Practical Aspects of Modern Cryptography

1. Practical Aspects of Modern Cryptography Josh Benaloh Brian LaMacchia Winter 2011

2. Some Tools We’ve Developed • Homomorphic Encryption • Secret Sharing • Verifiable Secret Sharing • Threshold Encryption • Interactive Proofs Practical Aspects of Modern Cryptography

3. Secret Sharing Homomorphisms Many secret sharing methods have an additional useful feature: If two secrets are separately shared amongst the same set of people in the same way, then the sum of the individual shares constitute shares of the sum of the secrets. Practical Aspects of Modern Cryptography

4. Secret Sharing Homomorphisms OR Secret: – Shares: , , …, Secret: – Shares: , , …, Secret sum: Share sums: , , …, Practical Aspects of Modern Cryptography

5. Secret Sharing Homomorphisms AND Secret: – Shares: , , …, Secret: – Shares: , , …, Secret sum: Share sums: , , …, Practical Aspects of Modern Cryptography

6. Secret Sharing Homomorphisms THRESHOLD Secret: – Shares:,, …, Secret: – Shares:,, …, Secret sum: Share sums:, , …, Practical Aspects of Modern Cryptography

7. Threshold Encryption I want to encrypt a secret message for a set of recipients such that • any of the recipients can uniquely decrypt the secret message , • but any set of fewer than recipients has no information whatsoever about the secret message . Practical Aspects of Modern Cryptography

8. Recall Diffie-Hellman • Alice • Randomly select a large integer and send . • Compute the key . • Bob • Randomly select a large integer and send . • Compute the key . Practical Aspects of Modern Cryptography

9. ElGamal Encryption Practical Aspects of Modern Cryptography

10. ElGamal Encryption • Alice selects a large random private key and computes an associated public key . Practical Aspects of Modern Cryptography

11. ElGamal Encryption • Alice selects a large random private key and computes an associated public key . • To send a message to Alice, Bob selects a random value and computes the pair . Practical Aspects of Modern Cryptography

12. ElGamal Encryption • Alice selects a large random private key and computes an associated public key . • To send a message to Alice, Bob selects a random value and computes the pair . • To decrypt, Alice computes . Practical Aspects of Modern Cryptography

13. ElGamal Re-Encryption If is a public key and the pair is an encryption of message , then for any value , the pair is an encryption of the same message , for any value . Practical Aspects of Modern Cryptography

14. Group ElGamal Encryption Practical Aspects of Modern Cryptography

15. Group ElGamal Encryption • Each recipient selects a large random private key and computes an associated public key . Practical Aspects of Modern Cryptography

16. Group ElGamal Encryption • Each recipient selects a large random private key and computes an associated public key . • The group key is . Practical Aspects of Modern Cryptography

17. Group ElGamal Encryption • Each recipient selects a large random private key and computes an associated public key . • The group key is . • To send a message to the group, Bob selects a random value and computes the pair . Practical Aspects of Modern Cryptography

18. Group ElGamal Encryption • Each recipient selects a large random private key and computes an associated public key . • The group key is . • To send a message to the group, Bob selects a random value and computes the pair . • To decrypt, each group member computes . The message . Practical Aspects of Modern Cryptography

19. Threshold Encryption (ElGamal) Practical Aspects of Modern Cryptography

20. Threshold Encryption (ElGamal) • Each recipient selects large random secret coefficients , , …, , and forms the polynomial Practical Aspects of Modern Cryptography

21. Threshold Encryption (ElGamal) • Each recipient selects large random secret coefficients , , …, , and forms the polynomial • Each polynomial is then verifiably shared with the other recipients by distributing each . Practical Aspects of Modern Cryptography

22. Threshold Encryption (ElGamal) • Each recipient selects large random secret coefficients , , …, , and forms the polynomial • Each polynomial is then verifiably shared with the other recipients by distributing each . • The joint (threshold) public key is . Practical Aspects of Modern Cryptography

23. Threshold Encryption (ElGamal) • Each recipient selects large random secret coefficients , , …, , and forms the polynomial • Each polynomial is then verifiably shared with the other recipients by distributing each . • The joint (threshold) public key is . • Any set of recipients can form the secret key to decrypt. Practical Aspects of Modern Cryptography

24. An Application Verifiable Elections Practical Aspects of Modern Cryptography

25. Verifiable Election Technologies As a voter, you can check that • your vote is correctly recorded • all recorded votes are correctly counted …even in the presence of malicious software, hardware, and election officials. Practical Aspects of Modern Cryptography

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33. Traditional Voting Methods Practical Aspects of Modern Cryptography

34. Traditional Voting Methods • Hand-Counted Paper Practical Aspects of Modern Cryptography

35. Traditional Voting Methods • Hand-Counted Paper • Punch Cards Practical Aspects of Modern Cryptography

36. Traditional Voting Methods • Hand-Counted Paper • Punch Cards • Lever Machines Practical Aspects of Modern Cryptography

37. Traditional Voting Methods • Hand-Counted Paper • Punch Cards • Lever Machines • Optical Scan Ballots Practical Aspects of Modern Cryptography

38. Traditional Voting Methods • Hand-Counted Paper • Punch Cards • Lever Machines • Optical Scan Ballots • Electronic Voting Machines Practical Aspects of Modern Cryptography

39. Traditional Voting Methods • Hand-Counted Paper • Punch Cards • Lever Machines • Optical Scan Ballots • Electronic Voting Machines • Touch-Screen Terminals Practical Aspects of Modern Cryptography

40. Traditional Voting Methods • Hand-Counted Paper • Punch Cards • Lever Machines • Optical Scan Ballots • Electronic Voting Machines • Touch-Screen Terminals • Various Hybrids Practical Aspects of Modern Cryptography

41. Vulnerabilities and Trust • All of these systems have substantial vulnerabilities. • All of these systems require trust in the honesty and expertise of election officials (and usually the equipment vendors as well). Can we do better? Practical Aspects of Modern Cryptography

42. The Voter’s Perspective Practical Aspects of Modern Cryptography

43. The Voter’s Perspective Practical Aspects of Modern Cryptography

44. The Voter’s Perspective Practical Aspects of Modern Cryptography

45. The Voter’s Perspective Practical Aspects of Modern Cryptography

46. The Voter’s Perspective Practical Aspects of Modern Cryptography

47. The Voter’s Perspective Practical Aspects of Modern Cryptography

48. The Voter’s Perspective Practical Aspects of Modern Cryptography

49. The Voter’s Perspective Practical Aspects of Modern Cryptography

50. The Voter’s Perspective Practical Aspects of Modern Cryptography