320 lines
10 KiB
Plaintext
320 lines
10 KiB
Plaintext
(*
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Olm + Megolm protocol; proving secrecy, authentication, PCS, and PFS
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Author: [redacted]
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*)
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free m1: bitstring [private].
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free m2: bitstring [private].
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set selFun = Nounifset.
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set redundancyElim = best.
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set simpEqAll = false.
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set redundantHypElim = true.
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set simplifyProcess = true.
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set stopTerm = false.
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free c: channel.
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free a: channel. (* channel for the attacker *)
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free p: channel [private]. (* For the distribution of public keys with integrity and authenticity - verification happens out of band. This is a standard assumption. *)
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(* Symmetric key encryption *)
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type key.
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free mra0: key [private].
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fun senc(key, bitstring): bitstring.
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reduc forall m: bitstring, k: key; sdec(k, senc(k,m)) = m.
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(* Asymmetric key encryption *)
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type skey.
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type pkey.
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fun rb(pkey): bitstring.
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fun pk(skey): pkey.
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fun aenc(bitstring, pkey): bitstring.
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reduc forall m: bitstring, sk: skey; adec(aenc(m,pk(sk)),sk) = m.
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(* Digital signatures *)
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fun sign(skey, bitstring): bitstring.
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fun okay():bitstring.
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reduc forall m: bitstring, sk: skey; checksign(pk(sk), m, sign(sk, m)) = okay.
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(* MACs *)
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fun mac(key, bitstring): bitstring.
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reduc forall k: key, m: bitstring; checkmac(k, m, mac(k, m)) = okay.
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(* Diffie-Hellman *)
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(* DH -> Public^Private *)
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fun dh(pkey, skey): bitstring.
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equation forall a: skey, b: skey; dh(pk(a), b) = dh(pk(b), a). (* symmetry of DH *)
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(* the concat functions *)
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fun khash(key): key.
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fun hkdf(bitstring): key [data].
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fun concat0(key): bitstring [data, typeConverter].
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fun concat1(bitstring, pkey, pkey): bitstring [data].
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reduc forall m1: bitstring, k1: pkey, k2: pkey; split1(concat1(m1, k1, k2)) = (m1, k1, k2).
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fun concat2(bitstring, pkey): bitstring [data].
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reduc forall m1: bitstring, k1: pkey; split2(concat2(m1, k1)) = (m1, k1).
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fun concat3(bitstring, bitstring): bitstring [data].
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reduc forall m1: bitstring, m2: bitstring; split3(concat3(m1, m2)) = (m1, m2).
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fun concat4(bitstring, bitstring, bitstring): bitstring [data].
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reduc forall m1: bitstring, m2: bitstring, m3:bitstring; split4(concat4(m1, m2, m3)) = (m1, m2, m3).
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equation forall k: key, a: skey, b: skey; hkdf(concat3(concat0(khash(k)), dh(pk(a), b))) = hkdf(concat3(concat0(khash(k)), dh(pk(b), a))).
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(* the concats *)
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(* events *)
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event sendOE1(pkey, key, pkey, pkey).
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event recvOE1(pkey, key, pkey, pkey).
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event sendOE2(pkey, key, pkey, pkey).
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event recvOE2(pkey, key, pkey, pkey).
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event sendME1(bitstring, key, pkey).
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event recvME1(bitstring, key, pkey).
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event sendME2(bitstring, key, pkey).
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event recvME2(bitstring, key, pkey).
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event compromiseOSKA(skey).
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event compromiseOSKB(skey).
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event start().
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free tag_oe1: bitstring [private].
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free tag_oe2: bitstring [private].
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free tag_me1: bitstring [private].
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free tag_me2: bitstring [private].
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free tag_b_eph: bitstring [private].
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let PeerA(OSK_A: skey, OPK_A: pkey, OPK_B: pkey, OLM_KEYS_STR:bitstring, OLM_ROOT_STR: bitstring, MSK_A: skey, MPK_A: pkey) =
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phase 1;
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new ao: skey;
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let gao = pk(ao) in
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(* generate amaster and enc msg (PHASE 1) *)
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(* in(c, gbo: pkey); *)
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in(c, (gbo: pkey, gbo_sig: bitstring));
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if checksign(OPK_B, rb(gbo), gbo_sig) = okay then
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(* =================== PHASE 2: DERIVE OLM MASTER + MEGOLM SESSION A -> B =================== *)
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let amaster = hkdf(concat4(dh(OPK_B, OSK_A), dh(gbo, OSK_A), dh(OPK_B, ao))) in
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let ra1 = hkdf(concat0(amaster)) in
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let ca1 = hkdf(concat0(amaster)) in
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new ta1: skey;
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let gta1 = pk(ta1) in
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let ak1 = khash(ca1) in
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let ak1_auth = hkdf(concat0(ak1)) in
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let ak1_enc = hkdf(concat0(ak1)) in
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new mra0: key; (* mra0 := megolm ratchet *)
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let x1 = senc(ak1_enc, (MPK_A, mra0)) in
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let x1_mac = mac(ak1_auth, concat1(x1, gao, gta1)) in
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event sendOE1(MPK_A, mra0, gao, gta1);
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out(c, (x1, x1_mac, gao, gta1));
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(* =================== PHASE 3: MEGOLM SESH B -> A =================== *)
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(* second stage: now, decrypt the received message from bob *)
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in(c, (x2: bitstring, x2_mac: bitstring, gtb2: pkey));
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let ca2 = hkdf(concat3(concat0(khash(ra1)), dh(gtb2, ta1))) in
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let ak2 = khash(ca2) in
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let ak2_auth = hkdf(concat0(ak2)) in
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if checkmac(ak2_auth, concat2(x2, gtb2), x2_mac) = okay then (
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let mak2_enc = hkdf(concat0(ak2)) in
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let (MPK_B: pkey, mrb0: key) = sdec(mak2_enc, x2) in
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event recvOE2(MPK_B, mrb0, gao, gta1);
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(* =================== PHASE 4: MEGOLM MSG A -> B =================== *)
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let mra1 = khash(mra0) in
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let mak1_auth = hkdf(concat0(mra1)) in
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let mak1_enc = hkdf(concat0(mra1)) in
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let mx1 = senc(mak1_enc, m1) in
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let mx1_mac = mac(mak1_auth, mx1) in
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let mx1_sig = sign(MSK_A, concat3(mx1, mx1_mac)) in
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event sendME1(m1, mra1, MPK_A);
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out(c, (mx1, mx1_mac, mx1_sig));
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(* =================== PHASE 5: MEGOLM MSG B -> A =================== *)
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in(c, (mx2: bitstring, mx2_mac: bitstring, mx2_sig: bitstring));
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let mrb1 = khash(mrb0) in
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let mbk1_auth = hkdf(concat0(mrb1)) in
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let mbk1_enc = hkdf(concat0(mrb1)) in
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if checksign(MPK_B, concat3(mx2, mx2_mac), mx2_sig) = okay then
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if checkmac(mbk1_auth, mx2, mx2_mac) = okay then
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let m2 = sdec(mbk1_enc, mx2) in
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event recvME2(m2, mrb1, MPK_B);
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phase 3; (* =================== PHASE 6: PCS =================== *)
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event compromiseOSKA(OSK_A);
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out(c, OSK_A)
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).
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let PeerB(OSK_B: skey, OPK_B: pkey, OPK_A: pkey, OLM_KEYS_STR:bitstring, OLM_ROOT_STR: bitstring, MSK_B: skey, MPK_B: pkey) =
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new bo: skey;
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let gbo = pk(bo) in
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let gbo_sig = sign(OSK_B, rb(gbo)) in
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out(c, (gbo, gbo_sig));
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phase 1;
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(* =================== PHASE 2: DERIVE OLM MASTER + MEGOLM SESSION A -> B =================== *)
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(* first stage: derive bmaster, verfiy a's msgs, decrypt prekey message, reply *)
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in(c, (x1: bitstring, x1_mac: bitstring, gao: pkey, gta1: pkey));
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let bmaster = hkdf(concat4(dh(OPK_A, OSK_B), dh(OPK_A, bo), dh(gao, OSK_B))) in
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let rb1 = hkdf(concat0(bmaster)) in
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let cb1 = hkdf(concat0(bmaster)) in
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let bk1 = khash(cb1) in
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let bk1_auth = hkdf(concat0(bk1)) in
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let bk1_enc = hkdf(concat0(bk1)) in
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if checkmac(bk1_auth, concat1(x1, gao, gta1), x1_mac) = okay then
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(
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let (MPK_A: pkey, mra0: key) = sdec(bk1_enc, x1) in
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event recvOE1(MPK_A, mra0, gao, gta1);
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(* =================== PHASE 3: MEGOLM SESH B -> A =================== *)
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new tb2: skey;
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let gtb2 = pk(tb2) in
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new tb2: skey;
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let gtb2 = pk(tb2) in
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let rb2 = hkdf(concat3(concat0(khash(rb1)), dh(gta1, tb2))) in
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let cb2 = hkdf(concat3(concat0(khash(rb1)), dh(gta1, tb2))) in
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let bk2 = khash(cb2) in
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let bk2_auth = hkdf(concat0(bk2)) in
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let bk2_enc = hkdf(concat0(bk2)) in
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new mrb0: key; (* mrb0 := megolm ratchet *)
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let x2 = senc(bk2_enc, (MPK_B, mrb0)) in
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let x2_mac = mac(bk2_auth, concat2(x2, gtb2)) in
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event sendOE2(MPK_B, mrb0, gao, gta1);
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out(c, (x2, x2_mac, gtb2));
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(* =================== PHASE 4: MEGOLM MSG A -> B =================== *)
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in(c, (mx1: bitstring, mx1_mac: bitstring, mx1_sig: bitstring));
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let mra1 = khash(mra0) in
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let mak1_auth = hkdf(concat0(mra1)) in
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let mak1_enc = hkdf(concat0(mra1)) in
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if checksign(MPK_A, concat3(mx1, mx1_mac), mx1_sig) = okay then (
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if checkmac(mak1_auth, mx1, mx1_mac) = okay then (
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let m1 = sdec(mak1_enc, mx1) in
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event recvME1(m1, mra1, MPK_A);
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phase 3;
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event compromiseOSKB(OSK_B);
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out(c, OSK_B);
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(* =================== PHASE 5: MEGOLM MSG B -> A =================== *)
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let mrb1 = khash(mrb0) in
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let mbk1_auth = hkdf(concat0(mrb1)) in
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let mbk1_enc = hkdf(concat0(mrb1)) in
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let mx2 = senc(mbk1_enc, m2) in
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let mx2_mac = mac(mbk1_auth, mx2) in
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let mx2_sig = sign(MSK_B, concat3(mx2, mx2_mac)) in
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event sendME2(m2, mrb1, MPK_B);
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out(c, (mx2, mx2_mac, mx2_sig))
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(* =================== PHASE 6: PCS =================== *)
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))).
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query event(start()). (* reachable from all possible executions *)
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(* pre-compromise security, aka forward secrecy. the only way
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m1 can be compromised is if alice's sk is compromised
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NOTE: if signed, this is trivially true since m1 is never compromised
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*)
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query sk: skey; attacker(m1) ==> event(compromiseOSKB(sk)).
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(* post-compromise security. even if the secret key is compromised, message two remains secret
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*)
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query sk1: skey, sk2: skey; (event(compromiseOSKB(sk1)) && event(compromiseOSKA(sk2)) && attacker(m1)) ==> false.
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(* auth *)
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query k1: pkey, rk: key, k2: pkey, k3: pkey; inj-event(recvOE1(k1, rk, k2, k3)) ==> inj-event(sendOE1(k1, rk, k2, k3)).
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query k1: pkey, rk: key, k2: pkey, k3: pkey; inj-event(recvOE2(k1, rk, k2, k3)) ==> inj-event(sendOE2(k1, rk, k2, k3)).
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query m: bitstring, rk: key, k1: pkey; inj-event(recvME1(m, rk, k1)) ==> inj-event(sendME1(m, rk, k1)).
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query m: bitstring, rk: key, k1: pkey; inj-event(recvME2(m, rk, k1)) ==> inj-event(sendME2(m, rk, k1)).
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(* secrecy *)
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query attacker(m1).
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query attacker(m2).
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(* reachability *)
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query k1: pkey, rk: key, k2: pkey, k3: pkey; event(sendOE1(k1,rk,k2,k3)).
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query k1: pkey, rk: key, k2: pkey, k3: pkey; event(recvOE1(k1,rk,k2,k3)).
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query k1: pkey, rk: key, k2: pkey, k3: pkey; event(sendOE2(k1,rk,k2,k3)).
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query k1: pkey, rk: key, k2: pkey, k3: pkey; event(recvOE2(k1,rk,k2,k3)).
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query m: bitstring, rk: key, k1: pkey; event(sendME1(m, rk, k1)).
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query m: bitstring, rk: key, k1: pkey; event(recvME1(m, rk, k1)).
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query m: bitstring, rk: key, k1: pkey; event(sendME2(m, rk, k1)).
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query m: bitstring, rk: key, k1: pkey; event(recvME2(m, rk, k1)).
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process
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new OLM_KEYS_STR:bitstring; out(a, OLM_KEYS_STR);
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new OLM_ROOT_STR:bitstring; out(a, OLM_ROOT_STR);
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new OSK_A: skey; let OPK_A = pk(OSK_A) in
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new OSK_B: skey; let OPK_B = pk(OSK_B) in
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new MSK_A: skey; let MPK_A = pk(MSK_A) in
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new MSK_B: skey; let MPK_B = pk(MSK_B) in
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out(a, OPK_A);
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out(a, OPK_B);
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(*
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out(a, MPK_A);
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out(a, MPK_B);
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*)
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event start();
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( (!PeerA(OSK_A, OPK_A, OPK_B, OLM_KEYS_STR, OLM_ROOT_STR, MSK_A, MPK_A)) |
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(!PeerB(OSK_B, OPK_B, OPK_A, OLM_KEYS_STR, OLM_ROOT_STR, MSK_B, MPK_B)))
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