Hiding Cliques for Cryptographic Security

Abstract

We demonstrate how a well studied combinatorial optimizationproblem may be used as a new cryptographic primitive. The problemin question is that of finding a "large" clique in a randomgraph. While the largest clique in a random graph with nvertices and edge probability p is very likely tobe of size about $2\log _{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 p}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$p$}}} n$, it is widely conjecturedthat no polynomial-time algorithm exists which finds a cliqueof size $\geqslant (1 + \varepsilon )\log _{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 p}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$p$}}} n$ with significantprobability for any constant ∈ > 0. We presenta very simple method of exploiting this conjecture by “hiding”large cliques in random graphs. In particular, we show that ifthe conjecture is true, then when a large clique—of size,say, $(1 + 2\varepsilon )\log _{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 p}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$p$}}} n{\mathbf{---}}$is randomlyinserted (“hidden”) in a random graph, finding a clique ofsize $\geqslant (1 + \varepsilon )\log _{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 p}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$p$}}} n$ remains hard.Our analysis also covers the case of high edge probabilitieswhich allows us to insert cliques of size up to $n^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {4 - \varepsilon }}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${4 - \varepsilon }$}}} (\varepsilon >0)$. Our result suggests several cryptographicapplications, such as a simple one-way function.