Non-committing encryption (NCE) is an advanced form of public-key encryption which guarantees the security of a Multi-Party Computation (MPC) protocol in the presence of an adaptive adversary. Brakerski et al. (TCC 2020) recently proposed an intermediate notion, termed Packed Encryption with Partial Equivocality (PEPE), which implies NCE and preserves ciphertext-rate (up to a constant factor). In this work, we propose three new constructions of rate-1 PEPE based on standard assumptions. In particular, we obtain the first constant ciphertextrate NCE construction from the LWE assumption with polynomial modulus, and from the Subgroup Decision assumption. We also propose an alternative DDH-based construction with guaranteed polynomial running time. We also make clarifications on the requirements of PEPE.