Untraceable electronic mail, return addresses, and digital pseudonyms
Communications of the ACM
A verifiable secret shuffle and its application to e-voting
CCS '01 Proceedings of the 8th ACM conference on Computer and Communications Security
An Efficient Scheme for Proving a Shuffle
CRYPTO '01 Proceedings of the 21st Annual International Cryptology Conference on Advances in Cryptology
A "Paradoxical" Indentity-Based Signature Scheme Resulting from Zero-Knowledge
CRYPTO '88 Proceedings of the 8th Annual International Cryptology Conference on Advances in Cryptology
Wallet Databases with Observers
CRYPTO '92 Proceedings of the 12th Annual International Cryptology Conference on Advances in Cryptology
Mix-Networks on Permutation Networks
ASIACRYPT '99 Proceedings of the International Conference on the Theory and Applications of Cryptology and Information Security: Advances in Cryptology
How to Make Personalized Web Browising Simple, Secure, and Anonymous
FC '97 Proceedings of the First International Conference on Financial Cryptography
Sharing Decryption in the Context of Voting or Lotteries
FC '00 Proceedings of the 4th International Conference on Financial Cryptography
Remarks on Mix-Network Based on Permutation Networks
PKC '01 Proceedings of the 4th International Workshop on Practice and Theory in Public Key Cryptography: Public Key Cryptography
A Verifiable Secret Shuffle of Homomorphic Encryptions
PKC '03 Proceedings of the 6th International Workshop on Theory and Practice in Public Key Cryptography: Public Key Cryptography
Public-key cryptosystems based on composite degree residuosity classes
EUROCRYPT'99 Proceedings of the 17th international conference on Theory and application of cryptographic techniques
Verifiable shuffle of large size ciphertexts
PKC'07 Proceedings of the 10th international conference on Practice and theory in public-key cryptography
Sub-linear zero-knowledge argument for correctness of a shuffle
EUROCRYPT'08 Proceedings of the theory and applications of cryptographic techniques 27th annual international conference on Advances in cryptology
A sender verifiable mix-net and a new proof of a shuffle
ASIACRYPT'05 Proceedings of the 11th international conference on Theory and Application of Cryptology and Information Security
Simple and efficient shuffling with provable correctness and ZK privacy
CRYPTO'05 Proceedings of the 25th annual international conference on Advances in Cryptology
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Some of the most efficient shuffling schemes employ the same main idea to prove validity of shuffling. However, the principle behind the idea has not been explained in a completely formal and precise way. So formal guarantee of soundness of the shuffling schemes is not complete. Especially, it is unknown exactly how large the probability of failure of soundness is and exactly how strong soundness is. In this paper, why the proof mechanism guarantees validity of shuffling is formally proved to provide a formal guarantee of soundness of those shuffling schemes. Especially, the exact upper bound of the probability of failure of soundness is given to convincingly ensure soundness. Although we do not doubt soundness of most of the shuffling schemes, a formal and precise proof of their soundness is still valuable as it strengthens security guarantee of the schemes and removes a potential obstacle for their application to critical environments requiring formally guaranteed and measurable soundness. Moreover, the formal and precise proof shows that some shuffling scheme has serious problem in soundness.