On the classification of graphs induced by a sequence of local complementations of Paley graphs

The power of the public key cryptosystem based on Paley graphs is due to several mathematical problems namely quadratic residuosity, local equivalence, and identification of the graphs induced by a sequence of local complementations of the Paley graphs. The classification in terms of degree of these induced graphs can be useful in the cryptanalysis part of the proposed public-key cryptosystem based on these algebraic graphs. This work aims to give the exact value of the minimum and maximum degree by local complementation, then the possible classifications in terms of degree to the graphs induced by a sequence of local complementations of Paley graphs of degree p less than or equal to 13 and some information about the equivalence problem.