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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 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 | import OAI.Computability.DepthThree.HashAffine
namespace OAI
namespace DepthThreeLowerBound.BinaryHash
open BinaryAlgebra
variable {d r : ℕ}
theorem hashBool_eq_iff (u : Fin (d + r - 1) → Bool)
(x : Fin d → Bool) (a : Fin r → Bool) :
hashBool u x = a ↔
hashF2 (bitsEquivF2 (Fin (d + r - 1)) u) (bitsEquivF2 (Fin d) x) =
bitsEquivF2 (Fin r) a := by
rw [← bitsEquivF2_hashBool]
exact (bitsEquivF2 (Fin r)).injective.eq_iff.symm
theorem hashBool_collision_iff (u : Fin (d + r - 1) → Bool)
(x y : Fin d → Bool) :
hashBool u x = hashBool u y ↔
hashF2 (bitsEquivF2 (Fin (d + r - 1)) u) (bitsEquivF2 (Fin d) x) =
hashF2 (bitsEquivF2 (Fin (d + r - 1)) u) (bitsEquivF2 (Fin d) y) := by
rw [← bitsEquivF2_hashBool, ← bitsEquivF2_hashBool]
exact (bitsEquivF2 (Fin r)).injective.eq_iff.symm
def hashBool_fiberEquiv (x : Fin d → Bool) (a : Fin r → Bool) :
{u : Fin (d + r - 1) → Bool // hashBool u x = a} ≃
{u : Seed d r // hashF2 u (bitsEquivF2 (Fin d) x) =
bitsEquivF2 (Fin r) a} :=
Equiv.subtypeEquiv (bitsEquivF2 (Fin (d + r - 1)))
(fun u => hashBool_eq_iff u x a)
theorem hashBool_fiber_card (x : Fin d → Bool) (hx : x ≠ fun _ => false)
(a : Fin r → Bool) :
Fintype.card {u : Fin (d + r - 1) → Bool // hashBool u x = a} = 2 ^ (d - 1) := by
classical
have hnonzero : bitsEquivF2 (Fin d) x ≠ 0 := by
intro h
apply hx
funext i
apply bitValue_injective
simpa using congrFun h i
exact (Fintype.card_congr (hashBool_fiberEquiv x a)).trans
(hashF2_fiber_card (bitsEquivF2 (Fin d) x) hnonzero (bitsEquivF2 (Fin r) a))
def hashBool_collisionEquiv (x y : Fin d → Bool) :
{u : Fin (d + r - 1) → Bool // hashBool u x = hashBool u y} ≃
{u : Seed d r // hashF2 u (bitsEquivF2 (Fin d) x) =
hashF2 u (bitsEquivF2 (Fin d) y)} :=
Equiv.subtypeEquiv (bitsEquivF2 (Fin (d + r - 1)))
(fun u => hashBool_collision_iff u x y)
theorem hashBool_collision_card (x y : Fin d → Bool) (hxy : x ≠ y) :
Fintype.card {u : Fin (d + r - 1) → Bool // hashBool u x = hashBool u y} =
2 ^ (d - 1) := by
classical
exact (Fintype.card_congr (hashBool_collisionEquiv (r := r) x y)).trans
(hashF2_collision_card (bitsEquivF2 (Fin d) x) (bitsEquivF2 (Fin d) y)
(fun h => hxy ((bitsEquivF2 (Fin d)).injective h)))
def boolCoordinateCube (T : Finset (Fin d)) (rho : {i : Fin d // i ∉ T} → Bool) :
Set (Fin d → Bool) :=
{x | ∀ i (hi : i ∉ T), x i = rho ⟨i, hi⟩}
theorem mem_boolCoordinateCube_iff (T : Finset (Fin d))
(rho : {i : Fin d // i ∉ T} → Bool) (x : Fin d → Bool) :
x ∈ boolCoordinateCube T rho ↔
bitsEquivF2 (Fin d) x ∈ coordinateCube T (fun i => bitValue (rho i)) := by
constructor
· intro hx i hi
change bitValue (x i) = bitValue (rho ⟨i, hi⟩)
exact congrArg bitValue (hx i hi)
· intro hx i hi
exact bitValue_injective (hx i hi)
theorem injOn_hashBool_iff (u : Fin (d + r - 1) → Bool)
(T : Finset (Fin d)) (rho : {i : Fin d // i ∉ T} → Bool) :
Set.InjOn (hashBool u) (boolCoordinateCube T rho) ↔
Set.InjOn (hashF2 (bitsEquivF2 (Fin (d + r - 1)) u))
(coordinateCube T (fun i => bitValue (rho i))) := by
constructor
· intro hinj x hx y hy hxy
obtain ⟨a, rfl⟩ := (bitsEquivF2 (Fin d)).surjective x
obtain ⟨b, rfl⟩ := (bitsEquivF2 (Fin d)).surjective y
have hab := hinj ((mem_boolCoordinateCube_iff T rho a).mpr hx)
((mem_boolCoordinateCube_iff T rho b).mpr hy)
((hashBool_collision_iff u a b).mpr hxy)
exact congrArg (bitsEquivF2 (Fin d)) hab
· intro hinj x hx y hy hxy
apply (bitsEquivF2 (Fin d)).injective
exact hinj ((mem_boolCoordinateCube_iff T rho x).mp hx)
((mem_boolCoordinateCube_iff T rho y).mp hy)
((hashBool_collision_iff u x y).mp hxy)
def hashBool_badSeedEquiv (T : Finset (Fin d))
(rho : {i : Fin d // i ∉ T} → Bool) :
{u : Fin (d + r - 1) → Bool // ¬ Set.InjOn (hashBool u) (boolCoordinateCube T rho)} ≃
{u : Seed d r // ¬ Set.InjOn (hashF2 u)
(coordinateCube T (fun i => bitValue (rho i)))} :=
Equiv.subtypeEquiv (bitsEquivF2 (Fin (d + r - 1)))
(fun u => not_congr (injOn_hashBool_iff u T rho))
open scoped Classical in
theorem hashBool_badSeed_card_le (T : Finset (Fin d))
(rho : {i : Fin d // i ∉ T} → Bool) :
Fintype.card
{u : Fin (d + r - 1) → Bool // ¬ Set.InjOn (hashBool u) (boolCoordinateCube T rho)} ≤
(2 ^ T.card - 1) * 2 ^ (d - 1) := by
classical
calc
Fintype.card
{u : Fin (d + r - 1) → Bool // ¬ Set.InjOn (hashBool u) (boolCoordinateCube T rho)} =
Fintype.card {u : Seed d r // ¬ Set.InjOn (hashF2 u)
(coordinateCube T (fun i => bitValue (rho i)))} :=
Fintype.card_congr (hashBool_badSeedEquiv (r := r) T rho)
_ = (badSeeds (r := r) T (fun i => bitValue (rho i))).card := by
simp only [badSeeds, Fintype.card_subtype]
_ ≤ (2 ^ T.card - 1) * 2 ^ (d - 1) := badSeeds_card_le T _
end DepthThreeLowerBound.BinaryHash
end OAI
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