File size: 2,985 Bytes
5d5048e | 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 | import OAI.Combinatorics.LowSpaceSubsetSum.Counting.PrimeHash
import OAI.Combinatorics.LowSpaceSubsetSum.Counting.ResidueCoverage
import Mathlib.Data.Int.ModEq
namespace OAI
/-!
# Prime hashing supplies the finite collision interface
This connects the actual finite uniform prime sampler to the previously proved
collision-to-occupancy theorem. Integer differences may have polynomially many
bits; the parameter `u` remains explicit.
-/
namespace LowSpaceSubsetSum
open scoped BigOperators
noncomputable def primeProbability (ell : ℕ) (_p : ℕ) : ℝ :=
1 / (primeRange ell).card
theorem primeRange_nonempty {ell : ℕ} (hell : 4 ≤ ell) :
(primeRange ell).Nonempty := by
apply Finset.card_pos.mp
have h := primeRange_card_lower hell
have hp : 0 < 2 ^ ell := pow_pos (by decide) _
nlinarith
theorem primeProbability_sum {ell : ℕ} (hell : 4 ≤ ell) :
(∑ p ∈ primeRange ell, primeProbability ell p) = 1 := by
have h : ((primeRange ell).card : ℝ) ≠ 0 := by
exact_mod_cast (primeRange_nonempty hell).card_pos.ne'
simp [primeProbability, h]
theorem prime_pairwise_collision_bound (X : Finset ℤ) {ell u : ℕ}
(hell : 4 ≤ ell) (hellu : ell ≤ u)
(hbits : ∀ x ∈ X, ∀ y ∈ X, (x - y).natAbs < 2 ^ (2 * u)) :
PairwiseCollisionBound (primeRange ell) (primeProbability ell) X
(fun p x => x % (p : ℤ)) (8 * (u : ℝ) ^ 2 / (2 : ℝ) ^ ell) := by
intro x hx y hy hxy
have hd : 0 < (x - y).natAbs := Int.natAbs_pos.mpr (sub_ne_zero.mpr hxy)
have hiff (p : ℕ) : x % (p : ℤ) = y % (p : ℤ) ↔ p ∣ (x - y).natAbs := by
rw [Int.emod_eq_emod_iff_emod_sub_eq_zero, ← Int.dvd_iff_emod_eq_zero,
Int.natCast_dvd]
have heq : (∑ p ∈ primeRange ell,
primeProbability ell p * if x % (p : ℤ) = y % (p : ℤ) then (1 : ℝ) else 0) =
(((primeRange ell).filter fun p => p ∣ (x - y).natAbs).card : ℝ) /
(primeRange ell).card := by
simp_rw [hiff, mul_ite, mul_one, mul_zero]
rw [← Finset.sum_filter]
simp [primeProbability, div_eq_mul_inv, mul_comm]
rw [heq]
exact prime_divisor_probability_le hell hellu hd (hbits x hx y hy)
/-- The abstract occupancy result now has a proved prime-hashing input. -/
theorem prime_occupied_probability (X : Finset ℤ) {ell u : ℕ}
(hell : 4 ≤ ell) (hellu : ell ≤ u)
(hbits : ∀ x ∈ X, ∀ y ∈ X, (x - y).natAbs < 2 ^ (2 * u))
{T : ℝ} (hT : 0 < T) :
1 - ((X.card : ℝ) + (8 * (u : ℝ) ^ 2 / (2 : ℝ) ^ ell) * (X.card : ℝ) ^ 2) / T ≤
∑ p ∈ (primeRange ell).filter
(fun p : ℕ => (X.card : ℝ) ^ 2 / T ≤ ((X.image (fun x => x % (p : ℤ))).card : ℝ)),
primeProbability ell p := by
apply occupied_probability_of_collision_bound
· intro p _
exact div_nonneg zero_le_one (Nat.cast_nonneg _)
· exact primeProbability_sum hell
· positivity
· exact hT
· exact prime_pairwise_collision_bound X hell hellu hbits
end LowSpaceSubsetSum
end OAI
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