IND CPA step Adversary signed Advantage Real eq hybrid Diff half
AsymmEncAlg.IND_CPA_stepAdversary_signedAdvantageReal_eq_hybridDiff_half
Plain-language statement
Planned adjacent-gap characterization for the extracted step adversary. Once IND_CPA_stepAdversary_game_eq_hybridBranch is proved, this is just the one-time analogue of IND_CPA_signedAdvantageReal_eq_lrDiff_half.
Exact Lean statement
theorem IND_CPA_stepAdversary_signedAdvantageReal_eq_hybridDiff_half
[Inhabited M]
(adversary : encAlg'.IND_CPA_adversary) (k : ℕ) :
IND_CPA_OneTime_signedAdvantageReal (encAlg := encAlg')
(IND_CPA_stepAdversary (encAlg' := encAlg') adversary k) =
((Pr[= true | encAlg'.IND_CPA_LR_hybridGame adversary (k + 1)]).toReal -
(Pr[= true | encAlg'.IND_CPA_LR_hybridGame adversary k]).toReal) / 2Formal artifact
Lean source
theorem IND_CPA_stepAdversary_signedAdvantageReal_eq_hybridDiff_half [Inhabited M] (adversary : encAlg'.IND_CPA_adversary) (k : ℕ) : IND_CPA_OneTime_signedAdvantageReal (encAlg := encAlg') (IND_CPA_stepAdversary (encAlg' := encAlg') adversary k) = ((Pr[= true | encAlg'.IND_CPA_LR_hybridGame adversary (k + 1)]).toReal - (Pr[= true | encAlg'.IND_CPA_LR_hybridGame adversary k]).toReal) / 2 := by unfold IND_CPA_OneTime_signedAdvantageReal rw [show (Pr[= true | IND_CPA_OneTime_Game_ProbComp (encAlg := encAlg') (IND_CPA_stepAdversary (encAlg' := encAlg') adversary k)]).toReal = (Pr[= true | do let bit ← ($ᵗ Bool) let z ← if bit then encAlg'.IND_CPA_LR_hybridGame adversary (k + 1) else encAlg'.IND_CPA_LR_hybridGame adversary k pure (bit == z)]).toReal from congrArg ENNReal.toReal <| (evalDist_ext_iff.mp (IND_CPA_stepAdversary_game_eq_hybridBranch (encAlg' := encAlg') adversary k)) true] exact probOutput_uniformBool_branch_toReal_sub_half (encAlg'.IND_CPA_LR_hybridGame adversary (k + 1)) (encAlg'.IND_CPA_LR_hybridGame adversary k)- Project
- VCVio
- License
- Apache-2.0
- Commit
- 2ceb2d825ee3
- Source
- VCVio/CryptoFoundations/AsymmEncAlg/INDCPA/GenericLift.lean:358-378
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Person-level attribution pending.