Biblio

We suggest using Fully Homomorphic Encryption (FHE) to be used, not only to keep the privacy of information but also, to verify computations with no additional significant overhead, using only part of the variables length for verification. This method supports the addition of encrypted values as well as multiplication of encrypted values by the addition of their logarithmic representations and is based on a separation between hardware functionalities. The computer/server performs blackbox additions and is based on the separation of server/device/hardware, such as the enclave, that may deal with additions of logarithmic values and exponentiation. The main idea is to restrict the computer operations and to use part of the variable for computation verification (computation fingerprints) and the other for the actual calculation. The verification part holds the FHE value, of which the calculated result is known (either due to computing locally once or from previously verified computations) and will be checked against the returned FHE value. We prove that a server with bit computation granularity can return consistent encrypted wrong results even when the public key is not provided. For the case of computer word granularity the verification and the actual calculation parts are separated, the verification part (the consecutive bits from the LSB to the MSB of the variables) is fixed across all input vectors. We also consider the case of Single Instruction Multiple Data (SIMD) where the computation fingerprints index in the input vectors is fixed across all vectors.

In a secret sharing scheme a dealer shares a secret s among n parties such that an adversary corrupting up to t parties does not learn s, while any t+1 parties can efficiently recover s. Over a long period of time all parties may be corrupted thus violating the threshold, which is accounted for in Proactive Secret Sharing (PSS). PSS schemes periodically rerandomize (refresh) the shares of the secret and invalidate old ones. PSS retains confidentiality even when all parties are corrupted over the lifetime of the secret, but no more than t during a certain window of time, called the refresh period. Existing PSS schemes only guarantee secrecy in the presence of an honest majority with less than n2 total corruptions during a refresh period; an adversary corrupting a single additional party, even if only passively, obtains the secret. This work is the first feasibility result demonstrating PSS tolerating a dishonest majority, it introduces the first PSS scheme secure against t passive adversaries without recovery of lost shares, it can also recover from honest faulty parties losing their shares, and when tolerating e faults the scheme tolerates t passive corruptions. A non-robust version of the scheme can tolerate t active adversaries, and mixed adversaries that control a combination of passively and actively corrupted parties that are a majority, but where less than n/2-e of such corruptions are active. We achieve these high thresholds with O(n4) communication when sharing a single secret, and O(n3) communication when sharing multiple secrets in batches.