Logarithmic‑degree rational approximation enabling O(log n)‑buffer BLTs

Prove the existence, for every n ∈ N, of a rational function r of degree O(log n) such that the two integral approximation bounds to 1/√(1−x) over the circle x = exp(iθ − 1/n) hold, thereby enabling buffered linear Toeplitz mechanisms with O(log n) memory buffers while maintaining near‑optimality.

Background

The main BLT construction achieves near‑optimality with space O(log2 n) by using rational approximations to the square root that require degree Θ(log2 n). The conjecture asserts that degree O(log n) suffices if one directly controls the relevant weighted Parseval integrals along the circle of radius exp(−1/n).

If true, this would improve the space complexity of BLT mechanisms from O(log2 n) to O(log n), significantly enhancing practicality for large‑scale streaming differential privacy settings.

References

We give the following conjecture which improves the results in Section 4. If true, this would imply that BLTs can work with d = O(log n) space complexity. Conjecture 9.1. For all n E N there exists a rational function r : C > C of degree d = O(log n) such that the following hold. 2π −π 1 r(x) 1 − x 2 dθ ≤ 1 n − − 1 where x = exp(iθ − 1/n). 2π −π 1 1 − x − r(x) 2 dθ ≤ 1 n where x = exp(iθ − 1/n).

— Efficient and Near-Optimal Noise Generation for Streaming Differential Privacy (2404.16706 - Dvijotham et al., 25 Apr 2024) in Section 9 (Conjecture)