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Complexity-Theoretical Quantum List Decoding and Applications to Quantum Hardcore Functions(New Trends in Theory of Computation and Algorithm)

Akinori Kawachi, Tomoyuki Yamakami

Published May 1, 2006• 0 citations
OpenAlex

Abstract

We poesent thoee new quantum hardcore functions for any quantum one-way function.We also give a quantum" solution to Damgdrd's question $(\mathrm{C}\mathrm{R}\mathrm{Y}\mathrm{F}\Gamma \mathrm{O}88)$ 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" crrorcorrecting codes (rather than ' quantum" error-correcting codes).which is deflned on the platform of computational complexity theory and cryptography (rather than infonnation theory).In panicular, we give a simple but powerful criterion that makes a polynomial-time computable code (seen as a $\mathrm{f}\mathrm{u}\mathrm{n}\mathrm{c}\dot{\mathrm{u}}\mathrm{o}\mathrm{n})$ a quantum hardcore for any quantum one-way function.On their own inteoest, we also give quantum list-decoding algorithms for codes whose associated quantum states (called codeword states) are ' almost" orthogonal using the technique of poetty good measurement.

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Quantum Hardcore Functions by Complexity-Theoretical Quantum List Decoding

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Abstract. We present three new quantum hardcore functions for any quantum one-way function. We also give a “quantum ” solution to Damg˚ard’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. 1 Introduction: From Hardcore to List-Decoding

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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 efficient quantum list-decoding algorithms for classical block codes whose associated quantum states (called codeword states) form a nearly phase-orthogonal basis.

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Complexity-Theoretical Quantum List Decoding and Applications to Quamtum Hardcore Functions (計算理論とアルゴリズムの新展開 RIMS研究集会報告集)

亮周 河内, 智幸 山上

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We poesent thoee new quantum hardcore functions for any quantum one-way function.We also give a quantum" solution to Damgdrd's question (CRYFΓO88)(\mathrm{C}\mathrm{R}\mathrm{Y}\mathrm{F}\Gamma \mathrm{O}88)(CRYFΓO88) 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" crrorcorrecting codes (rather than ' quantum" error-correcting codes).which is deflned on the platform of computational complexity theory and cryptography (rather than infonnation theory).In panicular, we give a simple but powerful criterion that makes a polynomial-time computable code (seen as a funcu˙on)\mathrm{f}\mathrm{u}\mathrm{n}\mathrm{c}\dot{\mathrm{u}}\mathrm{o}\mathrm{n})funcu˙on) a quantum hardcore for any quantum one-way function.On their own inteoest, we also give quantum list-decoding algorithms for codes whose associated quantum states (called codeword states) are ' almost" orthogonal using the technique of poetty good measurement.
Abstract. We present three new quantum hardcore functions for any quantum one-way function. We also give a “quantum ” solution to Damg˚ard’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. 1 Introduction: From Hardcore to List-Decoding
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 efficient quantum list-decoding algorithms for classical block codes whose associated quantum states (called codeword states) form a nearly phase-orthogonal basis.
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 Damgård's question [Advances in Cryptology, Lecture Notes in Comput. Sci. 403, Springer, Berlin, 1990, pp. 163–172] 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 efficient quantum list-decoding algorithms for classical block codes whose associated quantum states (called codeword states) form a nearly phase-orthogonal basis.
In classical coding theory, the quest for explicit codes achieving list decoding capacity has been an important driving force. While random codes are easily shown to achieve capacity, an explicit construction was only discovered decades later in the seminal work [Guruswami and Rudra, STOC 2006]. In quantum coding theory, it is highly desirable that a code family be LDPC. While good quantum codes were known for decades, obtaining the additional LDPC property was elusive. In fact, only very recently that good quantum LDPC codes were discovered in a breakthrough work [Panteleev and Kalachev, STOC 2022]. In this context, a natural question is to ask for an explicit family of quantum codes achieving list decoding capacity while also possessing the important LDPC property. In this work, we provide (to the best of our knowledge) the first explicit constructions of quantum LDPC codes achieving list decoding capacity, namely, with a list decoding radius approaching the quantum Singleton bound with constant list sizes. Furthermore, we provide near-linear time (in the block-length) list decoding algorithms approaching capacity. Our explicit code construction are based on expander graphs via the quantum analogue of Alon-Edmonds-Luby (AEL) amplification [Bergamaschi, Golowich and Gunn, STOC 2024], and this enables the important LDPC property. Our efficient list decoding algorithms are obtained by generalizing the classical expander-based weak-regularity list decoding algorithms [Srivastava and Tulsiani, FOCS 2025] [Jeronimo and Singh, 2025] to suitable instantiations of quantum AEL.
Quantum computation and information has become a central research area in theoretical computer science. It studies how information is encoded in nature according to the laws of quantum mechanics and what this means for its computational power. On one hand, the ability of a quantum system to be in a superposition of states provides a way to encode an exponential amount of information in a quantum state. However, this information is accessible only indirectly via measurements. The goal of this thesis is precisely to investigate the fundamental question of how information is encoded and processed in quantum mechanical systems. (1) We investigate their power relative to classical encodings in the model of one-way communication complexity and show that they can be exponentially more efficient, answering a long standing open question in the area of quantum communication complexity. (2) We show how the theory developed for the study of quantum information can be employed in order to answer questions about classical coding theory and complexity. We resolve a main open question about the efficiency of Locally Decodable Codes by reducing the problem to one about quantum codes and solving the latter using tools from quantum information theory. This is the first example where quantum techniques are essential in resolving a purely classical question. (3) We investigate the power of quantum encodings in the area of cryptography. We study certain cryptographic primitives, such as Private Information Retrieval, Symmetrically-Private Information Retrieval and Coin Flipping.