Datasets:
Tasks:
Text Generation
Modalities:
Text
Sub-tasks:
language-modeling
Languages:
English
Size:
100K - 1M
License:
File size: 7,302 Bytes
4365a98 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 |
(* ========================================================================= *)
(* 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]);;
|