Exploring Relationship Between Traditional Lattices and Graph Lattices of Topological Coding

It is known that there are no polynomial quantum algorithms to solve some lattice difficult problems. Uncolored graphic lattice and colored graphic lattice are the products of multidisciplinary intersection inspired by lattice theory. A uncolored graphic lattice or a colored graphic lattice in topological coding is based on some graph operations and a set of disjoint connected graphs or disjoint connected colored graphs. Based on password authentication or digital file encryption, this paper introduces the number-based string topological authentication problem, and gives an asymmetric encryption system by topological coding. Topological coding can form an asymmetric encryption system with one public key corresponding to two or more private keys and, more public keys corresponding to more private keys. Topology authentication in topology coding requires two different fields of mathematical knowledge and can produce exponential level algorithm. Based on the edge-joining operation and vertex-coinciding operation of graphs, the existence of colored graphic lattice admitting graceful total colorings is shown, and graphic lattice and F-graphic lattice are established with infinite elements closed to graceful total coloring. Topological vectors for special coloring graphs are defined, and a connection between graphic lattice and non-negative integer traditional lattice is built up to provide a feasible technique for quantum resistance calculation, since there is no polynomial algorithm for solving number-based strings up to now. Because graph isomorphism problem is NP-hard, topological coding lattice has the function of resisting supercomputer and quantum computer