**A Public Shuffle without Private Permutations**

*Myungsun Kim and Jinsu Kim and Jung Hee Cheon*

**Abstract: **In TCC 2007, Adida and Wikstr\"{o}m proposed a novel approach to
shuffle, called a public shuffle,
in which a shuffler can perform shuffle publicly without needing information kept secret.
Their scheme uses an encrypted permutation matrix to shuffle
ciphertexts publicly.
This approach significantly reduces the cost of constructing a mix-net
to verifiable joint decryption. Though their method is successful in making
shuffle to be a public operation, their scheme
still requires that some trusted parties should choose a permutation
to be encrypted and construct zero-knowledge proofs on the
well-formedness of this permutation.

In this paper, we propose a method to construct a public shuffle without relying on permutations and randomizers generated privately: Given an $n$-tuple of ciphertext $(c_1,\dots,c_n)$, our shuffle algorithm computes $f_i(c_1,\dots,c_n)$ for $i=1,\dots,\ell$ where each $f_i(x_1,\dots,x_n)$ is a symmetric polynomial in $x_1,\dots,x_n$. Depending on the symmetric polynomials we use, we propose two concrete constructions. One is to use ring homomorphic encryption with constant ciphertext complexity and the other is to use simple ElGamal encryption with linear ciphertext complexity in the number of senders. Both constructions are free of zero-knowledge proofs and publicly verifiable.

**Category / Keywords: **secret shuffle, public shuffle, private permutation, mix-net, ElGamal encryption

**Date: **received 29 May 2012, last revised 24 Jun 2012

**Contact author: **msunkim at snu ac kr, kjs2002@snu ac kr, jhcheon@snu ac kr

**Available format(s): **PDF | BibTeX Citation

**Version: **20120624:064629 (All versions of this report)

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