Common use of Further Improvements Clause in Contracts

Further Improvements. Note that instead of using the code as guaranteed in Proposition 1, we could have used a random linear code in this application (where the code is chosen by Xxxxx and a description is sent as communication). In this case, the resulting polarization method is very efficient, as only k · poly((α2 − β)−1) copies of the circuits are needed. If this method is used in a statistical zero-knowledge proof system however, the prover needs additional power since he needs to decode a random linear code. Finally, a statistical zero knowledge proof for the promise problem statistical difference (with parameters α and β) can be realized as follows: the two given circuits are sampled obliviously and uniformly at random by the verifier, sending the samples to the prover. The information which circuit was sampled is used as random variables X1, . . . , Xn in a one-way secret-key agreement protocol, whose communication is also sent to the prover. Now, if the given instance produces distributions with statistical distance at least α, then the prover gets the same information as Xxx does, and he can prove this to the verifier by sending back the secret key. If the circuits produce distributions with statistical distance at most β, the prover gets the same information as Eve does, and cannot find the secret key. Thus, it can be useful to use protocols which yield more than one secret bit, as this immediately reduces the error of the zero-knowledge proof.

Further Improvements. Note that instead of using the code as guaranteed in Proposition 1, we could have used a random linear code in this application (where the code is chosen by Xxxxx and a description is sent as communication). In this case, the resulting polarization method is very efficient, as only k · poly((α2 − β)−1) copies of the circuits are needed. If this method is used in a statistical zero-knowledge proof system however, the prover needs additional power since he needs to decode a random linear code. Finally, a statistical zero knowledge proof for the promise problem statistical difference (with parameters α and β) can be realized as follows: the two given circuits are sampled obliviously and uniformly at random by the verifier, sending the samples to the prover. The information which circuit was sampled is used as random variables X1, . . . ,... , Xn in a one-way secret-key agreement protocol, whose communication is also sent to the prover. Now, if the given instance produces distributions with statistical distance at least α, then the prover gets the same information as Xxx does, and he can prove this to the verifier by sending back the secret key. If the circuits produce distributions with statistical distance at most β, the prover gets the same information as Eve Xxx does, and cannot find the secret key. Thus, it can be useful to use protocols which yield more than one secret bit, as this immediately reduces the error of the zero-knowledge proof.

Further Improvements. Note that instead of using the code as guaranteed in Proposition 1, we could have used a random linear code in this application (where the code is chosen by Xxxxx and a description is sent as communication). In this case, the resulting polarization method is very efficient, as only k · poly((α2 − — β)−1) copies of the circuits are needed. If this method is used in a statistical zero-knowledge proof system however, the prover needs additional power since he needs to decode a random linear code. Finally, a statistical zero knowledge proof for the promise problem statistical difference (with parameters α and β) can be realized as follows: the two given circuits are sampled obliviously and uniformly at random by the verifier, sending the samples to the prover. The information which circuit was sampled is used as random variables X1, . . . , Xn in a one-way secret-key agreement protocol, whose communication is also sent to the prover. Now, if the given instance produces distributions with statistical distance at least α, then the prover gets the same information as Xxx does, and he can prove this to the verifier by sending back the secret key. If the circuits produce distributions with statistical distance at most β, the prover gets the same information as Eve Xxx does, and cannot find the secret key. Thus, it can be useful to use protocols which yield more than one secret bit, as this immediately reduces the error of the zero-knowledge proof.

Further Improvements. − Note that instead of using the code as guaranteed in Proposition 1, we could have used a random linear code in this application (where the code is chosen by Xxxxx and a description is sent as communication). In this case, the resulting polarization method is very efficientefficient, as only k · poly((α2 − β)−1) copies of the circuits are needed. If this method is used in a statistical zero-knowledge proof system however, the prover needs additional power since he needs to decode a random linear code. Finally, a statistical zero knowledge proof for the promise problem statistical difference difference (with parameters α and β) can be realized as follows: the two given circuits are sampled obliviously and uniformly at random by the verifierverifier, sending the samples to the prover. The information which circuit was sampled is used as random variables X1, . . . ,... , Xn in a one-way secret-key agreement protocol, whose communication is also sent to the prover. Now, if the given instance produces distributions with statistical distance at least α, then the prover gets the same information as Xxx does, and he can prove this to the verifier verifier by sending back the secret key. If the circuits produce distributions with statistical distance at most β, the prover gets the same information as Eve does, and cannot find find the secret key. Thus, it can be useful to use protocols which yield more than one secret bit, as this immediately reduces the error of the zero-knowledge proof.