Fully homomorphic encryption using ideal lattices
Proceedings of the forty-first annual ACM symposium on Theory of computing
Computing arbitrary functions of encrypted data
Communications of the ACM
Quantum Hardcore Functions by Complexity-Theoretical Quantum List Decoding
SIAM Journal on Computing
Efficient quantum algorithm for identifying hidden polynomials
Quantum Information & Computation
On solving systems of random linear disequations
Quantum Information & Computation
The quantum query complexity of learning multilinear polynomials
Information Processing Letters
Finding hidden Borel subgroups of the general linear group
Quantum Information & Computation
ACM Transactions on Computation Theory (TOCT) - Special issue on innovations in theoretical computer science 2012
Journal of Symbolic Computation
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Almost all of the most successful quantum algorithms discovered to date exploit the ability of the Fourier transform to recover subgroup structures of functions, especially periodicity. The fact that Fourier transforms can also be used to capture shift structure has received far less attention in the context of quantum computation. In this paper, we present three examples of “unknown shift” problems that can be solved efficiently on a quantum computer using the quantum Fourier transform. For one of these problems, the shifted Legendre symbol problem, we give evidence that the problem is hard to solve classically, by showing a reduction from breaking algebraically homomorphic cryptosystems. We also define the hidden coset problem, which generalizes the hidden shift problem and the hidden subgroup problem. This framework provides a unified way of viewing the ability of the Fourier transform to capture subgroup and shift structure.