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Classical and Quantum Polynomial Reconstruction via Legendre Symbol Evaluation

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arxiv quant-ph/0212016 v1 pith:EAXZRRG3 submitted 2002-12-02 quant-ph

Classical and Quantum Polynomial Reconstruction via Legendre Symbol Evaluation

classification quant-ph
keywords polynomialcaseclassicalcomplexityproblemquantumsomealgorithm
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved
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We consider the problem of recovering a hidden monic polynomial f(X) of degree d > 0 over the finite field F of p elements given a black box which, for any x in F, evaluates the quadratic character of f(x). We design a classical algorithm of complexity O(d^2 p^{d + c}), for any c > 0, and also show that the quantum query complexity of this problem is O(d). Some of our results extend those of Wim van Dam, Sean Hallgren and Lawrence Ip obtained in the case of a linear polynomial f(X) = X + s (with unknown s); some are new even in this case.

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