Hidden Cliques as Cryptographic Keys

We demonstrate in this paper a very simple method for ``hiding'''' large cliques in random graphs. While the largest clique in a random graph is very likely to be of size about $2 \log_2{n}$, it is widely conjectured that no polynomial-time algorithm exists which finds a clique of size $(1 + \epsilon)\log_2{n}$ with significant probability for any constant $\epsilon < 0$. We show that if this conjecture is true, then when a clique of size at most $(2-\delta) \log_2{n}$ for constant $\delta < 0$ is randomly inserted (``hidden'''') in a random graph, finding a clique of size $(1 + \epsilon)\log_2{n}$ remains hard. In particular, we show that if there exists a polynomial-time algorithm which finds cliques of size $(1 + \epsilon)\log_2{n}$ in such graphs with probability $\frac{1}{poly}$, then the same algorithm will find cliques in completely random graphs with probability $\frac{1}{poly}$. Given the conjectured hardness of finding large cliques in random graphs, we therefore show that hidden cliques may be used as cryptographic keys.