Identity Testing from High Powers of Polynomials of Large Degree over Finite Fields

08/30/2017

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by   Marek Karpinski, et al.

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We consider the problem of identity testing of two "hidden" monic polynomials f and g, given an oracle access to f(x)^e and g(x)^e for x∈ F_q, where F_q is the finite field of q elements (an extension fields access is not permitted). The naive interpolation algorithm needs de+1 queries, where d ={deg f, deg g} and thus requires de<q. For a prime q = p. we design an algorithm that is asymptotically better in certain cases, especially when d is large. The algorithm is based on a result of independent interest in spirit of additive combinatorics. It gives an upper bound on the number of values of a rational function of large degree, evaluated on a short sequence of consecutive integers, that belong to a small subgroup of F_p^*.