Hiding Cliques for Cryptographic Security

  • Authors:

  • Ari Juels;Marcus Peinado

  • Affiliations:

  • RSA Laboratories, 20 Crosby Dr., Bedford, MA 01730, USA;Microsoft, One Microsoft Way, Redmond, WA 98052, USA

  • Venue:
  • Designs, Codes and Cryptography
  • Year:
  • 2000

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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_{1/p}{n}, it is widely conjecturedthat no polynomial-time algorithm exists which finds a cliqueof size \geq (1 + \epsilon)\log_{1/p}n with significantprobability for any constant \epsilon 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 \epsilon) \log_{1/p}{n}—is randomlyinserted (``hidden'') in a random graph, finding a clique ofsize \geq (1 + \epsilon)\log_{1/p}{n} remains hard.Our analysis also covers the case of high edge probabilitieswhich allows us to insert cliques of size up to n^{1/4-\epsilon}( \epsilon0). Our result suggests several cryptographicapplications, such as a simple one-way function.