Common use of Proof sketch Clause in Contracts

Proof sketch. The proof of Theorem 3.1 is provided in Appendix D. In summary, this proof proceeds as follows: We build a CKE construction that internally uses a CGKA scheme to execute a CGKA execution schedule Seq. For establishing a CKE key to k public keys, this sequence Seq contains at least one collective update assistance for k passive users. The core idea of the CKE construction is that precisely the effective operations’ CGKA ciphertexts of this collective update assistance in the CGKA sequence are embedded in the committed CKE ciphertext. Hence, the total ciphertext size of these effective CGKA operations equals the size of the CKE ciphertext. All remaining operations in the CGKA sequence (i.e., pre-add phase, add operations, and ineffective pre- assistance operations) are, in different shapes, encoded in the CKE common reference string CRS. The complex but interesting idea of this construction, and hence of this proof, is the isolation of the effective operations from the remaining operations in the entire sequence as well as their encoding in the CKE ciphertext such that CKE functionality and security are reached. As part of the proof, we reduce the security of this CKE construction to the security of the underlying CGKA scheme. Finally, we show that a CGKA scheme that executes schedule Seq without inducing a communication overhead of Ω(k) for the effective operations implies a CKE construction with compact ciphertexts.

Proof sketch. The proof of Theorem 3.1 is provided in Appendix D. In summary, this proof proceeds as follows: We build a CKE construction that internally uses a CGKA scheme to execute a CGKA execution schedule Seq. For establishing a CKE key to k public keys, this sequence Seq contains at least one collective update assistance for k passive users. The core idea of the CKE construction is that precisely the effective operations’ CGKA ciphertexts operations of this collective update assistance in the CGKA sequence are embedded in the committed CKE ciphertext. Hence, the total ciphertext size of these effective CGKA operations equals the size of the CKE ciphertext. All remaining operations in the CGKA sequence (i.e., pre-add phase, add operations, and ineffective pre- pre-assistance operations) are, in different shapes, encoded in the CKE common reference string CRS. The complex but interesting idea of this construction, and hence of this proof, is the isolation of the effective operations from the remaining operations in the entire sequence as well as their encoding in the CKE ciphertext such that CKE functionality and security are reached. As part of the proof, we reduce the security of this CKE construction to the security of the underlying CGKA scheme. Finally, we show that a CGKA scheme that executes schedule Seq without inducing a communication overhead of Ω(k) for the effective operations implies a CKE construction with compact ciphertexts.