The Division Barrier: Optimal Bounds and Structural Limits in Toom-Cook Interpolation

Toom-Cook-k (2 ≤ k ∈ ℕ) is a family of fast algorithms for multiplying long integers using O ⁽n^{logk (2k−1)}) arithmetic operations, offering asymptotical improvement over the naïve quadratic-time schoolbook approach. Despite this advantage, Toom-Cook algorithms often involve nontrivial divisions, divisions by elements that are not powers of 2, which can be both computationally expensive and numerically unstable, especially in cryptography or quantum computing applications. Reducing or eliminating these divisions is therefore of significant theoretical and practical interest. Although previous work has sought to minimize the number of such divisions, Bodrato and Zanoni conjectured that any Toom-Cook-k algorithm must perform at least 2k − 5 nontrivial divisions. In this work, we prove a stronger and more general result; any linear interpolation scheme for a Toom-Cook-(k₁, k₂) algorithm with integer evaluation points must include at least (k₁ + k₂ − 2 − p) divisions by every prime p ≤ 2k − 3. In the balanced case (k₁ = k₂ = k), this yields (2k − 2 − p) divisions by each prime p ≤ 2k − 3. Our findings confirms and extends the Bodrato–Zanoni division conjecture. Furthermore, we establish the optimality of several known algorithms Toom-Cook-3, 3.5, 4, 4.5, 5, and 8. We also extend our bounds to polynomial multiplication over finite fields, where Toom-Cook-like algorithms are widely used. We show that any Toom-Cook-(k₁, k₂) algorithm over a finite field F must involve at least k₁ + k₂ − 2 − |F| nontrivial divisions, where such divisions are by non-monomial polynomials. More precisely, we show that at least k₁ + k₂ − 2 − |F|^d divisions are required for each irreducible polynomial of degree d. From this, we conclude the optimality of several known multiplication algorithms over GL(2): Toom-Cook-2.5, 3, 4, and 5. Finally, under a division-aware cost model where a division by a prime p cost O(log p), any integer interpolation algorithm requires Ω(n²) arithmetic operations, even when evaluation points are optimally chosen.