The Enhanced Privacy ID (EPID) scheme is currently used for hardware enclave attestation by an increasingly large number of platforms that implement Intel Software Guard Extensions (SGX). However, the scheme currently deployed by Intel is supported on Elliptic Curve Cryptography (ECC), and will become insecure should a large quantum computer become available. As part of National Institute of Standards and Technology (NIST)’s effort for the standardisation of post-quantum cryptography, there has been a great boost in research on lattice-based cryptography. As this type of cryptography is more widely used, one expects that hardware platforms start integrating specific instructions that accelerate its execution. In this article, a new EPID scheme is proposed, supported on lattice primitives, that may benefit not only from future research developments in post-quantum cryptography, but also from instructions that may extend Intel’s Instruction Set Architecture (ISA) in the future. This paper presents a new security model for EPID in the Universal Composability (UC) framework. The proposed Lattice-based EPID (LEPID) scheme is proved secure under the new model. Experimentally compared with a closely related Lattice-based Direct Anonymous Attestation (DAA) (LDAA) scheme from related art, it is shown that the private-key size is reduced 1.5 times, and that signature and verification times are sped up up to 1.4 and 1.1 times, respectively, for the considered parameters, when LEPID is compared with LDAA. Moreover, the signature size compares favourably to LDAA for small and medium-sized communities .