On Exact Division and Divisibility Testing for Sparse Polynomials

Abstract

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.

Sparse Polynomial Interpolation and Division in Soft-linear Time

“…For the Euclidean division of sparse polynomials, the case of exact division (when the remainder is known to be zero) was improved by similar techniques [19]. is led to the first algorithm that is quasi-linear in the sparsity, though not in the total bit size.…”

“…As shown in Giorgi et al [19] some sparse interpolation algorithms can be carefully adapted to produce division algorithms if there is no remainder. As the interpolation algorithms they rely on, these division algorithms are not quasi-linear in the input plus the output bit-size.…”

“…To overcome the first difficulty, we use the same method as Giorgi et al [18,19]. We guess a sparsity bound for the quotient, interpolate a candidate quotient assuming the bound, and check its correctness a posteriori with a probabilistic verification.…”

Sparse Polynomial Interpolation and Division in Soft-linear Time

“…Given two sparse polynomials 𝑓 and 𝑔 such that 𝑔 divides 𝑓 , the problem of computing 𝑓 /𝑔 can be seen as a sparse interpolation of a specific SLP that has a single division. As shown in Giorgi et al [19] some sparse interpolation algorithms can be carefully adapted to produce division algorithms if there is no remainder. As the interpolation algorithms they rely on, these division algorithms are not quasi-linear in the input plus the output bit-size.…”

“…For the Euclidean division of sparse polynomials, the case of exact division (when the remainder is known to be zero) was improved by similar techniques [19]. This led to the first algorithm that is quasi-linear in the sparsity, though not in the total bit size.…”