1. 1/11/2007 bswilson/eVote-PTCWS 1 Enhancing PTC based Secure E-Voting System (note: modification of Brett Wilson’s Paillier Threshold Cryptography Web Service view graph) by Justin Pohlmann Allen Liu

2. 21/11/2007bswilson/eVote-PTCWS Outline of the Talk Introduction/Motivation Related Work Paillier Threshold Cryptography Suggested Improvement Encryption/Decryption Optimization Encryption/Decryption Optimization User Interface User Interface Future Direction Conclusion

3. 31/11/2007bswilson/eVote-PTCWS Introduction/Motivation E-voting Requirements Privacy/Anonymity, Completeness, Soundness, Un-reusability, Eligibility, Fairness Privacy/Anonymity, Completeness, Soundness, Un-reusability, Eligibility, Fairness Robustness, Universal Verifiability, Receipt- Freeness, Incoercibility Robustness, Universal Verifiability, Receipt- Freeness, Incoercibility

4. 41/11/2007bswilson/eVote-PTCWS Introduction/Motivation Many new Secure Voting protocols Mathematical algorithms presented in literature Mathematical algorithms presented in literature Unable to identify/locate implementations of these algorithms Unable to identify/locate implementations of these algorithms UCCS effort to develop a secure e-voting application Using the concept of Paillier Threshold Cryptosystem to implement a voting system Using the concept of Paillier Threshold Cryptosystem to implement a voting system Find areas of improvement Find areas of improvement Encryption Optimization User Interface

5. 51/11/2007bswilson/eVote-PTCWS Related Work Basis for Implementation Sharing Decryption in the context of Voting or Lotteries (Fouque, Poupard, Stern) Sharing Decryption in the context of Voting or Lotteries (Fouque, Poupard, Stern) Closely related research A Generalization of Paillier’s Public Key Cryptosystem with Applications to Electronic Voting (Damgard, Jurik, Nielson) A Generalization of Paillier’s Public Key Cryptosystem with Applications to Electronic Voting (Damgard, Jurik, Nielson) Encryption and Decryption Optimization

6. 61/11/2007bswilson/eVote-PTCWS Related Work Other Techniques Used In E-voting Protocols A Secure and Optimally Efficient Multi-Authority Election Scheme (Cramer, Gennaro, Schoenmakers) A Secure and Optimally Efficient Multi-Authority Election Scheme (Cramer, Gennaro, Schoenmakers) Receipt-free: protocols where vote-buying or coercing is not possible because voters cannot prove to others how they voted. Non-Interactive Zero Knowledge Proofs Non-Interactive Zero Knowledge Proofs Proof does not require interaction Proof does not require interaction Proof does not reveal any other information Proof does not reveal any other information Prove vote is valid without revealing content of vote Prove vote is valid without revealing content of vote Prove two encryptions encrypt the same message without revealing message Prove two encryptions encrypt the same message without revealing message

7. 71/11/2007bswilson/eVote-PTCWS Uses of Paillier Cryptography Electronic Voting Electronic Voting Anonymous Mix Nets (due to self-blinding property) Anonymous Mix Nets (due to self-blinding property) Electronic Auctions Electronic Auctions Electronic Lotteries Electronic Lotteries

8. 81/11/2007bswilson/eVote-PTCWS Cryptographic Techniques Implemented Paillier CryptoSystem Trapdoor Discrete Logarithm Scheme Trapdoor Discrete Logarithm Scheme c = g M r n mod n 2 c = g M r n mod n 2 n is an RSA modulus (modulus of 2 safe primes) n is an RSA modulus (modulus of 2 safe primes) Safe prime - Safe prime - p = 2q + 1 where q is also prime g is an integer of order nα mod n 2 g is an integer of order nα mod n 2 r is a random number in Z n * r is a random number in Z n * M = L(c λ(n) mod n 2 )/L(g λ(n) mod n 2 ) mod n M = L(c λ(n) mod n 2 )/L(g λ(n) mod n 2 ) mod n L(u) = (u-1)/n, λ(n)=lcm((p-1)(q-1)) L(u) = (u-1)/n, λ(n)=lcm((p-1)(q-1)) Important Properties Important Properties Probabilistic (randomness of E(M)) Homomorphic E(M 1 + M 2 ) = E(M 1 ) x E(M 2 ), E(k x M) = E(M) k E(M 1 + M 2 ) = E(M 1 ) x E(M 2 ), E(k x M) = E(M) kSelf-blinding D(E(M) r n mod n 2 ) = m D(E(M) r n mod n 2 ) = m

9. 91/11/2007bswilson/eVote-PTCWS Suggested Encryption/Decryption Optimization Current Paillier System Suggested Paillier System Key Generation g = nα mod n 2 Key Generation g=n+1 Encryption: C = g M x n modn 2 Encryption: C = g M x n s modn s+1 Decryption M = L(c λ(n) mod n 2 )/L(g λ(n) mod n 2 ) mod n Decryption 1.c d modn s+1  (1+n) jMdmodn s 2.g d modn s+1  (1+n) jdmodn s 3.(jMd)(jd) -1 = Mmodn s

10. 101/11/2007bswilson/eVote-PTCWS Suggested Encryption/Decryption Optimization. From O(n s )  O(s)

11. 111/11/2007bswilson/eVote-PTCWS Cryptographic Techniques Implemented Threshold Encryption Public key encryption as usual Public key encryption as usual Distribute secret key “shares” among i participants Distribute secret key “shares” among i participants Decryption can only be accomplished if a threshold number t of the i participants cooperate Decryption can only be accomplished if a threshold number t of the i participants cooperate No information about m can be obtained with less than t participants cooperating Shamir Secret Sharing Lagrange Interpolation formula Lagrange Interpolation formula f(X) = Σ t i=0 a i X i f(X) = Σ t i=0 a i X i a 0 is secret, a i are random, f(X) are “secret shares” a 0 is secret, a i are random, f(X) are “secret shares” X is share index (1 to number of servers) If enough f(X) available it is possible to recover a 0 If enough f(X) available it is possible to recover a 0

12. 121/11/2007bswilson/eVote-PTCWS Voting Application PTC Use Election Admin PTC Web Service PTC CSP 2. SOAP/XML Request for PTC Parameters 3. SOAP/XML Response containing RSA encrypted PTC Parameters Election Authorities 1. Election Authorities’ RSA Public Keys 8. Partial Decryption Shares of Vote Tally/Proofs of Correct Decryption 4. RSA Encrypted Secret Key Shares PTC CSP 7. Paillier Encrypted Vote Tally 9. Vote Tally Voter PTC CSP 5. Paillier Public Key 6. Paillier- Encrypted Vote Voters Vote

13. 131/11/2007bswilson/eVote-PTCWS Preliminary Website Layout LoginPage AdminPage VoterPage VotePage ElectionCreation Encrypt/Decrypt

14. 141/11/2007bswilson/eVote-PTCWS Admin Page Election Creation Add User(s)/Group(s) allowed to vote Add User(s)/Group(s) allowed to vote Add Election Admins Add Election AdminsEncryption/Decryption Via Email Via Email Add group

15. 151/11/2007bswilson/eVote-PTCWS Voter Page Elections users can vote on Brings up voting page Allows vote, then doesn’t allow user to vote again Allows vote, then doesn’t allow user to vote again

16. 161/11/2007bswilson/eVote-PTCWS Database Schema Ballots Candidate Elections Votes Users Eligibility Encryptors EncryptionStatus

17. 171/11/2007bswilson/eVote-PTCWS Future Direction Implement the suggested Improvement Integrate other cryptosystem protocols (i.e. Receipt-Freeness and Zero Knowledge Proofs)

18. 181/11/2007bswilson/eVote-PTCWS Questions?