Simulate Q random Oracle sample Nonzero ZMod
Groups.simulateQ_randomOracle_sampleNonzeroZMod
Plain-language statement
Simulating the random oracle leaves the nonzero SRS trapdoor sampler unchanged.
Exact Lean statement
lemma simulateQ_randomOracle_sampleNonzeroZMod :
((simulateQ (unifSpec.randomOracle :
QueryImpl unifSpec (StateT unifSpec.QueryCache ProbComp))
(sampleNonzeroZMod (p := p) : ProbComp (ZMod p)) :
StateT unifSpec.QueryCache ProbComp (ZMod p))).run' ∅ =
sampleNonzeroZMod (p := p)Formal artifact
Lean source
lemma simulateQ_randomOracle_sampleNonzeroZMod : ((simulateQ (unifSpec.randomOracle : QueryImpl unifSpec (StateT unifSpec.QueryCache ProbComp)) (sampleNonzeroZMod (p := p) : ProbComp (ZMod p)) : StateT unifSpec.QueryCache ProbComp (ZMod p))).run' ∅ = sampleNonzeroZMod (p := p) := by haveI : NeZero (p - 1) := ⟨Nat.pos_iff_ne_zero.mp (Nat.sub_pos_of_lt (Nat.Prime.one_lt Fact.out))⟩ unfold sampleNonzeroZMod cases p with | zero => exact False.elim (Nat.not_prime_zero Fact.out) | succ p' => cases p' with | zero => exact False.elim (Nat.not_prime_one Fact.out) | succ p'' => exact simulateQ_randomOracle_map_uniformFin p'' (fun i : Fin (p'' + 1) => ((i : ℕ) + 1 : ZMod (p'' + 1 + 1)))- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Commitments/Functional/KZG/Sampling.lean:39-57
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affine_gaps_lifted_to_interleaved_codes
Project documentation
This lemma proves the final algebraic step in the DG25 Theorem 3.1 proof. It shows that if R > e + 1, then e * (R / (R - 1)) < e + 1. The intuition is that the fraction R / (R - 1) is always greater than 1, but as R gets larger, it gets closer to 1. The hypothesis R > e + 1 provides a strong enough bound to ensure the product e * (fraction) do...
Source project: ArkLib
Person-level attribution pending.
Gadget Decompose coeff
ArkLib.Lattices.Ajtai.gadgetDecompose_coeff
Plain-language statement
The k-th coefficient (k < deg φ) of a gadget-decomposition block is exactly the corresponding digit of the corresponding input coefficient.
Source project: ArkLib
Person-level attribution pending.
Gadget Decompose lawful
ArkLib.Lattices.Ajtai.gadgetDecompose_lawful
Plain-language statement
The base-b gadget decomposition is a lawful gadget decomposition.
Source project: ArkLib
Person-level attribution pending.