Memory hard functions (MHFs) are an important cryptographic primitive that are used to design egalitarian proofs of work and in the construction of moderately expensive key-derivation functions resistant to brute-force attacks. Broadly speaking, MHFs can be divided into two categories: data-dependent memory hard functions (dMHFs) and data-independent memory hard functions (iMHFs). iMHFs are resistant to certain side-channel attacks as the memory access pattern induced by the honest evaluation algorithm is independent of the potentially sensitive input e.g., password. While dMHFs are potentially vulnerable to side-channel attacks (the induced memory access pattern might leak useful information to a brute-force attacker), they can achieve higher cumulative memory complexity (CMC) in comparison than an iMHF. In this paper, we introduce the notion of computationally data-independent memory hard functions (ciMHFs). Intuitively, we require that memory access pattern induced by the (randomized) ciMHF evaluation algorithm appears to be independent from the standpoint of a computationally bounded eavesdropping attacker --- even if the attacker selects the initial input. We then ask whether it is possible to circumvent known upper bound for iMHFs and build a ciMHF with CMC $Ω(N^2)$. Surprisingly, we answer the question in the affirmative when the ciMHF evaluation algorithm is executed on a two-tiered memory architecture (RAM/Cache). See paper for the full abstract.