This paper extends the (k, n) threshold scheme and proposes the (k, L, n) threshold scheme.
![quantum error correction ramp quantum error correction ramp](https://upload.wikimedia.org/wikipedia/commons/0/04/David_Deutsch.jpg)
On the other hand, each subinformation requires the same number of bits as the original information X, which is very inefficient from the viewpoint of the coding efficiency. Thus, the (k, n) threshold scheme is suited to the distributed storage or transmission of information. However, no information can be obtained at all concerning X from any (k – 1) subinformation. If any k subinformation is obtained among n subinformation, the original information X can be recovered completely. In the (k, n) threshold scheme, the information X is partitioned and coded into subinformation. Copyright © 2012 The Institute of Electronics, Information and Communication Engineers.
![quantum error correction ramp quantum error correction ramp](https://venturebeat.com/wp-content/uploads/2020/05/IROF-explanation-1.png)
Then, it is clarified that secret sharing schemes based on linear codes can always achieve the a-strong security where the value a is precisely char-acterized by the RGHW. We define a secret sharing scheme achiev-ing the a-strong security as the one such that the mutual information be-tween any r elements of (s 1,⋯,s l) and any α - r+1 shares is always zero. Moreover, this paper characterizes the strong security in secret sharing schemes based on linear codes, by generalizing the definition of strongly-secure threshold ramp schemes. It is clarified that both characterizations for t 1 and t 2 are better than Chen et al.'s ones derived by the regular minimum Hamming weight. One shows that any set of at most t 1 shares leaks no information about s, and the other shows that any set of at least t 2 shares uniquely de-termines s. We also characterize two thresholds t 1 and t 2 in the secret sharing schemes by the RGHW of linear codes. We first describe the equivocation Δ m of the secret vector s = given m shares in terms of the RDLP of linear codes. This paper precisely characterizes secret sharing schemes based on arbitrary linear codes by using the relative dimension/length pro-file (RDLP) and the relative generalized Hamming weight (RGHW). In addition, we discuss the relation with the quantum version of maximum distance separable (MDS) codes.
![quantum error correction ramp quantum error correction ramp](https://blogs.sw.siemens.com/wp-content/uploads/sites/50/2016/08/MW_Synthesis_Fig-4-1024x608.png)
Further, we also introduce the EA setting of SPIR, which is shown to link to MMSP. Showing that the linear version of SS with the EA setting is directly linked to MMSP, we characterize linear quantum versions of SS with the CQ ad QQ settings via MMSP. We newly introduce the third setting, i.e., the entanglement-assisted (EA) setting, which is defined by modifying the CQ setting with allowing prior entanglement between the dealer and the end-user who recovers the secret by collecting the shares.
![quantum error correction ramp quantum error correction ramp](https://media.springernature.com/w200/springer-static/cover/journal/41745.jpg)
The other is the quantum-quantum (QQ) setting, in which the secret to be sent is a quantum state and the shares are quantum systems. In particular, two kinds of quantum extensions of SS are known One is the classical-quantum (CQ) setting, in which the secret to be sent is classical information and the shares are quantum systems. This paper unifiedly addresses two kinds of key quantum secure tasks, i.e., quantum versions of secret sharing (SS) and symmetric private information retrieval (SPIR) by using multi-target monotone span program (MMSP), which characterizes the classical linear protocols of SS and SPIR.