Le prob Output seeded Fork
OracleComp.le_probOutput_seededFork
Project documentation
Key bound of the forking lemma: the probability that both runs succeed with fork point s is at least Pr[cf(main) = s]² - Pr[cf(main) = s] / |Range i|.
Exact Lean statement
theorem le_probOutput_seededFork (s : Fin (qb i + 1)) :
let h : ℝ≥0∞Formal artifact
Lean source
theorem le_probOutput_seededFork (s : Fin (qb i + 1)) : let h : ℝ≥0∞ := ↑(Fintype.card (spec.Range i)) Pr[= s | cf <$> main] ^ 2 - Pr[= s | cf <$> main] / h ≤ Pr[ fun r => r.map (cf ∘ Prod.fst) = some (some s) | seededFork main qb js i cf] := by set h : ℝ≥0∞ := ↑(Fintype.card (spec.Range i)) rw [probEvent_seededFork_fst_eq_probEvent_pair main qb js i cf, probEvent_seededFork_pair_eq_probOutput_map_aux main qb js i cf s] let collisionComp : OracleComp spec (Option (Fin (qb i + 1))) := do let seed ← liftComp (generateSeed spec qb js) spec let x₁ ← (simulateQ seededOracle main).run' seed let u ← liftComp ($ᵗ spec.Range i) spec if (seed i)[↑s]? = some u then return cf x₁ else return none have hCollision : Pr[= (some s : Option (Fin (qb i + 1))) | collisionComp] ≤ Pr[= s | cf <$> main] / h := by simpa [h, collisionComp] using probOutput_collision_le_main_div main qb js i cf s exact le_trans (tsub_le_tsub (sq_probOutput_main_le_noGuardComp main qb js i cf s) hCollision) (tsub_le_iff_right.2 (probOutput_noGuardComp_le_seededFork_add_collision_aux main qb js i cf s))- Project
- VCVio
- License
- Apache-2.0
- Commit
- 2ceb2d825ee3
- Source
- VCVio/CryptoFoundations/SeededFork.lean:566-583
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