Sparse polynomial interpolation based on diversification - Huang, Qiao-Long (Science China. Mathematics, 2022)
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 (1988) for interpolating polynomials over fields with characteristic zero, we develop a new Monte Carlo algorithm over finite fields 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 ∼ ( n T log 2 q + n T 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 the previous algorithms over general finite field, our algorithm has better complexity in the parameters n, T and D , and is the first one to achieve the complexity of fractional power about D , while keeping linear in n and T . A key technique is a randomization which makes all the coefficients of the unknown polynomial distinguishable, producing a diverse polynomial. This approach, called diversification, was proposed by Giesbrecht and Roche (2011). Our algorithm interpolates each variable independently by 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 the coefficients by solving a linear system. We have implemented our algorithm in Maple. Experimental results show that our algorithm can be applied to the sparse polynomials with large degree within a few minutes. We also analyze the success rate of the algorithm.