Polynomial multiplication is known to have quasi-linear complexity in both the dense and the sparse cases. Yet no truly linear algorithm has been given in any case for the problem, and it is not clear whether it is even possible. This leaves room for a better algorithm for the simpler problem of verifying a polynomial product. While finding deterministic methods seems out of reach, there exist probabilistic algorithms for the problem that are optimal in number of algebraic operations. We study the generalization of the problem to the verification of a polynomial product modulo a sparse divisor. We investigate its bit complexity for both dense and sparse multiplicands. In particular, we are able to show the primacy of the verification over modular multiplication when the divisor has a constant sparsity and a second highest-degree monomial that is not too large. We use these results to obtain new bounds on the bit complexity of the standard polynomial multiplication verification. In particular, we provide optimal algorithms in the bit complexity model in the dense case by improving a result of Kaminski and develop the first quasi-optimal algorithm for verifying sparse polynomial product.

中文翻译：

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多项式模块化产品验证及其含义

已知多项式乘法在稠密和稀疏情况下都具有准线性复杂度。然而，在任何情况下都没有给出真正的线性算法来解决这个问题，而且还不清楚是否可能。这为更好的算法留出了余地，可以用于更简单的验证多项式乘积的问题。虽然发现确定性方法似乎遥不可及，但存在针对该问题的概率算法，这些算法在代数运算的数量上是最佳的。我们研究将问题推广到以稀疏除数为模的多项式乘积的验证。我们研究稠密和稀疏被乘数的位复杂度。特别是，当除数具有恒定的稀疏性且第二高次多项式不是太大时，我们能够证明验证优于模块乘法。我们使用这些结果来获得标准多项式乘法验证的位复杂度的新界限。特别地，我们通过改进Kaminski的结果，在稠密情况下的位复杂度模型中提供了最佳算法，并开发了第一个用于验证稀疏多项式乘积的准最优算法。