Sparse polynomial interpolation based on diversification

We consider the problem of interpolating a sparse multivariate polynomial over finite field, represented with black box. Building on algorithm Ben-Or and Tiwari (1988) for polynomials fields characteristic zero, we develop new Monte Carlo by doing additional probes. To interpolate $$f \in {\mathbb{F}_q}[{x_1}, \ldots ,{x_n}]$$ partial degree bound D term T, our costs $${O^ \sim }(nT{\log ^2}q + nT\sqrt \log q)$$ bit operations uses 2(n 1)T probes to If q ⩾ O(nT2D), it has constant success rate return correct polynomial. Compared previous algorithms general better complexity in parameters n, T D, is first one achieve fractional power about while keeping linear n T. A key technique randomization which makes all coefficients unknown distinguishable, producing diverse This approach, called diversification, was proposed Giesbrecht Roche (2011). Our interpolates each variable independently using O(T) probes, then diversification correlate terms different images. At last, get exponents solving discrete logarithms obtain system. have implemented Maple. Experimental results show that can be applied large within few minutes. also analyze algorithm.