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No polynomial-time algorithm is known to test whether a sparse polynomial divides another sparse polynomial . While computing the quotient = quo can be done in polynomial time with respect to the sparsities of , and , this is not yet sufficient to get a polynomial-time divisibility test in general. Indeed, the sparsity of the quotient can be exponentially larger than the ones of and . In the favorable case where the sparsity # of the quotient is polynomial, the best known algorithm to compute has a non-linear factor # # in the complexity, which is not optimal.In this work, we are interested in the two aspects of this problem. First, we propose a new randomized algorithm that computes the quotient of two sparse polynomials when the division is exact. Its complexity is quasi-linear in the sparsities of , and . Our approach relies on sparse interpolation and it works over any finite field or the ring of integers. Then, as a step toward faster divisibility testing, we provide a new polynomial-time algorithm when the divisor has a specific shape. More precisely, we reduce the problem to finding a polynomial such that is sparse and testing divisibility by can be done in polynomial time. We identify some structure patterns in the divisor for which we can efficiently compute such a polynomial . CCS CONCEPTS• Computing methodologies → Algebraic algorithms; • Theory of computation → Design and analysis of algorithms; • Mathematics of computing → Probabilistic algorithms.

We describe a straightforward method to generate a random prime q such that the multiplicative group F q also has a random large prime-order subgroup. The described algorithm also yields this order p as well as a p'th primitive root of unity ω. The methods here are efficient asymptotically, but due to large constants may not be very useful in practical settings.

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