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Assignment #8 – Solutions

Assignment #8 – Solutions. Problem 1. Each participant selects a random polynomial The joint secret is the sum of the original secrets How are shares of the joint secret formed?. Problem 1. 1. 2. 3. Problem 1. 1. 2. 3. Problem 1. 1. 2. 3. Problem 1.

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Assignment #8 – Solutions

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  1. Assignment #8 – Solutions

  2. Problem 1 Each participant selects a random polynomial The joint secret is the sum of the original secrets How are shares of the joint secret formed? Practical Aspects of Modern Cryptography

  3. Problem 1 1 2 3 Practical Aspects of Modern Cryptography

  4. Problem 1 1 2 3 Practical Aspects of Modern Cryptography

  5. Problem 1 1 2 3 Practical Aspects of Modern Cryptography

  6. Problem 1 Participant computes its share of the secret by taking the values (including the value which it can compute for itself) and forming the sum Practical Aspects of Modern Cryptography

  7. Problem 2 How does a set of participants use their respective values to decode an ElGamal encryption ? Practical Aspects of Modern Cryptography

  8. Problem 2 Lagrange Interpolation: Given distinct pairs with , form the interpolated polynomial by computing The joint secret can be computed as Practical Aspects of Modern Cryptography

  9. Problem 2 Each with can compute its own portion of the sum Practical Aspects of Modern Cryptography

  10. 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

  11. Problem 2 Each with computes . The ElGamal encryption can now be decrypted as . Practical Aspects of Modern Cryptography

  12. Problem 3 Given a set of ElGamal encryptions , create an equivalent set of ElGamal encryptions and prove the equivalence without revealing the correspondence. Practical Aspects of Modern Cryptography

  13. Problem 3 Use ElGamal re-encryption to create new encryptions of the same values and permute the results to create a new set. Interactively prove the equivalence by creating, say, 100 additional equivalent permuted “intermediate” sets in exactly the same way. Answer each challenge by associating each intermediate set with either the original set of the new derived set. Practical Aspects of Modern Cryptography

  14. A Verifiable Re-encryption Mix Practical Aspects of Modern Cryptography

  15. A Verifiable Re-encryption Mix Practical Aspects of Modern Cryptography

  16. A Verifiable Re-encryption Mix Practical Aspects of Modern Cryptography

  17. A Verifiable Re-encryption Mix Practical Aspects of Modern Cryptography

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  20. A Verifiable Re-encryption Mix Practical Aspects of Modern Cryptography

  21. A Verifiable Re-encryption Mix Practical Aspects of Modern Cryptography

  22. A Verifiable Re-encryption Mix Practical Aspects of Modern Cryptography

  23. A Verifiable Re-encryption Mix Practical Aspects of Modern Cryptography

  24. A Verifiable Re-encryption Mix Practical Aspects of Modern Cryptography

  25. A Verifiable Re-encryption Mix Practical Aspects of Modern Cryptography

  26. A Verifiable Re-encryption Mix Practical Aspects of Modern Cryptography

  27. A Verifiable Re-encryption Mix Practical Aspects of Modern Cryptography

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  29. A Verifiable Re-encryption Mix Practical Aspects of Modern Cryptography

  30. A Verifiable Re-encryption Mix Practical Aspects of Modern Cryptography

  31. A Verifiable Re-encryption Mix Practical Aspects of Modern Cryptography

  32. A Verifiable Re-encryption Mix Practical Aspects of Modern Cryptography

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  41. A Verifiable Re-encryption Mix

  42. Problem 3 The challenges for this re-encryption mix can be obtained by feeding all of the intermediate and final ballot sets into a cryptographic hash function such as SHA-1. The output bits of the hash can be used as the challenge bits in an interactive proof. Practical Aspects of Modern Cryptography

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