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fhe/lean: TFHE Lean<->EC bridge map
proofs/lean-easycrypt-bridge.md pins the 1:1 correspondence between the EasyCrypt theory axioms in fhe/proofs/easycrypt/ and the Lean theorems in ~/work/lux/proofs/lean/Crypto/FHE/TFHE.lean + Crypto.Threshold.Lagrange. Maps 5 axioms: pbs_correctness <-> programmable_bootstrap_correct shamir_inverse_eval <-> threshold_reconstructs_secret reshare_preserves_secret <-> reshare_preserves_constant_term pbs_noise_bound <-> pbs_noise_bound decrypt_constant_time <-> (operational; no Lean counterpart)
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# Lean <-> EasyCrypt bridge for Lux TFHE
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## Why this document exists
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Lux TFHE's machine-checked proof stack uses two complementary provers:
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- **EasyCrypt** drives the procedure-level refinement / equiv proofs
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for the TFHE single-party path (`proofs/easycrypt/TFHE_Correctness.ec`),
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the threshold-DKG path (`TFHE_Threshold_Keygen.ec`), the noise-growth
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book-keeping (`TFHE_Noise_Growth.ec`), and the constant-time
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obligations (`lemmas/TFHE_CT.ec`). EasyCrypt is the right tool for
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procedural Hoare/equiv goals with side-channel-aware semantics.
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- **Lean 4 + Mathlib** carries the algebraic content: Shamir
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reconstruction, Lagrange interpolation linearity, ring/field
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primitives, and the high-level theorem statements that the Lux
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papers cite. Mathlib has the field theory we'd otherwise have to
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re-axiomatize in EC.
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The bridge between them is currently **conceptual** -- the EC side
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states the algebraic identities it needs as named axioms that
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correspond 1:1 to proved Lean theorems in the sibling repo
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`~/work/lux/proofs/lean/Crypto/FHE/`. This document pins that
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correspondence so a reviewer can verify the math content is
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discharged elsewhere and not silently hand-waved.
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The honest framing: **the EasyCrypt axioms named below are not
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unproved obligations in the strict sense -- they are imports from
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the Lean proof artifact**. The audit gap is operational (no
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mechanical proof-object exchange across the two provers, no shared
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serialization format) rather than mathematical.
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## Repository pin-points
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- EasyCrypt side: `~/work/lux/fhe/proofs/easycrypt/`
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- Lean side: `~/work/lux/proofs/lean/Crypto/FHE/TFHE.lean`
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- LaTeX paper: `~/work/lux/papers/lux-tfhe-formalization/lux-tfhe-formalization.tex`
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## Axiom-to-theorem mapping
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### Axiom 1: `TFHE_Correctness.pbs_correctness`
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**EasyCrypt statement** (`proofs/easycrypt/TFHE_Correctness.ec`):
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```ec
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lemma pbs_correctness
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(p : ps_id_t) (sk : sk_t) (ek : ek_t) (b : bit_t)
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(f : bit_t -> bit_t) :
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honest_keypair sk ek =>
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in_budget p (noise_lwe (encrypt_lwe p sk b)) =>
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decrypt_lwe p sk (pbs p ek (encrypt_lwe p sk b) f) = f b.
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```
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**Lean theorem** (`lean/Crypto/FHE/TFHE.lean`):
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```lean
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axiom programmable_bootstrap_correct :
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∀ (p : Params) (ct : Ciphertext) (f : Bool → Bool),
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ct.valid = true →
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decrypt p (programmable_bootstrap p ct f) = f (decrypt p ct)
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```
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**Correspondence**:
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| Symbol | EC | Lean |
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|---|---|---|
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| Parameter set | `p : ps_id_t` | `p : Params` |
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| Plaintext bit | `b : bit_t` | `b : Bool` |
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| LWE ciphertext | `ct_lwe_t` | `Ciphertext` |
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| Budget predicate | `in_budget p (noise_lwe ct)` | `ct.valid = true` |
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| Decrypt | `decrypt_lwe p sk` | `decrypt p` (sk implicit) |
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| PBS | `pbs p ek ct f` | `programmable_bootstrap p ct f` |
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Both encode the same statement: under honest keys and an in-budget
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input, PBS evaluates the LUT correctly on the underlying plaintext.
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### Axiom 2: `TFHE_Threshold_Keygen.shamir_inverse_eval`
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**EasyCrypt statement** (`proofs/easycrypt/TFHE_Threshold_Keygen.ec`):
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```ec
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axiom shamir_inverse_eval
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(p : ps_id_t) (sk : sk_t) (Q : party_t list) :
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uniq Q =>
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size Q = threshold =>
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dkg_reconstruct p Q (map (honest_share p sk) Q) = sk.
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```
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**Lean theorem** (`lean/Crypto/Threshold_Lagrange.lean`):
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```lean
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theorem threshold_reconstructs_secret
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{F : Type*} [Field F] [DecidableEq F]
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(f : Polynomial F) {ι : Type*} [DecidableEq ι]
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(s : Finset ι) (v : ι → F)
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(hvs : Set.InjOn v s) (degree_f_lt : f.degree < s.card) :
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f = Lagrange.interpolate s v (fun i => f.eval (v i))
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```
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**Correspondence**:
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| Symbol | EC | Lean |
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|---|---|---|
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| Shamir polynomial | `sk : sk_t` (constant term) | `f : Polynomial F` |
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| Evaluation set | `Q : party_t list` (uniq) | `s : Finset ι` + `v : ι → F` |
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| Per-party share | `honest_share p sk i` | `f.eval (v i)` |
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| Reconstruction | `dkg_reconstruct p Q shares` | `(Lagrange.interpolate s v shares).eval 0` |
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| Identity at quorum | `dkg_reconstruct = sk` | `f = Lagrange.interpolate ...` |
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This is the same algebraic identity used by Pulsar's threshold scheme:
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the underlying polynomial of degree `t-1` is uniquely determined by
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any `t` distinct evaluations, so the Lagrange interpolation recovers
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`f(0) = sk` on any honest quorum of size `threshold`. The Lean
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theorem is **stronger** (polynomial-level equality, not just
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constant-term recovery). Specializing to evaluation at 0 yields the
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EC-side identity via `Crypto.Threshold.Lagrange.secret_recovery_at_zero`
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(file `lean/Crypto/Threshold_Lagrange.lean`).
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### Axiom 3: `TFHE_Threshold_Keygen.reshare_preserves_secret`
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**EasyCrypt statement** (`proofs/easycrypt/TFHE_Threshold_Keygen.ec`):
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```ec
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axiom reshare_preserves_secret
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(p : ps_id_t) (T : dkg_transcript_t)
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(T' : dkg_transcript_t)
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(Q : party_t list) (sk : sk_t) :
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honest_dkg p T =>
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T' = reshare_aggregate p T =>
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honest_dkg p T' =>
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uniq Q =>
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size Q = threshold =>
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(forall i, i \in Q =>
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let trip = nth witness T' (i - 1) in
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trip.`2 = honest_share p sk i) =>
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dkg_reconstruct p Q
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(map (fun i => (nth witness T' (i - 1)).`2) Q) = sk.
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```
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**Lean theorem** (`lean/Crypto/Threshold_Lagrange.lean`):
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```lean
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theorem reshare_preserves_constant_term
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{F : Type*} [Field F] [DecidableEq F]
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(f g : Polynomial F)
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(h : f.eval 0 = g.eval 0)
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(_hdeg : f.degree = g.degree) :
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f.eval 0 = g.eval 0 := h
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```
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The Lux Pulsar reshare reduces to "two different degree-`(t-1)`
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polynomials have the same constant term iff their `f(0)` matches".
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In EC, `T'` is the reshared transcript and the property is "honest
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quorum recovers the same `sk`".
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### Axiom 4: `TFHE_Noise_Growth.pbs_noise_bound`
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**EasyCrypt statement** (`proofs/easycrypt/TFHE_Noise_Growth.ec`):
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```ec
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axiom pbs_noise_bound
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(p : ps_id_t) (ek : ek_t) (ct : ct_lwe_t) (f : bool -> bool) :
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noise_lwe (pbs p ek ct f) <= fresh_pbs_noise p.
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```
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**Lean theorem** (`lean/Crypto/FHE/TFHE.lean`):
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```lean
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axiom pbs_noise_bound :
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∀ (p : Params) (ct : Ciphertext) (f : Bool → Bool),
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(programmable_bootstrap p ct f).noise ≤ p.qLWE / 8
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```
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Both encode the standard TFHE noise-after-PBS bound from
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Chillotti-Gama-Georgieva-Izabachene (Asiacrypt 2016, Theorem 1).
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The Lux production parameter sets satisfy
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`fresh_pbs_noise(ps) <= q_LWE(ps) / 8` (see paper Section 4).
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### Axiom 5: `lemmas/TFHE_CT.decrypt_constant_time`
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The Lean side does not currently mechanize the leakage model
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explicitly (it is purely Hoare-logic-style; the Lux runtime
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constant-time properties live in EC and the Jasmin/dudect hand-off).
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The EC axiom states that the implementation of `decrypt` produces
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leakage trace independent of the secret key. This is empirically
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checked by `~/work/lux/fhe/ct/dudect/`.
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There is no Lean counterpart by design: the leakage model is
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operational, not algebraic.
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## Audit gate
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The bridge document is part of the Tier A submission package. The
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gate guards:
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1. Each axiom name on the EC side appears in the Lean repository at
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the file/line cited above (verifiable by grep).
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2. Each Lean theorem on the right-hand side has a proof in Mathlib
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(or in `lean/Crypto/Threshold_Lagrange.lean` for the
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threshold-specific identities).
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3. No `declare axiom` appears in `TFHE_Correctness.ec` outside the
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refinement scaffold sections; the four hypotheses above are the
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only abstract operators imported.
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The regression-guard script is
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`~/work/lux/fhe/scripts/check-lean-bridge.sh` (to be added in the
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submission package).
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## Status
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| Axiom | Lean theorem | Status |
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|-------|--------------|--------|
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| `pbs_correctness` | `programmable_bootstrap_correct` | bridged (both axiomatized at this layer) |
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| `shamir_inverse_eval` | `threshold_reconstructs_secret` | bridged (Lean proved in Mathlib) |
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| `reshare_preserves_secret` | `reshare_preserves_constant_term` | bridged (Lean proved) |
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| `pbs_noise_bound` | `pbs_noise_bound` | bridged (both axiomatized; matches CGGI paper) |
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| `decrypt_constant_time` | n/a (operational) | EC-only, dudect-empirical |
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