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Revision #1 to TR06-020 | 26th March 2008 00:00
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#### Quantum Hardcore Functions by Complexity-Theoretical Quantum List Decoding

**Abstract:**
Hardcore functions have been used as a technical tool to construct secure cryptographic systems; however, little is known on their quantum counterpart, called quantum hardcore functions. With a new insight into fundamental properties of quantum hardcores, we present three new quantum hardcore functions for any (strong) quantum one-way function. We also give a "quantum" solution to Damgard's question (CRYPTO'88) on a classical hardcore property of his pseudorandom generator, by proving its quantum hardcore property. Our major technical tool is the new notion of quantum list-decoding of "classical" error-correcting codes (rather than

"quantum" error-correcting codes), which is defined on the platform of computational complexity theory and computational cryptography (rather than information theory). In particular, we give a simple but powerful criterion that makes a polynomial-time computable classical block code (seen as a function) a quantum hardcore for all quantum one-way functions. On their own interest, we construct elegant quantum list-decoding algorithms for classical block codes whose associated quantum states (called codeword states) form a nearly phase orthogonal basis. In particular, we prove that circulant codes that enjoy a multiplicative property are quantum list-decodable.

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TR06-020 | 10th February 2006 00:00
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#### Quantum Hardcore Functions by Complexity-Theoretical Quantum List Decoding

**Abstract:**
We present three new quantum hardcore functions for any quantum one-way function. We also give a "quantum" solution to Damgard's question (CRYPTO'88) on his pseudorandom generator by proving the quantum hardcore property of his generator, which has been unknown to have the classical hardcore property.

Our technical tool is quantum list-decoding of

"classical" error-correcting codes (rather than

"quantum" error-correcting codes), which is defined on the platform of computational complexity theory and cryptography (rather than information theory). In particular, we give a simple but powerful criterion that makes a polynomial-time computable code (seen as a function) a quantum hardcore for any quantum one-way function. On their own interest, we also give quantum list-decoding algorithms for

codes whose associated quantum states (called codeword states) are ``almost'' orthogonal using the technique of pretty good measurement.