CryptoVerif and Tamarin models, minor doc updates
Signed-off-by: Kamal Tufekcic <kamal@lo.sh>
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Kamal Tufekcic
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266
tamarin/LO_KEX.spthy
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266
tamarin/LO_KEX.spthy
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theory LO_KEX
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begin
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/*
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* LO-KEX: Session Establishment (Theorems 1, 2a, 2b)
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*
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* SCOPE: This model covers the OPK-PRESENT case only. All three KEM
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* encapsulations (IK, SPK, OPK) are unconditional.
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*
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* The code (kex/mod.rs) supports OPK-absent sessions where IKM = ss_IK ‖ ss_SPK
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* (2 shared secrets instead of 3). For those sessions, corruption of IK + SPK
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* alone suffices to derive the session key — the Theorem 1 lemmas' requirement
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* of all-three-corrupted does not apply. The OPK-absent case has strictly
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* weaker security (2-key threshold vs 3-key) and is not modeled here.
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*
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* This is NOT a soundness issue: the proofs are correct for the OPK-present
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* case they model. The OPK-absent case would need separate rules with 2-KEM
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* IKM and adjusted secrecy lemmas requiring only IK+SPK corruption.
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*/
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/*
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* 1. CRYPTOGRAPHIC PRIMITIVES
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*/
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builtins: asymmetric-encryption, signing, hashing
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// AEAD Abstraction
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functions: aead_enc/4, aead_dec/4, aead_verify/4, accept/0
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equations:
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aead_dec(k, n, aad, aead_enc(k, n, m, aad)) = m,
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aead_verify(k, n, aad, aead_enc(k, n, m, aad)) = accept()
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// Constant-time equality check helper
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restriction Eq:
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"All x y #i. Eq(x,y) @i ==> x = y"
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/*
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* 2. ADVERSARY MODEL & PKI (§8.2)
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*/
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rule Generate_IK:
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[ Fr(~sk) ] --[ LtkGen($P, ~sk) ]->[ !Ltk($P, ~sk), !Pk($P, pk(~sk)), Out(pk(~sk)) ]
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rule Corrupt_IK:
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[ !Ltk($P, sk) ] --[ CorruptIK($P) ]-> [ Out(sk) ]
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rule Corrupt_SPK:
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[ !SPK_Secret($P, id, sk) ] --[ CorruptSPK($P, id) ]-> [ Out(sk) ]
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rule Corrupt_OPK:[ !OPK_Secret($P, id, sk) ] --[ CorruptOPK($P, id) ]-> [ Out(sk) ]
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rule Corrupt_RNG_KEX:[ !RNG_KEX($P, r_IK, r_SPK, r_OPK) ] --[ CorruptRNG_KEX($P) ]->[ Out(<r_IK, r_SPK, r_OPK>) ]
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/*
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* 3. LO-KEX PROTOCOL (§4)
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*/
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// §4.1: Pre-Key Bundle Generation (Bob)
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rule LO_KEX_Publish_Bundle:
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let
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pk_SPK = pk(~sk_SPK)
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pk_OPK = pk(~sk_OPK)
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pk_IK = pk(~sk_IK)
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sig_SPK = sign(<'lo-spk-sig-v1', pk_SPK>, ~sk_IK)
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bundle = <'lo-crypto-v1', pk_IK, pk_SPK, ~id_SPK, sig_SPK, pk_OPK, ~id_OPK>
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in[ !Ltk($B, ~sk_IK), Fr(~sk_SPK), Fr(~sk_OPK), Fr(~id_SPK), Fr(~id_OPK) ]
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--[ PublishBundle($B, ~id_SPK, ~id_OPK) ]->[
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!SPK_Secret($B, ~id_SPK, ~sk_SPK), !SPK_Pk($B, ~id_SPK, pk_SPK),
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!OPK_Secret($B, ~id_OPK, ~sk_OPK), !OPK_Pk($B, ~id_OPK, pk_OPK),
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// OPK_Available is a LINEAR fact per §4.4 Step 6
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OPK_Available($B, ~id_OPK),
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Out(bundle)
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]
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// §4.3: Session Initiation (Alice)
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rule LO_KEX_Alice_Init:
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let
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pk_IK_A = pk(~sk_IK_A)
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// KEM using standard PKE (ss = ~r, c = aenc(~r, pk))
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c_IK = aenc(~r_IK, pk_IK_B)
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ss_IK = ~r_IK
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c_SPK = aenc(~r_SPK, pk_SPK_B)
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ss_SPK = ~r_SPK
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c_OPK = aenc(~r_OPK, pk_OPK_B)
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ss_OPK = ~r_OPK
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// HKDF derivation using built-in hashing
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ikm = <ss_IK, ss_SPK, ss_OPK>
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pk_EK = pk(~sk_EK)
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info = <'lo-kex-v1', 'lo-crypto-v1', pk_IK_A, pk_IK_B, pk_EK>
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rk = h(<ikm, info, 'rk'>)
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ek = h(<ikm, info, 'ek'>)
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fp_IK_A = h(pk_IK_A)
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fp_IK_B = h(pk_IK_B)
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SI = <'lo-crypto-v1', fp_IK_A, fp_IK_B, pk_EK, c_IK, c_SPK, id_SPK, c_OPK, id_OPK>
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sig_SI = sign(<'lo-kex-init-sig-v1', SI>, ~sk_IK_A)
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mk0 = h(<ek, '0'>)
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aad0 = <'lo-dm-v1', fp_IK_A, fp_IK_B, SI>
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c0 = aead_enc(mk0, ~n0, ~m0, aad0)
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in[
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!Ltk($A, ~sk_IK_A),
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!Pk($B, pk_IK_B),
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!SPK_Pk($B, id_SPK, pk_SPK_B),
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!OPK_Pk($B, id_OPK, pk_OPK_B),
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In(sig_SPK),
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Fr(~sk_EK), Fr(~r_IK), Fr(~r_SPK), Fr(~r_OPK), Fr(~n0), Fr(~m0)
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]
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--[
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Eq(verify(sig_SPK, <'lo-spk-sig-v1', pk_SPK_B>, pk_IK_B), true),
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Init_A($A, $B, rk, ek, id_SPK, id_OPK)
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]->[
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Out(<SI, sig_SI, ~n0, c0>),
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!RNG_KEX($A, ~r_IK, ~r_SPK, ~r_OPK)
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]
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// §4.4: Session Reception (Bob)
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rule LO_KEX_Bob_Recv:
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let
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pk_IK_B = pk(~sk_IK_B)
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pk_SPK_B = pk(~sk_SPK_B)
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pk_OPK_B = pk(~sk_OPK_B)
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fp_IK_A = h(pk_IK_A)
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fp_IK_B = h(pk_IK_B)
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SI = <'lo-crypto-v1', fp_IK_A, fp_IK_B, pk_EK, c_IK, c_SPK, id_SPK, c_OPK, id_OPK>
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// Decapsulate
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ss_IK = adec(c_IK, ~sk_IK_B)
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ss_SPK = adec(c_SPK, ~sk_SPK_B)
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ss_OPK = adec(c_OPK, ~sk_OPK_B)
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// HKDF derivation using built-in hashing
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ikm = <ss_IK, ss_SPK, ss_OPK>
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info = <'lo-kex-v1', 'lo-crypto-v1', pk_IK_A, pk_IK_B, pk_EK>
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rk = h(<ikm, info, 'rk'>)
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ek = h(<ikm, info, 'ek'>)
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mk0 = h(<ek, '0'>)
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aad0 = <'lo-dm-v1', fp_IK_A, fp_IK_B, SI>
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in[
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!Ltk($B, ~sk_IK_B),
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!Pk($A, pk_IK_A),
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!SPK_Secret($B, id_SPK, ~sk_SPK_B),
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!OPK_Secret($B, id_OPK, ~sk_OPK_B),
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OPK_Available($B, id_OPK),
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In(<SI, sig_SI, n0, c0>)
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]
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--[
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Eq(verify(sig_SI, <'lo-kex-init-sig-v1', SI>, pk_IK_A), true),
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Eq(aead_verify(mk0, n0, aad0, c0), accept()),
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Init_B($B, $A, rk, ek, id_SPK, id_OPK)
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]->
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[ ]
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/*
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* 4. LEMMAS (§8.6)
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*/
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// Sanity check
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lemma KEX_Exists:
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exists-trace
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"Ex A B rk ek id_S id_O #i #j.
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Init_A(A, B, rk, ek, id_S, id_O) @ i &
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Init_B(B, A, rk, ek, id_S, id_O) @ j"
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// Theorem 1 (A's perspective): Session Key Secrecy
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// The root key is secret UNLESS Adv corrupts Alice's RNG, OR ALL of Bob's keys.
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lemma Theorem1_Session_Key_Secrecy_A:
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"All A B rk ek id_S id_O #i.
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Init_A(A, B, rk, ek, id_S, id_O) @ i
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==>
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not (Ex #j. K(rk) @ j)
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| (Ex #j. CorruptRNG_KEX(A) @ j)
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| ( (Ex #j1. CorruptIK(B) @ j1) & (Ex #j2. CorruptSPK(B, id_S) @ j2) & (Ex #j3. CorruptOPK(B, id_O) @ j3) )
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"
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// Theorem 1 (B's perspective): Session Key Secrecy
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// If Bob accepts, the key is secret UNLESS Adv impersonated Alice (CorruptIK(A))
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// OR Adv corrupted ALL of Bob's keys.
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lemma Theorem1_Session_Key_Secrecy_B:
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"All B A rk ek id_S id_O #i.
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Init_B(B, A, rk, ek, id_S, id_O) @ i
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==>
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not (Ex #j. K(rk) @ j)
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| (Ex #j. CorruptIK(A) @ j)
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| (Ex #j. CorruptRNG_KEX(A) @ j) // <--- THE MISSING PREDICATE!
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| ( (Ex #j1. CorruptIK(B) @ j1) & (Ex #j2. CorruptSPK(B, id_S) @ j2) & (Ex #j3. CorruptOPK(B, id_O) @ j3) )
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"
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// Theorem 1 (A's perspective): Epoch Key Secrecy
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// Same structure as rk — ek is derived from the same IKM via a distinct KDF label.
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lemma Theorem1_EK_Secrecy_A:
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"All A B rk ek id_S id_O #i.
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Init_A(A, B, rk, ek, id_S, id_O) @ i
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==>
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not (Ex #j. K(ek) @ j)
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| (Ex #j. CorruptRNG_KEX(A) @ j)
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| ( (Ex #j1. CorruptIK(B) @ j1) & (Ex #j2. CorruptSPK(B, id_S) @ j2) & (Ex #j3. CorruptOPK(B, id_O) @ j3) )
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"
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// Theorem 1 (B's perspective): Epoch Key Secrecy
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lemma Theorem1_EK_Secrecy_B:
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"All B A rk ek id_S id_O #i.
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Init_B(B, A, rk, ek, id_S, id_O) @ i
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==>
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not (Ex #j. K(ek) @ j)
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| (Ex #j. CorruptIK(A) @ j)
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| (Ex #j. CorruptRNG_KEX(A) @ j)
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| ( (Ex #j1. CorruptIK(B) @ j1) & (Ex #j2. CorruptSPK(B, id_S) @ j2) & (Ex #j3. CorruptOPK(B, id_O) @ j3) )
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"
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// Theorem 2(a): Recipient Binding
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// If Alice initiates with B and Bob accepts the same session, then Alice
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// and Bob agree on each other's identity. This is explicit binding:
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// fp_IK_B = H(pk_IK_B) is embedded in SI, which Alice signs as σ_SI,
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// and Bob verifies σ_SI against pk_IK_A. If both Init_A and Init_B fire
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// for the same (rk, ek), the session names both parties correctly.
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// (No corruption exclusion needed — this is a structural property of the
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// signature + fingerprint binding, not a secrecy claim.)
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lemma Theorem2a_Recipient_Binding:
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"All A B rk ek id_S id_O #i #j.
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Init_A(A, B, rk, ek, id_S, id_O) @i &
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Init_B(B, A, rk, ek, id_S, id_O) @j
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==>
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i < j
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"
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// Theorem 2(b): Initiator Authentication
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// If Bob verifies the session init, Alice actually generated it,
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// UNLESS the adversary corrupted Alice's Identity Key.
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lemma Theorem2b_Initiator_Authentication:
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"All B A rk ek id_S id_O #i.
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Init_B(B, A, rk, ek, id_S, id_O) @ i
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==>
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(Ex #j. Init_A(A, B, rk, ek, id_S, id_O) @ j & j < i)
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| (Ex #j. CorruptIK(A) @ j)
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"
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// §4.4 Step 6: OPK single-use — the linear OPK_Available fact ensures a given
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// OPK id is consumed by at most one session reception.
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lemma OPK_Single_Use:
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"All B A1 A2 rk1 rk2 ek1 ek2 id_S1 id_S2 id_O #i #j.
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Init_B(B, A1, rk1, ek1, id_S1, id_O) @ i &
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Init_B(B, A2, rk2, ek2, id_S2, id_O) @ j
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==>
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#i = #j
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"
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// Key uniqueness: distinct Init_A events (with fresh KEM randomness) produce
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// distinct root keys. Two sessions sharing the same rk must be the same event.
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lemma Key_Uniqueness:
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"All A1 A2 B1 B2 rk ek1 ek2 id_S1 id_S2 id_O1 id_O2 #i #j.
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Init_A(A1, B1, rk, ek1, id_S1, id_O1) @ i &
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Init_A(A2, B2, rk, ek2, id_S2, id_O2) @ j
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==>
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#i = #j
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"
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end
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