Semantic Security and the Second-Largest Eigenvalue of Biregular Graphs
share
It is investigated how to achieve semantic security for the wiretap channel. It is shown that asymptotically, every rate achievable with strong secrecy is also achievable with semantic security if the strong secrecy information leakage decreases sufficiently fast. If the decrease is slow, this continues to hold with a weaker formulation of semantic security. A new type of functions called biregular irreducible (BRI) functions, similar to universal hash functions, is introduced. BRI functions provide a universal method of establishing secrecy. It is proved that the known secrecy rates of any discrete and Gaussian wiretap channel are achievable with semantic security by modular wiretap codes constructed from a BRI function and an error-correcting code. A concrete universal hash function given by finite-field arithmetic can be converted into a BRI function for certain parameters. A characterization of BRI functions in terms of edge-disjoint biregular graphs on a common vertex set is derived. New BRI functions are constructed from families of Ramanujan graphs. It is shown that BRI functions used in modular schemes which achieve the semantic security capacity of discrete or Gaussian wiretap channels should be nearly Ramanujan. Moreover, BRI functions are universal hash functions on average.
READ FULL TEXT