Project-declaredLean 4.32.0 · mathlib@81a5d257

Cma to nma advantage bound

FiatShamir.cma_to_nma_advantage_bound

program verification separation logic cryptography reductions

Plain-language statement

CMA-to-NMA reduction for Fiat-Shamir signatures built from a Sigma protocol. The reduction runs the CMA adversary with simulated signing transcripts and a managed random oracle, then appends a single explicit live random-oracle query for the forgery's hash point so that the verification challenge is part of the forkable transcript (the `nmaAdvFromCmaWithF...

Exact Lean statement

theorem cma_to_nma_advantage_bound
    [DecidableEq M] [DecidableEq Commit] [SampleableType Stmt] [SampleableType Wit]
    [Finite Stmt] [Finite Commit] [Finite Resp]
    [Finite Chal] [Inhabited Chal] [SampleableType Chal]
    (simTranscript : Stmt → ProbComp (Commit × Chal × Resp))
    (ζ_zk : ℝ) (hζ_zk : 0 ≤ ζ_zk)
    (hHVZK : σ.HVZK simTranscript ζ_zk)
    (β : ENNReal)
    (hPredSim : σ.simCommitPredictability simTranscript β)
    (adv : SignatureAlg.unforgeableAdv
      (FiatShamir (m := OracleComp (unifSpec + (M × Commit →ₒ Chal))) σ hr M))
    (qS qH : ℕ)
    (hQ : ∀ pk, signHashQueryBound (M := M) (Commit := Commit) (Chal := Chal)
      (S' := Commit × Resp) (oa := adv.main pk) qS qH) :
    ∃ nmaAdv : SignatureAlg.managedRoNmaAdv
        (FiatShamir (m := OracleComp (unifSpec + (M × Commit →ₒ Chal))) σ hr M),
      adv.advantage (runtime M) ≤
        Fork.advantage σ hr M nmaAdv qH +
          ENNReal.ofReal ((qS : ℝ) * ζ_zk) + (qS : ENNReal) * (qS + qH) * β

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

theorem cma_to_nma_advantage_bound    [DecidableEq M] [DecidableEq Commit] [SampleableType Stmt] [SampleableType Wit]    [Finite Stmt] [Finite Commit] [Finite Resp]    [Finite Chal] [Inhabited Chal] [SampleableType Chal]    (simTranscript : Stmt → ProbComp (Commit × Chal × Resp))    (ζ_zk : ℝ) (hζ_zk : 0 ≤ ζ_zk)    (hHVZK : σ.HVZK simTranscript ζ_zk)    (β : ENNReal)    (hPredSim : σ.simCommitPredictability simTranscript β)    (adv : SignatureAlg.unforgeableAdv      (FiatShamir (m := OracleComp (unifSpec + (M × Commit →ₒ Chal))) σ hr M))    (qS qH : ℕ)    (hQ : ∀ pk, signHashQueryBound (M := M) (Commit := Commit) (Chal := Chal)      (S' := Commit × Resp) (oa := adv.main pk) qS qH) :    ∃ nmaAdv : SignatureAlg.managedRoNmaAdv        (FiatShamir (m := OracleComp (unifSpec + (M × Commit →ₒ Chal))) σ hr M),      adv.advantage (runtime M) ≤        Fork.advantage σ hr M nmaAdv qH +          ENNReal.ofReal ((qS : ℝ) * ζ_zk) + (qS : ENNReal) * (qS + qH) * β :=  ⟨Stateful.nmaAdvFromCmaWithFinalQuery σ hr M adv simTranscript,    Stateful.cma_advantage_le_fork_bound_of_h1h2 σ hr M      simTranscript ζ_zk hζ_zk hHVZK β hPredSim adv qS qH hQ      (le_of_eq <| (Stateful.publicUnforgeableAdvantage_eq_statefulPostKeygenFreshAdvantage          (σ := σ) (hr := hr) (M := M) (Commit := Commit) (Chal := Chal) (Resp := Resp) adv).trans        (Stateful.statefulPostKeygenFreshAdvantage_eq_cmaRealRunProb_signedFreshAdv          (σ := σ) (hr := hr) (M := M) (Commit := Commit) (Chal := Chal) (Resp := Resp) adv))⟩

Project: VCVio
License: Apache-2.0
Commit: 2ceb2d825ee3
Source: VCVio/CryptoFoundations/FiatShamir/Sigma/Reductions.lean:56-81

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