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arXiv:2211.06910 · International Symposium on Turbo Codes and Iterative Information Processing

Concatenating Extended CSS Codes for Communication Efficient Quantum Secret Sharing

Kaushik Senthoor, Pradeep Kiran Sarvepalli

Published Sep 4, 2023• 4 citations
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Abstract

Recently, a class of quantum secret sharing schemes called communication efficient quantum threshold secret sharing schemes (CE-QTS) was introduced. These schemes reduced the quantum communication cost during secret recovery in quantum threshold secret sharing schemes. However, non-threshold schemes are known to have less storage cost compared to threshold schemes. This motivates us to extend the idea of communication efficiency to non-threshold schemes as well. In this paper, we introduce a general class of communication efficient quantum secret sharing schemes (CE-QSS) which include both threshold and non-threshold schemes. Specifically, we provide a framework for constructing CE-QSS schemes by concatenating a pair of CSS codes. The main component in this framework is a new class of Calderbank-Shor-Steane (CSS) codes called the extended CSS codes.

DOI
10.1109/istc57237.2023.10273451

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Recently, a class of quantum secret sharing schemes called communication efficient quantum threshold secret sharing schemes (CE-QTS) was introduced. These schemes reduced the quantum communication cost during secret recovery in quantum threshold secret sharing schemes. However, non-threshold schemes are known to have less storage cost compared to threshold schemes. This motivates us to extend the idea of communication efficiency to non-threshold schemes as well. In this paper, we introduce a general class of communication efficient quantum secret sharing schemes (CE-QSS) which include both threshold and non-threshold schemes. Specifically, we provide a framework for constructing CE-QSS schemes by concatenating a pair of CSS codes. The main component in this framework is a new class of Calderbank-Shor-Steane (CSS) codes called the extended CSS codes.
Recently, a class of quantum secret sharing schemes called communication efficient quantum threshold secret sharing schemes (CE-QTS) was introduced. These schemes reduced the communication cost during secret recovery. In this paper, we introduce a general class of communication efficient quantum secret sharing schemes (CE-QSS) which include both threshold and non-threshold schemes. We propose a framework for constructing CE-QSS schemes to generalize the earlier construction of CE-QTS schemes which was based on the staircase codes. The main component in this framework is a class of quantum codes which we call the extended Calderbank-Shor-Steane codes. These extended CSS codes could have other applications. We derive a bound on communication cost for CE-QSS schemes. Finally, we provide a construction of CE-QSS schemes meeting this bound using the proposed framework.
A((k,n))((k,n))((k,n))quantum threshold secret sharing (QTS) scheme is a quantum cryptographic protocol for sharing a quantum secret amongnnnparties such that the secret can be recovered by anykkkor more parties whilek−1k-1k−1or fewer parties have no information about the secret. Despite extensive research on these schemes, there has been very little study on optimizing the quantum communication cost during recovery. Recently, we initiated the study of communication efficient quantum threshold secret sharing (CE-QTS) schemes. These schemes reduce the communication complexity in QTS schemes by accessingd>kd>kd>kparties for recovery; heredddis fixed ahead of encoding the secret. In contrast to the standard QTS schemes which requirekkkqudits for recovering each qudit in the secret, these schemes have a lower communication cost ofdd−k+1\frac {d}{d-k+1}d−k+1d​. In this paper, we further develop the theory of communication efficient quantum threshold schemes. Here, we propose universal CE-QTS schemes which reduce the communication cost for alld>kd>kd>ksimultaneously. We provide a framework based on ramp quantum secret sharing to construct CE-QTS and universal CE-QTS schemes. We give another construction for universal CE-QTS schemes based on Staircase codes. We derived a lower bound on communication complexity and show that our constructions are optimal. Finally, an information theoretic model is developed to analyse CE-QTS schemes and the lower bound on communication complexity is proved again using this model.
Quantum secret sharing (QSS) is a cryptographic protocol in which a quantum secret is distributed among a number of parties where some subsets of the parties are able to recover the secret while some subsets are unable to recover the secret. In the standard ((k, n)) quantum threshold secret sharing scheme, any subset of k or more parties out of the total n parties can recover the secret while other subsets have no information about the secret. But recovery of the secret incurs a communication cost of at least k qudits for every qudit in the secret. Recently, a class of communication efficient QSS schemes were proposed which can improve this communication cost to dd−k+1\frac{d}{{d - k + 1}}d−k+1d​ by contacting d ≥ k parties where d is fixed prior to the distribution of shares. In this paper, we propose a more general class of ((k, n)) quantum secret sharing schemes with low communication complexity. In these schemes the combiner can contact any d parties at the time of recovery where k ≤ d ≤ n. This is the first such class of universal communication efficient quantum threshold schemes.
In this paper we investigate the use of quantum information to share classical secrets. While every quantum secret sharing scheme is a quantum error-correcting code, the converse is not true. Motivated by this we sought to find quantum codes which can be converted to secret sharing schemes. If we are interested in sharing classical secrets using quantum information, then we show that a class of pure [[n,1,d]]q{[[n,1,d]]}_{q}[[n,1,d]]q​ Calderbank-Shor-Steane (CSS) codes can be converted to perfect secret sharing schemes. These secret sharing schemes are perfect in the sense that the unauthorized parties do not learn anything about the secret. Gottesman had given conditions to test whether a given subset is an authorized or unauthorized set; they enable us to determine the access structure of quantum secret sharing schemes. For the secret sharing schemes proposed in this paper the access structure can be characterized in terms of minimal codewords of the classical code underlying the CSS code. This characterization of the access structure for quantum secret sharing schemes is thought to be notable.
A method for concatenating quantum error-correcting codes is presented. The method is applicable to a wide class of quantum error-correcting codes known as Calderbank-Shor-Steane (CSS) codes. As a result, codes that achieve a high rate in the Shannon theoretic sense and that are decodable in polynomial time are presented. The rate is the highest among those known to be achievable by CSS codes. Moreover, the best known lower bound on the greatest minimum distance of codes constructible in polynomial time is improved for a wide range.
A method for concatenating quantum error-correcting codes is presented. The method is applicable to a wide class of quantum error-correcting codes known as Calderbank-Shor-Steane (CSS) codes. As a result, codes that achieve a high rate in the Shannon-theoretic sense and that are decodable in polynomial time are presented. The rate is the highest among those known to be achievable by CSS codes. Moreover, the best known lower bound on the greatest minimum distance of codes constructible in polynomial time is improved for a wide range.