Spectrum of Adjacency Matrix of Graphs in Cryptography

Abstract Having access to an elliptic curve with a specified number of points over a specific finite field isa prerequisite for many contemporary cryptography systems. The isogenies, which are surjectivemorphisms, play a significant role as specific mappings between these curves. Isogeny volcanoesare graphs with elliptic curves as their vertices and l-isogenies as their edges. Two identical ellipticcurves over Fq share the same trace t and cardinality. David Kohel examines the relationshipsbetweenisogenies of degreel and curves in Ellt(Fq ), the collection of curves defined over Fq withtrace t, in his thesis. More precisely, he describes the structure of the graph of l-isogenies definedon Ellt(Fq ).There is a relation between Isogeny graphs and orders in OK and uses modular poly-nomials to find the conductor of End(E). The connected components of this graphs are isogenyvolcanoes and it is possible to travel through these structures using modular polynomials, evenwithout knowing the cardinality of the curve.By using this information computation of the l-adicvaluation of the trace t, for l|g possible and hence obtain some information on the cardinality of thecurve. Isogeny graphs of supersingular elliptic curves plays a major role in cryptography.Isogenybased cryptography, studies cryptosystems whose security is based on the difficulty of finding apath in isogeny graphs of supersingular elliptic curves. This study investigates the use of graphadjacency matrices in maintaining the security of isogeny based key exchange protocols.