In traditional e-voting protocols, privacy is often provided by a trusted authority that learns the votes and computes the tally. Some protocols replace the trusted authority by a set of authorities, and privacy is guaranteed if less than a threshold number of authorities are corrupt. For verifiability, stronger security guarantees are demanded. Typically, corrupt authorities that try to fake the result of the tally must always be detected. To provide verifiability, many e-voting protocols use Non-Interactive Zero-Knowledge proofs (NIZKs). Thanks to their non-interactive nature, NIZKs allow anybody, including third parties that do not participate in the protocol, to verify the correctness of the tally. Therefore, NIZKs can be used to obtain universal verifiability. Additionally, NIZKs also improve usability because they allow voters to cast a vote using a non-interactive protocol. The disadvantage of NIZKs is that their security is based on setup assumptions such as the common reference string (CRS) or the random oracle (RO) model. The former requires a trusted party for the generation of a common reference string. The latter, though a popular methodology for designing secure protocols, has been shown to be unsound. In this paper, we address the design of an e-voting protocol that provides verifiability without any trust assumptions, where verifiability here is meant without eligibility verification. We show that Non-Interactive Witness-Indistinguishable proofs (NIWI) can be used for this purpose. The e-voting scheme is private under the Decision Linear assumption, while verifiability holds unconditionally. To our knowledge, this is the first private e-voting scheme with perfect universal verifiability, i.e. one in which the probability of a fake tally not being detected is 0, and with {\em non-interactive} protocols that does not rely on trust assumptions.