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 3102011 Practical Aspects of Modern Cryptography ID: 587296
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Assignment #8 – SolutionsSlide2
Problem 1
Each participant
selects a random polynomialThe joint secret is the sum of the original secretsHow are shares of the joint secret formed?
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Problem 1
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Problem 1
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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
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Problem 2
How does a set of
participants use their respective values to decode an ElGamal encryption ? 3/10/2011Practical Aspects of Modern CryptographySlide8
Problem 2
Lagrange Interpolation:
Given distinct pairs with , form the interpolated polynomial by computingThe joint secret can be computed as
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Problem 2
Each
with can compute its own portion of the sum
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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
.
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Problem 2
Each
with computes .The ElGamal encryption can now be decrypted as .
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Problem 3
Given a set of
ElGamal encryptions , create an equivalent set of ElGamal encryptions and prove the equivalence without revealing the correspondence. 3/10/2011Practical Aspects of Modern CryptographySlide13
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. 3/10/2011Practical Aspects of Modern CryptographySlide14
A Verifiable Re-encryption Mix
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A Verifiable Re-encryption Mix
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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.3/10/2011Practical Aspects of Modern Cryptography