Plain-language statement
Almost completeness of the Fischlin transform: if the underlying Σ-protocol is perfectly complete, then the signature scheme verifies with probability at least 1 - completenessError ρ b S t where t = FinEnum.card Chal is the challenge space size. Unlike the Fiat-Shamir transform (which is perfectly complete), the Fischlin transform has a non-zero comp...
Exact Lean statement
theorem almostComplete (hρ : 0 < ρ) (hc : σ.PerfectlyComplete) (msg : M) :
Pr[= true | (runtime ρ b M).evalDist do
let (pk, sk) ←
(Fischlin (m := OracleComp (unifSpec + fischlinROSpec Stmt Commit Chal Resp ρ b M))
σ hr ρ b S M).keygen
let sig ←
(Fischlin (m := OracleComp (unifSpec + fischlinROSpec Stmt Commit Chal Resp ρ b M))
σ hr ρ b S M).sign pk sk msg
(Fischlin (m := OracleComp (unifSpec + fischlinROSpec Stmt Commit Chal Resp ρ b M))
σ hr ρ b S M).verify pk msg sig]
≥ 1 - completenessError ρ b S (FinEnum.card Chal)Formal artifact
Lean source
theorem almostComplete (hρ : 0 < ρ) (hc : σ.PerfectlyComplete) (msg : M) : Pr[= true | (runtime ρ b M).evalDist do let (pk, sk) ← (Fischlin (m := OracleComp (unifSpec + fischlinROSpec Stmt Commit Chal Resp ρ b M)) σ hr ρ b S M).keygen let sig ← (Fischlin (m := OracleComp (unifSpec + fischlinROSpec Stmt Commit Chal Resp ρ b M)) σ hr ρ b S M).sign pk sk msg (Fischlin (m := OracleComp (unifSpec + fischlinROSpec Stmt Commit Chal Resp ρ b M)) σ hr ρ b S M).verify pk msg sig] ≥ 1 - completenessError ρ b S (FinEnum.card Chal) := by rw [ge_iff_le, fischlin_game_eq_model σ hr ρ b S M msg] have hbound := model_reject_le σ hr ρ b S M hρ hc msg set P : ℝ≥0∞ := Pr[= true | modelGame σ hr ρ b S] with hP -- From `1 - P ≤ e` and `P ≤ 1` conclude `1 - e ≤ P`. have hP1 : P ≤ 1 := probOutput_le_one rw [tsub_le_iff_right] at hbound ⊢ rwa [add_comm] at hbound- Project
- VCVio
- License
- Apache-2.0
- Commit
- 2ceb2d825ee3
- Source
- VCVio/CryptoFoundations/Fischlin/Completeness.lean:1494-1511
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