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(* ========================================================================= *)
(* The three forms of curve25519 related via isomorphisms. *)
(* ========================================================================= *)
needs "EC/edmont.ml";;
needs "EC/montwe.ml";;
needs "EC/edwards25519.ml";;
needs "EC/curve25519.ml";;
needs "EC/wei25519.ml";;
(* ------------------------------------------------------------------------- *)
(* Extra scaling factor. *)
(* ------------------------------------------------------------------------- *)
(***
https://datatracker.ietf.org/doc/html/draft-ietf-lwig-curve-representations-23
In terms of the other parameters c_25519 = sqrt(-(A_25519+2)/B_25519)
***)
let c_25519 = define `c_25519 = 51042569399160536130206135233146329284152202253034631822681833788666877215207`;;
(* ------------------------------------------------------------------------- *)
(* Mappings between Edwards and Montgomery forms. *)
(* ------------------------------------------------------------------------- *)
let curve25519_of_edwards25519 = define
`curve25519_of_edwards25519 =
montgomery_of_edwards (integer_mod_ring p_25519) (&c_25519)`;;
let edwards25519_of_curve25519 = define
`edwards25519_of_curve25519 =
edwards_of_montgomery (integer_mod_ring p_25519) (&c_25519)`;;
let MCURVE_OF_ECURVE_25519 = prove
(`mcurve_of_ecurve (integer_mod_ring p_25519,&e_25519,&d_25519) (&c_25519) =
(integer_mod_ring p_25519,&A_25519,&1)`,
REWRITE_TAC[mcurve_of_ecurve; PAIR_EQ] THEN
REWRITE_TAC[p_25519; A_25519; e_25519; d_25519; c_25519] THEN
CONV_TAC(DEPTH_CONV INTEGER_MOD_RING_RED_CONV) THEN REWRITE_TAC[]);;
let GROUP_ISOMORPHISMS_EDWARDS25519_CURVE25519 = prove
(`group_isomorphisms
(edwards25519_group,curve25519_group)
(curve25519_of_edwards25519,edwards25519_of_curve25519)`,
MP_TAC(ISPECL
[`integer_mod_ring p_25519`; `&e_25519:int`; `&d_25519:int`; `&c_25519:int`]
GROUP_ISOMORPHISMS_EDWARDS_MONTGOMERY_GROUP) THEN
REWRITE_TAC[GSYM curve25519_of_edwards25519; GSYM edwards25519_of_curve25519;
GSYM curve25519_group; GSYM edwards25519_group;
MCURVE_OF_ECURVE_25519] THEN
DISCH_THEN MATCH_MP_TAC THEN REWRITE_TAC[EDWARD_NONSINGULAR_25519] THEN
REWRITE_TAC[FIELD_INTEGER_MOD_RING; PRIME_P25519] THEN
REWRITE_TAC[INTEGER_MOD_RING_CHAR; GSYM INT_OF_NUM_EQ] THEN
REWRITE_TAC[IN_INTEGER_MOD_RING_CARRIER; INTEGER_MOD_RING] THEN
REWRITE_TAC[p_25519; e_25519; d_25519; c_25519] THEN
CONV_TAC INT_REDUCE_CONV);;
let ISOMORPHIC_GROUPS_EDWARDS25519_CURVE25519 = prove
(`edwards25519_group isomorphic_group curve25519_group`,
MP_TAC GROUP_ISOMORPHISMS_EDWARDS25519_CURVE25519 THEN
MESON_TAC[isomorphic_group; GROUP_ISOMORPHISMS_IMP_ISOMORPHISM]);;
let ISOMORPHIC_GROUPS_CURVE25519_EDWARDS25519 = prove
(`curve25519_group isomorphic_group edwards25519_group`,
ONCE_REWRITE_TAC[ISOMORPHIC_GROUP_SYM] THEN
REWRITE_TAC[ISOMORPHIC_GROUPS_EDWARDS25519_CURVE25519]);;
let CURVE25519_OF_EDWARDS25519_E25519 = prove
(`curve25519_of_edwards25519 E_25519 = C_25519`,
REWRITE_TAC[curve25519_of_edwards25519; montgomery_of_edwards;
C_25519; E_25519; PAIR_EQ; p_25519; e_25519; d_25519; c_25519] THEN
CONV_TAC(DEPTH_CONV INTEGER_MOD_RING_RED_CONV) THEN
CONV_TAC INT_REDUCE_CONV);;
let CURVE25519_OF_EDWARDS25519_EE25519 = prove
(`curve25519_of_edwards25519 EE_25519 = CC_25519`,
REWRITE_TAC[curve25519_of_edwards25519; montgomery_of_edwards;
CC_25519; EE_25519; PAIR_EQ; p_25519; e_25519; d_25519; c_25519] THEN
CONV_TAC(DEPTH_CONV INTEGER_MOD_RING_RED_CONV) THEN
CONV_TAC INT_REDUCE_CONV);;
let EDWARDS25519_OF_CURVE25519_C25519 = prove
(`edwards25519_of_curve25519 C_25519 = E_25519`,
REWRITE_TAC[edwards25519_of_curve25519; edwards_of_montgomery;
C_25519; E_25519; PAIR_EQ; p_25519; e_25519; d_25519; c_25519] THEN
CONV_TAC(DEPTH_CONV INTEGER_MOD_RING_RED_CONV) THEN
CONV_TAC INT_REDUCE_CONV);;
let EDWARDS25519_OF_CURVE25519_CC25519 = prove
(`edwards25519_of_curve25519 CC_25519 = EE_25519`,
REWRITE_TAC[edwards25519_of_curve25519; edwards_of_montgomery;
CC_25519; EE_25519; PAIR_EQ; p_25519; e_25519; d_25519; c_25519] THEN
CONV_TAC(DEPTH_CONV INTEGER_MOD_RING_RED_CONV) THEN
CONV_TAC INT_REDUCE_CONV);;
(* ------------------------------------------------------------------------- *)
(* Mappings between Montgomery and Weierstrass forms. *)
(* ------------------------------------------------------------------------- *)
let wei25519_of_curve25519 = define
`wei25519_of_curve25519 =
weierstrass_of_montgomery (integer_mod_ring p_25519,&A_25519,&1)`;;
let curve25519_of_wei25519 = define
`curve25519_of_wei25519 =
montgomery_of_weierstrass (integer_mod_ring p_25519,&A_25519,&1)`;;
let WCURVE_OF_MCURVE_25519 = prove
(`wcurve_of_mcurve(integer_mod_ring p_25519,&A_25519,&1) =
(integer_mod_ring p_25519,&a_25519,&b_25519)`,
REWRITE_TAC[wcurve_of_mcurve; p_25519; PAIR_EQ;
a_25519; b_25519; A_25519] THEN
CONV_TAC(DEPTH_CONV INTEGER_MOD_RING_RED_CONV) THEN
REWRITE_TAC[]);;
let GROUP_ISOMORPHISMS_CURVE25519_WEI25519 = prove
(`group_isomorphisms
(curve25519_group,wei25519_group)
(wei25519_of_curve25519,curve25519_of_wei25519)`,
MP_TAC(ISPECL
[`integer_mod_ring p_25519`; `&A_25519:int`; `&1:int`]
GROUP_ISOMORPHISMS_MONTGOMERY_WEIERSTRASS_GROUP) THEN
REWRITE_TAC[GSYM wei25519_of_curve25519; GSYM curve25519_of_wei25519;
GSYM wei25519_group; GSYM curve25519_group;
WCURVE_OF_MCURVE_25519] THEN
DISCH_THEN MATCH_MP_TAC THEN REWRITE_TAC[montgomery_nonsingular] THEN
REWRITE_TAC[FIELD_INTEGER_MOD_RING; PRIME_P25519] THEN
REWRITE_TAC[INTEGER_MOD_RING_CHAR; GSYM INT_OF_NUM_EQ] THEN
REWRITE_TAC[IN_INTEGER_MOD_RING_CARRIER; INTEGER_MOD_RING_CLAUSES] THEN
REWRITE_TAC[p_25519; A_25519] THEN CONV_TAC INT_REDUCE_CONV);;
let ISOMORPHIC_GROUPS_CURVE25519_WEI25519 = prove
(`curve25519_group isomorphic_group wei25519_group`,
MP_TAC GROUP_ISOMORPHISMS_CURVE25519_WEI25519 THEN
MESON_TAC[isomorphic_group; GROUP_ISOMORPHISMS_IMP_ISOMORPHISM]);;
let ISOMORPHIC_GROUPS_WEI25519_CURVE25519 = prove
(`wei25519_group isomorphic_group curve25519_group`,
ONCE_REWRITE_TAC[ISOMORPHIC_GROUP_SYM] THEN
REWRITE_TAC[ISOMORPHIC_GROUPS_CURVE25519_WEI25519]);;
let WEI25519_OF_CURVE25519_C25519 = prove
(`wei25519_of_curve25519 C_25519 = G_25519`,
REWRITE_TAC[wei25519_of_curve25519; weierstrass_of_montgomery;
C_25519; G_25519; PAIR_EQ; p_25519; A_25519] THEN
CONV_TAC(DEPTH_CONV INTEGER_MOD_RING_RED_CONV) THEN
CONV_TAC INT_REDUCE_CONV);;
let WEI25519_OF_CURVE25519_CC25519 = prove
(`wei25519_of_curve25519 CC_25519 = GG_25519`,
REWRITE_TAC[wei25519_of_curve25519; weierstrass_of_montgomery;
CC_25519; GG_25519; PAIR_EQ; p_25519; A_25519] THEN
CONV_TAC(DEPTH_CONV INTEGER_MOD_RING_RED_CONV) THEN
CONV_TAC INT_REDUCE_CONV);;
let CURVE25519_OF_WEI25519_G25519 = prove
(`curve25519_of_wei25519 G_25519 = C_25519`,
REWRITE_TAC[curve25519_of_wei25519; montgomery_of_weierstrass;
C_25519; G_25519; PAIR_EQ; p_25519; A_25519] THEN
CONV_TAC(DEPTH_CONV INTEGER_MOD_RING_RED_CONV) THEN
CONV_TAC INT_REDUCE_CONV);;
let CURVE25519_OF_WEI25519_GG25519 = prove
(`curve25519_of_wei25519 GG_25519 = CC_25519`,
REWRITE_TAC[curve25519_of_wei25519; montgomery_of_weierstrass;
CC_25519; GG_25519; PAIR_EQ; p_25519; A_25519] THEN
CONV_TAC(DEPTH_CONV INTEGER_MOD_RING_RED_CONV) THEN
CONV_TAC INT_REDUCE_CONV);;
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