Solving the Discrete Logarithm Problem for Packing Candidate Preferences

Ranked elections are used in many places across the world, and a number of end-to-end verifiable voting systems have been proposed to handle these elections recently. One example is the vVote system designed for the Victorian State Election, Australia. In this system, many voters will give a full ranking of up to 38 candidates. The easiest way to do this is to ask each voter to reorder ciphertexts representing the different candidates, so that the ciphertext ordering represents the candidate ranking. But this requires sending 38 ciphertexts per voter through the mixnets, which will take a long time. In this paper, we explore how to “pack” multiple candidate preferences into a single ciphertext, so that these preferences can be represented in the least number of ciphertexts possible, while maintaining efficient decryption. Both the packing and the unpacking procedure are performed publicly: we still provide 38 ciphertexts, but they are combined appropriately before they enter the mixnets, and after decryption, a meet-in-the-middle algorithm can be used to recover the full candidate preferences despite the discrete logarithm problem.

[1]  Hugo Krawczyk,et al.  Secure Distributed Key Generation for Discrete-Log Based Cryptosystems , 1999, EUROCRYPT.

[2]  Josh Benaloh,et al.  Secret Sharing Homomorphisms: Keeping Shares of A Secret Sharing , 1986, CRYPTO.

[3]  Jean-Jacques Quisquater,et al.  Advances in Cryptology — EUROCRYPT ’95 , 2001, Lecture Notes in Computer Science.

[4]  Taher El Gamal.  A public key cryptosystem and a signature scheme based on discrete logarithms , 1984, IEEE Trans. Inf. Theory.

[5]  Lee Naish,et al.  Coercion-Resistant Tallying for STV Voting , 2008, EVT.

[6]  Peter Y. A. Ryan,et al.  Prêt à Voter with Re-encryption Mixes , 2006, ESORICS.

[7]  Zhe Xia,et al.  A Supervised Verifiable Voting Protocol for the Victorian Electoral Commission , 2012, Electronic Voting.

[8]  Hugo Krawczyk,et al.  Secure Distributed Key Generation for Discrete-Log Based Cryptosystems , 1999, Journal of Cryptology.

[9]  Kazue Sako,et al.  Receipt-Free Mix-Type Voting Scheme - A Practical Solution to the Implementation of a Voting Booth , 1995, EUROCRYPT.

[10]  David Chaum,et al.  A Practical Voter-Verifiable Election Scheme , 2005, ESORICS.

[11]  Ben Adida,et al.  Helios: Web-based Open-Audit Voting , 2008, USENIX Security Symposium.

[12]  Torben P. Pedersen.  A Threshold Cryptosystem without a Trusted Party (Extended Abstract) , 1991, EUROCRYPT.

[13]  C. Andrew Neff,et al.  A verifiable secret shuffle and its application to e-voting , 2001, CCS '01.

[14]  Zhe Xia,et al.  Using Prêt à Voter in Victoria State Elections , 2012, EVT/WOTE.

[15]  Josh Benaloh,et al.  Secret Sharing Homomorphisms: Keeping Shares of A Secret Sharing , 1986, CRYPTO.

[16]  Whitfield Diffie,et al.  Special Feature Exhaustive Cryptanalysis of the NBS Data Encryption Standard , 1977, Computer.

[17]  Andrew Odlyzko,et al.  Advances in Cryptology — CRYPTO’ 86 , 2000, Lecture Notes in Computer Science.

[18]  Lee Naish,et al.  Shuffle-Sum: Coercion-Resistant Verifiable Tallying for STV Voting , 2009, IEEE Transactions on Information Forensics and Security.

[19]  Donald W. Davies,et al.  Advances in Cryptology — EUROCRYPT ’91 , 2001, Lecture Notes in Computer Science.

[20]  Jacques Stern,et al.  Advances in Cryptology — EUROCRYPT ’99 , 1999, Lecture Notes in Computer Science.

[21]  Matthew K. Franklin,et al.  Multi-Autority Secret-Ballot Elections with Linear Work , 1996, EUROCRYPT.

[22]  Pascal Paillier,et al.  Public-Key Cryptosystems Based on Composite Degree Residuosity Classes , 1999, EUROCRYPT.

[23]  Donald Ervin Knuth,et al.  The Art of Computer Programming, Volume II: Seminumerical Algorithms , 1970 .

[24]  Aggelos Kiayias,et al.  Self Protecting Pirates and Black-Box Traitor Tracing , 2001, CRYPTO.

[25]  Martin E. Hellman,et al.  On the security of multiple encryption , 1981, CACM.

[26]  Ronald Cramer,et al.  A secure and optimally efficient multi-authority election scheme , 1997, Eur. Trans. Telecommun..

[27]  Dieter Gollmann,et al.  Computer Security - ESORICS 2006, 11th European Symposium on Research in Computer Security, Hamburg, Germany, September 18-20, 2006, Proceedings , 2006, ESORICS.

[28]  Donald E. Knuth,et al.  The art of computer programming. Vol.2: Seminumerical algorithms , 1981 .

[29]  Kazue Sako,et al.  An Efficient Scheme for Proving a Shuffle , 2001, CRYPTO.

[30]  Zhe Xia,et al.  PrÊt À Voter: a Voter-Verifiable Voting System , 2009, IEEE Transactions on Information Forensics and Security.

[31]  Walter Fumy,et al.  Advances in Cryptology — EUROCRYPT ’97 , 2001, Lecture Notes in Computer Science.

[32]  Ueli Maurer,et al.  Advances in Cryptology — EUROCRYPT ’96 , 2001, Lecture Notes in Computer Science.

[33]  Dieter Gollmann,et al.  Computer Security - ESORICS 2005, 10th European Symposium on Research in Computer Security, Milan, Italy, September 12-14, 2005, Proceedings , 2005, ESORICS.

[34]  Jacques Stern,et al.  Practical multi-candidate election system , 2001, PODC '01.