No abstract

In this paper, we propose new deterministic and Monte Carlo interpolation algorithms for sparse multivariate polynomials represented by straight-line programs. Let f be an n-variate polynomial given by a straight-line program, which has a degree bound D and a term bound T . Our deterministic algorithm is quadratic in n, T and cubic in log D in the Soft-Oh sense, which has better complexities than existing deterministic interpolation algorithms in most cases. Our Monte Carlo interpolation algorithms have better complexities than existing Monte Carlo interpolation algorithms and are the first algorithms whose complexities are linear in nT in the Soft-Oh sense. Since nT is a factor of the size of f , our Monte Carlo algorithms are optimal in n and T in the Soft-Oh sense.

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 ∈ F q [x 1 , . . . , x n ] with a partial degree bound D and a term bound T , our new algorithm costs O ∼ (nT log 2 q +nT √ D log q) bit operations and uses 2(n+1)T probes to the black box. If q ≥ O(nT 2 D), 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.