Threshold%20Paillier%20Encryption%20Web%20Service

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Threshold Paillier Encryption Web Service

• A Masters Project Proposal
• by
• Brett Wilson

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Motivation

• Secure Electronic Voting Research
• Interest in improving current voting process is
  high
• 2000 Presidential election snafu
• Improved access/availability (voter turnout)
• Cryptographic research has led to new solutions
  to problems with electronic voting
• Basic requirements for electronic voting
• Privacy All votes should be kept secret
• Completeness All valid votes should be counted
  correctly
• Soundness Any invalid vote should not be
  counted
• Unreusability No voter can vote twice
• Eligibility Only authorized voters can cast a
  vote
• Fairness Nothing can affect the voting
• Extended Requirements for electronic voting
• Robustness faulty behavior of any reasonably
  sized coalition of participants can be tolerated
• Universal Verifiability any party can verify
  the result of the voting
• Recipt-freeness Voters are unable to prove the
  content of his/her vote
• Incoercibility Voter cannot be coerced into
  casting a particular vote by a coercer

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Motivation

• Many of the proposed electronic voting protocols
  utilize threshold homomorhpic encryption schemes
  as part of the protocol
• Protects voter privacy
• Individual vote can not be decrypted without
  cooperation of t of l authorities
• Efficient, universally verifiable vote tallying
• Only sum of votes is decrypted
• Individuals can compute encrypted sum, verify
  proof of correct decryption of sum
• Implementations of threshold homomorphic
  encryption algorithms are not freely available

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Threshold Encryption

• Public key encryption as usual
• Distribute secret key shares among l
  participants
• Decryption can only be accomplished if a
  threshold number t of the l participants
  cooperate
• No information about m can be obtained with less
  than t participants cooperating
• Proof of valid decryption is provided

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Paillier Encryption

• Trapdoor Discrete Logarithm Scheme
• c gMrn mod n2
• n is an RSA modulus
• g is an integer of order na mod n2
• r is a random number in Zn
• M L(c?(n) mod n2)/L(g?(n) mod n2) mod n
• L(u) (u-1)/n, ?(n)lcm((p-1)(q-1))
• Important Properties
• Homomorphic
• E(M1 M2) E(M1) x E(M2), E(k x M) E(M)k
• Self-blinding
• Re-encryption with a different r doesnt change M

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Threshold Paillier Encryption

• Different public key and secret key generation
  algorithm
• Distribute key shares using RSA public key
  encryption
• Distribute secret key shares using Shamir Secret
  Sharing scheme
• Web Service will be an implementation of scheme
  proposed in Sharing Decryption in the Context of
  Voting or Lotteries Fouque, Poupard, and Stern
  2000

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Use of Threshold Paillier Encryption in Secure
Voting

• Ballot format pick 1 out of c candidates
• Let N be number of voters, k such that Nlt2k
• Vote 2ck where c is the desired candidate
  number (0c)
• All Paillier-encrypted votes could be publicly
  posted
• Votes include proof of validity (v lies in a
  given set of valid votes)
• At end of election, all invalid votes are
  removed, all encrypted votes are then multiplied
  together to get encrypted sum (publicly
  verifiable)
• With cooperation of the required threshold number
  of authorities, the final product could be
  decrypted to reveal the vote total (sum of
  individual votes).
• A threshold number of authorities would not agree
  to decrypt a single particular vote, and thus the
  individual votes would remain private
• All computations are publicly verifiable given
  the validity proofs that prove the decryption was
  done correctly

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Web Service Design Goals

• Platform Independent
• Use of web service
• XML input/output
• Extensible
• Additional encryption algorithms could be added
• Additional services could be offered
• Threshold signatures
• Verifiable Mix Net

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Implementation Tools

• Visual Studio 2005
• VB.NET
• Gnu Multiprecision Library (Gmp)
• Open source arbitrary precision numeric library
• Compiled under Visual Studio 2005
• NGmp
• Open source VB.NET binding of gmp.dll
• Enables calling of gmp library functions through
  VB.NET
• Compiled under Visual Studio 2005

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Threshold Paillier EncryptionWeb Service

• Key generation algorithm
• Input
• k size of key
• l number of shares to generate
• One RSA public key (of the designated
  participant) for each share
• t threshold parameter
• Output
• Public Key PK
• List SK1, , SKl of private key shares
• Encrypted with supplied RSA keys so only
  designated participant can recover the key share
• List of Verifier Keys VK, VK1, ,VKl
• Used for proving validity of decryption

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Threshold Paillier EncryptionWeb Service

• Encryption Algorithm
• Input
• Public Key PK
• Random string r
• Cleartext M
• Output
• Ciphertext c

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Threshold Paillier EncryptionWeb Service

• Share Decryption Algorithm
• Input
• Ciphertext c
• Private Key Share Ski
• Encrypted with public key of webservice
• Output
• Decryption share ci
• Validity proof pi

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Threshold Paillier EncryptionWeb Service

• Combining Algorithm
• Input
• Ciphertext c
• List of decryption shares c1,,cl
• List of verification keys VK, VK1VKl
• List of validity proofs P1,Pl
• Output
• M

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Project Deliverables

• A working prototype of Paillier Threshold
  Encryption Web Service (PTEWS)
• A simple demo of applying PTEWS in online voting
• A master project report documenting the research
  findings and lessons learned

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Tasks and Milestones

• Week 1 Proposal Briefing/Approval
• Week 2 WebService skeleton complete
• WebMethod stubs created, classes for passing
  parameters and return results complete
• Week 3 Encryption algorithms implemented
• WebMethod stubs completely implemented with
  encryption and utility algorithms
• Week 4 Testing Interface complete
• Windows application for testing of Web Service
• Simple test of voting application
• Week 5 Final Report complete
• Week 1 ends Oct 30, Week 5 ends Nov 27

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References

• Sharing Decryption in the Context of Voting or
  Lotteries P. Fouque, G. Poupard, and J. Stern,
  2000
• Public Key Cryptosystems Based on Composite
  Degree Residuosity Classes P. Paillier, 1999
• How to Share a Secret A. Shamir, 1979
• Big Number Libraries
• Gnu Multiprecision Library Opensource C
  language library
• http//swox.com/gmp/
• J BigInteger J library available from
  Microsoft
• http//msdn.microsoft.com/msdnmag/issues/05/12/NET
  Matters/default.aspx
• C BigInteger Opensource implementation of Java
  BigInteger
• http//www.codeproject.com/csharp/biginteger.asp
• NGmp .NET Mono Multiprecision Library (gmp
  binding to .NET)
• http//sourceforge.net/projects/ngmp
• Building Gmp with Visual Studio 2005
• http//fp.gladman.plus.com/computing/gmp4win.htm