On the Cryptanalysis via Approximation of Cryptographic Primitives Relying on the Planted Clique Conjecture

The core constructive task of the field of cryptography is that of creating cryptographic primitives (e.g. private-key encryption, public-key encryption, message authentication codes,...) with provable security guarantees. Where perfect (information theoretic) security is impossible, cryptographers must in practice necessarily rely on the assumption that some problem is hard to solve in nearly every case (a computational hardness assumption). Primitives in these cases generally provide guarantees of the following informal form: assuming that an enemy takes a long time to solve some problem Y , a system using X is secure. In general, the computational hardness assumptions used in the wild are derived from problems for which many have tried and failed to provide polynomial-time solutions; some of the most common examples include integer factorization, quadratic residuosity, the discrete log problem, and on. At best, our reliance on these problems nests security in the hope that these specific, at times disparate problems will continue to elude researchers: by and large, none of these assumptions are supported by a well-founded general description of hardness as we would see in e.g. complexity theory. As such, one of the so-called ‘holy grails’ of cryptography is to instead vest the hopes for security in a single well-founded, well-known, and well-studied assumption, such as the assumption that P ≠NP . While the reliable use of some NP-complete problem in tandem with the assumption that P ≠ NP has eluded cryptographers due to lack of results showing average-case hardness, one alternative which has been explored is reliance on assumptions that solving certain NP-hard optimization problems within some degree of accuracy is computationally difficult in specific instance classes. In this work, we explore one such example of this effort, [10], which attempts to construct cryptographic primitives by relying on the planted clique conjecture. More specifically, we (1) present [10] in summary, (2) propose a simple cryptanalytic method for the one-way function primitive suggested in [10] using approximation algorithms, and (3) consider the feasibility of such cryptanalysis in the context of existing approximation algorithms for the maximum clique problem.