Abstract. As quantum computing threatens the security of classical cryptosystems, multivariate cryptography is one of possible avenues for secure public key cryptography in a post-quantum world. The “2F” construction was created to adapt known multivariate cryptosystems (through modulus switching) in the hopes of defending these schemes against powerful rank-based attacks. However, the resulting structure (inherent to any 2F cryptosystem) yields new challenges in the form of lattice-based attacks. Still, the 2F construction claimed post-quantum encryption with ciphertext sizes much smaller than prominent lattice-based schemes. In this work, we investigate the security of 2F cryptosystems against lattice-based attacks, determining the viability of the construction and finding realistic parameters.  As a fair point of comparison, we investigate rectangular variations on NTRU to determine the extent to which lattice-based schemes can achieve small ciphertexts and whether 2F holds an advantage over lattice-based encryption in this characteristic. We propose new parameter sets with several aspect ratios to determine the extent to which ciphertext size can be adjusted in NTRU-like schemes.