# Secret sharing schemes based on graphical codes

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## Abstract

We study the access structure and multiplicativity of linear secret sharing schemes based on codes from complete graphs. First, we describe the access structure of the schemes based on cut-set and cycle codes. Second, we show that the class of access structures based on odd cycles cannot be realized by ideal multiplicative linear secret sharing schemes over any finite field. This can be seen as a contribution to the characterization of access structures of ideal multiplicative schemes. The access structure based on odd cycles corresponds to the scheme based on the dual of the extended cycle code. Finally, we show that we can obtain ideal multiplicative linear secret sharing scheme based on the dual of an augmented extended cycle code.

## Keywords

Secret sharing Linear code Matroid Graph## Mathematics Subject Classifications (2000)

94A62 94B05 05C50## Notes

### Acknowledgements

The work of Y. Gao is supported in part by the National Natural Science Foundation of China by Grant 11101019 and the Fundamental Research Funds for the Central Universities in China (No. YWF-10-02-072). Part of the work was done while she was visiting Nanyang Technological University. The work of R. dela Cruz is supported in part by the NTU PhD Research Scholarship and the Merlion PhD Grant of the French Embassy in Singapore. He would like to thank Telecom-ParisTech for its hospitality. The authors would like to thank Carles Padró and Huaxiong Wang for some helpful discussions, and the anonymous reviewers for their valuable comments and suggestions.

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