(* ========================================================================= *) (* The SECG-recommended elliptic curve secp192k1. *) (* ========================================================================= *) 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_192k1 = define `p_192k1 = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFEE37`;; let n_192k1 = define `n_192k1 = 0xFFFFFFFFFFFFFFFFFFFFFFFE26F2FC170F69466A74DEFD8D`;; let G_192K1 = define `G_192K1 = SOME(&0xDB4FF10EC057E9AE26B07D0280B7F4341DA5D1B1EAE06C7D:int,&0x9B2F2F6D9C5628A7844163D015BE86344082AA88D95E2F9D:int)`;; (* ------------------------------------------------------------------------- *) (* Primality of the field characteristic and group order. *) (* ------------------------------------------------------------------------- *) let P_192K1 = prove (`p_192k1 = 2 EXP 192 - 2 EXP 32 - 4553`, REWRITE_TAC[p_192k1] THEN CONV_TAC NUM_REDUCE_CONV);; let P_192K1_ALT = prove (`p_192k1 = 2 EXP 192 - 2 EXP 32 - 2 EXP 12 - 2 EXP 8 - 2 EXP 7 - 2 EXP 6 - 2 EXP 3 - 1`, REWRITE_TAC[p_192k1] THEN CONV_TAC NUM_REDUCE_CONV);; let PRIME_P192K1 = time prove (`prime p_192k1`, REWRITE_TAC[p_192k1] THEN CONV_TAC NUM_REDUCE_CONV THEN (CONV_TAC o PRIME_RULE) ["2"; "3"; "5"; "7"; "11"; "13"; "17"; "19"; "23"; "29"; "37"; "41"; "43"; "47"; "61"; "79"; "103"; "149"; "193"; "251"; "281"; "487"; "563"; "1559"; "2473"; "2683"; "3119"; "7057"; "393721"; "706151"; "3651619"; "8473813"; "14606477"; "2307823367"; "11113956389"; "16189543961"; "138580737803"; "1295233555201613"; "10489845818524887021689201254173392444641"]);; let PRIME_N192K1 = time prove (`prime n_192k1`, REWRITE_TAC[n_192k1] THEN CONV_TAC NUM_REDUCE_CONV THEN (CONV_TAC o PRIME_RULE) ["2"; "3"; "5"; "7"; "11"; "13"; "17"; "19"; "23"; "29"; "31"; "41"; "59"; "73"; "83"; "97"; "137"; "167"; "443"; "971"; "2341"; "4933"; "11519"; "29131"; "54151"; "169361"; "444791"; "445097"; "552913"; "815669"; "866417"; "1611297632578441"; "31767070186748510944261247684750677"; "434093022356392396149847294750353440317757907331"; "143250697377609490729449607267616635304860109419231"]);; (* ------------------------------------------------------------------------- *) (* Definition of the curve group and proof of its key properties. *) (* ------------------------------------------------------------------------- *) let p192k1_group = define `p192k1_group = weierstrass_group(integer_mod_ring p_192k1,&0,&3)`;; let P192K1_GROUP = prove (`group_carrier p192k1_group = weierstrass_curve(integer_mod_ring p_192k1,&0,&3) /\ group_id p192k1_group = NONE /\ group_inv p192k1_group = weierstrass_neg(integer_mod_ring p_192k1,&0,&3) /\ group_mul p192k1_group = weierstrass_add(integer_mod_ring p_192k1,&0,&3)`, REWRITE_TAC[p192k1_group] THEN MATCH_MP_TAC WEIERSTRASS_GROUP THEN REWRITE_TAC[FIELD_INTEGER_MOD_RING; INTEGER_MOD_RING_CHAR; PRIME_P192K1] THEN REWRITE_TAC[p_192k1; weierstrass_nonsingular] THEN SIMP_TAC[INTEGER_MOD_RING_CLAUSES; ARITH; IN_ELIM_THM] THEN CONV_TAC INT_REDUCE_CONV);; add_ecgroup [p_192k1] P192K1_GROUP;; let NO_ROOTS_192K1 = prove (`!x:int. ~((x pow 3 + &3 == &0) (mod &p_192k1))`, EXCLUDE_MODULAR_CUBIC_ROOTS_TAC PRIME_P192K1 [p_192k1]);; let GENERATOR_IN_GROUP_CARRIER_192K1 = prove (`G_192K1 IN group_carrier p192k1_group`, REWRITE_TAC[G_192K1] THEN CONV_TAC ECGROUP_CARRIER_CONV);; let GROUP_ELEMENT_ORDER_G192K1 = prove (`group_element_order p192k1_group G_192K1 = n_192k1`, SIMP_TAC[GROUP_ELEMENT_ORDER_UNIQUE_PRIME; GENERATOR_IN_GROUP_CARRIER_192K1; PRIME_N192K1] THEN REWRITE_TAC[G_192K1; el 1 (CONJUNCTS P192K1_GROUP); option_DISTINCT] THEN REWRITE_TAC[n_192k1] THEN CONV_TAC(LAND_CONV ECGROUP_POW_CONV) THEN REFL_TAC);; let FINITE_GROUP_CARRIER_192K1 = prove (`FINITE(group_carrier p192k1_group)`, REWRITE_TAC[P192K1_GROUP] THEN MATCH_MP_TAC FINITE_WEIERSTRASS_CURVE THEN REWRITE_TAC[FINITE_INTEGER_MOD_RING; FIELD_INTEGER_MOD_RING; PRIME_P192K1] THEN REWRITE_TAC[p_192k1] THEN CONV_TAC NUM_REDUCE_CONV);; let SIZE_P192K1_GROUP = prove (`group_carrier p192k1_group HAS_SIZE n_192k1`, MATCH_MP_TAC GROUP_ADHOC_ORDER_UNIQUE_LEMMA THEN EXISTS_TAC `G_192K1:(int#int)option` THEN REWRITE_TAC[GENERATOR_IN_GROUP_CARRIER_192K1; GROUP_ELEMENT_ORDER_G192K1; FINITE_GROUP_CARRIER_192K1] THEN REWRITE_TAC[P192K1_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_P192K1] THEN ANTS_TAC THENL [REWRITE_TAC[p_192k1] THEN CONV_TAC NUM_REDUCE_CONV; MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] LET_TRANS)] THEN SIMP_TAC[CARD_INTEGER_MOD_RING; p_192k1; ARITH] THEN REWRITE_TAC[n_192k1] 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_192k1; 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_192K1) THEN REWRITE_TAC[INT_MUL_LZERO; INT_ADD_LID] THEN REWRITE_TAC[GSYM INT_REM_EQ; p_192k1; 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_P192K1_GROUP = prove (`subgroup_generated p192k1_group {G_192K1} = p192k1_group`, SIMP_TAC[SUBGROUP_GENERATED_ELEMENT_ORDER; GENERATOR_IN_GROUP_CARRIER_192K1; FINITE_GROUP_CARRIER_192K1] THEN REWRITE_TAC[GROUP_ELEMENT_ORDER_G192K1; REWRITE_RULE[HAS_SIZE] SIZE_P192K1_GROUP]);; let CYCLIC_P192K1_GROUP = prove (`cyclic_group p192k1_group`, MESON_TAC[CYCLIC_GROUP_ALT; GENERATED_P192K1_GROUP]);; let ABELIAN_P192K1_GROUP = prove (`abelian_group p192k1_group`, MESON_TAC[CYCLIC_P192K1_GROUP; CYCLIC_IMP_ABELIAN_GROUP]);;