Uncertainty principles and signal recovery
SIAM Journal on Applied Mathematics
Finite fields
Testing shift-equivalence of polynomials by deterministic, probabilistic and quantum machines
Theoretical Computer Science
Modern computer algebra
Finite and infinite pseudorandom binary words
Theoretical Computer Science
The Design and Analysis of Computer Algorithms
The Design and Analysis of Computer Algorithms
Quantum algorithms for some hidden shift problems
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
On the Randomness of Legendre and Jacobi Sequences
CRYPTO '88 Proceedings of the 8th Annual International Cryptology Conference on Advances in Cryptology
Classical and Quantum Computation
Classical and Quantum Computation
Pattern distributions of Legendre sequences
IEEE Transactions on Information Theory
Quantum algorithms for highly non-linear Boolean functions
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Efficient quantum algorithm for identifying hidden polynomials
Quantum Information & Computation
Quantum period reconstruction of binary sequences
AAECC'06 Proceedings of the 16th international conference on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
Quantum noisy rational function reconstruction
COCOON'05 Proceedings of the 11th annual international conference on Computing and Combinatorics
Journal of Symbolic Computation
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We consider two natural instances of the problem of determining a function f : G→G defined on a group G by making repeated queries to the oracle O(x) = χ(f(x)), where χ is a known character of the group G.In particular, we consider the problem of recovering a hidden monic polynomial f(X) of degree d≥1 over a finite field Fp of p elements given a black box which, for any x ∈ Fp, evaluates the quadratic character of f(x). We design a classical algorithm of complexity O(d2pd+ε) 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.We then consider the related problem of determining an element s of a finite group G given the oracle O(x) = χ(sx), where χ is the character of a (known) faithful irreducible representation ρ of G. We show that if d is the dimension of ρ, then O(d2 log|G|) classical queries suffice to determine s. The proof involves development of a new "uncertainty principle" for the Fourier transform over finite non-Abelian groups.