Abstract
In 2015, Koch et al. proposed a five-card finite-runtime committed protocol to compute securely the AND function, showing that their protocol was optimal: there is no protocol computing the AND function with four cards in finite-runtime fashion and committed format. Thus, necessary and sufficient numbers of cards for computing single-bit output functions are known. However, as for two-bit output functions, such an exact characterization is unknown. This paper gives a six-card (or less) protocol for each of all two-bit output functions and proves that our finite-runtime committed protocols are optimal by providing a lower bound. In other words, we give the necessary and sufficient number of cards for any two-bit output function to be computed by a finite-runtime committed protocol. Our lower bound can also be applied to any function which outputs more than two bits.
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This paper completes the work of [7] proving that what has been suggested is optimal.
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A deterministic shuffle is just a permutation of a sequence.
References
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Acknowledgment
We thank the anonymous referees, whose comments have helped us to improve the presentation of the paper. This work was supported by JSPS KAKENHI Grant Number 26330001.
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Francis, D., Aljunid, S.R., Nishida, T., Hayashi, Yi., Mizuki, T., Sone, H. (2017). Necessary and Sufficient Numbers of Cards for Securely Computing Two-Bit Output Functions. In: Phan, RW., Yung, M. (eds) Paradigms in Cryptology – Mycrypt 2016. Malicious and Exploratory Cryptology. Mycrypt 2016. Lecture Notes in Computer Science(), vol 10311. Springer, Cham. https://doi.org/10.1007/978-3-319-61273-7_10
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