Abstract
It is well-known that a protocol for any function can be constructed using only cards and various shuffling techniques (this is referred to as a card-based protocol). In this paper, we propose a new type of cards called regular polygon cards. These cards enable a new encoding for multi-valued inputs while the previous works can only handle binary inputs. We furthermore propose a new technique for constructing a card-based protocol for any n-ary function with small shuffle complexity. This is the first general construction in which the shuffle complexity is independent of the complexity (size/depth) of the desired functionality, although being directly proportional to the number of inputs. The construction furthermore supports a wide range of cards and encodings, including previously proposed types of cards. Our techniques provide a method for reducing the number of shuffles in card-based protocols.
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Notes
- 1.
We stress that the two-card-per-bit encoding schemes are important since the one-card-per-bit encoding scheme [7] needs unnatural shuffle for computing the AND function. Thus, it is still meaningful to improve protocols under the two-cards-per-bit encoding schemes.
- 2.
We assume that the direction of flipping is predetermined.
- 3.
The number of cards in a stack would be revealed by the thickness of the stack.
- 4.
- 5.
We demonstrate how to securely obtain a cyclic shuffle. Let \(P_0, \cdots , P_{n-1}\) be the parties participating in the protocol. \(P_0\) chooses a uniformly random value \(r_0 \in \mathbb {Z}_k\) and applies \({\mathsf {cyc}}^{r_0}\) to \({\varvec{d}}\), and sends \({\mathsf {cyc}}^{r_0}({\varvec{d}})\) to \(P_1\). Similarly, \(P_i\) receives \({\varvec{d'}}\), chooses \(r_i \in \mathbb {Z}_k\), and sends \({\mathsf {cyc}}^{r_i}({\varvec{d'}})\) to \(P_{i+1}\). Finally, \(P_{n-1}\) outputs \({\mathsf {cyc}}^{r}({\varvec{d}})\) where \(r = r_0 + \cdots + r_{n-1}\). Nobody knows the uniform random value r except when all parties are corrupted by an adversary assuming parties are honest-but-curious.
- 6.
All operations except shuffles output a sequence in a deterministic way. However, shuffles output a sequence in a probabilistic way under a uniformly random value r.
- 7.
When there are no shuffles in a transcript, the vector \(r_i\) should be an empty vector.
- 8.
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Acknowledgment
The authors would like to thank members of the study group “Shin-Akarui-Angou-Benkyou-Kai” for the valuable discussions and helpful comments, and thank the anonymous reviewers for their comments. This work was partially supported by JSPS KAKENHI Grant Numbers 26330001 and 26330151, Kurata Grant from The Kurata Memorial Hitachi Science and Technology Foundation, and JSPS A3 Foresight Program.
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Shinagawa, K. et al. (2015). Multi-party Computation with Small Shuffle Complexity Using Regular Polygon Cards. In: Au, MH., Miyaji, A. (eds) Provable Security. ProvSec 2015. Lecture Notes in Computer Science(), vol 9451. Springer, Cham. https://doi.org/10.1007/978-3-319-26059-4_7
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