RING_CHAR; INTEGER_MOD_RING_POW; p_256k1; IN_INSERT; FIELD_INTEGER_MOD_RING; ARITH; p256k1_beta; IN_ELIM_THM; weierstrass_nonsingular; INTEGER_MOD_RING_OF_NUM] THEN REWRITE_TAC[REWRITE_RULE[p_256k1] PRIME_P256K1] THEN CONV_TAC INT_REDUCE_CONV);; let P256K1_TRIPLEX_BETA = prove (`!x. x IN group_carrier p256k1_group ==> weierstrass_triplex (integer_mod_ring p_256k1) (&p256k1_beta) x = group_pow p256k1_group x p256k1_lambda`, GEN_REWRITE_TAC (BINDER_CONV o LAND_CONV o ONCE_DEPTH_CONV) [GSYM GENERATED_P256K1_GROUP] THEN MATCH_MP_TAC GROUP_HOMOMORPHISMS_EQ_ON_GENERATORS THEN EXISTS_TAC `p256k1_group` THEN SIMP_TAC[ABELIAN_GROUP_HOMOMORPHISM_POWERING; ABELIAN_P256K1_GROUP] THEN REWRITE_TAC[GSYM group_endomorphism] THEN REWRITE_TAC[GROUP_ENDOMORPHISM_TRIPLEX_BETA; ETA_AX] THEN REWRITE_TAC[IMP_CONJ_ALT; IN_SING; FORALL_UNWIND_THM2] THEN DISCH_TAC THEN SIMP_TAC[weierstrass_triplex; p256k1_beta; p256k1_lambda; G_256K1] THEN CONV_TAC(RAND_CONV ECGROUP_POW_CONV) THEN REWRITE_TAC[option_INJ; PAIR_EQ] THEN REWRITE_TAC[INTEGER_MOD_RING; p_256k1] THEN CONV_TAC INT_REDUCE_CONV);;