Sparse Polynomial Interpolation Based on Diversification

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