File size: 2,593 Bytes
9dbc64e | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 | import OAI.Computability.DepthThree.HashTriangular
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Logic.Equiv.Basic
namespace OAI
namespace DepthThreeLowerBound
namespace BinaryHash
noncomputable section
variable {d r : ℕ}
def hashF2_fiberEquiv (w : Data d) (hw : w ≠ 0) (a : Output r) :
{u : Seed d r // hashF2 u w = a} ≃ (Nonpivot (r := r) w hw → F2) where
toFun u := (triangularEquiv (r := r) w hw u.1).2
invFun b := ⟨(triangularEquiv (r := r) w hw).symm (a, b), by
rw [← triangularEquiv_fst w hw]
exact congrArg Prod.fst
((triangularEquiv (r := r) w hw).apply_symm_apply (a, b))⟩
left_inv u := by
apply Subtype.ext
apply (triangularEquiv (r := r) w hw).injective
change (triangularEquiv (r := r) w hw)
((triangularEquiv (r := r) w hw).symm
(a, (triangularEquiv (r := r) w hw u.1).2)) =
triangularEquiv (r := r) w hw u.1
rw [Equiv.apply_symm_apply]
exact Prod.ext ((triangularEquiv_fst w hw u.1).trans u.2).symm rfl
right_inv b := by
change ((triangularEquiv (r := r) w hw)
((triangularEquiv (r := r) w hw).symm (a, b))).2 = b
rw [Equiv.apply_symm_apply]
theorem hashF2_fiber_card (w : Data d) (hw : w ≠ 0) (a : Output r) :
Fintype.card {u : Seed d r // hashF2 u w = a} = 2 ^ (d - 1) := by
classical
calc
Fintype.card {u : Seed d r // hashF2 u w = a} =
Fintype.card (Nonpivot (r := r) w hw → F2) :=
Fintype.card_congr (hashF2_fiberEquiv w hw a)
_ = 2 ^ (d - 1) := by
change Fintype.card (Nonpivot (r := r) w hw → ZMod 2) = 2 ^ (d - 1)
rw [Fintype.card_fun, ZMod.card, card_nonpivot (r := r) w hw]
theorem hashF2_surjective (w : Data d) (hw : w ≠ 0) :
Function.Surjective (fun u : Seed d r => hashF2 u w) := by
intro a
let u : {u : Seed d r // hashF2 u w = a} :=
(hashF2_fiberEquiv w hw a).symm (fun _ => 0)
exact ⟨u.1, u.2⟩
def hashF2_collisionEquiv (x y : Data d) :
{u : Seed d r // hashF2 u x = hashF2 u y} ≃
{u : Seed d r // hashF2 u (x - y) = 0} :=
Equiv.subtypeEquivRight fun u => by
rw [hashF2_sub_data, sub_eq_zero]
theorem hashF2_collision_card (x y : Data d) (hxy : x ≠ y) :
Fintype.card {u : Seed d r // hashF2 u x = hashF2 u y} = 2 ^ (d - 1) := by
calc
Fintype.card {u : Seed d r // hashF2 u x = hashF2 u y} =
Fintype.card {u : Seed d r // hashF2 u (x - y) = 0} :=
Fintype.card_congr (hashF2_collisionEquiv (r := r) x y)
_ = 2 ^ (d - 1) :=
hashF2_fiber_card (x - y) (sub_ne_zero.mpr hxy) 0
end
end BinaryHash
end DepthThreeLowerBound
end OAI
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